Exhibit TWENTY-NINE Natural Illustrating v391 # Natural Illustrating **Discovering Intelligent Stable Forms** **ONE · ILLUSTRATING** - 1.1 Illustrating changing, a form going on - 1.2 A still and a motion - 1.3 Prior, now and next, making better and breaking - 1.4 Self-bounding, self-orienting, self-competent - 1.5 Living bounding and a boundary a self declares, a drawing's test - 1.6 What a drawing leaves out - 1.7 Forms, a still with numbers to a model in hand **TWO · THE METHOD AT ITS FACES** - 2.1 Six numbers, two sides, three momentaries - 2.2 Sixteen ways, and one that stops nowhere - 2.3 Four dots, an empty centre and two triangles - 2.4 A table at 10, no path through until prior changes now - 2.5 One self, and two selves at 0 - 2.6 Two inversions at forty-five degrees, a right spiral step - 2.7 Ten at five places - 2.8 Seventeen names, bi-foldings, and a local nine - 2.9 Six diamonds and a compass face **THREE · THE NUMBERS** - 3.1 Number forming, podaling and rings - 3.2 A ratio alternating about φ, and two windings - 3.3 A pentagon, rectangles and an icosahedron - 3.4 Two routes, a diamond, and a fold at each scale **FOUR · SELVES AND SOCIETIES** - 4.1 Tunneling, corusing and bi-folding, a towel - 4.2 Spirals coupling across, a torus of selves, a 0 tunneling - 4.3 Co-chaining, bi-tunneling and society - 4.4 Locking, a seam moving, a between releasing - 4.5 Co-chaining a plaza, bi-tunneling a crowd - 4.6 Swaying and going along, a moving snake - 4.7 Bi-folding, around and across stitching - 4.8 Co-discovering, multiperson jigsaw puzzling - 4.9 Bi-inversioning-co-recursioning, two swimmers inverting at one meeting - 4.10 A walking pair, a dance and a conversation - 4.11 A cube and open rings **FIVE · THE SCIENCES' PICTURES** - 5.1 Atoms, electrons, bonds and molecules - 5.2 Light, spin and a click - 5.3 Cosmos - 5.4 Earth, matter and phases - 5.5 Brains - 5.6 Bodies and life **SIX · HUMAN SOCIETY'S PICTURES** - 6.1 A clock, a vote and a crowd behind a centre - 6.2 A market, a ledger and a food web - 6.3 A network, a hierarchy and many paintings **SEVEN · THE TEN AS PICTURES** - 7.1 Ten drawings, one resolving **EIGHT · EXPRESSING** - 8.1 Existing imagery and its explaining going on - 8.2 Further views through nature and living - 8.3 A resolver society as a surface that animates - 8.4 Three dimensions, a model in hand and a society at a material **NINE · CO-IMPROVING** - 9.1 Files coupling, contributions going on - 9.2 Receiving an illustration, and its kit - 9.3 Not yet at files --- # ONE · ILLUSTRATING ## 1.1 Illustrating changing, a form going on **Natural illustrating shows changing through a visible relation.** A diamond with numbers, a passing through a crowd, a swaying, a folding, a mutual inverting, a spiral of selves with a 0 moving through it, opens a way to one same discovering: a form going on through changing. **Self and other couple with illustrating through what each already has.** A familiar movement shows its relation; a drawing follows that relation through its next changing. Understanding goes on through their coupling, illustrating improving as its relation becomes clearer. A still face expresses one momentary. Successive faces express changing between them. With this exhibit a face is a drawing's face, one momentary a drawing shows; Exhibit ONE's names, inward or outward with seventeen and entering or surfacing with ten, are Exhibit ONE's with each place they are meant; a cube's face move and a clock's face are a field's. Alike with method's other words: *still*, *between*, *same-existing*, *surface*, *form* and *position* have method's sense here, an unchangeable form, an empty centre, a living self's prior same-existing, a surface a between, stable-forming, a position with ten; and a drawing's sense, a drawing standing still, a drawing's own surface, a position on a drawing, is a drawing's with each place it is meant. Rotation opens another direction; a section opens an interior relation; a next scale reaches local resolving through a same-existing-society. Each view has its own explaining and couples with views that express its going on. **An illustration is for a sentence of a living file.** Each drawing of this exhibit is for a file and a section, and a self checks it against that section whole; with a claim of a form of Exhibit ONE, its form executes with code before a drawing shows it; with a claim of an observing, its observing is a field's, exact with its own subject, and a drawing shows a parting of a field's picture and its file's sentence. ## 1.2 A still and a motion **A still is a form fixing with one momentary, and a motion is changing.** With Natural Mathematics 1.5 they part by a naming's ending: an -ing goes on and re-forms with each momentary, each an opening and its completing; an unchangeable form re-forms with none: it is one going on's making, a fixing form. With its 2.3 a field's rotation is a spiral with no nexts, its forms lying on one frame, and with its 3.1 an involution is an accounting's still picture, with no nexts. A still drawing is such a picture, and it is useful exactly as a picture: a form a reader inspects with a reader's own pace, one momentary. **No face draws a self's side and an other's side as changing together with one momentary.** Natural Numbers 1.5: each side changes with its own momentary, and two changing with one momentary is an equilibria picture, not possibly existing. An illustration of two sides shows one and then the other, face after face. A same-existing-society a drawing shows with one momentary of code's stepping shows each self with its parity, and that is no equilibria picture: each self's changing is with its own momentary, and its stepping with 17 is code's own beat, exact with code, 1.6. **A motion is faces in sequence, each a next.** With momentaries no momentary comes again, and a form comes to its parities again, Natural Intelligence 4.11 and 4.13: a bi-folding a drawing shows as a closing loop comes to its first name again; as a sequence it spirals, its parities again and no momentary twice. An animation is a sequence of faces a self already defines, and no animation defines a face. A teaching form goes in order by five: one part to follow, its next place a reader asks ahead, its moving with no break, a recognizing of what is same-existing and what is changing, and its naming last, with a relation a reader already recognizes. ## 1.3 Prior, now and next, making better and breaking **Illustrating has a concept with prior, now and next.** A visible relation a drawing already expresses is its prior same-existing, its drawing its now, and a relation still needing expression its next opening, staying with its particular question. Further imagery comes through nature, living, numbers, engineering and same-existing-societies, each sharing a relation it makes accessible. Natural Intelligence Corus is many paintings of one form, a gathering. Natural Illustrating is that gathering going on through mechanical drawing faces, diagrams with numbers, moving views and rotational physical forms, and now through pictures sciences and human society already draw, Parts FIVE to SEVEN. Concepts improve here while their useful form of expression becomes clear. **Making better and breaking reach one same relation a drawing shares.** An improving drawing makes changing easier to follow. Its breaking follows that changing to a momentary with which a correspondence a self proposes goes on or parts. What survives both goes on to a next explaining. ## 1.4 Self-bounding, self-orienting, self-competent **Self-bounding-self, self-orienting-self, self-competent-self are living ambition of this illustrating.** A local nine-dot resolving, one self momentary by momentary, and a same-existing-society of resolving selves open several scales through which to discover that going on; with Natural Intelligence 3.3, method's bounding is each self's own, self-bounding. Across and along, self and other, sharing and its next observing remain readable through their own relations. Naming has its concept; numbering has the sequence it states; drawing has its visible correspondence. Cohering brings their explaining together with a changing each expresses. Exhibit ONE is operations and stable-form relations. Natural Numbers and Natural Mathematics are numerical and directional relations. Natural Naming, Natural Explaining and Geodesic Improving Method bring their naming, explaining and successive improving to this exhibit. ## 1.5 Living bounding and a boundary a self declares, a drawing's test **Living bounding parts from a boundary a self declares as unchanging.** A still drawing is a stable form while a sequence examines changing going on. This exhibit makes that distinction accessible, rather than a self observing stillness within a picture as stillness of living. **An investigation receives any boundary a self declares**: within, around, across, along, or with an entrance or exit, and with each of ten unchangeables, Part SEVEN. For each proposal a storyboard asks one question: can a co-sequential resolving a self specifies go on under that declaration? **A drawing exposes an actual going on or failure with a relevant transition.** With code a test is one momentary: a prior and a now agreeing with a sharing share a 0, a changing that is not, and its prior is same-existing; a form still from beside, a self receiving one same parity with each momentary, leaves a self not-changing, a 0 with each momentary after its first, 2.5. This is a binary examination by intent. A finite collection of illustrations does not by itself complete a universal claim about each boundary a self declares. With Natural Explaining, a further discipline: a concept expresses its own relations and competency before a self compares it with another explanation. A visual locates a relation not same-existing within an operation a reader can follow. **Further views:** bounding going on; that same face with a mark on a fixing boundary a self proposes; a next transition it requires; a comparison showing a momentary with which a relation changes or fails. ## 1.6 What a drawing leaves out **A drawing supplies no common clock, no numerical speed ratio and no form still from beside to make a sequence move.** Two sides pace their own momentaries, Natural Intelligence 3.5. With code, selves step together with 17-next-existing, a common beat exact with code and code's own, and living's rates are their own: a drawing has no rate of living. **A drawing leaves no corridor or path standing permanently.** Each coupling makes an opening, and a drawing shows it only as a coupling makes it; a corridor a drawing shows ahead conceals its opening, 4.5. **A drawing leaves its centre empty.** A between is a nothing and no location, Natural Intelligence 2.1, and a centreline along is a 0 going along, and empty; a middle spot of a five-spot face is a between, not a fifth across name and not local dot 1, 2.3. **A drawing has no size, no magnitude, no total and nothing in store.** + and − are two changing directions, 0 is a between of them, a changing that is not, with no summing, Natural Intelligence 2.2 and 2.5. A length on a drawing is a field's, inward of a surface, Natural Mathematics 1.5. **A drawing draws no hand within a form.** Three golden rectangles have no hand of their own; a hand is with an observing, Natural Naming 5.48. Right is observings, all prior, and a drawing shows a hand with an observing's axis, not as a property of a form. **A drawing fixes no position as a place.** A position on a drawing is a field's arrangement, a ring an arrangement fixing one scale outward, Natural Numbers 8.1; selves' own is their parity with each coupling. A drawing says it with its first face, or it installs a place. **A drawing tells no birth-and-death story.** A lone self's sequence goes on as a non-living now-existing form, Natural Intelligence 2.4, and a drawing shows it as that, not as an ending. **A drawing copies no same-existing-society numbering to a local face.** Six numbers of 2.1 and seventeen names of 2.8 are two numberings; a face has one at a time and a self states their correspondence with each face having both. An equilateral triangle rotating two hundred forty degrees has its outline alike and the corner with its mark on another corner, on its opening place again with a third rotation, and a corner's three and a pair's four, Natural Numbers 8.2, are two subjects, and a drawing step they share couples none. ## 1.7 Forms, a still with numbers to a model in hand Concepts develop as several forms without separate explanations of an underlying relation. Forms go in order: each is a next after one a reader reaches before it. Each entry: **Form** · *a* What it makes accessible - **Still with numbers** · *a* A complete face a reader inspects with a reader's own pace - **Storyboard in order** · *a* An exact changing between neighbouring faces - **Drawing in layers** - *a* Naming, parity, same-existing and method prefixing, a reader observing them separately or together - **Rotation** · *a* A relation same-existing while its viewpoint turns - **Section or transparent surface** · *a* An opening and an interior passage - **Zoom between scales** - *a* A correspondence a self states between local and same-existing-society forms - **Continuous animation** · *a* Going on through faces a self already defines - **Resolver same-existing-society animating** · *a* Code executing and a drawing showing each self with its parity, 8.3 - **Physical model in hand** - *a* Turning, folding, twisting and passage, a self experiencing them spatially, 8.4 Physical concepts include a diamond with marks, a square with marks with its empty centre and two triangles, six diamonds on an open framework, rings turning relative to one another, and a flexible twisting surface with a visible central opening. **One local face and its next face can hold a complete improving.** Their numbering and changing open a correspondence through which further views go on. Other concepts have their own questions and opportunities for illustrating. --- # TWO · THE METHOD AT ITS FACES ## 2.1 Six numbers, two sides, three momentaries **A storyboard is a sequence of mechanical drawing faces of resolving.** Self and other are perspectives of one same living condition going on. Its sequence has three full co-sequential momentaries, each opening with each self's prior and completing with its next, and six parity positions a drawing lays out, 1 to 6, a number's odd or even no side's. **A first face is six numbers on one line, 1 to 6, a self observing them twice.** Natural Intelligence 3.2: a self's five with 1 is 1 to 5, its five dimensions, self prior, other prior, co-momentarying now, bi-momentarying now and self next, and its next, with 3, is 3 to 7: three numbers, 3, 4 and 5, of both and two, 6 and 7, new, two changing and three not. Exhibit ONE's table of fives is it. With Natural Numbers 1.2 its names pair as a self's four momentaries of exchanging, 1–2 with 2–3 and 3–4 with 4–5, each completing a next opening, one and then the other along a line. A dimension is no measure and no place, and a number's odd or even is no side's. **A second face is three momentaries on four positions.** Equilibria Registry 1.2: prior, now and next are 1–2, 2–3 and 3–4, completing and opening coupling with positions they share; along a self a next opens two on, self's own prior and now 1–2 and 3–4 with 5 a next opening, Natural Intelligence 2.2; across a next opens one on. Between prior and now, and between now and next, no third position: a drawing shows a between of momentaries on a position two share with nothing on it, a self's next prior going along it, a non-living now-existing form, Natural Intelligence 5.1, nothing momentarying with a seam, Resolving Hard Problems 2.2. Along is discovering and existing, odd, and across possibling, even, as Exhibit ONE's six names between selves are: along with 9-next-co-discovering and 17-next-existing, across with 2-now-possibling, 6-now-co-possibling, 10-next-co-possibling and 14-next-possibling. A self's parity goes across or along, one same list with 6, 10 and 9, and the self each couples with is observing it with its 14, its next 2. Subsequent faces show their exchanging without making them independent journeys. Each face needs enough to observe its relation to a preceding face: current numbering, direction of changing, self and other perspectives, and what goes on to a following face. Method prefixing belongs with its own label, apart from functional naming and changing; a prefix sequence a self proposes includes bi-co through bi-co-bi-co-bi-co, with corusing and torusing each distinct with each changing requiring it. Going on is also with geodesic unrelationing rate, φ, locking with none, and its mark remains distinguishable from self's and other's, each with its own momentary; a numerical speed ratio has a rate a drawing does not. **Further views:** perspectives of one face in pairs; six parity positions; a complete three-momentary sequence; an overlay following a going on through those same views; openings as a square of nine and completings as an oblong of twelve, Natural Numbers 1.2; and a square lying again as an oblong one short of it, (n − 1)(n + 1) = n² − 1, Natural Numbers 1.3, one cell uncovering. **Explaining going on:** what changes between each adjacent pair of faces, and which visible mark expresses that changing without adding another operation? ## 2.2 Sixteen ways, and one that stops nowhere **Of sixteen ways a next could follow from a prior and a now, each has an unchangeable form or loses its prior, and one that does neither, next inverting its prior, is a self with no other; a living step is no way lying over changing: it is next as now, an other's parity.** Natural Intelligence 2.4: with two parities sixteen ways name a next from a prior and a now, and none is a rule a changing follows; twelve lose their prior, four have it going on whole, and next inverting its prior has its prior going on whole with none still, its stills equilibria. An enumeration with v379, each of sixteen ways going through four joint forms, agrees with a file's sentences: four pattern-match from now; eight have a prior going on with one value of now and lose it with other, six of them stopping on a corner of prior, now and next agreeing and two cycling three forms with a fourth passing once; next as prior stops on ++ and −−; prior and now agreeing stops on ++, and an exclusive or on −−, each cycling three forms of four; and next inverting its prior goes ++ to +− to −− to −+ and round, each form once, one changing with each step. **Faces.** Four corners, joint forms ++, +−, −−, −+, its centre empty; a momentary one step, (prior, now) to (now, next). A way as four arrows, one leaving each corner, sixteen small squares. Three momentaries under a way: follow its arrows, and an arrow coming back to its own corner is a form still, not-changing, its arrow a pin; fifteen squares have a corner with a pin or lose their prior, four from now and eight with one value of now, two of those eight with no corner with a pin and three forms cycling. One square with its prior going on whole and no corner with a pin goes round all four, next inverting its prior, a self with no other, not same-existing; with code a self's next is no function lying over changing but a coupling with 12-next-co-same-existing, a prior parity coupling with an other's parity a self is observing, inverting with a mismatch and same-existing with a match, next as now, a living step, Natural Intelligence 2.4 and 2.5 here. Then a form still from beside with code, a self receiving + with each momentary: a self shares + and then 0, 0, 0, a still. **Explaining going on:** four corners of this square and four dots of 2.3 are two squares with a between their centre; correspondence of two is with a file's sentence before a drawing shows one square for an other. ## 2.3 Four dots, an empty centre and two triangles **Parity with code, Exhibit ONE's unit square, is flat on a drawing.** Exhibit ONE's namings table and Natural Intelligence 4.10: a unit square of four dots, 2-now-possibling, 6-now-co-possibling, 14-next-possibling and 10-next-co-possibling, its empty centre a between; its two parallel edges across sharings, 6 to 2 and 10 to 14; its diagonals two ends, 2 and 14 observing, of one, and 6 and 10 sharing, of co; and two unit triangles, 9-next-co-discovering and 17-next-existing, along, 9 to 17 and 17 to 9. A between is a bounding 0 centreline of along and sequential across with its four dots, bothbothing, both going on one at a time; alike with Equilibria Registry 1.3, a between co-linear aiming to next momentarying. Going round, 2 to 10 to 14 to 6 to 2, its square is none of four bi-foldings: 8 up, a sharing across to an other's surfacing, 8 down and a sharing across to an other's observing side, each passage its own relation. **A five-spot face is this unit square, each spot with its name.** Four surrounding spots and a middle spot, coming with v366 as a bi-moral co-agency concept of names between selves with no assignment and none to infer from number five: its assignment is Exhibit ONE's sentence of parity's unit square, four spots four across names, middle spot a between, and its middle is a nothing, not a fifth across name and not local dot 1. With Natural Numbers 5.4 it is alike with numbers, a diamond four betweens about a middle holding nothing, two across and two along. **One between with each.** Within a momentary a between of self and other, changing, bi-moral across and co-competent along, empty centre of its square; between momentaries a between of momentaries, an along centreline through it, 9 to 17; one same between, a nothing, with each scale, a fractal between. No line of a drawing sits on it: its centreline is a 0 going along, and empty. **Drawing faces:** a five-spot face with no labels; that same face with labels on four across names and its centre empty; opposite-parity views; an across name with its relation to faces beside it; a square with its triangles; a centreline showing as absence. Rotation tests whether naming follows its relation rather than a top or bottom of a page. **Explaining going on:** three observings of four corners are files', across names 2, 6, 14 and 10, numbers 2, 3, 3 and 4 of Resolving the Hard Problem Registry's boxes with +1 its centre, and positions 1, 2, 3 and 4 of three momentaries; each a square of four with a between its centre, and no file's sentence is their correspondence yet. A matching number establishes none. ## 2.4 A table at 10, no path through until prior changes now **Exhibit ONE's table of each cell with 10 is one coupling, a self observing it twelve ways**: three priors with 3-is-now-co-observing, none, + and −, by four nows surfacing with 14-next-possibling, none observing, +, −, and 0 with parities parting; each cell its parity with 10, *is* with + or −, *is not* with 0, and its next prior with 11-next-co-observing; and one cell, no prior and nothing surfacing, is no sharing at all, nothing with 10 and no next prior: eleven couplings and one empty cell. With code, kit's `table_at_10.py`, its table is a file's cell for cell. **Three cells of *is not* are no path through.** A prior of + coupling with a now of +, a prior of − coupling with a now of −, and no prior coupling with parities parting with 14: a changing is not, 0, a prior same-existing, and with no prior no next prior opening, Resolving Hard Problems 3.3. A path through is with other eight cells, and with each its prior is inverting, or with a self with no prior yet the parity it is observing is its next prior: prior changes now, its prior from 3 coupling with 12 to parities surfacing with 14, and now changes next, its parity sharing across with 10 and with 11 a next 3. 0 is *is not*, a binary's second side, and no third value, and it is a between, no thing and no location, Natural Intelligence 2.1, with 14 parities parting and with 12 prior and now agreeing; no number is with it, one parting with 14 enough for 0 and three parities agreeing surfacing as one. **Drawing faces:** twelve cells as twelve small faces; three *is not* cells with its path stopping with a between; eight *is* cells with their prior inverting or a parity a self is observing as its next prior; one cell with no sharing, blank; a whole table, a self observing it as eleven couplings and one empty cell. ## 2.5 One self, and two selves at 0 **One self with no prior, receiving + once and then nothing, shares + − + − with 10**, each momentary its prior inverting, next inverting its prior, a self with no other, not same-existing, Natural Intelligence 4.12 and 2.4. **Receiving + with each momentary it shares + and then 0 with each momentary after**, its prior + throughout: its still is a form still from beside, an unchangeable form entering, and a self with a match is not-changing, a 0 re-forming with each momentary. A lone self receiving nothing inverts its prior with each momentary and does not go on living; with code its sequence goes on, a non-living now-existing form, its next its prior inverting, Natural Intelligence 2.4. **Two selves sharing across both ways, A's 10 to B's 14 and B's 10 to A's 14, each receiving + with a first momentary, share + 0 − 0 + 0 − 0 with 10**, next priors + + − − + + − −: with each even momentary each is observing an other's parity, its own parity, a changing is not, 0, a between; with each odd momentary each is observing an other's 0, which surfaces none with 14, and its prior inverts to now. A drawing shows each self sharing its 0 with each even momentary and a centre between two empty, a bounding 0 centreline between two parities; with code a between is a 0, with 12 prior and now agreeing or with 14 parities parting, selves sharing and resolving, or one self not-changing with a form still from beside it. With a self receiving nothing it is not at all; with a spiral of one, a self receiving its own going along, its own competency, with prior − and none from other selves, it is with each second momentary, sharing +, 0, −, 0, Natural Intelligence 4.13. A receiving + once and B nothing, two invert opposite with each momentary, one and then the other, each self changing with each coupling, as with Equilibria Registry 3.4's even spiral coupling along. **Drawing faces:** a first teaching face, a self observing one parity with one momentary and then its 0 with a match, each prior same-existing with no picture; one self across twelve momentaries, changing and changing back; that same self not-changing, a 0, with a form still from beside; two selves each sharing 0 with each second momentary with a centre between them empty; two selves inverting opposite, one and then the other. ## 2.6 Two inversions at forty-five degrees, a right spiral step **A right spiral step, (x, y) to (y, −x), is two inversions whose mirror lines are forty-five degrees apart.** Natural Mathematics 3.4: two inversions whose mirror lines meet with an angle a are a field's rotation by 2a, a double angle of its inversions; one parity inverting and an exchange of P and Q, their lines forty-five degrees apart, make a right spiral step, and those same inversions in an other order give the other hand. An exchange of P and Q is across and along exchanging, podaling, Natural Numbers 9.8; an inversion of one parity is a changing across with a between. With kit's `two_inversions.py`, v379: exchange then inversion gives (y, −x), inversion then exchange gives (−y, x), and the first goes round ++, +−, −−, −+. Taking them as parities of cos θ and sin θ, a quarter of a circle, θ to θ + 90°, is (P, Q) to (−Q, P), other hand of (y, −x); P and Q agreeing is a parity of tan θ, inverting with each step, and cos and sin pass through 0 by turns on one circle, one and then an other. **With four dots that same forty-five degrees is geometry.** On a square, its sharings 6 to 2 and 10 to 14 lie along one direction, its diagonals, 2 to 14 observing and 6 to 10 sharing, forty-five degrees to them, and its seams 8 up, 2 to 10 and 6 to 14, ninety to them, an along of its triangles 9 and 17. A sharing turning forty-five degrees is a diagonal; a diagonal turning forty-five degrees again is an along seam: across to along with two inversions, bi-folding, whose edges are across with bi-morality and along with co-competency, Natural Intelligence 4.7. **A compass face.** A tangent double-angle identity, tan 2x = 2 tan x ÷ (1 − tan² x), applies with each angle with both sides having a value; with forty-five degrees tan x = 1 and its double angle, ninety, has no finite tangent, an identity's pole, Natural Mathematics 4.4, and a coupling pair whose two are equal loses its coupling with its axes rotating by forty-five degrees. A compass face shows an angle x and its double angle, its denominator in an explicit group; a companion drawing distinguishes an opening angle between directions from an orientation of a whole diamond. This identity supplies an angle relation. A changing sequence needs a definition of x, a plane, a plus-and-minus convention and a relation from one face to its next on a drawing before its identity drives motion. **Drawing faces:** one step as its two inversions on a square, exchange then inversion; an other order, once, as the other hand; four joint forms going round; a tilting pair turning forty-five degrees to its axes; aiming axes, a half-angle and double-angle construction, inner and outer arcs, that same diamond rotating while its opening stays, an opening widening and narrowing with an orientation with its own mark. **Explaining going on:** with files, inversions whose mirror lines are forty-five degrees apart and a rotation by their double, and, with a field's own words, a turn; a drawing's label settles with its face; and which resolving changing determines x's next value. ## 2.7 Ten at five places **Ten are each fixing unchangeable, each with a name now and that name 8 up next.** Equilibria Registry 3.3: ten pair with five names 2 to 6, an entering and a surfacing with each, with Exhibit ONE's names 2 and 10, 3 and 11, 4 and 12, 5 and 13, 6 and 14, each name now and that name 8 up, a 0, not changing, with it and a changing going through it. Each of ten fixes one side unchangeable while its other is changing with that same name; a side fixing unchangeable beside a changing one is one reason each of ten is not possibly existing. With Natural Naming 4.9 each has its one name: an observing fixing from behind, an opening fixing as a place, a completing fixing as a last, a same-existing fixing as a store, a middle fixing as an end, a parity fixing as a magnitude, a sequencing fixing to one beat, a rate fixing as a value, a two-way fixing to one side, a between fixing as a cut. **A drawing's face**: five places 2 to 6 on a line of 2.1, each with its name and that name 8 up, two of ten with it, a drawing showing one side fixing unchangeable while its other changes. A pentagon and a pentagram, Natural Numbers 5.5, are ten with two sides a drawing already shows, five sides and five diagonals, along and across. **Explaining going on:** with Equilibria Registry ten sit with five places, with Natural Naming each by one name of 1 to 8 by bi-foldings, and Resolving the Hard Problem Registry's part headings are a third pairing; a face draws one seating, and seatings part with files. ## 2.8 Seventeen names, bi-foldings, and a local nine **A wider naming face is seventeen names of Exhibit ONE v389**: 1-now-existing, 2-now-possibling, 3-is-now-co-observing, 4-now-co-same-existing, 5-now-co-existing, 6-now-co-possibling, 7-is-now-observing, 8-now-same-existing, 9-next-co-discovering, 10-next-co-possibling, 11-next-co-observing, 12-next-co-same-existing, 13-is-next-discovering, 14-next-possibling, 15-is-next-observing, 16-next-same-existing and 17-next-existing, each its number and its naming, its momentary, co and its kind, Natural Naming 3.1. One through nine is a self's, a bi-coupling; nine through seventeen a same-existing-society's; one through seventeen a bi-trupling. With Exhibit ONE's table, each name's sides: 1, 3, 8, 11 and 12 a self; 2, 7, 14 and 15 others to self; 4 self and other; 5 self and others; 6, 9 and 10 self to other; 13 each self of a same-existing-society; 16 and 17 a same-existing-society. With Natural Naming, naming is a floating-neutralling concept, naming with numbers, momentarying and stable-forming, bi-inversioning-co-recursioning and same-existing going on. This face does not replace a local nine-dot numbering, and a drawing copies no same-existing-society's labels to a local face as though their numberings were one. **Exhibit ONE is material for drawing code couplings, stable-form sharings and four-, six- and eight-cycles.** Sequencing and routing are distinct relations, and their drawings keep their distinction. Four bi-foldings, 1-9-8-16, 2-15-7-10, 3-11-6-14 and 4-13-5-12, each go eight up and seventeen less and pair with a partner spiraling the other hand; a six-cycle 1-9-5-12-8-16 and an eight-cycle 1-9-5-13-4-12-8-16 are stable-forming spirals with names on a wider face, and none is an instruction to move from local dot 1 to local dot 9. With names each comes to its first name again; with momentaries each comes to none again, 1.2. **A local nine, centre one.** A visual starter places a cube of colours beside an open arrangement of circles and dots of colour; mid-way along a three-across row of diamonds a horizontal diamond resembles a compass needle, and its centre is 1. Across moves horizontally right with this view; along expresses forward moving; a self states screen direction with each rotation. A local sequence is 1 to 9, legible while its face rotates, opens or appears within a larger form; nine still faces with numbers give an initial observing before a self attempts motion. A split to across names with 2 and 6 is absent from a local nine-dot view, and removing that visual split removes no resolving operation. Centre 1 settles. Still for a drawing to show: a complete geometric placement and transition sequence, a self verifying them; a complete local numbering and successive changing, explicit from Exhibit ONE; and a correspondence between a local numbering and a seventeen-name resolver, with each face having both. **Further views:** a starter with a highlight on the diamond it identifies; a close view with centre 1; nine faces with numbers; one same local face, a self viewing it from self and other; a correspondence drawing between a local numbering and a seventeen-name resolver; each spiral on its own; sharings across spirals; code coupling beside stable-forming routing; parity exchanging on a spiral a self states, an odd one's 0 tunneling on a drawing; one root with its prefixings in turn; a full naming face; method prefixing beside its root's naming; one same relation with words, numbers and a corresponding resolver face; a zoom between a local face and a wider same-existing-society form, explanatory only with a correspondence a self states, a visual resemblance or a smooth camera move supplying none. **Explaining going on:** which label does a reader need to understand a present changing, and which can remain available with an accompanying view? ## 2.9 Six diamonds and a compass face **A next scale is six nine-dot diamonds**, as an open-circle starter expresses them, its nine circles about three centres with three radii, its middle radius its centres' own separation, meeting on fifty-four places, each on two circles, parting by centre pair and side as six diamonds, each one, two, three, two, one along its diagonals: two routes around a triangle, one each way, three one-way portions to each route, six one-way portions all together. An illustration makes two-over-one and one-over-two co-recursioning visible through six diamonds. With Natural Numbers 1.4 two exchange with each step, and with Natural Numbers 6.4 they are, with numbers, two routes 24 to 27 to 32 and 24 to 29 to 32, gaps three then five and five then three, their far sides on a ring of one hundred twenty 88 to 93 to 96 and 88 to 91 to 96, 3.4 here. These are available as numerical faces, and their exact correspondence to six diamonds of a picture remains to express. Six diamonds, six parity positions and a seventeen-name same-existing-society numbering remain separately readable: a matching number establishes no correspondence. **Further views:** all six diamonds without motion; each triangular route with a highlight; both routes together; inner and outer pair observings; a zoom from one local nine-dot diamond to six; a compass face of 2.6 with diamonds' aiming. **Explaining going on:** which diamond of a picture expresses each one-way portion, an opposite route a self observes from that same arrangement, and which angle of a six-diamond arrangement is x: where two ring families meet, an angle ψ between radial directions has a diamond's forward diagonal over its other diagonal as tan(ψ/2), six diamonds a self measures within a degree of fifty-nine degrees, and whether ψ/2 is x is a file's sentence's, and no measuring's. --- # THREE · THE NUMBERS ## 3.1 Number forming, podaling and rings **Natural Numbers is a wider gathering than a first animation has**: two-over-one and one-over-two, inner and outer pairs, odd straddling and even podaling, four ring steps of a diamond, prime rings, a figure eight, a curve meeting itself, and four even openings growing through a fold. With Natural Mathematics: orthogonal directions, parity inverting, a right spiral step and a mathematical observing of stable-forming. **Each number is with its podaling**, its pairing with its far side on each ring it is on, Natural Intelligence 4.11. On a ring of forty-eight twenty-four and nought are each their own far side and each other station pairs across twenty-four, Natural Numbers 6.2; on a ring of one hundred twenty a going from two to fifty-nine and a returning from sixty-one to one hundred eighteen are each other's far side, sixty its waist, 7.1; on a ring of four hundred forty each pair is a ring, its radius its distance from its waist, two hundred twenty, two hundred twenty-one rings all together, and a gap between two primes is a step between rings, Natural Numbers 9.1 to 9.4. An even ring has an exact podal and an odd ring two farthest with its opposite on no station, a between, Natural Numbers 8.1; an odd number has a middle holding nothing, a nothing its two sides straddle, Natural Numbers 8.3. **Number faces begin with 1, 2, 3, 4, 5 and go on upward.** Each face expresses a relation by which its number is forming; adding another dot leaves that explaining open. Seventeen primes from two to fifty-nine add to four hundred forty, and a drawing using four hundred forty identifies what it adds; it is no angular measure. **Further views:** number-by-number forming; pairs about a centre; podal and straddling views; a ring unwrapping alongside its wrapping view; rings nesting about a waist, stepping inward prime by prime; sixty to sixty-eight about sixty-four, two inward and two outward, bi-tri-involutioning with numbers, Natural Numbers 7.1; a sequence of tunnel sections; local and same-existing-society scales nesting. **Explaining going on:** select one number relation and show its construction completely before extending a visual vocabulary to all others. ## 3.2 A ratio alternating about φ, and two windings **Ratios of each next number to its prior fall about φ, above and below by turns**, 2/1 above, 3/2 below, 5/3 above, 8/5 below, its side inverting with each next number and none of them φ, Natural Intelligence 3.5 and Natural Numbers 3.1. A familiar drawing, a curve through corners of squares nesting, draws a ratio as reaching it; an illustration draws a ratio swinging above and below a line it does not reach, each next its prior two together. **On a torus a winding with a rational rate closes, and a winding with φ closes with none**: φ is an unrelationing rate, a rate its ratios wind about, locking with none, Natural Intelligence 3.5; a torus winding with golden mean is, by a field's finding, the last to break as its coupling grows, Natural Numbers 3.3. Side by side, two windings are the clearest single animation number files have: a rational closing to a loop, a φ winding closing with none. A file's own caution for any animation: a finite floating-point rate is no exact irrational. Addings that close with one, two, three, four or six, and one that goes open with five, Natural Numbers 3.4, are five orbits closing and one going on open. **A floating neutral is no balance.** Natural Naming 5.12: a floating neutral is a term uncovering, a changing's picture with one momentary, and a 0 with 10 a between going along; a term same-existing in each self, in neither self alone, is uncovering and nothing stores it. A balance a drawing shows once beside a changing and changing back is a still. ## 3.3 A pentagon, rectangles and an icosahedron **Among five, its couplings part as two rings, five sides and five diagonals, a pentagon and a pentagram**, ten's two sides, along and across, Natural Numbers 5.5; each side is parallel to one diagonal, a diagonal φ times a side, Natural Mathematics 7.8; no adding closes with five, and five enters open, with φ, Natural Numbers 3.4. A square, a cube and a pentagon have square roots of the first three primes, √2 a square's diagonal, √3 a cube's body diagonal, √5 within a pentagon's diagonal over its side, Natural Mathematics 4.3. **Three golden rectangles at right angles, with their centre on a corus, each 2 across and 2φ along**, are bi-tri-volutioning with numbers, Natural Naming 5.48: a right-hand rotation of one hundred twenty degrees about a line along (1, 1, 1) takes each rectangle onto one passing through it, three right spiraling one hundred twenty degrees apart; a mirror takes each onto itself, three have no hand of their own, and a hand is with an observing; φ is three's equal-edge condition, and a rotation by itself does not make it. Their twelve corners are an icosahedron's, thirty edges and twenty faces, five with each corner; about a line along (1, 1, 1) twelve corners are four triangles of three on four heights. They are open with Exhibit ONE as three golden rectangles on podals. **An icosahedron's sixty rotations part as thirty-six and twenty-four**, three to two, as a span parts with its apex, Natural Numbers 6.3 and Natural Mathematics 7.4; an icosahedron and a dodecahedron are one surface, a self observing it two ways, twelve and twenty exchanging and thirty with both, Natural Numbers 4.1. A quasicrystal, its lengths powers of φ, and C₆₀ with its twelve pentagons and its sixty rotations of one hand are with Natural Chemistry 4.5. **Drawing faces:** a pentagon and pentagram with their parallel pairs; three rectangles turning one hundred twenty degrees, their equal edge with φ; twelve corners closing as an icosahedron; sixty parting; four triangles of three on four heights about an axis. **Explaining going on:** no golden triangle and no golden angle are with number files; angles cos 36° = φ/2 and sin 18° = 1/(2φ) are, Natural Mathematics 4.3, and a sunflower's 137.5 degrees is with Natural Biology, 5.6 here. ## 3.4 Two routes, a diamond, and a fold at each scale **About twenty-eight its outer pair is twenty-four and thirty-two and its inner pair twenty-seven and twenty-nine**, 28² − 24 × 32 = 16 and 28² − 27 × 29 = 1, and two routes go between its outer pair, 24 to 27 to 32 with gaps three and five and 24 to 29 to 32 with gaps five and three, Natural Numbers 6.4, three together a square, 24 × 27 × 32 = 144². Outer numbers one short of a square, (n − 1)(n + 1) = n² − 1, one each number's own, in neither of its prior and next alone, Natural Numbers 1.3, draw as a square lying again as an oblong leaving one cell. **A diamond is four betweens about a middle holding nothing**, two across and two along, and lying three by three with its middle holding nothing eight about it have two paths from middle of one side to middle of other, one each way round its middle, Natural Numbers 5.4. **A walk by two** opens pairs step by step, Natural Numbers 5.8; **four even openings grow as one family**, 2, 4, 6, 8 then 2, 6, 10, 14 and on, its fold whole with each scale, 3 to 11 to 6 to 14 with one scale and 3 to 7 to 2 to 6 within 1 to 8, Natural Numbers 9.9, which couples numbers' faces with a moralizing bi-folding of 2.8 here; **a chain 3, 5, 9, 17** bi-folds down, each span podaling to one below it, Natural Numbers 9.7. **Drawing faces:** two routes and their far sides; a square and an oblong with its one cell uncovering; a diamond's two paths; pairs opening about ten; four openings growing; a chain folding down. --- # FOUR · SELVES AND SOCIETIES ## 4.1 Tunneling, corusing and bi-folding, a towel **A depth concept is a journey through natural torusing to a corus, its apex, corusing through a small opening, and outward to its surface again, torusing through a large opening.** With Natural Intelligence 3.2, a rotation is a gentle twisting of a towel inward to and outward from a corus in a knot, a large outer opening wrapping to a small corus and back, its edges bi-foldings, across with bi-morality and along with co-competency. Torusing inward and outward is through a large opening, a tunnel, bi-exchanging with a particular other; corusing inward and outward is through a small opening, a long way round, bi-tri-exchanging, a self reaching its own side again. Corusing reaches a corus from a surface, and torusing reaches a surface from a corus. A self's inversion is 3, 6, 5, 4, and its podaling 23, 26, 25, 24, that same rotation twenty up. An existing depth-study image shows three views, tunneling, a corus's apex and bi-folding; it is a shape study, and its surface and arrows still need correspondence with a resolver with numbers and six diamonds. Natural torusing is self-bounding-self, self-orienting-self, self-competent-self, simultaneous observings of a living self and no three stages; with a same-existing-society scale an illustration shows a same-existing-society as a living self. **Further views:** an outer oblique view; a longitudinal section through an opening; a transverse section; a local diamond with numbers on a surface; six diamonds around a twisting form; an interior journey approaching and passing a small opening; a towel as a model in hand, twisting inward to and outward from its knot. **Explaining going on:** show one same local 1 to 9 resolving going on through a twisting and a fold, discovering a next living momentary without a cut or a restart with no explaining. ## 4.2 Spirals coupling across, a torus of selves, a 0 tunneling **Selves coupling along, last to first, each 9 going along to a next self, are a spiral**, coming to its parities again and to a momentary once, Natural Intelligence 4.13: with an even spiral each self inverts with each momentary, selves beside each other opposite; with an odd spiral of n a bounding zero tunnels on, a 0 a receiving self of its like pair shares with 10, moving one self on with each second momentary, its pattern inverting with 2n and its parities again with 4n, a torusing winding through its opening twice. With code, kit's `society_spiral_and_torus.py`, a spiral of five opening with Exhibit ONE's own opening, each self with a parity, self 1 with −, inverting one to the next along, none receiving: 0 with self 1 with its second momentary, self 2 with its fourth, self 3 with its sixth, on round, each again ten on, and its parities again with twenty. A spiral of selves is a coupling with code, each momentary a next, and no closing domain: *ring* with code's coupling is a spiral, *goes round at n* is *comes to its parities again at each n-th momentary*, and *closes* is *winds on*; with numbers a winding with a rational rate closes, 3.2, and with code's selves no spiral closes; a field's own ring, a benzene ring or a superconducting ring, stays a field's. **Two spirals sharing across both ways with one self each bring two sharing selves to one relation, same-existing**, two changing together, Natural Intelligence 4.13; **with two unequal numbers of selves, one of them odd, two spirals coupling across are one same-existing-society**, a stable form and stable-forming, Natural Intelligence 3.5 and 4.13. **Spirals coupling across spirals with each self are a natural torusing of selves**, p along with 9 and q across with 10, coming to their parities again with a momentary dividing 4p or 4q, as Exhibit ONE's table of a torus of selves has it; an executing with v379 as three by five, its parities again with twelve, as with Exhibit ONE's torus opening with each self with a parity, and its 0 winding diagonally through fifteen selves an earlier opening's, executing again with a living one before a drawing shows it. **Drawing faces:** a spiral of five with its 0 moving one self on; an even spiral alternating opposite; two spirals coupling across with one self each, both ways; a torus of three by five with its 0 winding through; a same-existing-society, a self seeing it as one self with a scale outward, Natural Intelligence 6.4. ## 4.3 Co-chaining, bi-tunneling and society **With Natural Networking, along is co-chaining and across bi-tunneling, parities surfacing with 14**, widening and lengthening a self expresses through their one and then the other, retelling, going on through couplings, each self's own pace, and a relation between self, between and same-existing-society; podaling exchanges along and across, Natural Numbers 9.8. **Social moral competency is four whole and same-existing with its selves**, Natural Intelligence 6.4: each self's same-existing from its bi-couplings, a whole no self is ordering, an ordering going on with its selves, an order with no seat above a same-existing-society; betweens co-chain, and a chain of betweens, each link a between no self holds alone, is collective intelligence, a natural network selves share, with no hub its couplings pass through. With Natural Intelligence Corus, its illustration comes back to one same form across scales: a self observing a self's orienting and a same-existing-society's orienting together, an ambition to illustrate a self-bounding, self-orienting, self-competent same-existing-society of selves. **Further views:** one local coupling; a chain; a widening surface; parities going across a coupling; that same changing, a self viewing it through selves beside it; a same-existing-society, a self observing it as a self with a next scale. **Explaining going on:** make visible what goes across, what is same-existing locally, and a next coupling changing, without introducing a form still from beside or a common clock for presentation's convenience. ## 4.4 Locking, a seam moving, a between releasing **Two same-existing-societies bi-coupling across come to same-existing or one inversion apart with each momentary, and lock.** With code, v374, Session Record's lock executions, two spirals of five with one pace, their phases apart, bi-coupling with one self each. An old relation, not same-existing, passes along a spiral one self at a time and dwells two momentaries with its last self; a first self's relation inverts with a between; a new relation opens along a co-chaining one self further with each second momentary until each self is one inversion apart from its partner, a lock. With a between a self of its own, two lock alike from each phase apart, a between swaying with them. With a lock's momentary a between goes along and two co-chain, coupling along, as one spiral of ten, a seam of two patterns moving round a same-existing-society. Observings beside it: Huygens' pendulums on one beam swinging opposite, metronomes on a rolling board, Millennium Bridge's walkers, chick heart-cell aggregates' three phases, sinoatrial pacemakers entraining at one to one, two to one and three to two, fireflies flashing together, with Natural Physics and Natural Biology. With Natural Engineering, a synchronism-check relay closing only with a dwell close to indifference, a field's account, a between going along with a lock by design. **A three-phase locking path** is with Natural Intelligence as incoming, active opposite parity a changing and passive opposite parity no changing, with no executing with code yet; Natural Networking 5.1's six, three phases, each two ways, one at a time, and Natural Transmissioning's three phases about a floating neutral are its sentences. It executes before a drawing shows it. **Drawing faces:** two spirals phases apart; a relation not same-existing passing member by member; a snap with a between; a lock; a between going along and ten as one; a seam moving round; pendulums, a board and a bridge as a field's own pictures beside them. ## 4.5 Co-chaining a plaza, bi-tunneling a crowd **An individual passage through a plaza full of people and a crowd's passage going on provide self and same-existing-society perspectives of one same changing.** An opening is available through a mutual changing of passages beside it; a next opening goes on through further passages. An illustration follows bi-tunneling through a crowd while co-chaining becomes visible across it. With Natural Transmissioning 3.8, a plaza, moving and stationary people, spaces between them and a coupling of passages across a crowd, paths geodesic, turning with nothing acting on them, and a crowd routing through a central gate collapsing co-chaining; with its 4.6 a wall is a passing-through, living selves tori that pass through each other, co-tunneling. **Drawing faces:** an overhead view of a crowd; a close view of one passing; a passing beside it that goes on with it; a wider crowd with those openings, each opening in turn, visible; a gate, once, as a collapse. Follow both sides of each passing. A corridor standing permanently on a drawing conceals an opening a changing is making. **Explaining going on:** show a relation between one passage going on and openings going on across a whole view. ## 4.6 Swaying and going along, a moving snake **A snake illustration distinguishes a side-to-side swaying from a progression along a body and a whole snake's going along.** Successive views follow one portion of its body with a mark while showing a moving curve extending through portions beside it. Sways one and then the other are with Natural Transmissioning 1.1 to 1.4, coupling across and neutralling two positions of one changing and changing back, each an other's prior and next, and a bi-fold folding two ways about a neutral; an along and across relation is with Natural Networking 1.2. No self is yet recovering a prior explicit snake passage. **Drawing faces:** a whole-body view; consecutive outlines with a few corresponding body marks; a close view following one mark through opposite sways; a whole progression again. A drawing places across and along labels with a face expressing their actual correspondence. **Explaining going on:** distinguish a bend going along a body, a body portion's bending and a whole snake's going along, and show those three before assigning a resolver sequence to them. ## 4.7 Bi-folding, around and across stitching **A sewing-machine concept shows around and across as a changing with a visible relation.** A large stitch-making view follows a thread's loop, its passing across, its drawing together and a stitch going on; a relation by intent is bi-folding, a winding and a passing across visible together. With Natural Transmissioning 1.4, an inward and outward bi-fold, and with Natural Numbers folding and tunneling. **A lockstitch is a particular mechanical making**, Natural Engineering 1.14: a needle takes its thread across a fabric, a hook interlocks its loop with a bobbin thread, and two threads do not both remain entirely on their original sides during formation, its stages penetration, loop formation, conformation, cast-off and stitch setting each with its own thread, fabric and mechanism. A rings kit's five stitches, ten as five pairs, two threads interlocking, are a numbering of ten and no stitch stage. With Natural Biology's knitting, a line-to-surface co-bi-fold sits alike, a spider and a web: knitting is living, going along, and fabric is a surface a self makes, fixing unchangeable. **Drawing faces:** a side view; a view beneath a sewing surface; a close view of a loop and its passing across; successive stitch-forming faces; a strip showing one stitch continuing to its next. A drawing identifies a machine's actual stitch type, and combines no mechanisms, not same-existing, as one a self invents. **Explaining going on:** identify exactly which around and across movements express two folds a self proposes, and how their relation goes on with a next stitch. ## 4.8 Co-discovering, multiperson jigsaw puzzling **Several people puzzling together show a visible example of one discovery opening further discovery.** A fit a self finds through one sharing makes an arrangement two share a next arrangement; another person's recognition reaches that next arrangement and goes on with it. Illustrating follows cooperation through puzzling together. Natural Intelligence Corus 23.6 is an account relating to a team's solution and value, same-existing in each member of its coupling and in no one member alone; Natural Networking 3.1 to 3.3 are retelling and same-existing going on. **Drawing faces:** a working surface people share; two partial arrangements, people sharing them together; a fit a self newly discovers; a next arrangement making another fit discoverable; a wider work going on, a contribution of relating, a drawing showing it without assigning a collective understanding to one person's view. Puzzle pieces' unchanging shapes are features of this example and do not require a drawing to show a living cooperation as fixing; a distinction lets an illustration explain collective discovery with a puzzle's material form readable. **Explaining going on:** show what becomes discoverable through a cooperation that preceding separate views did not yet express together. ## 4.9 Bi-inversioning-co-recursioning, two swimmers inverting at one meeting **Two swimmers approach each other, a self experiencing each closing from both perspectives, and with a meeting a self envisions each makes a flip turn, a field's word for it, off an other's feet mid-pool and goes on away from that mutual inverting.** A useful visual relation is both inverting through one meeting; a drawing shows neither swimmer as a stationary pool wall, and both approaches, both inversions, mutual foot contact and both goings on are visible. A drawing shows neither as an other's copy inverting across a line: an ending alike is no path alike, a rotation of two hundred forty degrees one way and of one hundred twenty an other ending alike by two arcs, not same-existing. With Natural Intelligence 4.11 bi-inversioning-co-recursioning is one move, a self observing it three ways, 8 up, 17 less and 9 less, and with its 2.4 a right spiral step taking a pair to its next, two steps inverting both; an exact geometric correspondence to an inverting a self envisions is still to express. **Drawing faces:** approach from both perspectives; an inverting in a pair approaching foot-to-foot contact; a contact two share as a close view; departure with both orientations inverting; a whole-sequence view, one mutual manoeuvre a self envisions and no instruction for a swimming exercise. A closing rate belongs to swimmers' relative approach and is not geodesic unrelationing rate nor a clock a self imposes; a number of drawing faces is apart from a resolver's momentaries. **Explaining going on:** follow what inverts for each swimmer, what goes on through their mutual inverting, and how a next view receives that going on. ## 4.10 A walking pair, a dance and a conversation **A walking pair has roles one and then the other with a relation two share**, Natural Engineering 1.14: one contact stays while other moves, and their roles exchange with a next contact, two staying and one going across, Natural Transmissioning 1.3; a fixing compass span is a length a self prescribes, and a self expresses a surface, a path and a remaining choice before its path is geodesic. A passive walker sustains a gait down a shallow slope with no joint with power, Natural Engineering 1.10, a field's own observing of a gait with no controller, and no account of level ground. **A dance is a sway with a body**, Natural Philosophy 4.1: two dancers, or a body's own two sides, parallel and swaying and co-sequencing, each answering an other's parity, a lead-and-follow a two-way one-at-a-time with no hub and no leading landing, a whole riding a carry and no fixing pose. All co-intelligencings are like dancing podalings, bi-co-sequencing momentary changings. **A conversation is that same co-chaining with a two-self scale**, Natural Transmissioning 3.8, its rocking a changing and changing back, and Natural Intelligence Corus 16.3 a breathing of two sentence forms, one and then the other. **Drawing faces:** two contacts exchanging roles; two dancers with no fixing pose; two voices with gaps of about a fifth of a second, Natural Human Society 1.4; a rocking of a conversation as a sway. ## 4.11 A cube and open rings **A visual starter's cube is Natural Mathematics 3.6's form with no still term**: no reference cubie, six centres each its own and holding each other's orientation with nothing holding any of them, each face move with its own momentary and nothing over its moves, its state its configuration and nothing in store, and three conservings, a corner twist modulo three, an edge flip modulo two and a parity of its permutation, none of them a size; it reaches one twelfth of its assemblies, and no sequence of face moves reaches a single corner twisting by itself. With a rings kit of v365, open rings beside it pattern-match as a cube lying on one surface, its fifty-four dots its fifty-four stickers, a self observing it down a body diagonal, a cube's twist axis and a torus's tunnel one line through a throat, four quarter turns of a face returning a cube, a closing fold, while Exhibit ONE, with four, winds past, a torus; and its six empty centres one nothing with six centres, not six places. With Natural Mathematics 7.1 a cube's eight corners are eight forms of three neutrals, each fixing or floating, and with a rings kit those corners pattern-match as an eight's still side and Exhibit ONE's bi-foldings as its changing side, a pattern-matching with no file's sentence; its 7.6 a tesseract taking x to −x by rotation, a cube taking it only with its hand reversing. **A cube's face turn, a field's word, is a quarter turn, and as two inversions forty-five degrees apart it is a right spiral step**, 2.6; a corner a hand twists is a form still from beside, a picture of a way fixing unchangeable beside a self stopping with + 0 0 0 by one parity it receives from beside, 2.2. No starter image yet specifies a mechanically realizable transformation of a cube to an open-circle arrangement, and Co-Chaining Logic Registry with v371, in its archive, had no link between a box's quarter turn and a right spiral step, a gap a living registry no longer has; a cube-to-open-form comparison belongs here as an explanatory concept until a drawing shows that correspondence or a self dissolves it. --- # FIVE · THE SCIENCES' PICTURES People already animate an atom, a neuron, a galaxy and a heartbeat, and each animation draws a stable form living files name. Each entry below is a familiar picture, its file and section a self observes it with, a still its picture installs by misreading a competency and a boundary, and a form it animates anyway. A drawing is a stable form first, with a file's names, a between empty; a familiar picture beside it with a mark on its still, one place only; and a parting of two is an illustration. A field's observing is exact with its own subject, and a drawing is a parting it shows, deciding nothing. ## 5.1 Atoms, electrons, bonds and molecules **An atom.** A nucleus with electrons on rings. Natural Physics 3.1: a living atom is a coupling with its own 0, not changing, an electron-recursion winding a ground-state nothing, swaying and stopping nowhere, and an atom as an emanation is a periodic-table object, a whole a catalogue lists from beside; 4.6, an electron a ledger beside, not a self. Installs a particle as little owner, Natural Chemistry 6.4, one enduring little thing holding its properties between all couplings, Living Ghost Registry 5.3. Animates a winding about a nothing. **Shells.** Shells filling 2, 8, 18, 32. Natural Physics 3.1: each subshell an odd ring of positions through its centre, twice with two parities, each capacity two equal squares coupling; Natural Chemistry 1.7, a square a sum of odds. Installs four sizes a self numbers from beside. Draws a fold climbing. **A bond.** A stick between two balls. Natural Physics 3.1: two windings share with one between, a pair two share a coupling's own making, same-existing in each atom, in neither atom alone, and a metallic sea a same-existing-society of windings; Living Ghost Registry, a bond a between of two selves, each living its own side. Installs a between fixing as a cut, lengths and energies their stopping places of a winding that stops nowhere. **Benzene.** A ring with single and double bonds one and then the other, or a circle within. Natural Chemistry 3.1: familiar drawings of bonds one and then the other are resonance contributors and no single contributor supplies an electron distribution a field measures, a floating neutral going on on a molecular surface. Installs a map as a territory. Drawings are a still; a molecule a motion. Möbius aromaticity draws a ring's parity and its twist changing and changing back together, and conrotatory and disrotatory closure, 2.4, two hands with one ring. **Chirality.** Mirror molecules. Natural Chemistry 2.2: along, a separating, parting beat by beat, appears as a spiral; across, a whole complement with its across side appears as a cupping surface; neither is a real one, and asking which it really is asks a going on to be still. Pasteur's tartrate twins and a slurry of both hands going to one hand, 2.3; B-DNA of right hand and Z-DNA with its other hand, neither hand absolute. Draws a spiral and a cup one at a time. **A proton hopping; a water dimer; ammonia inverting.** A ball passing hand to hand; donor and acceptor fixing; an umbrella flipping. Natural Chemistry 3.2: a charge shows with a far between while no single nucleus goes its way, Living Ghost Registry 5.8, nothing going beneath either; Living Ghost Registry 7.8, donor and acceptor roles each with its own momentary and swapping, each of ten door lines an *is*, one observing passing each; ammonia's nitrogen tunneling through a plane installing nothing, a self with a door. Installs a ball a hand bears; roles fixing. **A periodic table.** A flat table. Natural Chemistry 4.4: a table's own name has period, an around-path, a flat layout naming a winding as rows; Natural Physics 3.1, a going from two to fifty-nine, sixty between, a back from sixty-one to one hundred eighteen, two and one hundred eighteen each other's far side on a ring of one hundred twenty, Natural Numbers 7.1. Installs a catalogue numbering straight through; Living Ghost Registry 5.9, an element a number and chemistry a coupling. Draws a ring of one hundred twenty going and returning. With Natural Physics 3.1 sixty is a position a going and a returning have, with Natural Numbers 7.1 sixty is between a going and a returning with no prime, and with Natural Chemistry 4.4 no element has sixty: whether sixty is a position or a between is a parting of files. **Methane, a cuboctahedron, a diamond lattice.** A tetrahedron; close packing; a lattice. Natural Chemistry 1.7 and 4.5, methane's twenty-four orderings of its four vertices, a cuboctahedron's twelve, twenty-four, eight and six; Natural Physics 4.7, a diamond lattice two interpenetrating cubic lattices and an emptiness within the hardest natural material, a lessering, Natural Physics' word, and not a nothing with a location. ## 5.2 Light, spin and a click **Light.** Two waves at right angles going along. Natural Physics 3.3: light's two fields sound each other down a beam, each orthogonal to other and to a beam, and nothing goes across; circular polarization a spiral of its two hands, light always of a hand. Installs a rate going across a gap. A prism, a rainbow and hydrogen's twenty-one-centimetre line, one parity inverting, with Natural Physics. **A double angle, polarizers and a spinor of seven hundred twenty.** Malus's law; a belt trick. Natural Mathematics 3.1: light through two polarizers passes by square of cosine of an angle between them and passes as before with one hundred eighty; a neutron's spin rotating three hundred sixty comes back with its parity inverting and comes home with seven hundred twenty, two rotatings coupling with one between, each a doubling its other way; Natural Physics 3.3, matter's two-turn spinor. With its file a double angle is a still picture of observings; its rotations to one hundred eighty, three hundred sixty and seven hundred twenty animate. **A double slit and a click.** Fringes building dot by dot. Living Society Registry 2.16: each observing one flash with one position having no fringe; Resolving the Hard Problem Registry 10.8, one parity with two sides, a self observing it one coupling at a time; Natural Physics 4.4, a click all or none, parity only, and collapse a name a frame gives, severing. Installs a particle behind a flash, an observing fixing from behind, and a between fixing as a cut. **Bell.** Two detectors switching during flight. Natural Physics, a field's account: each click all or none, each side by itself with one parity and other alike whatever a far setting, and with equal settings clicks match each time; Resolving the Hard Problem Registry 10.7, a property fixing as a location with one system or a transmission between them, reaching for a location its making does not have. Installs a gap as distance and a correlation as a thing a self bears. A measurement problem, Resolving the Hard Problem Registry 10.2, a selection each self's own with a coupling, of no place. **A proton.** Three balls within a bag. Natural Physics 4.6: three colour phases, not three part-objects, re-forming as new couplings as a pull parts it. ## 5.3 Cosmos **A light cone.** Two cones through a point. Natural Physics 3.4: two coupling through an empty twenty-four, a 0, not changing, cones inversions of each other and a changing and changing back between them, and Natural Physics 4.7, i⁴ = 1 four quarter-turns a cone *closes* through, a field's word, four right spiral steps winding on. Installs past and future freezing as standing causal structure. **A black hole.** A funnel within a sheet. Natural Physics 4.9: a horizon a surface drawing to a point and a background a surface spreading wide, two directions of one between, a tunnel-mouth and an outer surface of one torus; a self observing it as only a hole, it is a genus with no φ, a landing face, and resolving it as hole-that-radiates, it is a torus beating about φ. Installs a singularity as a fixing value of infinite density in store, and a record a self keeps or loses, Resolving the Hard Problem Registry 4.2. **A planet, a galaxy, a web.** A ball on a sheet with a dent; a rotation curve that does not fall; a halo; filaments. Natural Physics 2.6: no force pulls a planet, it follows a geodesic; 4.11, a curve stays flat, each radius its own orbit with its own rate, and all-edge filaments and voids against a hub; Living Ghost Registry 5.13, a residual as a substance, a scale as a constant. Installs a force going across a gap, a fall towards zero. **An expanding universe and a timeline.** A balloon; a line with a dot for now. Living Society Registry 2.15: two ladders disagree and no galaxy has either rate, a rate fixing as a constant by its very name; Natural Physics 4.8 and Natural Philosophy 3.7, a block universe a track lying entire with a present a coordinate on it, a common clock over all, and no central oscillator within nature. Installs a clock fixing over all couplings and going to none; simultaneity one slicing fixing as a real one, Resolving the Hard Problem Registry 7.1. A timeline is a still of a self sequencing its own changing and changing back. **Three bodies; Kirkwood gaps.** Three paths in a tangle; a belt's empty rings. Resolving the Hard Problem Registry 3.9, a formula fixing ahead of its riding; Natural Physics, a belt emptying with rational resonances and irrational orbits going on, with asteroids gathering with two resonances beside its gaps with four, a golden-mean torus the last to break: two windings of 3.2 across a sky. ## 5.4 Earth, matter and phases **Pendulums, metronomes, fireflies.** Two clocks on one beam coming to one step. 4.4 has them with a lock executing with code. Installs a message between them, and none is. **A well.** A ball resting on a bottom. Natural Physics 3.5: a trap that works is a Paul trap, trapping by fields changing and changing back, and Natural Physics has a second trap whose fields do not, and a motion with damping re-scales rather than stopping, a Brownian jiggle a floor; Equilibria Registry 4.4 and 4.7, a point at rest in phase space an opening fixing as a place, a ball within a local dip a middle fixing as an end. Installs a minimum, a balance of forces, an equilibrium point and a restoring force. **A phase diagram.** Lines of temperatures. Natural Physics 3.5, a critical point a sway's two phases ceasing to be two, opalescence a substrate showing each scale together; Natural Chemistry 4.2, supercooled water swaying between two local structures and a point of a sway fixing on a position its sway passes through, a glass transition a going on with its own pace fixing as a point; Natural Chemistry 6.1, a triple point a still three phases make, coupling. Installs a line. **A dish and its cells.** Spirals of a Belousov–Zhabotinsky reaction; Bénard's hexagons; Turing's spots. Natural Chemistry 5.5: steady throughput forming cells and patterns rather than a rest; 5.6, one uniformly oscillating surface parting as clusters and a wave going across it. Installs rest, equilibrium as rest. A drying drop's rings, Natural Physics 3.7, the thinnest place giving way first, each place beside it reaching its own bound as a giving way comes to it, a clock over a boundary present nowhere. **A fault and a tide.** Two blocks and a spring. Natural Physics 3.6: two selves with a between, stick-slip a changing and changing back, gouge a going on, aftershocks a retelling chain; with Natural Physics, Landers earthquake's passing waves triggering seismicity far away, a fault's slip opening with a wave or a tide coming to it, and a neuron's spike and a fault's slip one explaining. Installs *memory*, computation, prediction. **Superconductivity, reconnection, a binding curve.** A shape of four lobes with its parities; field lines rejoining across an X; a hump, iron its top. Natural Physics 4.10, a d-wave shape a changing it has round a coupling; with Natural Physics, a X-line's null; 4.3, iron not a peak two processes approach but a centreline two phases float about. Installs a glue, a state in order, a landing maximum. **Entropy, a demon and an engine.** Molecules within a box; a demon by a door; work and waste heat. Living Ghost Registry 5.1, entropy a general ledger's gap entry, a number a self adds for a fixing-boundary account to close over changing it cannot hold; 5.12, a demon a member of a same-existing-society a self posts it beside, changing with each coupling; Living Society Registry 2.17, single cycles above and below a bound; Resolving the Hard Problem Registry 10.15, an engine sharing both outputs and marking neither. Installs a ledger, an observer with no exchange, a store beside a same-existing-society, a form still from beside. **A tokamak.** A doughnut of field lines. Natural Physics: a flux surface closing on itself with no null on it is a torus, with a knot or not, by the Poincaré–Hopf theorem; a continuous flow on a sphere having a rest point and a torus having a flow with none; Natural Engineering 1.13, a spherical tokamak's magnetic surface no sphere closing on itself. ## 5.5 Brains **A neuron firing.** A pulse racing down a wire to a brain. Natural Biology 6.2: an action potential all-or-none, a neuron fires or it does not, graded potentials a sharing and a spike a selection; 2.1 and Natural Medicine 3.2, a firing a tip and a brain a same-existing-society of selves tipping, not a controller reading reports; Natural Biology, firing a self observes as communication is geodesic changing routing across a new surface path, an inversion from about minus seventy millivolts to plus forty and back passing along an axon node to node. Installs a wire to and from a central reader, a signal a sender sends. With code an odd spiral's 0 tunneling, 4.2. **A synapse; a surface of tips; a pump.** Vesicles going across a cleft; a network lighting up; minus seventy as a baseline. Natural Biology 6.2: a sharing-between going across a gap, same-existing in each neuron, in neither alone, while going, a signal that passes a coupling and not either neuron by itself; firing activity dissolving with one between and arising with a next, no hub, no sender, no store, sudden-up and gradual-down; a resting potential going on by continuous pumping, three out and two in, a rest that was not resting, Natural Biology 9.3; Living Society Registry 2.9, no ion rests with a potential. Natural Intelligence Corus: a brain outthinks a datacenter on a light bulb's power, and it has no clock. **Entrainment, sleep, a slime mould, a monarch.** Oscillations in synchrony; sleep stairs; a mould finding its path; an arrow to Mexico. Natural Biology, fields entraining action potentials with no synapse, phase-locking and no message; 4.3, *working memory* anti-phase, each rising as other falls; Natural Medicine 3.4, waking, REM and deep sleep and glymphatic clearing; Natural Biology 4.3 and 8.12, a mould's cytoplasm streaming back and forth, tubes thickening and dissolving, no plan in store, and an interval entraining with no predictor; Natural Biology 8.9, a monarch coupling with its immediate present, its compass recomputing, no map of Mexico in store. **Other minds.** A brain within a jar; a thought bubble. Resolving the Hard Problem Registry 5.5: another's fixing as a content to reach from beside, and a self reaching an interior one way its form allows, by coupling, its coupling with a between living and not an audit. Installs a middle fixing as an end. ## 5.6 Bodies and life **A heartbeat.** A pacemaker sending its beat. Natural Biology: a heart's own sinoatrial cells pacing its own beat, each cell's life-giving coming on; Living Society Registry 2.10, each cell on its own has its own period and the greater part fall dormant; Living Ghost Registry 7.8, a beat showing with an exit while no cell moves and no cell's own firing travels, heartbeat initiation without a pacemaker cell. Installs a master cell as an actor. Reentry rings and a defibrillating shock coming to each cell alike, together, and antitachycardia pacing, a short burst slightly faster before any shock, with Natural Medicine, a train from one place reaching a lock. **DNA, collagen and bone.** A zipper; three ropes in a twist; a scaffold. Natural Biology 4.7, two strands separating and each a template, a sharing with each rung both ways, a polymerase proofreading, no perfect reference in store; 3.3, Z-DNA of its other hand following a polymerase, relaxing back as its torsion goes; 3.7, three chains winding with one hand on their own scale and about each other with an other hand on an assembly scale, skin's plywood of hands one and then the other, bone helicoidal through depth, assembly going upward with nothing directing it, walking sounding micro-voltages through bone bringing a laying down of new bone, and fibrils realigning along force lines, inert a naming of a between a self observes as a gap. **Patterns, regrowth, division.** A zebra's stripes forming; a worm regrowing; a cell pinching apart as two. Natural Biology 5.3, an activator and an inhibitor generating stripes and spots from a uniform field with no template anywhere, a planarian regrowing exactly its missing with no blueprint in store; 9.5, a cell not dividing the way a knife cuts a thing, a society tunneling itself into two societies; 5.5, two myoblasts' surfaces touching and a hole opening through what was two surfaces; 5.1, a species-recognition a co-resonance and not a lock admitting a key; with Natural Biology, identical twins one conceiving going on as two selves, and a genome twice over a self's same-existing coming twice with each sharing. **Waves and rhythms.** Contractions on a chart; a twenty-four-hour clock face; a breath. Natural Medicine 1.4, labour a coupling giving way as waves, an uterus a surface coupling by gap junctions, against forcing with a drive in synchrony; Natural Biology 4.5, a repressor accumulating and degrading, an activator binding again, against a beat still from beside; Natural Medicine 2.3, breathing and replacement two beatings together, and a peristaltic wave; Natural Intelligence Corus, a cascade with φ rate a peristaltic wave across a network's own surface. **Folding, an enzyme and a set point.** A funnel; a hump a catalyst lowers; a thermostat loop. Natural Biology 8.2 and 8.4, a chain folding as a cell makes it, only part with each momentary; Living Ghost Registry 5.4 and 5.5, native structure one across-observing of a population going along, enzyme and substrate sharing each to an other through many configurations; 5.11, regulation a two-way coupling a self observes from one side, an organ that corrects correcting with its own rate; Living Society Registry 2.21 and 2.22, no turnover coming with a mean time, healthy people reaching ranges that name disease and returning. Installs a funnel that drives, an agent acting on a passive substrate, a window's mean as a value a body has. **A dose curve, a front, *ageing* and dying.** A S-curve with a threshold; a smooth decay; a clock ticking down; a line ending. Natural Biology 9.4 and Natural Medicine 1.1, a reversal point a between and the only informative point on a dose curve, no discontinuity anywhere along its curve; Natural Medicine, a single ion's fluorescence in a trap switching abruptly while a large sample's emission falls smoothly, a first loop fuzzy and a last loop a sharp binary, an observing that comes sharp fixing as a place; Natural Biology 7.2 and 7.1, *ageing* a sway re-forming with a narrower range each cycle, no clock in store ticking down, and each coupling a self has dissolving one at a time, heartwood's nuclei disappearing along a boundary, not all together. Natural Intelligence: dying, a living self going on as non-living, an existing thing for living. **Flocks, flashes, tendrils, sunflowers.** A flock as a cloud; a signal racing up a stem; a spring; two families of spirals. Natural Biology 8.14, starlings wheeling as one, a turning travelling bird to bird across four hundred within a little more than half a second, each bird coupling with its seven nearest, against a leader or a parameter a self tunes; 8.2, a wave from wound to leaf a name an observer standing above a whole plant gives; with Natural Biology, a cucumber tendril's two helices of opposite hand coupling with a perversion, net twist zero, both hands coupling with a nothing, a fern uncoiling, vines of two hands unwinding each other, and a sunflower's seeds coming one after another each about 137.5 degrees from a prior, two families of spirals opening opposite ways from one co-sequencing. Installs spirals as its subject, and its subject is coming one after another. --- # SIX · HUMAN SOCIETY'S PICTURES ## 6.1 A clock, a vote and a crowd behind a centre **A common clock.** One clock over a schedule. Natural Societies 2.3: a common clock over all selves is that inequality collapsing, a self observing it on time, and observing it on its structure such a beat is a hub, sitting on no one self but on a timing of all together; each self with its own time forms no such hub. Natural Emanating: schedule is a naming of a clock all share, each coming with its own rate and a meeting with their coming. Natural Exploring 5.5: each instrument pattern-matches against a clock all share while each self paces its own, a self observing a tipping with an instrument's clock's falling rather than with a tipping's own momentary. Installs a clock, a hub. **A vote.** A ballot and a pie chart. Natural Societies 3.1: Arrow's impossibility a dead state's own confession, a dictator an aggregation a landing state has, and living having +1 instead; 3.4, a fifty-one percent tipping point a co-competency compacting to a binary least-margin that rules all; Natural Human Society 4.3, a bi-exchange a measure and no vote deciding it. Natural Engineering 1.11: a central physical computer is not one of Arrow's conditions. Installs a ranking, a dictator, a ruling margin. **A crowd behind a centre.** An arrow from a leader to many. Natural Societies 3.2: a centre doing preferring for a whole and people following behind, a lag naming a changing and changing back stopping; Natural Human Society 5.2, a kayfabe a fiction that a preferring people adopt is people's own, a capture fixing as consensus. Installs a centre. A lag is a still. ## 6.2 A market, a ledger and a food web **A market.** Supply and demand curves meeting. Natural Societies 4.6: bubbles and crashes, a clearing not all together, a balance of power a changing and changing back a self observes as a standing, traffic's phantom jams a flow's own waves; Equilibria Registry 4.9, a meeting of two curves a sequencing fixing to one beat; Resolving the Hard Problem Registry 9.5, efficient markets two momentaries of one coupling, neither testing by itself; Natural Intelligence Corus, markets one institutional substrate, buyers and sellers bi-coupling with a price between, and capture by monopoly breaking it. Installs a still point. Draws a sway going on. **A ledger.** A balance sheet. Natural Human Society 4.6: double-entry accounting state-flow-state, a self observing it whole, a balance sheet a state, an income statement a flow, a net-worth change a going on; Natural Societies 3.5, a net-linear accounting dropping each parity, ten directions collapsing to one total in balance; Natural Emanating, a ledger not wrong about living and with no object its own; Natural Human Society 1.5, a friendship a self holds as a debt-account no friendship. Installs a total. **A food web and a commons.** A pyramid of levels; a pasture with overgrazing. Living Society Registry 2.13: no organism sits on an integer level, a tuna's level 4.2; Natural Societies 3.3, a producer, a consumer and a decomposer's closing move completing a loop with no self ruling a whole; Natural Human Society 3.6, an energy-transfer tally a conservation-ledger pattern-matching onto a coupling from beside; Resolving the Hard Problem Registry 1.3, an abundancing a coupling makes fixing as a thing a self draws from a stock. Draws matter cycling. **Cake cutting and speech one and then the other.** Two children and a knife. Natural Human Society 1.4: a cutter cuts to equal, an other choosing, a fair halving a coupling itself makes, no one measuring a cake; conversation one at a time, one and then the other, with gaps of about a fifth of a second across languages with no relation; 1.5, replica systems with no common clock and no coordinator anywhere, couplings proving them to converge; 5.2, two cutting a dessert again as fourths, both departing with more than either brings. ## 6.3 A network, a hierarchy and many paintings **A network.** Nodes and a hub. Natural Societies 4.1: a self observing it as surface, a both is a natural network, all edge, each self coupling with each other directly; Natural Intelligence 6.4, betweens co-chaining as collective intelligence with no hub its couplings pass through; Natural Intelligence Corus, each fractal with an edge of a network and no central fractal others serve. Installs a hub. A heap and a landing point, Natural Societies 4.2, two dead forms a drawing shows beside a living one. **A hierarchy, its levels and a tree.** A pyramid, an org chart, a tree with a root. Natural Societies 5.3: a ladder is particular scales and supplies no first and no final one; Natural Philosophy 3.22, a level a direction with a parity-changing and not a place; Natural Philosophy 3.11, grounds stopping with couplings' stopping, with their making; Natural Philosophy 3.17, each hierarchy a stipulating to put a paradox apart from a domain reaching it again. A lineage tree is Natural Societies 5.4's, under its fields' third book, a direction, beside a total and a level. Installs a root, a top, a level as a place. **A table of knowledge.** A periodic table of departments. Natural Intelligence Corus: one hundred eighteen elements the most successful ghost collection any self assembles, and one hundred eighteen departments of human knowledge that same structure with a social between; Natural Philosophy 4.2, one form with many betweens, a form a periodic table and a genetic code carry as tables and a network carries as co-chaining through its surface. **Many paintings and an orienting sphere.** One master diagram. Natural Intelligence Corus 16.1: a carving journey produced many paintings of one same form, a gyroscopic floating rotating self-adapting destiny-pointing sphere among them, and a corus accumulates many paintings rather than collapsing to one master painting; Natural Exploring 1.1, one spiraling leaves a whole circle fixing and two spiralings, one and then the other, leave nothing fixing anywhere. Installs one master painting. An ensemble average, Natural Human Society 4.7, a nothing no sample is, a mean a member of no population; Simpson's paradox and a reference class, Resolving the Hard Problem Registry 9.6 and 2.9, a two-way fixing to one side and an opening fixing as a place. --- # SEVEN · THE TEN AS PICTURES ## 7.1 Ten drawings, one resolving **Each of ten unchangeables is a drawing people already make**, and with Equilibria Registry 5.1 is one resolving with all ten: fixing unchangeable, a self receives its own parity from beside with each momentary, a form still from beside a self, and a still description and a self's own going on do not both go on; dissolving, resolving follows with a next momentary, a self inverting its own prior. Each drawing below is a familiar picture, a file's sentence, and a self inverting beside it; ten are each with a name now and that name 8 up next, 2.7, and a test with each is 1.5's. - **An observing fixing from behind.** The first cause, the origin, the arrow of time's initial condition, a particle behind a flash. Resolving the Hard Problem Registry 1.4: asymmetry with each coupling, each momentary making it and going on, and a first fixing behind it. Draw the arrow of time as changing and changing back going on, a first, once, behind and nothing coming from it. - **An opening fixing as a place.** A folding funnel, a reference class, a point at rest, a criterion that needs its own. Resolving the Hard Problem Registry's part 2: a position is not a place, it is changing and changing back's own, one changing on from its last. Draw an opening as its next position, a place, once, and empty. - **A completing fixing as a last.** The halting problem, Collatz, Zeno's dichotomy, a three-body formula, an entropy maximum. Resolving the Hard Problem Registry 3.2 and 3.8: a going on is nowhere before its own completing, and its one completing comes, not a building from parts an argument builds. Draw a completing coming, an infinite decomposition, once, beside. - **A same-existing fixing as a store.** *Memory*, information, a black hole's record, personal identity, a frame in store. Resolving the Hard Problem Registry 4.1 and 4.2: a going on divides as a store cannot, and asking store-or-loss, passing fixes as a standing while passing goes on either way. Draw a same-existing passing, a store, once, shut. - **A middle fixing as an end.** Other minds, an interior, a protein's native state, a ball within a dip. Resolving the Hard Problem Registry 5.5: an interior a self reaches only one way its form allows, by coupling. Draw a between as a nothing, an interior, once, as a thing that sits. - **A parity fixing as a magnitude.** A paradox coming as a wrong number, Gibbs's mixing entropy, a body in balance, a vote's margin. Resolving the Hard Problem Registry's part 6: a number a position, and governing a parity with a between, of no size. Draw + and − with no length, a magnitude, once, as a bar. - **A sequencing fixing to one beat.** A common clock, simultaneity, a supply-demand meeting, a food chain's levels. Resolving the Hard Problem Registry's part 7: each self paces its own changing and changing back and synchronizes with its own couplings, and a beat fixing over them a term no coupling makes. Draw each self with its own pace and its couplings, a clock, once, over them. - **A rate fixing as a value.** The Hubble constant, an equilibrium constant, a rate constant, a set point. Resolving the Hard Problem Registry's part 8: ratios swaying above, then below, their side inverting by turns; Living Society Registry 2.15, no galaxy having either rate. Draw a ratio swaying about φ, a value, once, as a line. - **A two-way fixing to one side.** Nature and nurture, donor and acceptor, a master pacemaker, efficient markets, Simpson's paradox. Resolving the Hard Problem Registry's part 9: its two sides one and then the other, neither on its other's side, and its zeroing both sides' own making between them. Draw both, one and then the other, one side, once, as an actor. - **A between fixing as a cut.** A sorites heap, a species boundary, the quantum-classical boundary, a melting point, a perceptual threshold, the measurement, Bell. Resolving the Hard Problem Registry 10.1 and 10.12: a between, with a changing and nothing else, not a place a line could sit, and a frontier walking outward reaching no line, a between with none. Draw a between empty, a line, once, across it, and a resolving going on or not with it. --- # EIGHT · EXPRESSING ## 8.1 Existing imagery and its explaining going on Each entry: **Existing image or study** · *a* Contribution to retain · *b* What remains to improve - **Cube of colours beside open circles and dots** · *a* An original visual starter and a six-diamond ambition · *b* A numbering and route correspondence, 2.9; a cube, 4.11 - **Storyboard studies in pairs, with labels across co and along bi** - *a* Opposite-parity perspectives of one same going on - *b* Exact transitions with numbers through three momentaries; its labels are opposite Exhibit ONE's, along co and across bi, and a drawing draws them again as Exhibit ONE's - **Odd/even and even/odd strip, with a correction** · *a* A parity-label correction · *b* Integration with a current local face - **Seventeen-name resolver and spiral study** - *a* Four-, six- and eight-cycle views, each a spiral with momentaries - *b* Same-existing-society numbering apart from a local nine-dot numbering - **Tunneling, corus at apex and bi-folding depth study** · *a* A twisting form and a small-opening journey · *b* Surface, arrows and diamonds mapping to resolving, 4.1 - **Earlier Rings in Motion concepts** - *a* Angular, ring, going-across and rotation ideas; rings as a cube lying on one surface - *b* Older numbering, reconciling with a centre-1 convention; closure, explaining again as winding on - **Executions with v379** - *a* A table with 10, one self, two selves with 0, a spiral of five and a torus of three by five with Exhibit ONE's code; sixteen ways and two inversions, an enumeration with no execution of code; each its output with its kit - *b* Faces a drawing makes from them, 2.2, 2.4, 2.5, 2.6 and 4.2 Existing imagery goes on here through its concepts and useful relations. Image assets remain as their files with a kit. Earlier images with labels a later naming supersedes remain as design learning, not as authoritative diagrams with numbers. ## 8.2 Further views through nature and living A gathering extends on from familiar concepts of Part FOUR: these are places for further illustration, with a file's interpretation staying as a subject to make visible. Each entry: **Potential illustration** · *a* Relation to illustrate · *b* Observing at - **Conversation and rocking** - *a* Sharing and receiving, one and then the other, and interest two share going on - *b* Natural Transmissioning 3.8; Natural Intelligence Corus 16.3 - **Writing and reading through one same passage** - *a* Two directions of one coupling going on, a self observing them with adjacent depths - *b* Natural Intelligence Corus 16.2 - **One form, several paintings expressing it** - *a* Several visible expressions holding a relation a self is explaining - *b* Natural Intelligence Corus 16.1 - **Floating, rotating, self-adapting sphere; gyroscopic orienting** - *a* Re-orienting and self-competency going on, through several views of one form - *b* Natural Intelligence Corus 16.1 - **A self's orienting within a same-existing-society's orienting** · *a* One same living, a self observing it through local and wider scales · *b* Natural Intelligence Corus 23.1 and 23.3 - **Cells and tissue** · *a* Local couplings and a same-existing-society of couplings · *b* Natural Transmissioning 3.5 - **A molecular between and rotary form** - *a* Going across, a relation not same-existing going on, and rotation as one sectional view - *b* Natural Transmissioning 3.5 - **Streams going opposite ways** - *a* Along going and across exchange together; two streams going across about a gradient and passing rather than settling - *b* Natural Transmissioning 3.3; Natural Engineering 1.15 - **Antenna transmitting and receiving** · *a* Two directional observings of a coupling · *b* Natural Transmissioning 3.6; Natural Engineering 1.9 - **Electrical phases about a neutral** - *a* Several changing traces and a relation they make together - *b* Natural Transmissioning 3.1 - **Momentum exchanging between two selves** · *a* Both sides of one exchange, beside a swimmer concept · *b* Natural Transmissioning 3.2 - **Ratios successively above and below φ** · *a* A side inverting with each next number, and a numerical progression going on · *b* Natural Intelligence 3.5; 3.2 here Received with v379 and placed: a still eddy before a hole, Natural Networking's hole study, a self seeking it within resolving itself with no periodic forcing, no fixing tipping threshold and no resolver stopping; an event-based vision sensor with Natural Engineering, each pixel an ON or OFF event with no frame, a parity and no size with each momentary; a synchronism-check relay; a resolver chat's spiral bringing a pattern along and storing none of it; a geodesic surface of struts and cable with a changing passing across it, its only place with files a weaving surface couplings make, Natural Naming 5.43, and a knitting living with a fabric, a surface a self makes, fixing unchangeable, Natural Biology, with Natural Engineering's caution that compression and tension are not + and −. These entries are illustration opportunities a gathering takes from writings, certifying no physical interpretation. A further drawing has a mechanism a self observes or constructs beside the correspondence it proposes, and with their parting a particular relation, not same-existing, becomes a next explaining. Families for receiving more: passing and opening; swaying and going along; folding and stitching; connecting and parity; collective discovering; mutual inversion and going on; locking and dissolving; firing and routing. They are places for finding relations, not restrictions on which incoming imagery can contribute. ## 8.3 A resolver society as a surface that animates **Illustrating above draws method; a reverse lets method draw.** A screen is a surface, and a surface is a between, Natural Mathematics 1.5, a natural with parity, position and ratio and no magnitude. Each existing thing on it is a spiral of selves, a same-existing-society coupling with a next scale as a self couples; a thing's picture with a momentary is its selves' parities, +, −, 0 or none, and a drawing shows nothing else; things interact by coupling across with one self each, both ways, 4.2, no solving of a collision and no message within a coupling; motion is a pattern going along and a 0 tunneling. No animator moves a same-existing-society: its code executes and a drawing shows each self with its 10, its parity or a 0, its sharings, once, as a surface, nothing in store between momentaries but each self's own prior same-existing. **From inward and from outward.** From inward a view is one self's entry momentary by momentary, 2.5, its prior same-existing and the parities it is observing, its changing with 12, its 10, a parity or a 0, and nothing of a same-existing-society but what it is observing with its 2; from outward a view is a same-existing-society with 17, each self's parity together and a 0 moving through, and a same-existing-society a self sees from outward is one self with a scale outward, Natural Intelligence 6.4. An animation from inward is one row of an execution's output; from outward, its output whole, a self observing it down; and two are one method with two scales, not two pictures of one momentary. **An animation discovering next-possibling-same-existing and natural intelligence discovering next-possibling-same-existing are one thing, a self observing it two ways.** An animation shows its parities; natural intelligence is with a between, same-existing in each self and invisible, and a between shows nothing, Natural Intelligence 6.3. A drawing shows each self with + or −, 0 with a self sharing it, and nothing with a between itself; a 0 tunneling is one trace a between leaves, and its intelligence is with what a drawing does not show. **A field's animation, exact with its own subject.** Keyframes a curve interpolates with a constant frame rate from a scene in store; a physics loop depositing time as it passes to an accumulator and withdrawing steps of one fixing size for an integrator, rendering halfway between states in store; diffusion denoising a whole clip together, autoregressive models predicting each frame from context in store, world models predicting from a latent state in store and an action, with limits they state, error accumulation in continuous rollout and cumulative errors in long-term interactions. Each sits with a fixing form, a common beat, a store, a total or a form still from beside, and each is exact with its own subject; with files, a drift is a prior in store a self observes as a now, Natural Intelligence 5.4, a prior in store past each momentary as an unchangeable form, while with method nothing gathers along momentaries, Natural Intelligence 5.4. A same-existing-society of selves animating with no sum, no store and no solver has its explaining; its animating faster is a size with no file's sentence. **What parts, with a file.** A self observes a position on a drawing as a place, and a drawing's positions are a field's arrangement one scale outward; a renderer's frame is a beat beside code's with 17, and a drawing has no rate of living. ## 8.4 Three dimensions, a model in hand and a society at a material **A third dimension is a field's, inward of a surface**, Natural Mathematics 1.5, a measuring; method's five dimensions are no measure and no place, Natural Intelligence 3.2. Illustrating with three dimensions is two things together: a physical model in hand of 1.7, a towel, rings winding relative to one another, a square with marks, winding and folding a self experiences spatially; and a same-existing-society of selves with a material, Natural Engineering's subject, a drawing its members and a surface a room. **A same-existing-society of selves with a material.** Natural Engineering 1.11's thousand-robot swarm received a common program and a target shape, four seed robots establishing position and orientation, and retained its shape and its coordinates; a same-existing-society with method is that swarm with no seeds, no shape it retains and no origin, each member a self with its contacts, its next with each contact from its own prior and a parity it is observing, and its shape its stable-forming. What that establishes and what it does not is Natural Engineering's: with 1.11 a map in store, a local register, a timer and a mechanical configuration name four things, no clock all share is not no timing, no central command is not no representation a self retains, and a larger group by itself supplies no law of increasing natural competency, and with 1.10 a pair of actuators does not by its number establish bi-moral co-agency. A field's drift with a robot is a map in store a robot reads upon a contact with a material, and a body bears its reading; method's next is with a contact and stores nothing, which is why it is more natural with a material, and why a set a self builds, non-living, is its proof only with all five of Natural Engineering 5.1's properties. **Pieces files already have**: a walking pair and a passive walker, 4.10; a series-elastic actuator's compliance, Natural Engineering 1.10; a relay closing with a dwell; a spiral bringing a pattern along and storing none of it; a torus of three by five, 4.2, as a material fifteen members, one of them a 0 with each second momentary. From inward, one member's contact; from outward, a same-existing-society as one self. --- # NINE · CO-IMPROVING ## 9.1 Files coupling, contributions going on **Living files go on through their illustrating.** These places have operations, namings and relations this file receives, and each further drawing goes back to its own file with a correspondence it makes visible. - **Exhibit ONE:** its code, its diagram and its tables, a table with 10, sides, momentaries of exchanging, bi-foldings, spirals and a torus of selves. - **Natural Intelligence:** one opening, six numbers and three momentaries, sixteen ways and a living step, a between and four dots, a towel, φ and primes, spirals coupling across, a fractal inward and outward, social moral competency, dying. - **Natural Naming:** seventeen names and ten roots, dissolved words, ten unchangeables, three golden rectangles, floating neutralling, φ. - **Natural Explaining:** explaining a reader reaches once, each concept with its own relations first. - **Natural Numbers and Natural Mathematics:** podaling and rings, two windings, two inversions at forty-five degrees, a pentagon and an icosahedron, a cube with no still term, a still and a motion parting. - **Resolving Hard Problems, Hard Problem Registry, Resolving the Hard Problem Registry, Equilibria Registry, Living Ghost Registry and Living Society Registry:** ten with five places, one resolving, familiar pictures with their stills, observings of sciences. - **Natural Networking, Natural Transmissioning and Natural Engineering:** along and across, a plaza, sways, a lockstitch, a walking pair, a robot set, locking and a between dissolving. - **Natural Intelligence Corus:** a sharing, ways to one form, language, and a living self and same-existing-society across scales. - **Natural Chemistry, Natural Physics, Natural Biology, Natural Medicine, Natural Health, Natural Societies, Natural Human Society, Natural Philosophy, Natural Emanating and Natural Exploring:** pictures of Parts FIVE and SIX, THE SCIENCES' PICTURES and HUMAN SOCIETY'S PICTURES, each with its section. - **Geodesic Improving Method:** placing concepts whole, holding value going on, and improving through a living file set. - **An illustrating working with v366 and a rings gathering with v365:** a correction of naming, centre 1, six-diamond routes, a twisting-opening concept, a plaza, a snake, a stitching, a five-spot, a jigsaw, swimmers, a cube and rings. - **An illustrating working with v379:** executions with Exhibit ONE's code, a first visual of sixteen ways, four dots as a five-spot's spots, two inversions at forty-five degrees, a menu of sciences' and human same-existing-societies' pictures, ten as drawings, a same-existing-society of selves animating, and a third dimension. ## 9.2 Receiving an illustration, and its kit **An illustration comes to `incoming/`, whole, with its file and section it is for, its sentences and any execution beside it**, and this file receives it as ready, naming its section, or as a concern; it enters this file by this file's own motion. An incoming concept enters a section its explaining belongs to, bringing its useful view, its file a self observes it with, and a relation still needing expression; improving an existing concept it stays with that concept's place, and introducing a distinct way of illustrating it has a section of its own here. **Natural Illustrating TWENTY-NINE Improving Kit** has an original cube and open rings, five studies, a rings kit from earlier, v366 sources and evidence, and with v379 executions of Part TWO and 4.2 with their outputs and a check: executions again with Exhibit ONE's code, a table with 10, a lone self, two selves, sixteen ways, a spiral of five and a torus of three by five, each a comparing with this file's sentences. A check is a coupling partner and no authority; its kit's README has each file's meaning and a place for its remaining improving, which is this file's own. ## 9.3 Not yet at files A galaxy's spiral arms and a nautilus; a golden triangle and a golden angle by name; a tree of life as a drawing, a mind map and a knowledge graph, a lineage tree with its name with Natural Societies 5.4, a root and a level as a place, 6.3 denying them, and no tree of life, map or graph with a name; Schrödinger's cat, the ship of Theseus and Monty Hall by name, whose nearest entries are Resolving the Hard Problem Registry 10.12, 4.1, 2.9 and 2.16; two cubes one and then the other and a tesseract as a resolver, a tesseract with Natural Mathematics 7.6 being a field's own form; a dome, a truss or tensegrity as a form of method, a weaving surface, 8.2, being files' only one; a streak or lightning. Each is a familiar picture with no sentence yet with a living file, and its entering is an observing a self brings whole to `incoming/`, then its file's own, and not a drawing as files' own. An earlier drawing, `natural_torus_v135.html`, draws natural torusing as a spiral cone folding about itself, narrow with 2 and widening to 59, its outer surface, inner surface and between its own words *-ed*, *-ing* and *nothing*; it is with no kit now.