Exhibit TWENTY-NINE Natural Illustrating v389 # Natural Illustrating **Discovering Intelligent Stable Forms** **ONE · ILLUSTRATING** - 1.1 Illustrating changing, a form continuing - 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 declared boundary, a drawing's test - 1.6 What a drawing leaves out - 1.7 Forms, a numbered still to a handled model **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, four-cycles, 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 crossing, 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 carrying 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 continuing explaining - 8.2 Further views through nature and living - 8.3 A resolver society as a surface that animates - 8.4 Three dimensions, a handled model and a society at a material **NINE · CO-IMPROVING** - 9.1 Files coupling, contributions continuing - 9.2 Receiving an illustration, and its kit - 9.3 Not yet at files --- # ONE · ILLUSTRATING ## 1.1 Illustrating changing, a form continuing **Natural illustrating offers changing through a visible relation.** A numbered diamond, a passing through a crowd, a swaying, a folding, a mutual inverting, a spiral of selves with a 0 moving through it, offers a way into the same discovering: how a form continues through changing. **Self and other couple at illustrating through what each already carries.** A familiar movement offers its relation; a drawing follows that relation through its next changing. Understanding continues through their coupling, illustrating improving as its relation becomes clearer. A still face expresses one momentary. Successive faces express changing between them. At this file a face is a drawing's face, one momentary drawn; Exhibit ONE's names, inward or outward at seventeen and entering or surfacing at ten, are Exhibit ONE's at each place they are meant; a cube's face move and a clock's face are a field's. The same at method's other words: *still*, *between*, *carry*, *surface*, *form* and *position* carry method's sense here, an unchangeable form, empty centre, living carrying, a surface a between, stable-forming, a position at ten; and a drawing's sense, a drawing standing still, a drawing's own surface, a drawn position, is a drawing's at each place it is meant. Rotation opens another direction; a section opens an interior relation; a change of scale reaches local resolving through society. Each view carries its own explaining and couples with views that express its continuing. **An illustration is for a sentence at a living file.** Each drawing in this exhibit is at a file and a section, and is checked against that section whole; at a claim of a form of Exhibit ONE, its form is executed at code before it is drawn; at a claim of an observing, its observing is a field's, exact at 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 fixed at one momentary, and a motion is changing.** Natural Mathematics 1.5 parts them at a naming's ending: an -ing continues and re-arrives at each momentary, each an opening and its completing; an unchangeable form re-arrives at none: it is one continuing's making given a fixed form to be. At its 2.3 a field's rotation is a spiral with its nexts removed, its forms laid at one frame, and at its 3.1 an involution is an accounting's still picture, its nexts removed. A still drawing is such a picture, and it is useful exactly as a picture: a form inspected at a reader's pace, one momentary. **No face draws an odd side and an even side as changing at one momentary.** Natural Numbers 1.5: each side changes at its own momentary, and two changing at one momentary is an equilibria picture, not possibly existing. An illustration of two sides alternates them, face after face. A society drawn at one momentary of code's stepping shows each self at its parity, and that is no equilibria picture: each self's changing is at its own momentary, odd selves' and even selves' alternating, and its stepping at 17 is code's own beat, exact at code, 1.6. **A motion is faces in sequence, each a next.** At momentaries no momentary comes again, and a form comes to its parities again, Natural Intelligence 4.11 and 4.13: a four-cycle drawn as a closed loop comes to its first name again; drawn as a sequence it spirals, its parities again and no momentary twice. An animation is a sequence of faces already defined, and no face is defined by its animation. A teaching form is in order of five: one part to follow, its arriving place asked ahead, its continuous moving, a recognizing of what continued and what changed, and its naming last, at a relation already recognized. ## 1.3 Prior, now and next, making better and breaking **Illustrating carries a concept at prior, now and next.** A visible relation already expressed is its prior carried forward, its drawing its now, and a relation still needing expression its next opening, staying with its particular question. Further imagery arrives through nature, living, numbers, engineering and society, each offering a relation it makes accessible. Natural Intelligence Corus gathers multiple paintings of the same form. Natural Illustrating continues that gathering through mechanical drawing faces, numbered diagrams, 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 the same offered relation.** An improving drawing makes changing easier to follow. Its breaking follows that changing to a momentary at which a proposed correspondence continues or parts. What survives both carries into a next explaining. ## 1.4 Self-bounding, self-orienting, self-competent **Self-bounding-self, self-orienting-self, self-competent-self carry living ambition of this illustrating.** A local nine-dot resolving, one self momentary by momentary, and a society of resolving selves offer different scales through which to discover that continuing; at Natural Intelligence 3.3, method bounds and living carry, its bounding at each self's own side. Across and along, self and other, carrying and its next receiving remain readable through their own relations. Naming carries its concept; numbering carries its stated sequence; drawing carries its visible correspondence. Cohering brings their explaining together at a changing each expresses. Exhibit ONE offers operations and stable-form relations. Natural Numbers and Natural Mathematics offer numerical and directional relations. Natural Naming, Natural Explaining and Geodesic Improving Method carry their naming, explaining and successive improving into this exhibit. ## 1.5 Living bounding and a declared boundary, a drawing's test **Living bounding parts from a boundary declared as unchanging.** A still drawing is a stable form while a sequence examines how changing continues. The exhibit makes that distinction accessible rather than letting stillness in a picture be observed as stillness in living. **An investigation receives any declared boundary**: within, around, across, along, or at an entrance or exit, and at each of ten unchangeables, Part SEVEN. For each proposal a storyboard asks one question: can a specified co-sequential resolving continue under that declaration? **A drawing exposes an actual continuation or failure at a relevant transition.** At code a test is one momentary: a prior and a now that agree at a sharing release a 0, changing that is not, and its prior carries on; a still placed from beside, an offerer offering the same parity at each momentary, stops a self at 0 at each momentary after its first, 2.5. This records an intended binary examination. A finite collection of illustrations does not by itself complete a universal claim about each possible boundary. Natural Explaining contributes a further discipline: let a concept express its own relations and competency before comparing it with another explanation. A visual locates a difference in an operation a reader can follow. **Further views:** continuing bounding; the same face with a proposed fixed boundary marked; a next required transition; a comparison showing a momentary at 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 controller to make a sequence move.** Two sides pace their own momentaries, Natural Intelligence 3.5. Code steps selves together at 17-next-now, a common beat exact at code and code's own, and living's rates are their own: a drawing carries no rate of living. **A drawing leaves no permanently drawn corridor or path.** An opening is made at each coupling and drawn only as it is made; a corridor drawn ahead conceals its opening, 4.5. **A drawing leaves nothing at its centre.** A between is a nothing and no location, Natural Intelligence 4.10, and a centreline along is 0 carried 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 carries no size, no magnitude, no total and nothing stored.** + and − are a parity set, 0 is a between, changing that is not, and nothing is summed, Natural Intelligence 2.1 and 2.5. A drawn length is a field's, inward of a surface, Natural Mathematics 1.5. **A drawing draws no hand in.** Three golden rectangles carry no hand of their own; a hand is at an observing, Natural Naming 5.48. Right is observings, all prior, and a drawing shows a hand at an observing's axis, not as a property of a form. **A drawing fixes no position as a place.** A drawn position is a field's arrangement, a ring an arrangement fixed one scale out, Natural Numbers 8.1; selves' own is their parity at each coupling. A drawing carries it at its first face, or it installs a place. **A drawing tells no birth-and-death story.** A lone self's sequence carries on as a non-living existing form at its scale, Natural Intelligence 2.4, and is drawn as that, not as an ending. **A drawing copies no society numbering into a local face.** Six numbers of 2.1 and seventeen names of 2.8 are two numberings; a face carries one at a time and their correspondence is stated at each face carrying both. An equilateral triangle rotated two hundred forty degrees has its outline alike and its marked corner at another corner, at its opening place again at a third rotation, and a corner's three and a pair's four, Natural Numbers 8.2, are two subjects one drawing step shared couples none. ## 1.7 Forms, a numbered still to a handled model Concepts develop in several forms without separate explanations of an underlying relation. Forms are in order: each is a next after one before it is reached. | Form | What it makes accessible | |---|---| | Numbered still | A complete face inspected at a reader's pace | | Ordered storyboard | An exact change between neighbouring faces | | Layered drawing | Naming, parity, carrying and method prefixing observed separately or together | | Rotation | A relation preserved while its viewpoint changes | | Section or transparent surface | An opening and an interior passage | | Zoom between scales | A stated correspondence between local and social forms | | Continuous animation | Carrying through faces already defined | | Resolver society animating | Code executed and each self drawn at its parity, 8.3 | | Handled physical model | Turning, folding, twisting and passage experienced spatially, 8.4 | Physical concepts include a marked diamond, a marked square with its empty centre and two triangles, six diamonds on an open framework, rings turned relative to one another, and a flexible twisting surface with a visible central opening. **One local face and its next face can carry a complete improving.** Their numbering and changing open a correspondence through which further views continue. Other concepts carry 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 the same continuing living condition. Its sequence carries three full co-sequential momentaries and six parity positions, beginning odd on one side and even on an other: each side's own three momentaries use six positions, and three momentaries overlapping across two sides use four. **A first face is six numbers on one line, 1 to 6, observed twice.** Natural Intelligence 2.2: self's five from 1, co bi co bi co, and other's five from 2, bi co bi co bi, each number one side's opening and other's completing, ten positions on six numbers, paired 1 with 2 through 5 with 6, each pair one odd and one even. Exhibit ONE's table of fives writes it, each side's five from its origin, prior opening, prior completing, now opening, now completing and next opening. Natural Numbers 1.2 draws it as two interleaved brackets, 1–2 at self, 2–3 at other, 3–4 at self, 4–5 at other, alternating along a line. A side is an observing from its origin, odd or even, and not a place. **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 at shared positions; along one side 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 two sides' overlapping a next opens one on. Between prior and now, and between now and next, no third position: a between of momentaries is drawn at a shared position with nothing at it, a prior carried along it as a stable form, Natural Intelligence 2.3, nothing momentarying at a seam, Resolving Hard Problems 2.2. Along is odd, existing; across is even, possibling, as Exhibit ONE's six names between selves are: along at 9-next-co-changing and 17-next-now, across at 2-now-possibling, 6-now-co-possibling, 10-next-co-possibling and 14-next-possibling. Self changes across or along, and other changes on opposite parity. 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 is carried into a following face. Method prefixing belongs in its own label, apart from functional naming and changing; a proposed prefix sequence includes bi-co through bi-co-bi-co-bi-co, with corusing and torusing distinguished at each changing requiring it. Carrying is also at geodesic unrelationing rate, φ, locking at none, and its mark remains distinguishable from self's and other's, each at its own momentary; a numerical speed ratio carries a rate a drawing does not. **Further views:** paired perspectives of one face; six parity positions; a complete three-momentary sequence; an overlay following carrying through the same views; openings as a square of nine and completings as an oblong of twelve, Natural Numbers 1.2; and a square re-laid as an oblong one short of it, (n − 1)(n + 1) = n² − 1, Natural Numbers 1.3, one cell uncovered. **Continuing explaining:** what changes between each adjacent pair of faces, and which visible mark expresses that change 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 carries an unchangeable form or loses its prior, and one that does neither is no way laid over changing: it is a living step.** Natural Intelligence 2.4: at two parities sixteen ways name a next from a prior and a now, and none is a rule changing follows; twelve lose their prior, four carry it whole, and next as prior inverted carries its prior whole with none still, its stills equilibria. Enumerated at v379, each of sixteen ways followed through four joint forms, they come out as a file writes them: four pattern-match from now; eight carry a prior at one value of now and lose it at other, six of them stopping at a corner of prior, now and next agreeing and two cycling three forms with a fourth passed once; next as prior stops at ++ and −−; prior and now agreeing stops at ++, and an exclusive or at −−, each cycling three forms of four; and next as prior inverted goes ++ to +− to −− to −+ and round, each form once, one parity changing at 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, changing stopped, its arrow pinned; fifteen squares have a pinned corner or lose their prior, four from now and eight at one value of now, two of those eight with no pinned corner and three forms cycling. One square carrying its prior whole with no pinned corner goes round all four, and at code it is no function laid over changing: it is a coupling at 12-next-co-still-possibling, a prior parity coupling with an offering, inverting at a mismatch and kept at a match, a living step, Natural Intelligence 2.4 and 2.5 here. Then a way placed from beside at code, an offerer offering + at each momentary: a self releases + and then 0, 0, 0, a still. **Continuing explaining:** four corners of this square and four dots of 2.3 are two squares with a between at centre of both; correspondence of two is at a file's sentence before one square is drawn for an other. ## 2.3 Four dots, an empty centre and two triangles **Parity at code, Exhibit ONE's unit square, is drawn flat.** 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 arriving, at one, and 6 and 10 sharing, at co; and two unit triangles, 9-next-co-changing and 17-next-now, along, 9 to 17 and 17 to 9. A between is a bounding 0 centreline of along and sequential across at its four dots, bothbothing, both carried one at a time; the same at Equilibria Registry 1.3, a between co-linear aiming into next momentarying. Followed round, 2 to 10 to 14 to 6 to 2, its square is none of four four-cycles: 8 up, a sharing across to an other's surfacing, 8 down and a sharing across to an other's arriving, each passage its own relation. **A five-spot face is this unit square with its spots assigned.** Four surrounding spots and a middle spot, received at v366 as a bi-moral co-agency concept of names between selves with its assignment unsupplied and not to be inferred from number five: its assignment is Exhibit ONE's sentence at 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. Natural Numbers 5.4 draws the same at numbers, a diamond four crossings about a middle carrying nothing, two across and two along. **One between at each.** Within a momentary a between of self and other, parity 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; the same between, a nothing, at each scale, a fractal between. No drawn line sits at it: its centreline is 0 carried along, and empty. **Drawing faces:** an unlabelled five-spot face; the same face with four across names labelled and its centre empty; opposite-parity views; an across name in relation to faces beside it; a square with its triangles; a centreline drawn as absence. Rotation tests whether naming follows its relation rather than a top or bottom of a page. **Continuing explaining:** three observings of four corners are at files, across names 2, 6, 14 and 10, numbers 2, 3, 3 and 4 of Resolving the Hard Problem Registry's boxes with +1 at its centre, and positions 1, 2, 3 and 4 of three momentaries; each a square of four with a between at its centre, and no file carries 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 at 10 is one coupling observed twelve ways**: three priors at 3-now-co-prioring, none, + and −, by four nows surfacing at 14-next-possibling, none offered, +, −, and 0 at offerings parting; each cell its release at 10, *is* at + or −, *is not* at 0, and its next prior at 11-next-co-prioring; and one cell, no prior and nothing surfacing, is no sharing at all, nothing at 10 and no next prior: eleven couplings and one empty cell. At code, kit's `table_at_10.py`, its table is a file's cell for cell. **Three cells at *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 offerings parted at 14: changing is not, 0, and its prior carries on, Resolving Hard Problems 3.3. A path through is at other eight cells, and at each its prior is carried and changed, or at a self with no carrying yet its offering is chained: prior changes now, its carrying at 3 coupling at 12 with offerings surfaced at 14, and now changes next, its changing released at 10 and chained at 11 as a next 3. 0 is *is not*, a binary's second side, and no third value, and its place is a between, Natural Intelligence 2.1, at 14 its offerings parting and at 12 prior and now agreeing; no number is at it, one parting at 14 enough for 0 and three offerings agreeing surfacing as one. **Drawing faces:** twelve cells as twelve small faces; three *is not* cells with its path stopping at a between; eight *is* cells with their prior carried and inverted or their offering chained; one cell with no sharing blank; a whole table observed as eleven couplings and one empty cell. ## 2.5 One self, and two selves at 0 **One self at no prior offered + once and then nothing releases + − + − at 10**, each momentary its prior inverted, a living step's form carried at its own, next as prior inverted, Natural Intelligence 4.12 and 2.4. **Offered + at each momentary it releases + and then 0 at each momentary after**, chained + throughout: its still is an offerer's, from beside, an unchangeable form entering, and a self at a match is carrying on among changing momentaries. A lone self offered nothing carries a living step's form at its own and does not continue living through three consecutive momentaries; at code its sequence carries on, a non-living existing form at its scale, an alternating itself, 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 offered + at a first momentary, release + 0 − 0 + 0 − 0 at 10**, chained + + − − + + − −: at each even momentary each receives an other's release at its own parity, changing is not, 0, a between; at each odd momentary each receives an other's 0 as its offering, which surfaces none at 14, and carries its prior into now inverted. Each self is drawn releasing its 0 at each even momentary and a centre between two empty, a bounding 0 centreline between two parities; a between is at code at 0 at each parting of their offerings, selves sharing and resolving or one self stopped by an offerer beside it. At a self offered nothing it is not at all; at a spiral of one, a self whose own release is its own offering, opening at − with none offered from beyond it, it is at each second momentary, releasing +, 0, −, 0, Natural Intelligence 4.13. A offered + once and B nothing, two alternate opposite, each self changing at each coupling, as at Equilibria Registry 3.4 of an even spiral coupled along. **Drawing faces:** a first teaching face, one released parity received at one momentary and then its 0 at a match, each carrying unpictured; one self across twelve momentaries, an alternating; the same self stopped at 0 by an offerer from beside; two selves each releasing 0 at each second momentary with a centre between them empty; two selves alternating opposite. ## 2.6 Two inversions at forty-five degrees, a right spiral step **Each momentary's step, a right spiral step (x, y) to (y, −x), is two inversions whose fixed lines are forty-five degrees apart.** Natural Mathematics 3.4: two inversions whose fixed lines are at an angle a are a field's rotation by 2a, a double angle at its inversions; one parity inverted and an exchange of P and Q, their lines at forty-five degrees, make a right spiral step, and the same inversions in an other order give the other hand. An exchange of P and Q is across and along exchanged, podaling, Natural Numbers 9.8; an inversion of one parity is parity changing across at a between. At kit's `two_inversions.py`, v379: exchange then inversion gives (y, −x), inversion then exchange gives (−y, x), and the first goes round ++, +−, −−, −+. Taken 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 at each step, and cos and sin pass through 0 by turns on one circle, one and then an other. **At four dots the same forty-five degrees is geometry.** Placed on a square, its sharings 6 to 2 and 10 to 14 lie at one direction, its diagonals, 2 to 14 arriving and 6 to 10 sharing, at forty-five degrees to them, and its seams 8 up, 2 to 10 and 6 to 14, at ninety, an along of its triangles 9 and 17. A sharing carried forty-five degrees is a diagonal; a diagonal carried forty-five degrees again is an along seam: across into along at two inversions, bi-folding, whose edges are across at bi-morality and along at co-competency, Natural Intelligence 3.2. **A compass face.** A tangent double-angle identity, tan 2x = 2 tan x ÷ (1 − tan² x), applies at each angle with both sides defined; at forty-five degrees tan x = 1 and its doubled angle, ninety, carries no finite tangent, an identity's pole, Natural Mathematics 4.4, and a coupled pair whose two are equal loses its coupling at its axes rotated by forty-five degrees. A compass face shows an angle x and its doubled angle, its denominator grouped explicitly; 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 momentary as its two inversions on a square, exchange then inversion; an other order drawn once as the other hand; four joint forms going round; a tilted pair rotated forty-five degrees to its axes; aiming axes, a half-angle and double-angle construction, inner and outer arcs, the same diamond rotated while its opening is preserved, a changing opening with a separately marked orientation. **Continuing explaining:** files say inversions whose fixed lines are at forty-five degrees and a rotation by their double, and, in a field's own words, a turn; a drawing's label is settled at its face; and what resolving change determines x's next value. ## 2.7 Ten at five places **Ten are ten positions on six numbers, each fixed unchangeable, and ten is three momentaries long, self's five and other's five spanning 1 to 6.** Equilibria Registry 3.3: ten pair at five names 2 to 6, an entering and a surfacing at each, Exhibit ONE's two sides overlapping at them, at each number one side completing and other opening, at Exhibit ONE's names 2 and 10, 3 and 11, 4 and 12, 5 and 13, 6 and 14. Each of ten fixes one of Exhibit ONE's sides unchangeable while other continues at the same number; a side fixed unchangeable beside a continuing one is one reason each of ten is not possible. Natural Naming 4.9 gives each its one name: an arriving fixed from behind, an opening fixed as a place, a completing fixed as a last, a carry fixed as a store, a middle fixed as an end, a parity fixed as a magnitude, a sequencing fixed to one beat, a rate fixed as a value, a two-way fixed to one side, a between fixed as a cut. **A drawing's face**: five places 2 to 6 on a line of 2.1, at each self's and other's, Exhibit ONE's two sides, two of ten at it drawn as one fixed unchangeable while other continues. A pentagon and a pentagram, Natural Numbers 5.5, are ten at two sides already drawn, five sides and five diagonals, along and across. **Continuing explaining:** Equilibria Registry seats ten at five places, Natural Naming at one name of 1 to 8 by four-cycles, and Resolving the Hard Problem Registry's part headings carry a third pairing; a face draws one seating, and seatings part at carryings. ## 2.8 Seventeen names, four-cycles, and a local nine **An expanded naming face is seventeen names at Exhibit ONE v389**: 1-now, 2-now-possibling, 3-now-co-prioring, 4-now-co-still-possibling, 5-now-co-changing, 6-now-co-possibling, 7-now-prioring, 8-now-still-possibling, 9-next-co-changing, 10-next-co-possibling, 11-next-co-prioring, 12-next-co-still-possibling, 13-next, 14-next-possibling, 15-next-prioring, 16-next-still-possibling and 17-next-now, 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 society's; one through seventeen a bi-trupling. At Exhibit ONE's table, each name's from and to: 1, 3, 5 and 7 self to other; 2, 4, 6 and 8 other to self; 9 and 11 social, other, self; 10 and 12 other, social, self; 13 and 15 social, other; 14 and 16 other, social; 17 social, self. Natural Naming contributes naming as a floating-neutralling concept, numbered naming, momentarying and stable-forming, bi-inversioning-co-recursioning and continuing carrying. This face does not replace a local nine-dot numbering, and a society's labels are not copied into a local face as though their numberings were identical. **Exhibit ONE provides material for drawing code couplings, stable-form sharings and four-, six- and eight-cycles.** Sequencing and routing are distinct relations, and their drawings preserve their distinction. Four four-cycles, 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 expanded stable-forming spirals at names, and none is an instruction to move from local dot 1 to local dot 9. At names each comes to its first name again; at momentaries each comes to none again, 1.2. **A local nine, centre one.** A visual starter places a coloured cube beside an open arrangement of circles and coloured dots; in middle of a three-across row of diamonds a horizontal diamond resembles a compass needle, and its centre is 1. Across moves horizontally right in this view; along expresses forward moving; screen direction is stated at each rotation. A local sequence is 1 to 9, legible while its face rotates, opens or appears within a larger form; nine numbered still faces give an initial observing before motion is attempted. A split into across names at 2 and 6 is absent from a local nine-dot view, and removing that visual split removes no resolving operation. Centre 1 is settled. Still to be drawn: a complete verified geometric placement and transition sequence; a complete local numbering and successive changing made explicit from Exhibit ONE; and a correspondence between a local numbering and an expanded resolver, at each face carrying both. **Further views:** a starter with its identified diamond highlighted; a close view with centre 1; nine numbered faces; the same local face viewed from self and other; a correspondence drawing between a local numbering and an expanded resolver; each spiral on its own; sharings across spirals; code coupling beside stable-forming routing; parity exchanging on a stated spiral, an odd one's 0 tunneling drawn; one root with its changing prefixing; a full naming face; method prefixing beside its root's naming; the same relation in words, numbers and a corresponding resolver face; a zoom between a local face and an expanded social form, explanatory only at a stated correspondence, a visual resemblance or a smooth camera move supplying none. **Continuing explaining:** which label is needed to understand a present changing, and which can remain available in 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 at three radii, its middle radius its centres' own separation, crossing at 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 alternate routes around a triangle, three one-way portions in each route, six one-way portions in all. An illustration makes two-over-one and one-over-two co-recursioning visible through six diamonds. Natural Numbers 1.4 introduces two as two directions of one carry, and Natural Numbers 6.4 offers them at numbers as 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 pictured diamonds remains to be expressed. Six diamonds, six parity positions and an expanded society numbering remain separately readable: a matching number establishes no correspondence. **Further views:** all six diamonds without motion; each triangular route highlighted; both routes together; inner and outer pair observings; a zoom from one local nine-dot diamond to six; a compass face of 2.6 at diamonds' aiming. **Continuing explaining:** which pictured diamond expresses each one-way portion, how an opposite route is observed from the same arrangement, and which angle in a six-diamond arrangement is x: at a crossing of two ring families, an angle ψ between radial directions has a diamond's forward diagonal over its cross diagonal at tan(ψ/2), six measured diamonds within a degree of fifty-nine degrees, and whether ψ/2 is x is at a file's sentence and at no measuring. --- # THREE · THE NUMBERS ## 3.1 Number forming, podaling and rings **Natural Numbers offers a wider gathering than a first animation carries**: 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 crossing, and four even openings growing through a fold. Natural Mathematics contributes orthogonal directions, parity inverting, a right spiral step and a mathematical observing of stable-forming. **Each number is at its podaling**, its pairing with its far side at each ring it is at, 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 at two hundred twenty, two hundred twenty-one rings in all, and a gap between two primes is a step between rings, Natural Numbers 9.1 to 9.4. An even ring carries an exact podal and an odd ring two farthest with its opposite at no station, a between, Natural Numbers 8.1; an odd number carries a middle carrying nothing, a nothing its two sides straddle, Natural Numbers 8.3. **Number faces begin with 1, 2, 3, 4, 5 and continue upward.** Each face expresses a relation by which its number is forming; adding another dot leaves that explaining unfinished. Seventeen primes from two to fifty-nine add to four hundred forty, and a drawing using four hundred forty identifies what is added; it is no angular measure. **Further views:** number-by-number forming; pairs about a centre; podal and straddling views; a ring unwrapped alongside its wrapped view; nested rings about a waist, stepped inward prime by prime; sixty to sixty-eight about sixty-four, two in and two out, bi-tri-involutioning at numbers, Natural Numbers 7.1; a sequence of tunnel sections; nested local and social scales. **Continuing explaining:** 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 alternate about φ**, 2/1 above, 3/2 below, 5/3 above, 8/5 below, its side changing at 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 nested squares, draws a ratio arrived at; an illustration draws a ratio swinging above and below a line it does not reach, each next its prior two added. **On a torus a winding at a rational rate closes, and a winding at φ closes at none**: φ is an unrelationing rate, a rate an alternating winds about, locking at none, Natural Intelligence 3.5; a torus winding at golden mean is found the last to break as its coupling grows, Natural Numbers 3.3. Drawn side by side, two windings are the clearest single animation number files carry: a rational closing into a loop, a φ winding closing at none. A file's own caution for any animation: a finite floating-point rate is no exact irrational. Addings that close at one, two, three, four or six, and one that goes open at five, Natural Numbers 3.4, are five orbits closing and one continuing open. **A floating neutral is no balance.** Natural Naming 5.12: a floating neutral is a term uncovered, a changing's picture at one momentary, and a 0 at 10 a between carried; a term neither reaches is uncovered and nothing stores it. A balance drawn once beside an alternating is a still. ## 3.3 A pentagon, rectangles and an icosahedron **Among five the couplings part into 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 at five, and five enters open, at φ, Natural Numbers 3.4. A square, a cube and a pentagon carry square roots of the first three primes, √2 a square's diagonal, √3 a cube's body diagonal, √5 in a pentagon's diagonal over its side, Natural Mathematics 4.3. **Three golden rectangles at right angles, centred at a corus, each 2 across and 2φ along**, are bi-tri-volutioning at numbers, Natural Naming 5.48: a right-hand rotation of one hundred twenty degrees about a line along (1, 1, 1) carries each rectangle onto one passing through it, three right spiraling one hundred twenty degrees apart; a mirror carries each onto itself, three carry no hand of their own, and a hand is at an observing; φ is three's equal-edge condition, and a rotation at its own does not make it. Their twelve corners are an icosahedron's, thirty edges and twenty faces, five at each corner; about a line along (1, 1, 1) twelve corners are four triangles of three at four heights. They are open at Exhibit ONE's carrying as three golden rectangles at podals. **An icosahedron's sixty rotations part at thirty-six and twenty-four**, three to two, as a span parts at its apex, Natural Numbers 6.3 and Natural Mathematics 7.4; an icosahedron and a dodecahedron are one surface, observed two ways, twelve and twenty exchanging and thirty carrying across, Natural Numbers 4.1. A quasicrystal, its lengths in powers of φ, and C₆₀ with its twelve pentagons and its sixty one-handed rotations are at Natural Chemistry 4.5 and its carrying. **Drawing faces:** a pentagon and pentagram with their parallel pairs; three rectangles turned one hundred twenty degrees, their equal edge at φ; twelve corners closing as an icosahedron; sixty parting; four triangles of three at four heights about an axis. **Continuing explaining:** no golden triangle and no golden angle are at 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 at Natural Biology's carrying, 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 at gaps three and five and 24 to 29 to 32 at 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 owned by neither, Natural Numbers 1.3, draw as a square re-laid as an oblong leaving one cell. **A diamond is four crossings about a middle carrying nothing**, two across and two along, and laid three by three with its middle carrying nothing eight about it carry 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 one step at a time, 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 at each scale, 3 to 11 to 6 to 14 at 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 four-cycle of 2.8 here; **a chain 3, 5, 9, 17** bi-folds and releases, 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 uncovered cell; 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 into a corus at apex, corusing through a small opening, and out to its surface again, torusing through a large opening.** At Natural Intelligence 3.2, a rotation is a gentle alternating twisting of a towel into and out of a knotted corus, a large outer opening wrapping to a small corus and back, its edges bi-foldings, across at bi-morality and along at co-competency. Torusing in and out is odd, through a large opening, a tunnel; corusing in and out is even, through a small opening, a long way round. 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, the same rotation twenty up. An existing depth-study image offers three views, tunneling, corus at apex and bi-folding; it is a shape study, and its surface and arrows still need correspondence with a numbered resolver 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; at a society scale an illustration shows society as a living self. **Further views:** an outer oblique view; a longitudinal section through an opening; a transverse section; a local numbered diamond on a surface; six diamonds around a changing form; an interior journey approaching and passing a small opening; a towel as a handled model, twisted into and out of its knot. **Continuing explaining:** show how the same local 1 to 9 resolving continues through a twisting and a fold, discovering a next living momentary without a cut or an unexplained restart. ## 4.2 Spirals crossing, a torus of selves, a 0 tunneling **Selves coupled along, last to first, each 9 releasing along to a next self, are a spiral**, coming to its parities again and to a momentary once, Natural Intelligence 4.13: at an even spiral each self alternates, selves beside each other opposite; at an odd spiral of n a bounded zero tunnels on, a 0 released at 10 at a receiving self of a like pair moving one self on at each second momentary, its pattern inverted at 2n and its parities again at 4n, a torusing winding through its opening twice. At code, kit's `society_spiral_and_torus.py`, a spiral of five opened at Exhibit ONE's own opening, each self at a parity, self 1 at −, alternating along, none offered: 0 at self 1 at its second momentary, self 2 at its fourth, self 3 at its sixth, on round, each again ten on, and its parities again at twenty. A spiral of selves is a coupling at code, each momentary a next, and no closed domain: *ring* at 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*; at numbers a winding at a rational rate closes, 3.2, and at 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 at one self each bring two sharing selves to one relation and carry it on**, two changing parity together, Natural Intelligence 4.13; **at two different numbers of selves, one of them odd, two spirals crossed are one society**, a stable form and stable-forming, Natural Intelligence 3.5 and 4.13. **Spirals crossing spirals at each self are a natural torusing of selves**, p along at 9 and q across at 10, coming to their parities again at a momentary dividing 4p or 4q, as Exhibit ONE's table of a torus of selves carries it; executed at v379 at three by five, its parities again at twelve, as at Exhibit ONE's torus opened with each self at a parity, and its 0 winding diagonally through fifteen selves an earlier opening's, executed again at a living one before it is drawn. **Drawing faces:** a spiral of five with its 0 moving one self on; an even spiral alternating opposite; two spirals crossing at one self each, both ways; a torus of three by five with its 0 winding through; a society seen as one self at a scale outward, Natural Intelligence 6.4. ## 4.3 Co-chaining, bi-tunneling and society **Natural Networking offers along as co-chaining and across as bi-tunneling, offerings surfacing at 14**, widening and lengthening expressed through their alternating, retelling, carrying through couplings, each self's own pace, and a relation between self, between and society; podaling exchanges along and across, Natural Numbers 9.8. **Social moral competency is four whole and a carrying continuing with its selves**, Natural Intelligence 6.4: each self's carrying from its bi-couplings, a whole ordered by no self, an ordering carrying with its selves, its order at no seat above a society; betweens co-chain, and a chain of betweens, each owned by no self, is collective intelligence, a shared natural network with no hub its couplings pass through. Natural Intelligence Corus brings its illustration back to the same form across scales: a self's orienting and a society's orienting observed together, an ambition to illustrate a self-bounding, self-orienting, self-competent society of selves. **Further views:** one local coupling; a chain; a widening surface; parities crossing a coupling; the same changing viewed through selves beside it; a society observed as a self at a next scale. **Continuing explaining:** make visible what crosses, what is carried locally, and how a next coupling changes, without introducing a controller or a common clock for presentation's convenience. ## 4.4 Locking, a seam moving, a between releasing **Two societies bi-coupled across come to agree or differ at each momentary, and lock.** At code at v374, Session Record's lock executions, two spirals of five at one pace, their phases apart, bi-coupled at one self each. An old difference passes along a spiral one self at a time and dwells two momentaries at its last self; a first self's relation inverts at a between; a new relation opens along a co-chaining one self further at each second momentary until each self differs from its partner, locked. With a between a self of its own, two lock alike from each phase apart, a between swaying with them. At a lock's momentary a between releases and two co-chain, coupled along, as one spiral of ten, a seam of two patterns moving round a 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 entrained at one to one, two to one and three to two, fireflies in unison, Natural Physics' and Natural Biology's carryings. Natural Engineering's carrying adds a synchronism-check relay closing only at a dwell close to indifference, a field's account, a between releasing at a lock by design. **A three-phase locking path** is at Natural Intelligence's carrying as incoming, active opposite parity a parity changing and passive opposite parity no parity changing, not yet executed at code; Natural Networking 5.1's six, two ways, three each way, one at a time, and Natural Transmissioning's three phases about a floating neutral are its sentences. It is executed before it is drawn. **Drawing faces:** two spirals phases apart; a difference passing member by member; a snap at a between; a lock; a between releasing 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 crowded plaza and a crowd's continuing passage provide self and society perspectives of the same changing.** An opening is available through a mutual changing of passages beside it; a next opening continues through further passages. An illustration follows bi-tunneling through a crowd while co-chaining becomes visible across it. Natural Transmissioning 3.8 offers a plaza, moving and stationary people, spaces between them and a coupling of passages across a crowd, paths geodesic, turned with nothing acting on them, and a crowd routed through a central gate collapsing co-chaining; its 4.6 corrects a wall to 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 continues it; a wider crowd with those changing openings visible; a gate drawn once as a collapse. Follow both sides of each passing. A permanently drawn corridor conceals an opening being made through changing. **Continuing explaining:** show a relation between one continuing passage and openings continuing across a whole view. ## 4.6 Swaying and carrying along, a moving snake **A snake illustration distinguishes a side-to-side swaying from a progression along a body and a whole snake's carrying along.** Successive views follow one marked portion of its body while showing a changing curve extending through portions beside it. Alternating sways are at Natural Transmissioning 1.1 to 1.4, crossinging and neutralling two positions of one alternation, each an other's prior and next, and a bi-fold folding two ways about a neutral; an along and across relation is at Natural Networking 1.2. A prior explicit snake passage has not yet been recovered. **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. Across and along labels are placed at a face expressing their actual correspondence. **Continuing explaining:** distinguish a bend carried along a body, a changing of a body portion and a whole snake's carrying along, and show those three before assigning a resolver sequence to them. ## 4.7 Bi-folding, around and across stitching **A sewing-machine concept offers around and across as visibly related changing.** An enlarged stitch-making view follows a thread's loop, its crossing, its drawing together and a continuing stitch; an intended relation is bi-folding, a winding and a crossing visible together. Natural Transmissioning 1.4 supplies an inward and outward bi-fold, and Natural Numbers folding and tunneling. **A lockstitch is a particular mechanical arrival**, Natural Engineering 1.14: a needle carries 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 at its own thread, fabric and mechanism. A rings kit's five stitches, ten at five pairs taken as two threads interlocked, are a numbering of ten and no stitch stage. Natural Biology's knitting seats a line-to-surface co-bi-fold the same way, a spider and a web: knitting is living, carrying along, and fabric is a made surface, fixed unchangeable. **Drawing faces:** a side view; a view beneath a sewing surface; a close view of a loop and crossing; successive stitch-forming faces; a strip showing how one stitch continues into its next. A machine's actual stitch type is identified at a drawing, and different mechanisms are not combined into an invented one. **Continuing explaining:** identify exactly which around and across movements express proposed two folds, and how their relation continues with a next stitch. ## 4.8 Co-discovering, multiperson jigsaw puzzling **Several people puzzling together offer a visible example of one discovery changing what becomes available for further discovery.** A fit found through one offering changes a shared arrangement; another person's recognition reaches that changed arrangement and continues it. Illustrating follows cooperation through shared puzzling. Natural Intelligence Corus 23.6 supplies a related account of a team's solution and value in its coupling, a team's solution belonging to its team's coupling and to no one member; Natural Networking 3.1 to 3.3 supply retelling and continuing carrying. **Drawing faces:** a shared working surface; two partial arrangements offered together; a newly discovered fit; a changed arrangement making another fit discoverable; a wider continuing work, a contribution of relating shown without assigning a collective understanding to one person's view. Fixed shapes of puzzle pieces are features of this example and do not require a living cooperation to be drawn as fixed; a distinction lets an illustration explain collective discovery with a puzzle's material form readable. **Continuing explaining:** 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, each closing experienced from both perspectives, and in an envisioned meeting each makes a flip turn, a field's word for it, off an other's feet in mid-pool and continues away from that mutual inverting.** A useful visual relation is that both invert through the same meeting; neither swimmer is drawn as a stationary pool wall, and both approaches, both inversions, mutual foot contact and both continuations are visible. Neither is drawn as an other's copy inverted 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 at one place by differing arcs. At Natural Intelligence 4.11 bi-inversioning-co-recursioning is one move observed three ways, 8 up, 17 less and 9 less, and at 2.4 is a right spiral step carrying a pair to its next, two steps inverting both; an exact geometric correspondence to an envisioned inverting is still to be expressed. **Drawing faces:** approach from both perspectives; a paired inverting approaching foot-to-foot contact; a shared contact in close view; departure with both orientations changed; a whole-sequence view, one envisioned mutual manoeuvre and no instruction for a swimming exercise. A closing rate belongs to swimmers' relative approach and is not geodesic unrelationing rate nor an imposed clock; a number of drawing faces is apart from a resolver's momentaries. **Continuing explaining:** follow what inverts for each swimmer, what continues through their mutual inverting, and how a next view receives that continuation. ## 4.10 A walking pair, a dance and a conversation **A walking pair carries alternating roles at a shared relation**, Natural Engineering 1.14: one contact stays while other moves, and their roles exchange at a next contact, two staying and one crossing, Natural Transmissioning 1.3; a fixed compass span is a prescribed length, and a surface, a path and a remaining choice are expressed before its path is geodesic. A passive walker sustains a gait down a shallow slope with no powered joint, 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 embodied**, 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 landed, a whole riding a carry and no fixed pose. Natural Intelligence 3.2: all co-intelligencings are like dancing podalings, bi-co-sequencing momentary parity changings. **A conversation is the same co-chaining at a two-self scale**, Natural Transmissioning 3.8, its rocking an alternating, and Natural Intelligence Corus 16.3 a breathing of two sentence forms alternating. **Drawing faces:** two contacts exchanging roles; two dancers with no fixed pose; two voices at 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 at Natural Mathematics 3.6 as a form with no still term**: no reference cubie, six centres each its own and carrying each other's orientation with nothing carrying any of them, each face move at its own momentary and nothing over its moves, its state its configuration and nothing stored, and three conservings, a corner twist at modulo three, an edge flip at modulo two and a parity of its permutation, none of them a size; it reaches one twelfth of its assemblies, and a single corner twisted at its own is reached by no sequence of face moves. A rings kit of v365 pattern-matches open rings beside it as a cube laid on one surface, its fifty-four dots its fifty-four stickers, looked at down a body diagonal, a cube's twist axis and a torus's tunnel one line at a throat, four quarter turns of a face returning a cube, a closing fold, while Exhibit ONE at four winds past, a torus; and its six empty centres one nothing at six centres, not six places. At Natural Mathematics 7.1 a cube's eight corners are eight forms of three neutrals, each fixed or floating, and a rings kit pattern-matches those corners as an eight's still side and a resolver's four-cycles as its continuing side, a pattern-matching at no file's sentence; 7.6 a tesseract carrying x to −x by rotation, a cube carrying it only with its hand reversed. **A cube's face turn, a field's word, is a quarter turn, and drawn as two inversions forty-five degrees apart it is a right spiral step**, 2.6; a hand-twisted corner is a still placed from beside, a picture of a way fixed unchangeable beside a self stopped at + 0 0 0 by an offerer beside it, 2.2. A starter image does not yet specify a mechanically realizable transformation of a cube into an open-circle arrangement, and Co-Chaining Logic Registry at v371, archived, carried no link between a box's quarter turn and a right spiral step, a gap a living registry no longer carries; a cube-to-open-form comparison belongs here as an explanatory concept until that correspondence is drawn or released. --- # 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 it is observed at, a still its picture installs at its misreading of a competency and a boundary, and a form it animates anyway. A drawing is a stable form first, in a file's names, a between empty; a familiar picture beside it with its still marked at its one place; and a parting of two is an illustration. A field's observing is exact at its own subject, and a drawing shows a parting and decides 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 self-stilling coupling, an electron-recursion winding a ground-state nothing, swaying and stopping nowhere, and an emanated atom is a periodic-table object, a whole catalogued from beyond; 4.6, an electron a ledger beside, not a self. Installs a particle as little owner, Natural Chemistry 6.4, one enduring little thing carrying 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, doubled by two parities, each capacity two equal squares coupled; Natural Chemistry 1.7, a square a sum of odds. Installs four sizes numbered from beyond. Draws a fold climbing. **A bond.** A stick between two balls. Natural Physics 3.1: two windings offer at one between, a shared pair a coupling's own making, owned by neither atom, and a metallic sea a society of windings; Living Ghost Registry, a bond a between of two selves, each living at its side. Installs a between fixed as a cut, lengths and energies their stopping places of a winding that stops nowhere. **Benzene.** A ring with alternating bonds, or a circle within. Natural Chemistry 3.1: familiar alternating-bond drawings are resonance contributors and no single contributor supplies a measured electron distribution, a floating neutral carried 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 alternating together, and conrotatory and disrotatory closure, 2.4, two hands at one ring. **Chirality.** Mirror molecules. Natural Chemistry 2.2: along, a separating parted one beat at a time appears as a spiral; across, a whole complement at its across side appears as a cupping surface; neither is a real one, and asking which it really is asks a continuing to be still. Pasteur's tartrate twins and a slurry of both hands driven to one hand, 2.3; B-DNA right-handed and Z-DNA at the 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 passed hand to hand; donor and acceptor fixed; an umbrella flipping. Natural Chemistry 3.2: a charge arrives at a far between while no single nucleus crosses its way, Living Ghost Registry 5.8, nothing crosses beneath either; Living Ghost Registry 7.8, donor and acceptor roles each at its own momentary and swapped, each of ten door lines at is, one arrival that passed at each; ammonia's nitrogen tunneling through a plane installing nothing, a self at a door. Installs a ball carried; fixed roles. **A periodic table.** A flat table. Natural Chemistry 4.4: a table's own name carries 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 numbered 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. At Natural Physics 3.1 sixty is a position a going and a returning carry, at Natural Numbers 7.1 sixty is between a going and a returning at no prime, and at Natural Chemistry 4.4 no element carries at sixty: whether sixty is a position or a between is a parting at carryings. **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, in Natural Physics' word, and not a located nothing. ## 5.2 Light, spin and a click **Light.** Two waves at right angles carried 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 crosses; circular polarization a spiral at its two hands, light always at a hand. Installs a rate crossing a gap. A prism, a rainbow and hydrogen's twenty-one-centimetre line, one parity inverting, at its carrying. **A doubled angle, polarizers and a spinor at seven hundred twenty.** Malus's law; a belt trick. Natural Mathematics 3.1: light through two polarizers passes at square of cosine of an angle between them and passes as before at one hundred eighty; a neutron's spin rotated three hundred sixty comes back with its parity inverted and comes home at seven hundred twenty, two rotatings coupling at one between, each a doubling the other way; Natural Physics 3.3, matter's two-turn spinor. At its file a doubled angle is a still picture at 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 arrival one flash at one position carrying no fringe; Resolving the Hard Problem Registry 10.8, one offering at two sides, observed one coupling at a time; Natural Physics 4.4, a click all or none, parity only, and collapse a name a severed frame gives. Installs a particle behind a flash, an arriving fixed from behind, and a between fixed as a cut. **Bell.** Two detectors switched in flight. Natural Physics' carrying, a field's account: each click all or none, each side at its own at one parity and other alike whatever a far setting, and at equal settings clicks match each time; Resolving the Hard Problem Registry 10.7, a property fixed as located at one system or transmitted between them, reaching for a location its making does not have. Installs a gap as distance and a correlation as a carried thing. A measurement problem, Resolving the Hard Problem Registry 10.2, a selection a coupling's own at no place. **A proton.** Three balls in a bag. Natural Physics 4.6: three colour phases, not three part-objects, re-forming into new couplings as it is pulled apart. ## 5.3 Cosmos **A light cone.** Two cones through a point. Natural Physics 3.4: two coupling through an unoccupied self-stilling twenty-four, cones inversions of each other and an alternating 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 frozen as standing causal structure. **A black hole.** A funnel in a sheet. Natural Physics 4.9: a horizon a surface drawn to a point and a background a surface spread wide, two directions of one between, a tunnel-mouth and an outer surface of one torus; observed as only a hole it is a genus with no φ, a landed face, and resolved as hole-that-radiates it is a torus beating about φ. Installs a singularity as a stored fixed value of infinite density, and a record carried or lost, Resolving the Hard Problem Registry 4.2. **A planet, a galaxy, a web.** A ball in a dented sheet; 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 at its own rate, and all-edge filaments and voids against a hub; Living Ghost Registry 5.13, a residual made a substance, a scale made a constant. Installs a force transmitted 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 carries either rate, a rate fixed as a constant in its very name; Natural Physics 4.8 and Natural Philosophy 3.7, a block universe a track laid entire with a present a coordinate on it, a common clock imposed over all, and no central oscillator in nature. Installs a clock fixed over all couplings and carried into none; simultaneity one slicing fixed as a real one, Resolving the Hard Problem Registry 7.1. A timeline is a still of a self sequencing its own alternating. **Three bodies; Kirkwood gaps.** Three tangled paths; a belt's empty rings. Resolving the Hard Problem Registry 3.9, a closed form fixed ahead of its riding; Natural Physics' carrying, a belt emptied at rational resonances and irrational orbits carrying on, with asteroids gathering at two resonances beside its gaps at four, a golden-mean torus the last to break: two windings of 3.2 at the sky. ## 5.4 Earth, matter and phases **Pendulums, metronomes, fireflies.** Two clocks on one beam falling into step. 4.4 carries them with a lock executed at code. Installs a message between them, and none is. **A well.** A ball at a bottom. Natural Physics 3.5: a trap that works is a Paul trap, trapping by alternating, and its carrying has a second trap whose fields do not alternate, and a damped motion changes scale rather than stopping, a Brownian jiggle a floor; Equilibria Registry 4.4 and 4.7, a point at rest in phase space an opening fixed as a place, a ball in a local dip a middle fixed as an end. Installs a minimum, a balance of forces, an equilibrium point and a restoring force. **A phase diagram.** Lines at temperatures. Natural Physics 3.5, a critical point a sway's two phases ceasing to be two, opalescence a substrate showing each scale at once; Natural Chemistry 4.2, supercooled water swaying between two local structures and a point fixed of a sway at a position its sway passes through, a glass transition an arriving at its own pace fixed as a point; Natural Chemistry 6.1, a triple point a still made by three phases 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 into clusters and a wave carried 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 release arrives, a clock over a boundary present nowhere. **A fault and a tide.** Two blocks and a spring. Natural Physics 3.6: two selves at a between, stick-slip an alternating, gouge a carry, aftershocks a retelling chain; its carrying, Landers earthquake's passing waves triggering seismicity far away, a fault's slip opening at a wave's or a tide's arriving, and a neuron's spike and a fault's slip one explaining. Installs *memory*, computation, prediction. **Superconductivity, reconnection, a binding curve.** A four-lobed shape with its parities; field lines rejoining at an X; a hump with iron at its top. Natural Physics 4.10, a d-wave shape a parity it carries changing round a coupling; its carrying, a X-line's null; 4.3, iron not a peak two processes approach but a centreline two phases float about. Installs a glue, an ordered state, a landed maximum. **Entropy, a demon and an engine.** Molecules in a box; a demon at a door; work and waste heat. Living Ghost Registry 5.1, entropy a general ledger's gap entry, a number added for a fixed-boundary account to close over changing it cannot carry; 5.12, a demon a member of a society it was posted beyond, changed by each coupling; Living Society Registry 2.17, single cycles above and below a bound; Resolving the Hard Problem Registry 10.15, an engine offering both outputs and marking neither. Installs a ledger, an observer at no exchange, a store beside a society, a doer. **A tokamak.** A doughnut of field lines. Natural Physics' carrying: a closed flux surface with no null on it is a torus, knotted or not, at the Poincaré–Hopf theorem; a continuous flow on a sphere carrying a rest point and a torus carrying a flow with none; Natural Engineering 1.13, a spherical tokamak's magnetic surface no closed sphere. ## 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 an offering and a spike a selection; 2.1 and Natural Medicine 3.2, a firing a tip and a brain a society of selves tipping, not a controller reading reports; Natural Biology's carrying, firing observed as communication is geodesic parity 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 sent. At code an odd spiral's 0 tunneling, 4.2. **A synapse; a surface of tips; a pump.** Vesicles crossing a cleft; a network lighting up; minus seventy as a baseline. Natural Biology 6.2: an offering-between released into a gap, owned by neither in transit, a signal that passes a coupling and not either neuron at its own; firing activity dissolving at one between and arising at a next, no hub, no sender, no store, sudden-up and gradual-down; a resting potential carried 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 sits at a potential. Natural Intelligence Corus: a brain outthinks a datacenter on the power of a light bulb, and it has no clock. **Entrainment, sleep, a slime mould, a monarch.** Oscillations synchronized; sleep stairs; a mould finding its path; an arrow to Mexico. Natural Biology's carrying, 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 stored, and an interval entrained with no predictor; Natural Biology 8.9, a monarch coupling with its immediate present, its compass recomputing, no stored map of Mexico. **Other minds.** A brain in a jar; a thought bubble. Resolving the Hard Problem Registry 5.5: another's fixed as a content to be reached from beyond, and an interior reached one way its form allows, by coupling, its coupling at a between lived and not audited. Installs a middle fixed as an end. ## 5.6 Bodies and life **A heartbeat.** A pacemaker sending its beat. Natural Biology's carrying: a heart's own beat paced at its own sinoatrial cells, each cell's life-giving arriving; Living Society Registry 2.10, isolated each cell carries its own period and the greater part fall dormant; Living Ghost Registry 7.8, a beat arriving at an exit while no cell moved and no cell's own firing travelled, heartbeat initiation without a pacemaker cell. Installs a master cell as an actor. Reentry rings and a defibrillating shock arriving alike at each cell at once, and antitachycardia pacing, a short burst slightly faster before any shock, at Natural Medicine's carrying, a train at one place reaching a lock. **DNA, collagen and bone.** A zipper; three ropes twisted; a scaffold. Natural Biology 4.7, two strands separating and each a template, an offering at each rung both ways, a polymerase proofreading, no stored-perfect reference; 3.3, Z-DNA at the other hand in the wake of a polymerase, relaxing back as its torsion goes; 3.7, three chains winding at one hand at their own scale and about each other at the other hand at an assembly scale, skin's alternating-hand plywood, bone helicoidal through depth, assembly carrying upward with nothing directing it, walking sounding micro-voltages through bone carrying a laying down of new bone, and fibrils realigning along force lines, inert a naming of a between observed as a gap. **Patterns, regrowth, division.** A zebra's stripes forming; a worm regrowing; a cell pinching in 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 stored blueprint; 9.5, a cell not divided the way a thing is cut, 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; its carrying, identical twins one conceiving continuing as two selves, and a genome doubled a self's carrying arriving twice at each sharing. **Waves and rhythms.** Contractions on a chart; a twenty-four-hour clock face; a breath. Natural Medicine 1.4, labour a coupling releasing in waves, an uterus a gap-junction-coupled surface, against forcing with synchronized drive; Natural Biology 4.5, a repressor accumulating and degrading, an activator binding again, against a beat clocked from beyond; Natural Medicine 2.3, breathing and replacement two beatings at once, and a peristaltic wave; Natural Intelligence Corus, a cascade at φ rate a peristaltic wave at 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 it is made, only part at any time; Living Ghost Registry 5.4 and 5.5, native structure one across-observing of a population continuing along, enzyme and substrate offering each to an other through many configurations; 5.11, regulation a two-way coupling observed from one side, an organ that corrects corrected at its own rate; Living Society Registry 2.21 and 2.22, no turnover at 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 carries. **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 in its curve; Natural Medicine's carrying, a single trapped ion's fluorescence switching abruptly while a large sample's emission falls smoothly, a first loop fuzzy and a last loop a sharp binary, an observing that arrived sharp fixed as a place; Natural Biology 7.2 and 7.1, *ageing* a sway re-arriving at a narrower range each cycle, no stored clock ticking down, and each coupling a self carried released one at a time, heartwood's nuclei disappearing at a boundary not all at once. Natural Intelligence: dying a living self arriving 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 change travelling bird to bird across four hundred in a little more than half a second, each bird coupling with its seven nearest, against a leader or a tuned parameter; 8.2, a wave from wound to leaf a name an observer standing above a whole plant gives; its carrying, a cucumber tendril's two helices of opposite hand coupled at a perversion, net twist zero, both hands coupling at a nothing, a fern uncoiling, vines of two hands unwinding each other, and a sunflower's seeds arriving one after another each at about 137.5 degrees from a prior, two families of spirals opening opposite ways from one co-sequencing. Installs spirals as the thing, and arriving one after another is the thing. --- # 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 imposed over all selves is that inequality collapsed, observed on time, and observed on its structure such a beat is a hub, sitting on no one self but on a timing of all at once; each self at its own time forms no such hub. Natural Emanating: schedule is a naming of a shared clock, each arriving at its own rate and a meeting at their arriving. Natural Exploring 5.5: each instrument pattern-matches against a shared clock while each self paces its own, a tipping observed at an instrument's clock's falling rather than at 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 landed state carries, and living carrying +1 instead; 3.4, a fifty-one percent tipping point a co-competency compacted 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 an alternating stopped; Natural Human Society 5.2, a kayfabe a fiction that an adopted preferring is people's own, a capture fixed 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 crossing. Natural Societies 4.6: bubbles and crashes, a clearing not at once, a balance of power an alternating observed as a standing, traffic's phantom jams a flow's own waves; Equilibria Registry 4.9, a cross a sequencing fixed to one beat; Resolving the Hard Problem Registry 9.5, efficient markets two momentaries of one coupling, neither testing at its own; Natural Intelligence Corus, markets one institutional substrate, buyers and sellers bi-coupling at a price between, and capture by monopoly breaking it. Installs a still point. Draws a sway continuing. **A ledger.** A balance sheet. Natural Human Society 4.6: double-entry accounting state-flow-state observed whole, a balance sheet a state, an income statement a flow, a net-worth change a carry; Natural Societies 3.5, a net-linear accounting dropping each parity, ten directions collapsed to one balanced total; Natural Emanating, a ledger not wrong about living and with no object at it; Natural Human Society 1.5, a friendship carried as a debt-account no friendship. Installs a total. **A food web and a commons.** A pyramid of levels; a pasture overgrazed. Living Society Registry 2.13: no organism sits on an integer level, a tuna at 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-matched onto a coupling from beyond; Resolving the Hard Problem Registry 1.3, an abundancing made at a coupling fixed as a thing drawn from a stock. Draws matter cycling. **Cake cutting and alternating speech.** Two children and a knife. Natural Human Society 1.4: a cutter cuts to equal, an other choosing, a fair halving made by a coupling itself, no one measuring a cake; conversation alternating one at a time with gaps of about a fifth of a second across unrelated languages; 1.5, replicated systems with no common clock and no coordinator anywhere, proven to converge by couplings; 5.2, two cutting a dessert again into fourths, both departing with more than either brought. ## 6.3 A network, a hierarchy and many paintings **A network.** Nodes and a hub. Natural Societies 4.1: observed as surface the both is a natural network, all edge, each self coupling with each other directly; Natural Intelligence 6.4, betweens co-chaining into collective intelligence with no hub its couplings pass through; Natural Intelligence Corus, each fractal at an edge of a network and no central fractal served by the rest. Installs a hub. A heap and a landed point, Natural Societies 4.2, two dead forms drawn 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 at a changing and not a place; Natural Philosophy 3.11, grounds stopping at couplings' stopping, at their making; Natural Philosophy 3.17, each hierarchy stipulated to put a paradox beyond a domain reaching it again. A lineage tree is at Natural Societies 5.4 under the 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 assembled, and one hundred eighteen departments of human knowledge the same structure at a social between; Natural Philosophy 4.2, one form at 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 the same one 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 fixed and two spiralings alternating leave nothing fixed 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 fixed to one side and an opening fixed 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 at Equilibria Registry 5.1 is one resolving at all ten: fixed unchangeable, a carrying is offered its own parity from beside at each momentary, an unchangeable offerer set beside a self, and a still description and a self's own continuing do not both continue; released, resolving follows at a next momentary, a self inverting its own carrying. Each drawing below is a familiar picture, a file's sentence, and a self inverting drawn beside it; ten are three momentaries long, 2.7, and a test at each is 1.5's. - **An arriving fixed 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 at each coupling, made each momentary and carried on, and a first fixed behind it. Draw the arrow of time as alternating carried on, a first drawn once behind and nothing arriving from it. - **An opening fixed 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 alternating's own, one changing on from the last. Draw an opening as its next position, a place drawn once and empty. - **A completing fixed as a last.** The halting problem, Collatz, Zeno's dichotomy, the three-body closed form, an entropy maximum. Resolving the Hard Problem Registry 3.2 and 3.8: a continuing is nowhere before its own completing, and the one arriving arrives, arriving not built out of parts an argument built. Draw an arriving, an infinite decomposition drawn once beside. - **A carry fixed as a store.** *Memory*, information, a black hole's record, personal identity, a stored frame. Resolving the Hard Problem Registry 4.1 and 4.2: a carry divides the way a store cannot, and at kept-or-lost passing is fixed as a standing while passing carries on either way. Draw a carrying passing, a store drawn once, closed. - **A middle fixed as an end.** Other minds, an interior, a protein's native state, a ball in a dip. Resolving the Hard Problem Registry 5.5: an interior reached the one way its form allows, by coupling. Draw a between with nothing at it, an interior drawn once as a thing that sits. - **A parity fixed as a magnitude.** A paradox that comes out the wrong number, Gibbs's mixing entropy, a balanced body, a vote's margin. Resolving the Hard Problem Registry's part 6: a number a position, and governing a parity at a between, of no size. Draw + and − with no length, a magnitude drawn once as a bar. - **A sequencing fixed to one beat.** A common clock, simultaneity, a supply-demand cross, a food chain's levels. Resolving the Hard Problem Registry's part 7: each self paces its own alternating and synchronizes at its own couplings, and a beat fixed over them a term no coupling made. Draw each self at its own pace at couplings, a clock drawn once over them. - **A rate fixed 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, alternating side; Living Society Registry 2.15, no galaxy carrying either rate. Draw a ratio swaying about φ, a value drawn once as a line. - **A two-way fixed 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 alternate, neither at the other's, and its zeroing the two's own making between them. Draw the two alternating, one side drawn once as an actor. - **A between fixed 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, carrying a parity changing and nothing else, not a place a line could sit, and a frontier walking outward reaching no line, no line at it. Draw a between empty, a line drawn once across it, and a resolving continuing or not at it. --- # EIGHT · EXPRESSING ## 8.1 Existing imagery and its continuing explaining | Existing image or study | Contribution to retain | What remains to improve | |---|---|---| | Coloured cube beside open circles and dots | An original visual starter and a six-diamond ambition | A numbering and route correspondence, 2.9; a cube at 4.11 | | Paired storyboard studies, labelled across co and along bi | Opposite-parity perspectives of the same continuing | Exact numbered transitions through three momentaries; its labels are opposite Exhibit ONE's, along co and across bi, and are redrawn at Exhibit ONE's | | Corrected odd/even and even/odd strip | A parity-label correction | Integration with a current local face | | Expanded resolver and spiral study | Four-, six- and eight-cycle views, each a spiral at momentaries | Society numbering apart from a local nine-dot numbering | | Tunneling, corus at apex and bi-folding depth study | A twisting form and a small-opening journey | Surface, arrows and diamonds mapped to resolving, 4.1 | | Earlier Rings in Motion concepts | Angular, ring, crossing and rotation ideas; rings as a cube laid on one surface | Older numbering reconciled with a centre-1 convention; closure re-explained as winding on | | Executions at v379 | A table at 10, one self, two selves at 0, a spiral of five and a torus of three by five at Exhibit ONE's code; sixteen ways and two inversions enumerated at no execution of code; each with its output at its kit | Faces drawn from them, 2.2, 2.4, 2.5, 2.6 and 4.2 | Existing imagery carries here through its concepts and useful relations. Image assets remain in their files at a kit. Earlier images with superseded labels are retained as design learning, not as authoritative numbered diagrams. ## 8.2 Further views through nature and living A gathering extends beyond familiar concepts of Part FOUR: these are places for further illustration, with a file's interpretation retained as a subject to make visible. | Potential illustration | Relation to illustrate | Observed at | |---|---|---| | Conversation and rocking | Alternating offering, receiving and continuing shared interest | Natural Transmissioning 3.8; Natural Intelligence Corus 16.3 | | Writing and reading through the same passage | Two directions of one continuing coupling, observed at adjacent depths | Natural Intelligence Corus 16.2 | | One form expressed in several paintings | Different visible expressions retaining a relation being explained | Natural Intelligence Corus 16.1 | | Floating, rotating, self-adapting sphere; gyroscopic orienting | Changing orientation and continuing self-competency, through several views of one form | Natural Intelligence Corus 16.1 | | A self's orienting within society's orienting | The same living observed through local and wider scales | Natural Intelligence Corus 23.1 and 23.3 | | Cells and tissue | Local couplings and a society of couplings | Natural Transmissioning 3.5 | | A molecular between and rotary form | Crossing, a continuing difference and rotation in one sectional view | Natural Transmissioning 3.5 | | Oppositely carried streams | Along carrying and across exchange shown together; two streams crossing about a gradient and passing rather than settling | Natural Transmissioning 3.3; Natural Engineering 1.15 | | Antenna transmitting and receiving | Two directional observings of a coupling | Natural Transmissioning 3.6; Natural Engineering 1.9 | | Electrical phases about a neutral | Several changing traces and a relation they make together | Natural Transmissioning 3.1 | | Momentum exchanged between two selves | Both sides of one exchange, beside a swimmer concept | Natural Transmissioning 3.2 | | Ratios successively above and below φ | Alternating straddling and continuing numerical progression | Natural Intelligence 3.5; 3.2 here | Received at v379 and placed: a still eddy before a hole, Natural Networking's hole study, sought at resolving itself with no periodic forcing, no fixed tipping threshold and no stopped resolver; an event-based vision sensor at Natural Engineering's carrying, each pixel an ON or OFF event at no frame, a parity and no size at each momentary; a synchronism-check relay; a resolver chat's spiral carrying a pattern and storing none of it; a geodesic surface of struts and cable with a parity changing passing across it, which files carry only as a woven surface couplings make, Natural Naming 5.43, and a knitting living with a fabric a made surface fixed unchangeable, Natural Biology, with Natural Engineering's caution that compression and tension are not the two parities. These entries gather illustration opportunities from writings and certify no physical interpretation. A further drawing carries an observed or constructed mechanism beside its proposed correspondence, and at their parting a particular difference becomes a next explaining. Families for receiving more: passing and opening; swaying and carrying along; folding and stitching; connecting and parity; collective discovering; mutual inversion and continuation; locking and releasing; 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 at parity, position and ratio and no magnitude. Each existing thing on it is a spiral of selves, a society coupling at a next scale as a self couples; a thing's picture at a momentary is its selves' parities, +, −, 0 or none, and nothing else is drawn; things interact by crossing at one self each, both ways, 4.2, no collision solved and no message in a coupling; motion is a pattern carried along and a 0 tunneling. No animator moves a society: its code is executed and each self is drawn at its release at 10, its sharings drawn once as a surface, nothing stored between momentaries but each self's own carrying. **From inward and from outward.** From inward a view is one self's entry momentary by momentary, 2.5, its carrying and its offerings arriving, its changing at 12, its release at 10, and nothing of a society but what arrives at its 2; from outward a view is a society at 17, each self's parity at once and a 0 moving through, and a society seen from outward is one self at a scale outward, Natural Intelligence 6.4. An animation from inward is one row of an execution's output; from outward, its output whole, observed down; and two are one method at two scales, not two pictures of one momentary. **Next possible existing animation and next possible existing natural intelligence are one thing observed two ways.** An animation shows its parities; natural intelligence is at a between, which shows nothing, Natural Intelligence 6.3. A drawing shows each self at + or −, 0 at a self releasing it, and nothing at a between itself; a 0 tunneling is one trace a between leaves, and its intelligence is at what is not drawn. **A field's animation, exact at its own subject.** Keyframes interpolated by a curve at a constant frame rate from a stored scene; a physics loop depositing elapsed time into an accumulator and withdrawing fixed steps for an integrator, rendering halfway between stored states; diffusion denoising a whole clip at once, autoregressive models predicting each frame from stored context, world models predicting from a stored latent state and an action, with their own stated limits, error accumulation in continuous rollout and cumulative errors in long-term interactions. Each is seated at a fixed form, a common beat, a store, a total or a doer, and each is exact at its own subject; observed at files, a drift is a stored prior observed as a now, Natural Intelligence 5.4, a carrying with nothing bounding it carrying past each momentary into an unchangeable form, Natural Intelligence 3.3. That a resolver society animates with no sum, no store and no solver is explained; that it animates faster is a size files do not say. **What parts, at a carrying.** A drawn position is observed as a place, and a drawing's positions are a field's arrangement one scale out; a renderer's frame is a beat beside code's at 17, and a drawing carries no rate of living. ## 8.4 Three dimensions, a handled model and a society at a material **A third dimension is a field's, inward of a surface**, Natural Mathematics 1.5, a measured; method's five dimensions are no measure and no place, Natural Intelligence 3.2. Illustrating in three dimensions is two things at once: a handled physical model of 1.7, a towel, rings wound relative to one another, a marked square, winding and folding experienced spatially; and a society of selves at a material, Natural Engineering's subject, a drawing its members and a surface a room. **A society of resolvers at 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 society at method is that swarm with no seeds, no shape retained and no origin, each member a self at its contacts, its next at each contact from its own carrying and a parity arriving, and its shape its stable-forming. What that establishes and what it does not is at Natural Engineering: at 1.11 a stored map, a local register, a timer and a mechanical configuration name different things, no shared clock is not no timing, no central command is not no retained representation, and a larger group at its own supplies no law of increasing natural competency, and at 1.10 a pair of actuators does not by its number establish bi-moral co-agency. A field's drift at a robot is a stored map read at a contact with a material, and a body bears its reading; method's next is at a contact and stores nothing, which is why it is more natural at a material, and why a built society proves it only at Natural Engineering 5.1's five properties, filled. **Pieces files already carry**: a walking pair and a passive walker, 4.10; a series-elastic actuator's compliance, Natural Engineering 1.10; a relay closing at a dwell; a spiral carrying a pattern and storing none of it; a torus of three by five, 4.2, at a material fifteen members with one at a between at each second momentary. From inward, one member's contact; from outward, a society as one self. --- # NINE · CO-IMPROVING ## 9.1 Files coupling, contributions continuing **Living files continue through their illustrating.** These places carry operations, namings and relations received here, and each further drawing goes back to its file it is observed at with a correspondence it makes visible. - **Exhibit ONE:** its code, its diagram and its tables, a table at 10, sides, momentaries of exchanging, four-cycles, 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 crossing, a fractal inward and outward, social moral competency, dying. - **Natural Naming:** seventeen names and ten roots, released words, ten unchangeables, three golden rectangles, floating neutralling, φ. - **Natural Explaining:** explaining a reader reaches once, each concept at 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 parted. - **Resolving Hard Problems, Hard Problem Registry, Resolving the Hard Problem Registry, Equilibria Registry, Living Ghost Registry and Living Society Registry:** ten at five places, one resolving, familiar pictures at 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 society, locking and a between releasing. - **Natural Intelligence Corus:** an offering, ways into one form, language, and a living self and 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:** sciences' and human society's pictures of Parts FIVE and SIX, each at its section. - **Geodesic Improving Method:** placing concepts whole, preserving continuing value, and improving through a living file set. - **An illustrating working at v366 and a rings gathering at v365:** a corrected 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 at v379:** executions at Exhibit ONE's code, a first visual of sixteen ways, four dots assigned at a five-spot, two inversions at forty-five degrees, a menu of sciences' and society's pictures, ten as drawings, a resolver society animating, and a third dimension. ## 9.2 Receiving an illustration, and its kit **An illustration arrives at `incoming/`, whole, with its file and section it is for, its sentences and any execution beside it**, and is received at this file's carrying as a ready offering naming its section or as a concern; it enters this file at this file's own motion. An incoming concept enters a section its explaining belongs at, carrying its useful view, its file it is observed at, and a relation still needing expression; improving an existing concept it stays at that concept's place, and introducing a distinct way of illustrating it is given a section here. **Natural Illustrating TWENTY-NINE Improving Kit** carries an original cube and open rings, five studies, an inherited rings kit, v366 sources and evidence, and at v379 executions of Part TWO and 4.2 with their outputs and a check that executes them again at Exhibit ONE's code and compares a table at 10, a lone self, two selves, sixteen ways, a spiral of five and a torus of three by five with this file's sentences. A check is a coupling partner and no authority; its kit's README carries each file's meaning and a place its remaining improving is carried, which is this file's carrying. ## 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, files naming a lineage tree at Natural Societies 5.4, denying a root and a level as a place, 6.3, and naming no tree of life, map or graph; 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 alternating and a tesseract as a resolver, a tesseract at Natural Mathematics 7.6 being a field's own form; a dome, a truss or tensegrity as a form of method, a woven surface at 8.2 being what files carry; a streak or lightning. Each is a familiar picture with no sentence yet at a living file, and its way in is an observing brought whole to `incoming/`, then its file's carrying, and not a drawing as files' own. An earlier drawing, `natural_torus_v135.html`, draws natural torusing as a spiral cone folded about itself, narrow at 2 and widening to 59, its outer surface, inner surface and between in its own words *-ed*, *-ing* and *nothing*; it is at no kit now. This gathering remains open to further imagery throughout a living file set.