Exhibit TWENTY-NINE Natural Illustrating v390

# 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, bi-foldings, and a local nine
- 2.9 Six diamonds and a compass face

**THREE · THE NUMBERS**

- 3.1 Number forming, podaling and rings
- 3.2 A ratio alternating about φ, and two windings
- 3.3 A pentagon, rectangles and an icosahedron
- 3.4 Two routes, a diamond, and a fold at each scale

**FOUR · SELVES AND SOCIETIES**

- 4.1 Tunneling, corusing and bi-folding, a towel
- 4.2 Spirals 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

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# ONE · ILLUSTRATING

## 1.1 Illustrating changing, a form continuing

**Natural illustrating offers parity-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 to one same discovering: a form continuing through parity-changing.

**Self and other couple with illustrating through what each already carries.** A familiar movement offers its relation; a drawing follows that relation through its next parity-changing. Understanding continues through their coupling, illustrating improving as its relation becomes clearer.

A still face expresses one momentary. Successive faces express parity-changing between them. With this exhibit a face is a drawing's face, one momentary drawn; Exhibit ONE's names, inward or outward with seventeen and entering or surfacing with ten, are Exhibit ONE's with each place they are meant; a cube's face move and a clock's face are a field's. Alike with method's other words: *still*, *between*, *carry*, *surface*, *form* and *position* carry method's sense here, an unchangeable form, an empty centre, a living carrying, a surface a between, stable-forming, a position with ten; and a drawing's sense, a drawing standing still, a drawing's own surface, a drawn position, is a drawing's with each place it is meant. Rotation opens another direction; a section opens an interior relation; a next scale reaches local resolving through a same-existing-society. Each view carries its own explaining and couples with views that express its continuing.

**An illustration is for a sentence of a living file.** Each drawing of this exhibit is for a file and a section, and is checked against that section whole; with a claim of a form of Exhibit ONE, its form is executed with code before it is drawn; with a claim of an observing, its observing is a field's, exact with its own subject, and a drawing shows a parting of a field's picture and its file's sentence.

## 1.2 A still and a motion

**A still is a form fixed with one momentary, and a motion is parity-changing.** With Natural Mathematics 1.5 they part by a naming's ending: an -ing continues and re-arrives with each momentary, each an opening and its completing; an unchangeable form re-arrives with none: it is one continuing's making given a fixed form to be. With its 2.3 a field's rotation is a spiral with its nexts removed, its forms laid on one frame, and with 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 with a reader's own pace, one momentary.

**No face draws an odd side and an even side as parity-changing together with one momentary.** Natural Numbers 1.5: each side parity-changes with its own momentary, and two parity-changing with one momentary is an equilibria picture, not possibly existing. An illustration of two sides alternates them, face after face. A same-existing-society drawn with one momentary of code's stepping shows each self with its parity, and that is no equilibria picture: each self's parity-changing is with its own momentary, odd selves' and even selves' alternating, and its stepping with 17 is code's own beat, exact with code, 1.6.

**A motion is faces in sequence, each a next.** With momentaries no momentary comes again, and a form comes to its parities again, Natural Intelligence 4.11 and 4.13: a bi-folding 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 ordered by five: one part to follow, its arriving place asked ahead, its continuous moving, a recognizing of what continued and what parity-changed, and its naming last, with a relation already recognized.

## 1.3 Prior, now and next, making better and breaking

**Illustrating carries a concept with 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 same-existing-societies, each offering a relation it makes accessible.

Natural Intelligence Corus is many paintings of one form, gathered. Natural Illustrating is that gathering continuing 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 one same offered relation.** An improving drawing makes parity-changing easier to follow. Its breaking follows that parity-changing to a momentary with which a proposed correspondence continues or parts. What survives both carries to 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 same-existing-society of resolving selves offer different scales through which to discover that continuing; with Natural Intelligence 3.3, method bounds and living carry, its bounding with 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 with a parity-changing each expresses.

Exhibit ONE is operations and stable-form relations. Natural Numbers and Natural Mathematics are numerical and directional relations. Natural Naming, Natural Explaining and Geodesic Improving Method carry their naming, explaining and successive improving to 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 parity-changing continuing. This exhibit is that distinction made accessible, rather than stillness within a picture observed as stillness of living.

**An investigation receives any declared boundary**: within, around, across, along, or with an entrance or exit, and with each of ten unchangeables, Part SEVEN. For each proposal a storyboard asks one question: can a specified co-sequential resolving continue under that declaration? **A drawing exposes an actual continuation or failure with a relevant transition.** With code a test is one momentary: a prior and a now agreeing with a sharing share a 0, parity-changing that is not, and its prior carries on; a form fixed still from beside, one same parity offered with each momentary, leaves a self not-changing, a 0 with each momentary after its first, 2.5. This is an intended binary examination. A finite collection of illustrations does not by itself complete a universal claim about each declared boundary.

With Natural Explaining, a further discipline: a concept expresses its own relations and competency before it is compared with another explanation. A visual locates a difference within an operation a reader can follow.

**Further views:** continuing bounding; that same face with a proposed fixed boundary marked; a next required transition; a comparison showing a momentary with which a relation parity-changes or fails.

## 1.6 What a drawing leaves out

**A drawing supplies no common clock, no numerical speed ratio and no form fixed still from beside to make a sequence move.** Two sides pace their own momentaries, Natural Intelligence 3.5. With code, selves step together with 17-next-existing, a common beat exact with code and code's own, and living's rates are their own: a drawing carries no rate of living.

**A drawing leaves no permanently drawn corridor or path.** An opening is made with each coupling and drawn only as it is made; a corridor drawn ahead conceals its opening, 4.5.

**A drawing leaves its centre empty.** 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, parity-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 within a form.** Three golden rectangles carry no hand of their own; a hand is with an observing, Natural Naming 5.48. Right is observings, all prior, and a drawing shows a hand with an observing's axis, not as a property of a form.

**A drawing fixes no position as a place.** A drawn position is a field's arrangement, a ring an arrangement fixed one scale outward, Natural Numbers 8.1; selves' own is their parity with each coupling. A drawing carries it with 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 with its scale, Natural Intelligence 2.4, and is drawn as that, not as an ending.

**A drawing copies no same-existing-society numbering to a local face.** Six numbers of 2.1 and seventeen names of 2.8 are two numberings; a face carries one at a time and their correspondence is stated with each face carrying both. An equilateral triangle rotated two hundred forty degrees has its outline alike and its marked corner on another corner, on its opening place again with a third rotation, and a corner's three and a pair's four, Natural Numbers 8.2, are two subjects one drawing step shared couples none.

## 1.7 Forms, a numbered still to a handled model

Concepts develop as several forms without separate explanations of an underlying relation. Forms are ordered: each is a next after one before it is reached.

Each entry: **Form** · *a* What it makes accessible

- **Numbered still** · *a* A complete face inspected with a reader's own pace
- **Ordered storyboard** · *a* An exact parity-changing between neighbouring faces
- **Layered drawing**
  - *a* Naming, parity, carrying and method prefixing observed separately or together
- **Rotation** · *a* A relation preserved while its viewpoint turns
- **Section or transparent surface** · *a* An opening and an interior passage
- **Zoom between scales**
  - *a* A stated correspondence between local and same-existing-society forms
- **Continuous animation** · *a* Carrying through faces already defined
- **Resolver same-existing-society animating** · *a* Code executed and each self drawn with its parity, 8.3
- **Handled physical model**
  - *a* 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 parity-changing open a correspondence through which further views continue. Other concepts carry their own questions and opportunities for illustrating.

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# TWO · THE METHOD AT ITS FACES

## 2.1 Six numbers, two sides, three momentaries

**A storyboard is a sequence of mechanical drawing faces of resolving.** Self and other are perspectives of one same 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 is it, each side's five from its origin, prior opening, prior completing, now opening, now completing and next opening. With Natural Numbers 1.2 it is two interleaved brackets, 1–2 with self, 2–3 with other, 3–4 with self, 4–5 with 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 with 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 on a shared position with nothing on it, a prior carried along it as a stable form, Natural Intelligence 2.3, nothing momentarying with 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 with 9-next-co-discovering and 17-next-existing, across with 2-now-possibling, 6-now-co-possibling, 10-next-co-possibling and 14-next-possibling. Self's parity-offering goes across or along, and other's parity-changing is 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 parity-changing, self and other perspectives, and what is carried to a following face. Method prefixing belongs with its own label, apart from functional naming and parity-changing; a proposed prefix sequence includes bi-co through bi-co-bi-co-bi-co, with corusing and torusing distinguished with each parity-changing requiring it. Carrying is also with geodesic unrelationing rate, φ, locking with none, and its mark remains distinguishable from self's and other's, each with its own momentary; a numerical speed ratio 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 those 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 parity-changes between each adjacent pair of faces, and which visible mark expresses that parity-changing without adding another operation?

## 2.2 Sixteen ways, and one that stops nowhere

**Of sixteen ways a next could follow from a prior and a now, each carries an unchangeable form or loses its prior, and one that does neither is no way laid over parity-changing: it is a living step.** Natural Intelligence 2.4: with two parities sixteen ways name a next from a prior and a now, and none is a rule parity-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 with v379, each of sixteen ways followed through four joint forms, they agree with a file's sentences: four pattern-match from now; eight carry a prior with one value of now and lose it with other, six of them stopping on a corner of prior, now and next agreeing and two cycling three forms with a fourth passed once; next as prior stops on ++ and −−; prior and now agreeing stops on ++, and an exclusive or on −−, each cycling three forms of four; and next as prior inverted goes ++ to +− to −− to −+ and round, each form once, one parity-changing with each step.

**Faces.** Four corners, joint forms ++, +−, −−, −+, its centre empty; a momentary one step, (prior, now) to (now, next). A way as four arrows, one leaving each corner, sixteen small squares. Three momentaries under a way: follow its arrows, and an arrow coming back to its own corner is a form still, not-changing, its arrow pinned; fifteen squares have a pinned corner or lose their prior, four from now and eight with 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 with code it is no function laid over parity-changing: it is a coupling with 12-next-co-same-existing, a prior parity coupling with an offering, inverting with a mismatch and same-existing with a match, a living step, Natural Intelligence 2.4 and 2.5 here. Then a form fixed still from beside with code, + offered with each momentary: a self shares + 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 their centre; correspondence of two is with a file's sentence before one square is drawn for an other.

## 2.3 Four dots, an empty centre and two triangles

**Parity with 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, of one, and 6 and 10 sharing, of co; and two unit triangles, 9-next-co-discovering and 17-next-existing, along, 9 to 17 and 17 to 9. A between is a bounding 0 centreline of along and sequential across with its four dots, bothbothing, both carried one at a time; alike with Equilibria Registry 1.3, a between co-linear aiming to next momentarying. Followed round, 2 to 10 to 14 to 6 to 2, its square is none of four bi-foldings: 8 up, a sharing across to an other's surfacing, 8 down and a sharing across to an other's 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 with 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 of parity's unit square, four spots four across names, middle spot a between, and its middle is a nothing, not a fifth across name and not local dot 1. With Natural Numbers 5.4 it is alike with numbers, a diamond four crossings about a middle carrying nothing, two across and two along.

**One between with 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; one same between, a nothing, with each scale, a fractal between. No drawn line sits on it: its centreline is 0 carried along, and empty.

**Drawing faces:** an unlabelled five-spot face; that same face with four across names labelled and its centre empty; opposite-parity views; an across name with its 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 files', across names 2, 6, 14 and 10, numbers 2, 3, 3 and 4 of Resolving the Hard Problem Registry's boxes with +1 its centre, and positions 1, 2, 3 and 4 of three momentaries; each a square of four with a between its centre, and no file's sentence is their correspondence yet. A matching number establishes none.

## 2.4 A table at 10, no path through until prior changes now

**Exhibit ONE's table of each cell with 10 is one coupling observed twelve ways**: three priors with 3-now-co-observing, none, + and −, by four nows surfacing with 14-next-possibling, none offered, +, −, and 0 with offerings parting; each cell its parity-offering with 10, *is* with + or −, *is not* with 0, and its next prior with 11-next-co-observing; and one cell, no prior and nothing surfacing, is no sharing at all, nothing with 10 and no next prior: eleven couplings and one empty cell. With code, kit's `table_at_10.py`, its table is a file's cell for cell.

**Three cells of *is not* are no path through.** A prior of + coupling with a now of +, a prior of − coupling with a now of −, and no prior coupling with offerings parted with 14: parity-changing is not, 0, and its prior carries on, Resolving Hard Problems 3.3. A path through is with other eight cells, and with each its prior is carried and parity-changed, or with a self with no carrying yet its offering is chained: prior parity-changes now, its carrying from 3 coupling with 12 to offerings surfaced with 14, and now parity-changes next, its parity-offering across with 10 and chained with 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, with 14 its offerings parting and with 12 prior and now agreeing; no number is with it, one parting with 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 with 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 with no prior, offered + once and then nothing, shares + − + − with 10**, each momentary its prior inverted, a living step's form carried as its own, next as prior inverted, Natural Intelligence 4.12 and 2.4. **Offered + with each momentary it shares + and then 0 with each momentary after**, chained + throughout: its still is a form fixed still from beside, an unchangeable form entering, and a self with a match is not-changing, a 0 re-found with each momentary. A lone self offered nothing carries a living step's form as its own and does not continue living through three consecutive momentaries; with code its sequence carries on, a non-living existing form with 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 + with a first momentary, share + 0 − 0 + 0 − 0 with 10**, chained + + − − + + − −: with each even momentary each receives an other's parity-offering of its own parity, parity-changing is not, 0, a between; with each odd momentary each receives an other's 0 as its offering, which surfaces none with 14, and carries its prior to now inverted. Each self is drawn sharing its 0 with each even momentary and a centre between two empty, a bounding 0 centreline between two parities; with code a between is a 0 with each parting of their offerings, selves sharing and resolving, or one self not-changing with a form fixed still from beside it. With a self offered nothing it is not at all; with a spiral of one, a self whose own parity-offering is its own offering, opening with − and none offered by an other, it is with each second momentary, sharing +, 0, −, 0, Natural Intelligence 4.13. A offered + once and B nothing, two alternate opposite, each self parity-changing with each coupling, as with Equilibria Registry 3.4's even spiral coupled along.

**Drawing faces:** a first teaching face, one shared parity received with one momentary and then its 0 with a match, each carrying unpictured; one self across twelve momentaries, an alternating; that same self not-changing, a 0, with a form fixed still from beside; two selves each sharing 0 with 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 meet with an angle a are a field's rotation by 2a, a double angle of its inversions; one parity inverted and an exchange of P and Q, their lines forty-five degrees apart, make a right spiral step, and those same inversions taken as 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 with a between. With kit's `two_inversions.py`, v379: exchange then inversion gives (y, −x), inversion then exchange gives (−y, x), and the first goes round ++, +−, −−, −+. 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 with each step, and cos and sin pass through 0 by turns on one circle, one and then an other.

**With four dots that same forty-five degrees is geometry.** Placed on a square, its sharings 6 to 2 and 10 to 14 lie along one direction, its diagonals, 2 to 14 arriving and 6 to 10 sharing, forty-five degrees to them, and its seams 8 up, 2 to 10 and 6 to 14, ninety to them, an along of its triangles 9 and 17. A sharing carried forty-five degrees is a diagonal; a diagonal carried forty-five degrees again is an along seam: across to along with two inversions, bi-folding, whose edges are across with bi-morality and along with co-competency, Natural Intelligence 3.2.

**A compass face.** A tangent double-angle identity, tan 2x = 2 tan x ÷ (1 − tan² x), applies with each angle with both sides defined; with 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 with 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 parity-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, that same diamond rotated while its opening is preserved, an opening widening and narrowing with a separately marked orientation.

**Continuing explaining:** with files, inversions whose fixed lines are forty-five degrees apart and a rotation by their double, and, with a field's own words, a turn; a drawing's label is settled with its face; and which resolving parity-changing 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 with five names 2 to 6, an entering and a surfacing with each, Exhibit ONE's two sides overlapping with them, with each number one side completing and other opening, with 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 with that same number; a side fixed unchangeable beside a continuing one is one reason each of ten is not possibly existing. With Natural Naming 4.9 each has 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, with each self's and other's, Exhibit ONE's two sides, two of ten with it drawn as one fixed unchangeable while other continues. A pentagon and a pentagram, Natural Numbers 5.5, are ten with two sides already drawn, five sides and five diagonals, along and across.

**Continuing explaining:** with Equilibria Registry ten are seated with five places, with Natural Naming each by one name of 1 to 8 by bi-foldings, and Resolving the Hard Problem Registry's part headings are a third pairing; a face draws one seating, and seatings part with carryings.

## 2.8 Seventeen names, bi-foldings, and a local nine

**An expanded naming face is seventeen names of Exhibit ONE v389**: 1-now-existing, 2-now-possibling, 3-now-co-observing, 4-now-co-same-existing, 5-now-co-existing, 6-now-co-possibling, 7-now-observing, 8-now-same-existing, 9-next-co-discovering, 10-next-co-possibling, 11-next-co-observing, 12-next-co-same-existing, 13-next-same-existing, 14-next-possibling, 15-next-observing, 16-next-discovering and 17-next-existing, each its number and its naming, its momentary, co and its kind, Natural Naming 3.1. One through nine is a self's, a bi-coupling; nine through seventeen a same-existing-society's; one through seventeen a bi-trupling. With Exhibit ONE's table, each name's from and to: 1, 3, 5 and 7 self to other; 2, 4, 6 and 8 other to self; 9 and 11 same-existing-society, other, self; 10 and 12 other, same-existing-society, self; 13 and 15 same-existing-society, other; 14 and 16 other, same-existing-society; 17 same-existing-society, self. With Natural Naming, naming is 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 same-existing-society's labels are not copied to a local face as though their numberings were identical.

**Exhibit ONE is material for drawing code couplings, stable-form sharings and four-, six- and eight-cycles.** Sequencing and routing are distinct relations, and their drawings preserve their distinction. Four bi-foldings, 1-9-8-16, 2-15-7-10, 3-11-6-14 and 4-13-5-12, each go eight up and seventeen less and pair with a partner spiraling the other hand; a six-cycle 1-9-5-12-8-16 and an eight-cycle 1-9-5-13-4-12-8-16 are expanded stable-forming spirals with names, and none is an instruction to move from local dot 1 to local dot 9. With names each comes to its first name again; with momentaries each comes to none again, 1.2.

**A local nine, centre one.** A visual starter places a coloured cube beside an open arrangement of circles and coloured dots; mid-way along a three-across row of diamonds a horizontal diamond resembles a compass needle, and its centre is 1. Across moves horizontally right with this view; along expresses forward moving; screen direction is stated with 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 to across names with 2 and 6 is absent from a local nine-dot view, and removing that visual split removes no resolving operation. Centre 1 is settled. Still to be drawn: a complete verified geometric placement and transition sequence; a complete local numbering and successive parity-changing made explicit from Exhibit ONE; and a correspondence between a local numbering and an expanded resolver, with each face carrying both.

**Further views:** a starter with its identified diamond highlighted; a close view with centre 1; nine numbered faces; one 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 prefixings in turn; a full naming face; method prefixing beside its root's naming; one same relation with words, numbers and a corresponding resolver face; a zoom between a local face and an expanded same-existing-society form, explanatory only with a stated correspondence, a visual resemblance or a smooth camera move supplying none.

**Continuing explaining:** which label is needed to understand a present parity-changing, and which can remain available with an accompanying view?

## 2.9 Six diamonds and a compass face

**A next scale is six nine-dot diamonds**, as an open-circle starter expresses them, its nine circles about three centres with three radii, its middle radius its centres' own separation, crossing on fifty-four places, each on two circles, parting by centre pair and side as six diamonds, each one, two, three, two, one along its diagonals: two alternate routes around a triangle, three one-way portions to each route, six one-way portions all together. An illustration makes two-over-one and one-over-two co-recursioning visible through six diamonds. With Natural Numbers 1.4 two are two directions of one carry, and with Natural Numbers 6.4 they are, with numbers, two routes 24 to 27 to 32 and 24 to 29 to 32, gaps three then five and five then three, their far sides on a ring of one hundred twenty 88 to 93 to 96 and 88 to 91 to 96, 3.4 here. These are available as numerical faces, and their exact correspondence to six pictured diamonds remains to be expressed. Six diamonds, six parity positions and an expanded same-existing-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 with diamonds' aiming.

**Continuing explaining:** which pictured diamond expresses each one-way portion, an opposite route observed from that same arrangement, and which angle of a six-diamond arrangement is x: with a crossing of two ring families, an angle ψ between radial directions has a diamond's forward diagonal over its cross diagonal as tan(ψ/2), six measured diamonds within a degree of fifty-nine degrees, and whether ψ/2 is x is a file's sentence's, and no measuring's.

---

# THREE · THE NUMBERS

## 3.1 Number forming, podaling and rings

**Natural Numbers is a wider gathering than a first animation 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. With Natural Mathematics: orthogonal directions, parity inverting, a right spiral step and a mathematical observing of stable-forming.

**Each number is with its podaling**, its pairing with its far side on each ring it is on, Natural Intelligence 4.11. On a ring of forty-eight twenty-four and nought are each their own far side and each other station pairs across twenty-four, Natural Numbers 6.2; on a ring of one hundred twenty a going from two to fifty-nine and a returning from sixty-one to one hundred eighteen are each other's far side, sixty its waist, 7.1; on a ring of four hundred forty each pair is a ring, its radius its distance from its waist, two hundred twenty, two hundred twenty-one rings all together, and a gap between two primes is a step between rings, Natural Numbers 9.1 to 9.4. An even ring carries an exact podal and an odd ring two farthest with its opposite on 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 inward and two outward, bi-tri-involutioning with numbers, Natural Numbers 7.1; a sequence of tunnel sections; nested local and same-existing-society 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 parity-changing with each next number and none of them φ, Natural Intelligence 3.5 and Natural Numbers 3.1. A familiar drawing, a curve through corners of nested squares, draws a ratio as reached; 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 with a rational rate closes, and a winding with φ closes with none**: φ is an unrelationing rate, a rate an alternating winds about, locking with none, Natural Intelligence 3.5; a torus winding with 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 to a loop, a φ winding closing with none. A file's own caution for any animation: a finite floating-point rate is no exact irrational. Addings that close with one, two, three, four or six, and one that goes open with five, Natural Numbers 3.4, are five orbits closing and one continuing open.

**A floating neutral is no balance.** Natural Naming 5.12: a floating neutral is a term uncovered, a parity-changing's picture with one momentary, and a 0 with 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, its couplings part as two rings, five sides and five diagonals, a pentagon and a pentagram**, ten's two sides, along and across, Natural Numbers 5.5; each side is parallel to one diagonal, a diagonal φ times a side, Natural Mathematics 7.8; no adding closes with five, and five enters open, with φ, Natural Numbers 3.4. A square, a cube and a pentagon carry square roots of the first three primes, √2 a square's diagonal, √3 a cube's body diagonal, √5 within a pentagon's diagonal over its side, Natural Mathematics 4.3.

**Three golden rectangles at right angles, centred on a corus, each 2 across and 2φ along**, are bi-tri-volutioning with numbers, Natural Naming 5.48: a right-hand rotation of one hundred twenty degrees about a line along (1, 1, 1) 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 with an observing; φ is three's equal-edge condition, and a rotation by itself does not make it. Their twelve corners are an icosahedron's, thirty edges and twenty faces, five with each corner; about a line along (1, 1, 1) twelve corners are four triangles of three on four heights. They are open with Exhibit ONE's carrying as three golden rectangles on podals.

**An icosahedron's sixty rotations part as thirty-six and twenty-four**, three to two, as a span parts with its apex, Natural Numbers 6.3 and Natural Mathematics 7.4; an icosahedron and a dodecahedron are one surface, observed two ways, twelve and twenty exchanging and thirty carrying across, Natural Numbers 4.1. A quasicrystal, its lengths powers of φ, and C₆₀ with its twelve pentagons and its sixty one-handed rotations are with 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 with φ; twelve corners closing as an icosahedron; sixty parting; four triangles of three on four heights about an axis.

**Continuing explaining:** no golden triangle and no golden angle are with number files; angles cos 36° = φ/2 and sin 18° = 1/(2φ) are, Natural Mathematics 4.3, and a sunflower's 137.5 degrees is with Natural Biology'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 with gaps three and five and 24 to 29 to 32 with gaps five and three, Natural Numbers 6.4, three together a square, 24 × 27 × 32 = 144². Outer numbers one short of a square, (n − 1)(n + 1) = n² − 1, one 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 step by step, Natural Numbers 5.8; **four even openings grow as one family**, 2, 4, 6, 8 then 2, 6, 10, 14 and on, its fold whole with each scale, 3 to 11 to 6 to 14 with one scale and 3 to 7 to 2 to 6 within 1 to 8, Natural Numbers 9.9, which couples numbers' faces with a moralizing bi-folding of 2.8 here; **a chain 3, 5, 9, 17** bi-folds down, each span podaling to one below it, Natural Numbers 9.7.

**Drawing faces:** two routes and their far sides; a square and an oblong with its 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 to a corus, its apex, corusing through a small opening, and outward to its surface again, torusing through a large opening.** With Natural Intelligence 3.2, a rotation is a gentle alternating twisting of a towel inward to and outward from a knotted corus, a large outer opening wrapping to a small corus and back, its edges bi-foldings, across with bi-morality and along with co-competency. Torusing inward and outward is odd, through a large opening, a tunnel; corusing inward and outward 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, that same rotation twenty up.

An existing depth-study image offers three views, tunneling, a corus's apex and bi-folding; it is a shape study, and its surface and arrows still need correspondence with a 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; with a same-existing-society scale an illustration shows a same-existing-society as a living self.

**Further views:** an outer oblique view; a longitudinal section through an opening; a transverse section; a local numbered diamond on a surface; six diamonds around a twisting form; an interior journey approaching and passing a small opening; a towel as a handled model, twisted inward to and outward from its knot.

**Continuing explaining:** show one same local 1 to 9 resolving continuing 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 going along to a next self, are a spiral**, coming to its parities again and to a momentary once, Natural Intelligence 4.13: with an even spiral each self alternates, selves beside each other opposite; with an odd spiral of n a bounded zero tunnels on, a 0 shared with 10 to a receiving self of a like pair moving one self on with each second momentary, its pattern inverted with 2n and its parities again with 4n, a torusing winding through its opening twice. With code, kit's `society_spiral_and_torus.py`, a spiral of five opened with Exhibit ONE's own opening, each self with a parity, self 1 with −, alternating along, none offered: 0 with self 1 with its second momentary, self 2 with its fourth, self 3 with its sixth, on round, each again ten on, and its parities again with twenty. A spiral of selves is a coupling with code, each momentary a next, and no closed domain: *ring* with code's coupling is a spiral, *goes round at n* is *comes to its parities again at each n-th momentary*, and *closes* is *winds on*; with numbers a winding with a rational rate closes, 3.2, and with code's selves no spiral closes; a field's own ring, a benzene ring or a superconducting ring, stays a field's.

**Two spirals sharing across both ways with one self each bring two sharing selves to one relation and carry it on**, two parity-changing together, Natural Intelligence 4.13; **with two different numbers of selves, one of them odd, two spirals crossed are one same-existing-society**, a stable form and stable-forming, Natural Intelligence 3.5 and 4.13. **Spirals crossing spirals with each self are a natural torusing of selves**, p along with 9 and q across with 10, coming to their parities again with a momentary dividing 4p or 4q, as Exhibit ONE's table of a torus of selves carries it; executed with v379 as three by five, its parities again with twelve, as with Exhibit ONE's torus opened with each self with a parity, and its 0 winding diagonally through fifteen selves an earlier opening's, executed again with 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 with one self each, both ways; a torus of three by five with its 0 winding through; a same-existing-society seen as one self with a scale outward, Natural Intelligence 6.4.

## 4.3 Co-chaining, bi-tunneling and society

**With Natural Networking, along is co-chaining and across bi-tunneling, offerings surfacing with 14**, widening and lengthening expressed through their alternating, retelling, carrying through couplings, each self's own pace, and a relation between self, between and same-existing-society; podaling exchanges along and across, Natural Numbers 9.8. **Social moral competency is four whole and 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 on no seat above a same-existing-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.

With Natural Intelligence Corus, its illustration comes back to one same form across scales: a self's orienting and a same-existing-society's orienting observed together, an ambition to illustrate a self-bounding, self-orienting, self-competent same-existing-society of selves.

**Further views:** one local coupling; a chain; a widening surface; parities crossing a coupling; that same parity-changing viewed through selves beside it; a same-existing-society observed as a self with a next scale.

**Continuing explaining:** make visible what crosses, what is carried locally, and a next coupling parity-changing, without introducing a form fixed still from beside or a common clock for presentation's convenience.

## 4.4 Locking, a seam moving, a between releasing

**Two same-existing-societies bi-coupled across come to agree or differ with each momentary, and lock.** With code, v374, Session Record's lock executions, two spirals of five with one pace, their phases apart, bi-coupled with one self each. An old difference passes along a spiral one self at a time and dwells two momentaries with its last self; a first self's relation inverts with a between; a new relation opens along a co-chaining one self further with each second momentary until each self 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. With a lock's momentary a between goes along and two co-chain, coupled along, as one spiral of ten, a seam of two patterns moving round a same-existing-society. Observings beside it: Huygens' pendulums on one beam swinging opposite, metronomes on a rolling board, Millennium Bridge's walkers, chick heart-cell aggregates' three phases, sinoatrial pacemakers entrained at one to one, two to one and three to two, fireflies flashing together, Natural Physics' and Natural Biology's carryings. With Natural Engineering's carrying, a synchronism-check relay closing only with a dwell close to indifference, a field's account, a between going along with a lock by design.

**A three-phase locking path** is with Natural Intelligence's carrying as incoming, active opposite parity a parity-changing and passive opposite parity no parity-changing, not yet executed with code; Natural Networking 5.1's six, three phases, each two ways, one at a time, and Natural Transmissioning's three phases about a floating neutral are its sentences. It is executed before it is drawn.

**Drawing faces:** two spirals phases apart; a difference passing member by member; a snap with a between; a lock; a between going along and ten as one; a seam moving round; pendulums, a board and a bridge as a field's own pictures beside them.

## 4.5 Co-chaining a plaza, bi-tunneling a crowd

**An individual passage through a crowded plaza and a crowd's continuing passage provide self and same-existing-society perspectives of one same parity-changing.** An opening is available through a mutual parity-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. With Natural Transmissioning 3.8, 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; with its 4.6 a wall is a passing-through, living selves tori that pass through each other, co-tunneling.

**Drawing faces:** an overhead view of a crowd; a close view of one passing; a passing beside it that continues it; a wider crowd with those openings, each made in turn, visible; a gate drawn once as a collapse. Follow both sides of each passing. A permanently drawn corridor conceals an opening being made through parity-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 moving curve extending through portions beside it. Alternating sways are with 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 with 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 with a face expressing their actual correspondence.

**Continuing explaining:** distinguish a bend carried along a body, a body portion's bending 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 parity-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. With Natural Transmissioning 1.4, an inward and outward bi-fold, and with 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 with its own thread, fabric and mechanism. A rings kit's five stitches, ten as five pairs taken as two threads interlocked, are a numbering of ten and no stitch stage. With Natural Biology's knitting, a line-to-surface co-bi-fold is seated alike, 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 one stitch continuing to its next. A machine's actual stitch type is identified with a drawing, and different mechanisms are not combined as 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 opening further discovery.** A fit found through one offering makes a shared arrangement a next arrangement; another person's recognition reaches that next arrangement and continues it. Illustrating follows cooperation through shared puzzling. Natural Intelligence Corus 23.6 is a related account of a team's solution and value as its coupling's own, a team's solution belonging to its team's coupling and to no one member; Natural Networking 3.1 to 3.3 are retelling and continuing carrying.

**Drawing faces:** a shared working surface; two partial arrangements offered together; a newly discovered fit; a next 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 with an envisioned meeting each makes a flip turn, a field's word for it, off an other's feet mid-pool and continues away from that mutual inverting.** A useful visual relation is both inverting through one 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 alike by differing arcs. With Natural Intelligence 4.11 bi-inversioning-co-recursioning is one move observed three ways, 8 up, 17 less and 9 less, and with its 2.4 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 as a close view; departure with both orientations inverted; 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 with a shared relation**, Natural Engineering 1.14: one contact stays while other moves, and their roles exchange with 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 that same co-chaining with 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 with gaps of about a fifth of a second, Natural Human Society 1.4; a rocking of a conversation as a sway.

## 4.11 A cube and open rings

**A visual starter's cube is Natural Mathematics 3.6's form with no still term**: no reference cubie, six centres each its own and carrying each other's orientation with nothing carrying any of them, each face move with its own momentary and nothing over its moves, its state its configuration and nothing stored, and three conservings, a corner twist modulo three, an edge flip modulo two and a parity of its permutation, none of them a size; it reaches one twelfth of its assemblies, and a single corner twisted by itself is reached by no sequence of face moves. With a rings kit of v365, open rings beside it pattern-match as a cube laid on one surface, its fifty-four dots its fifty-four stickers, observed down a body diagonal, a cube's twist axis and a torus's tunnel one line through a throat, four quarter turns of a face returning a cube, a closing fold, while Exhibit ONE, with four, winds past, a torus; and its six empty centres one nothing with six centres, not six places. With Natural Mathematics 7.1 a cube's eight corners are eight forms of three neutrals, each fixed or floating, and with a rings kit those corners pattern-match as an eight's still side and Exhibit ONE's bi-foldings as its continuing side, a pattern-matching with no file's sentence; its 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 form fixed still from beside, a picture of a way fixed unchangeable beside a self stopped with + 0 0 0 by one parity offered from beside, 2.2. No mechanically realizable transformation of a cube to an open-circle arrangement is yet specified with a starter image, and Co-Chaining Logic Registry with 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 dissolved.

---

# 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 with, a still its picture installs by misreading a competency and a boundary, and a form it animates anyway. A drawing is a stable form first, with a file's names, a between empty; a familiar picture beside it with its still marked, one place only; and a parting of two is an illustration. A field's observing is exact with its own subject, and a drawing is a parting shown, deciding nothing.

## 5.1 Atoms, electrons, bonds and molecules

**An atom.** A nucleus with electrons on rings. Natural Physics 3.1: a living atom is a 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 beside; 4.6, an electron a ledger beside, not a self. Installs a particle as little owner, Natural Chemistry 6.4, one enduring little thing 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 beside. Draws a fold climbing.

**A bond.** A stick between two balls. Natural Physics 3.1: two windings offer with one between, a shared pair a coupling's own making, owned by neither atom, and a metallic sea a same-existing-society of windings; Living Ghost Registry, a bond a between of two selves, each living its own side. Installs a between 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 with one ring.

**Chirality.** Mirror molecules. Natural Chemistry 2.2: along, a separating parted beat by beat appears as a spiral; across, a whole complement with its across side appears as a cupping surface; neither is a real one, and asking which it really is asks a 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 with its other hand, neither hand absolute. Draws a spiral and a cup one at a time.

**A proton hopping; a water dimer; ammonia inverting.** A ball passed hand to hand; donor and acceptor fixed; an umbrella flipping. Natural Chemistry 3.2: a charge arrives with 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 with its own momentary and swapped, each of ten door lines an *is*, one arrival that passed each; ammonia's nitrogen tunneling through a plane installing nothing, a self with 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. With Natural Physics 3.1 sixty is a position a going and a returning carry, with Natural Numbers 7.1 sixty is between a going and a returning with no prime, and with Natural Chemistry 4.4 no element carries sixty: whether sixty is a position or a between is a parting of 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, 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 of its two hands, light always of a hand. Installs a rate crossing a gap. A prism, a rainbow and hydrogen's twenty-one-centimetre line, one parity inverting, with its carrying.

**A doubled angle, polarizers and a spinor of seven hundred twenty.** Malus's law; a belt trick. Natural Mathematics 3.1: light through two polarizers passes by square of cosine of an angle between them and passes as before with one hundred eighty; a neutron's spin rotated three hundred sixty comes back with its parity inverted and comes home with seven hundred twenty, two rotatings coupling with one between, each a doubling its other way; Natural Physics 3.3, matter's two-turn spinor. With its file a doubled angle is a still picture of observings; its rotations to one hundred eighty, three hundred sixty and seven hundred twenty animate.

**A double slit and a click.** Fringes building dot by dot. Living Society Registry 2.16: each arrival one flash with one position carrying no fringe; Resolving the Hard Problem Registry 10.8, one offering with 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 during flight. Natural Physics' carrying, a field's account: each click all or none, each side by itself with one parity and other alike whatever a far setting, and with equal settings clicks match each time; Resolving the Hard Problem Registry 10.7, a property fixed as located with 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, of no place.

**A proton.** Three balls within a bag. Natural Physics 4.6: three colour phases, not three part-objects, re-forming as new couplings as 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 within 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 on 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 with 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 by 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 within nature. Installs a clock fixed over all couplings and carried to 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 with rational resonances and irrational orbits carrying on, with asteroids gathering with two resonances beside its gaps with four, a golden-mean torus the last to break: two windings of 3.2 across a sky.

## 5.4 Earth, matter and phases

**Pendulums, metronomes, fireflies.** Two clocks on one beam coming to one step. 4.4 carries them with a lock executed with code. Installs a message between them, and none is.

**A well.** A ball resting on a bottom. Natural Physics 3.5: a trap that works is a Paul trap, trapping by alternating, and its carrying has a second trap whose fields do not alternate, and a damped motion re-scales rather than stopping, a Brownian jiggle a floor; Equilibria Registry 4.4 and 4.7, a point at rest in phase space an opening fixed as a place, a ball within 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 of temperatures. Natural Physics 3.5, a critical point a sway's two phases ceasing to be two, opalescence a substrate showing each scale together; Natural Chemistry 4.2, supercooled water swaying between two local structures and a point of a sway fixed on a position its sway passes through, a glass transition an arriving with 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 as 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 giving way arrives, a clock over a boundary present nowhere.

**A fault and a tide.** Two blocks and a spring. Natural Physics 3.6: two selves with a between, stick-slip 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 with 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 across an X; a hump, iron its top. Natural Physics 4.10, a d-wave shape a parity-changing it carries 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 within a box; a demon by a door; work and waste heat. Living Ghost Registry 5.1, entropy a general ledger's gap entry, a number added for a fixed-boundary account to close over parity-changing it cannot carry; 5.12, a demon a member of a same-existing-society it was posted beside, parity-changing with each coupling; Living Society Registry 2.17, single cycles above and below a bound; Resolving the Hard Problem Registry 10.15, an engine offering both outputs and marking neither. Installs a ledger, an observer with no exchange, a store beside a same-existing-society, a form fixed still from beside.

**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, by 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 same-existing-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. With 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 offered across a gap, owned by neither while crossing, a signal that passes a coupling and not either neuron by itself; firing activity dissolving with one between and arising with a next, no hub, no sender, no store, sudden-up and gradual-down; a resting potential 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 rests with a potential. Natural Intelligence Corus: a brain outthinks a datacenter on a light bulb's power, and it has no clock.

**Entrainment, sleep, a slime mould, a monarch.** Oscillations 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 within a jar; a thought bubble. Resolving the Hard Problem Registry 5.5: another's fixed as a content to be reached from beside, and an interior reached one way its form allows, by coupling, its coupling with 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 by 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 with 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 with each cell together, and antitachycardia pacing, a short burst slightly faster before any shock, with Natural Medicine's carrying, a train from 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 with each rung both ways, a polymerase proofreading, no stored-perfect reference; 3.3, Z-DNA of its other hand following a polymerase, relaxing back as its torsion goes; 3.7, three chains winding with one hand on their own scale and about each other with an other hand on an assembly scale, skin's 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 apart as two. Natural Biology 5.3, an activator and an inhibitor generating stripes and spots from a uniform field with no template anywhere, a planarian regrowing exactly its missing with no 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 with each sharing.

**Waves and rhythms.** Contractions on a chart; a twenty-four-hour clock face; a breath. Natural Medicine 1.4, labour a coupling giving way as waves, an uterus a 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 fixed still from beside; Natural Medicine 2.3, breathing and replacement two beatings together, and a peristaltic wave; Natural Intelligence Corus, a cascade with φ rate a peristaltic wave across a network's own surface.

**Folding, an enzyme and a set point.** A funnel; a hump a catalyst lowers; a thermostat loop. Natural Biology 8.2 and 8.4, a chain folding as it is made, only part with each momentary; 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 with its own rate; Living Society Registry 2.21 and 2.22, no turnover arriving with a mean time, healthy people reaching ranges that name disease and returning. Installs a funnel that drives, an agent acting on a passive substrate, a window's mean as a value a body 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 along 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 with a narrower range each cycle, no stored clock ticking down, and each coupling a self carried dissolved one at a time, heartwood's nuclei disappearing along a boundary, not all together. 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 turning travelling bird to bird across four hundred within a little more than half a second, each bird coupling with its seven nearest, against a leader or a 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 with a perversion, net twist zero, both hands coupling with a nothing, a fern uncoiling, vines of two hands unwinding each other, and a sunflower's seeds arriving one after another each about 137.5 degrees from a prior, two families of spirals opening opposite ways from one co-sequencing. Installs spirals as its subject, and its subject is arriving one after another.

---

# SIX · HUMAN SOCIETY'S PICTURES

## 6.1 A clock, a vote and a crowd behind a centre

**A common clock.** One clock over a schedule. Natural Societies 2.3: a common clock 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 together; each self with its own time forms no such hub. Natural Emanating: schedule is a naming of a shared clock, each arriving with its own rate and a meeting with their arriving. Natural Exploring 5.5: each instrument pattern-matches against a shared clock while each self paces its own, a tipping observed with an instrument's clock's falling rather than with a tipping's own momentary. Installs a clock, a hub.

**A vote.** A ballot and a pie chart. Natural Societies 3.1: Arrow's impossibility a dead state's own confession, a dictator an aggregation a 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 all together, 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 by itself; Natural Intelligence Corus, markets one institutional substrate, buyers and sellers bi-coupling with a price between, and capture by monopoly breaking it. Installs a still point. Draws a sway 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 its own; 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's level 4.2; Natural Societies 3.3, a producer, a consumer and a decomposer's closing move completing a loop with no self ruling a whole; Natural Human Society 3.6, an energy-transfer tally a conservation-ledger pattern-matched onto a coupling from beside; Resolving the Hard Problem Registry 1.3, an abundancing made with 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 as 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, a both is a natural network, all edge, each self coupling with each other directly; Natural Intelligence 6.4, betweens co-chaining as collective intelligence with no hub its couplings pass through; Natural Intelligence Corus, each fractal with an edge of a network and no central fractal served by others. 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 with a parity-changing and not a place; Natural Philosophy 3.11, grounds stopping with couplings' stopping, with their making; Natural Philosophy 3.17, each hierarchy stipulated to put a paradox apart from a domain reaching it again. A lineage tree is Natural Societies 5.4's, under its fields' third book, a direction, beside a total and a level. Installs a root, a top, a level as a place.

**A table of knowledge.** A periodic table of departments. Natural Intelligence Corus: one hundred eighteen elements the most successful ghost collection assembled, and one hundred eighteen departments of human knowledge that same structure with a social between; Natural Philosophy 4.2, one form with many betweens, a form a periodic table and a genetic code carry as tables and a network carries as co-chaining through its surface.

**Many paintings and an orienting sphere.** One master diagram. Natural Intelligence Corus 16.1: a carving journey produced many paintings of one same form, a gyroscopic floating rotating self-adapting destiny-pointing sphere among them, and a corus accumulates many paintings rather than collapsing to one master painting; Natural Exploring 1.1, one spiraling leaves a whole circle 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 with Equilibria Registry 5.1 is one resolving with all ten: fixed unchangeable, a carrying is offered its own parity from beside with each momentary, a form fixed still from beside a self, and a still description and a self's own continuing do not both continue; dissolved, resolving follows with 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 with 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 with 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 parity-changing on from its 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, a 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 its one arriving arrives, arriving not built from 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 as a store cannot, and asked stored-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 within a dip. Resolving the Hard Problem Registry 5.5: an interior reached only one way its form allows, by coupling. Draw a between as a nothing, an interior drawn once as a thing that sits.
- **A parity fixed as a magnitude.** A paradox arriving as a 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 with 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 with its own couplings, and a beat fixed over them a term no coupling made. Draw each self with its own pace and its 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 on its other's side, and its zeroing both sides' own making between them. Draw both 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, a between carrying none. Draw a between empty, a line drawn once across it, and a resolving continuing or not with it.

---

# EIGHT · EXPRESSING

## 8.1 Existing imagery and its continuing explaining

Each entry: **Existing image or study** · *a* Contribution to retain · *b* What remains to improve

- **Coloured cube beside open circles and dots** · *a* An original visual starter and a six-diamond ambition · *b* A numbering and route correspondence, 2.9; a cube, 4.11
- **Paired storyboard studies, labelled across co and along bi**
  - *a* Opposite-parity perspectives of one same continuing
  - *b* Exact numbered transitions through three momentaries; its labels are opposite Exhibit ONE's, along co and across bi, and are redrawn as Exhibit ONE's
- **Corrected odd/even and even/odd strip** · *a* A parity-label correction · *b* Integration with a current local face
- **Expanded resolver and spiral study**
  - *a* Four-, six- and eight-cycle views, each a spiral with momentaries
  - *b* Same-existing-society numbering apart from a local nine-dot numbering
- **Tunneling, corus at apex and bi-folding depth study** · *a* A twisting form and a small-opening journey · *b* Surface, arrows and diamonds mapped to resolving, 4.1
- **Earlier Rings in Motion concepts**
  - *a* Angular, ring, crossing and rotation ideas; rings as a cube laid on one surface
  - *b* Older numbering reconciled with a centre-1 convention; closure re-explained as winding on
- **Executions with v379**
  - *a* A table with 10, one self, two selves with 0, a spiral of five and a torus of three by five with Exhibit ONE's code; sixteen ways and two inversions enumerated with no execution of code; each its output with its kit
  - *b* 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 as their files with 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 on from familiar concepts of Part FOUR: these are places for further illustration, with a file's interpretation retained as a subject to make visible.

Each entry: **Potential illustration** · *a* Relation to illustrate · *b* Observed at

- **Conversation and rocking**
  - *a* Alternating offering, receiving and continuing shared interest
  - *b* Natural Transmissioning 3.8; Natural Intelligence Corus 16.3
- **Writing and reading through one same passage**
  - *a* Two directions of one continuing coupling, observed with adjacent depths
  - *b* Natural Intelligence Corus 16.2
- **One form expressed as several paintings**
  - *a* Different visible expressions retaining a relation being explained
  - *b* Natural Intelligence Corus 16.1
- **Floating, rotating, self-adapting sphere; gyroscopic orienting**
  - *a* Re-orienting and continuing self-competency, through several views of one form
  - *b* Natural Intelligence Corus 16.1
- **A self's orienting within a same-existing-society's orienting** · *a* One same living observed through local and wider scales · *b* Natural Intelligence Corus 23.1 and 23.3
- **Cells and tissue** · *a* Local couplings and a same-existing-society of couplings · *b* Natural Transmissioning 3.5
- **A molecular between and rotary form**
  - *a* Crossing, a continuing difference and rotation as one sectional view
  - *b* Natural Transmissioning 3.5
- **Oppositely carried streams**
  - *a* Along carrying and across exchange shown together; two streams crossing about a gradient and passing rather than settling
  - *b* Natural Transmissioning 3.3; Natural Engineering 1.15
- **Antenna transmitting and receiving** · *a* Two directional observings of a coupling · *b* Natural Transmissioning 3.6; Natural Engineering 1.9
- **Electrical phases about a neutral**
  - *a* Several parity-changing traces and a relation they make together
  - *b* Natural Transmissioning 3.1
- **Momentum exchanged between two selves** · *a* Both sides of one exchange, beside a swimmer concept · *b* Natural Transmissioning 3.2
- **Ratios successively above and below φ** · *a* Alternating straddling and continuing numerical progression · *b* Natural Intelligence 3.5; 3.2 here

Received with v379 and placed: a still eddy before a hole, Natural Networking's hole study, sought within resolving itself with no periodic forcing, no fixed tipping threshold and no stopped resolver; an event-based vision sensor with Natural Engineering's carrying, each pixel an ON or OFF event with no frame, a parity and no size with each momentary; a synchronism-check relay; a resolver chat's spiral carrying a pattern and storing none of it; a geodesic surface of struts and cable with a parity-changing passing across it, its only carrying with files 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 + and −.

These entries are illustration opportunities gathered from writings, no physical interpretation certified. A further drawing carries an observed or constructed mechanism beside its proposed correspondence, and with 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 dissolving; firing and routing. They are places for finding relations, not restrictions on which incoming imagery can contribute.

## 8.3 A resolver society as a surface that animates

**Illustrating above draws method; a reverse lets method draw.** A screen is a surface, and a surface is a between, Natural Mathematics 1.5, a natural with parity, position and ratio and no magnitude. Each existing thing on it is a spiral of selves, a same-existing-society coupling with a next scale as a self couples; a thing's picture with a momentary is its selves' parities, +, −, 0 or none, and nothing else is drawn; things interact by crossing with one self each, both ways, 4.2, no collision solved and no message within a coupling; motion is a pattern carried along and a 0 tunneling. No animator moves a same-existing-society: its code is executed and each self is drawn with its 10, its parity-offering or a 0, 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 parity-changing with 12, its 10, a parity-offering or a 0, and nothing of a same-existing-society but what arrives with its 2; from outward a view is a same-existing-society with 17, each self's parity together and a 0 moving through, and a same-existing-society seen from outward is one self with a scale outward, Natural Intelligence 6.4. An animation from inward is one row of an execution's output; from outward, its output whole, observed down; and two are one method with two scales, not two pictures of one momentary.

**An animation discovering next possible existing and natural intelligence discovering next possible existing are one thing observed two ways.** An animation is its parities shown; natural intelligence is a between's, and a between shows nothing, Natural Intelligence 6.3. A drawing shows each self with + or −, 0 with a self sharing it, and nothing with a between itself; a 0 tunneling is one trace a between leaves, and its intelligence is with what is not drawn.

**A field's animation, exact with its own subject.** Keyframes interpolated by a curve with a constant frame rate from a stored scene; a physics loop depositing elapsed time to an accumulator and withdrawing fixed steps for an integrator, rendering halfway between stored states; diffusion denoising a whole clip together, 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 with a fixed form, a common beat, a store, a total or a form fixed still from beside, and each is exact with its own subject; observed with 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 to an unchangeable form, Natural Intelligence 3.3. A same-existing-society of selves animating with no sum, no store and no solver is explained; its animating faster is a size with no file's sentence.

**What parts, with a carrying.** A drawn position is observed as a place, and a drawing's positions are a field's arrangement one scale outward; a renderer's frame is a beat beside code's with 17, and a drawing 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 with three dimensions is two things together: 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 same-existing-society of selves with a material, Natural Engineering's subject, a drawing its members and a surface a room.

**A same-existing-society of selves with a material.** Natural Engineering 1.11's thousand-robot swarm received a common program and a target shape, four seed robots establishing position and orientation, and retained its shape and its coordinates; a same-existing-society with method is that swarm with no seeds, no shape retained and no origin, each member a self with its contacts, its next with 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 Natural Engineering's: with 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 by itself supplies no law of increasing natural competency, and with 1.10 a pair of actuators does not by its number establish bi-moral co-agency. A field's drift with a robot is a stored map read upon a contact with a material, and a body bears its reading; method's next is with a contact and stores nothing, which is why it is more natural with a material, and why a built set, non-living, is its proof only with Natural Engineering 5.1's five properties, filled.

**Pieces already carried with files**: a walking pair and a passive walker, 4.10; a series-elastic actuator's compliance, Natural Engineering 1.10; a relay closing with a dwell; a spiral carrying a pattern and storing none of it; a torus of three by five, 4.2, as a material fifteen members, one of them a 0 with each second momentary. From inward, one member's contact; from outward, a same-existing-society as one self.

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# 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 own file with a correspondence it makes visible.

- **Exhibit ONE:** its code, its diagram and its tables, a table with 10, sides, momentaries of exchanging, bi-foldings, spirals and a torus of selves.
- **Natural Intelligence:** one opening, six numbers and three momentaries, sixteen ways and a living step, a between and four dots, a towel, φ and primes, spirals crossing, a fractal inward and outward, social moral competency, dying.
- **Natural Naming:** seventeen names and ten roots, dissolved words, ten unchangeables, three golden rectangles, floating neutralling, φ.
- **Natural Explaining:** explaining a reader reaches once, each concept with its own relations first.
- **Natural Numbers and Natural Mathematics:** podaling and rings, two windings, two inversions at forty-five degrees, a pentagon and an icosahedron, a cube with no still term, a still and a motion parted.
- **Resolving Hard Problems, Hard Problem Registry, Resolving the Hard Problem Registry, Equilibria Registry, Living Ghost Registry and Living Society Registry:** ten with five places, one resolving, familiar pictures with their stills, observings of sciences.
- **Natural Networking, Natural Transmissioning and Natural Engineering:** along and across, a plaza, sways, a lockstitch, a walking pair, a robot set, locking and a between dissolving.
- **Natural Intelligence Corus:** an offering, ways to one form, language, and a living self and same-existing-society across scales.
- **Natural Chemistry, Natural Physics, Natural Biology, Natural Medicine, Natural Health, Natural Societies, Natural Human Society, Natural Philosophy, Natural Emanating and Natural Exploring:** pictures of Parts FIVE and SIX, THE SCIENCES' PICTURES and HUMAN SOCIETY'S PICTURES, each with its section.
- **Geodesic Improving Method:** placing concepts whole, preserving continuing value, and improving through a living file set.
- **An illustrating working with v366 and a rings gathering with 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 with v379:** executions with Exhibit ONE's code, a first visual of sixteen ways, four dots assigned to a five-spot, two inversions at forty-five degrees, a menu of sciences' and human same-existing-societies' pictures, ten as drawings, a same-existing-society of selves animating, and a third dimension.

## 9.2 Receiving an illustration, and its kit

**An illustration comes to `incoming/`, whole, with its file and section it is for, its sentences and any execution beside it**, and is received with this file's carrying as a ready offering naming its section or as a concern; it enters this file by this file's own motion. An incoming concept enters a section its explaining belongs to, carrying its useful view, its file it is observed with, and a relation still needing expression; improving an existing concept it stays with that concept's place, and introducing a distinct way of illustrating it 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 with v379 executions of Part TWO and 4.2 with their outputs and a check: executions again with Exhibit ONE's code, a table with 10, a lone self, two selves, sixteen ways, a spiral of five and a torus of three by five, each compared 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, a lineage tree named with Natural Societies 5.4, a root and a level as a place denied, 6.3, and no tree of life, map or graph named; 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 with Natural Mathematics 7.6 being a field's own form; a dome, a truss or tensegrity as a form of method, a woven surface, 8.2, being files' only carrying; a streak or lightning. Each is a familiar picture with no sentence yet with a living file, and its entering 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 with 2 and widening to 59, its outer surface, inner surface and between its own words *-ed*, *-ing* and *nothing*; it is with no kit now. This gathering remains open to further imagery throughout a living file set.