The Shape of a Rule
Take a closed sheet made of small cells, each joined to its neighbours by springs, and fill it with a fluid that cannot be squeezed. Now give every cell a local rule that says how long its springs should be and how sharply it should bend. Change the rule and the whole sheet takes a different three-dimensional shape. The card asked whether a rule written into the cells can decide a body's form.
Turing showed how a local chemical rule can paint spots and stripes onto a surface that stays put. This panel lets the surface move. The rule's pattern sets how long each spring wants to be and how much each hinge wants to bend. The sheet cannot satisfy all of those at once while staying a sphere, so it settles into whatever shape comes closest. Same sheet, same volume, same starting sphere: four rules, four different bodies.
The classical picture below is drawn from Canham (1970), whose abstract was read, and from three works identified by title and venue but not read in full: Turing (1952), Klein, Efrati and Sharon (2007), and Mietke, Jülicher and Sbalzarini (2019). The panel is a teaching model built for this page. Every number quoted from it was measured from the code before the page was written. It was not fitted to any cell.
Layer 1 of 2 · the classical picture
Why a red blood cell is shaped like a disc
A human red blood cell is a flattened disc, thinner in the middle than at the rim. In 1970 Canham proposed that this shape needs no special machinery: it is the shape with the least bending for the cell's amount of membrane and its amount of contents. He computed the bending energy of many candidate shapes "having the same area and volume" and found that the one with the least energy "closely approximated the bending energy and geometry of the actual red-cell profile". He also noted that "the existence of cup shapes frequently observed in hypotonic media fits within the framework".
Two numbers do most of the work. The membrane's area is nearly fixed, because a membrane can bend easily but stretches very little. The volume is fixed, because the fluid inside cannot be squeezed. For a given area, a sphere holds the most volume, so a membrane whose contents exactly fill a sphere of its area has only one possible shape, the sphere. Shapes other than a sphere become possible only when the contents fill less than that. This page calls that ratio how full the sheet is:
how full = enclosed volume ÷ volume of a sphere with the same surface area
At a fullness of 1 the only shape is a sphere. Below 1 there is spare area, and the bending energy decides where it goes.
Shape written into the sheet
A flat elastic sheet whose cells are told to grow by different amounts in different places cannot stay flat: no flat arrangement gives every cell the length it wants, so the sheet buckles out of its plane. Klein, Efrati and Sharon (2007) shaped elastic sheets this way, by prescribing the local lengths and letting each sheet find its own three-dimensional form (the page draws that from the paper's title; the paper itself was not read). The shape is not specified anywhere. Only the local lengths are.
The panel's sheet works the same way. Each spring has a rest length, each hinge between two neighbouring triangles has a preferred angle, and the sheet's energy counts how far every spring and hinge is from what it wants:
E = Σsprings ½ ks (length − rest length)2 + Σhinges ½ kb (angle − preferred angle)2, with the volume held fixed
The sheet moves downhill on that energy until the forces balance. A local rule changes the rest lengths and preferred angles, and so changes the shape the sheet settles into.
Why twelve cells have five neighbours
A flat sheet can be tiled by cells that each have exactly six neighbours, like a honeycomb. A closed sphere cannot. Euler's formula for any closed surface shaped like a sphere, vertices minus edges plus faces equals 2, forces the neighbour counts to fall short of six by exactly twelve in total. The simplest way to pay that is twelve cells with five neighbours, which is how a football is stitched. Seven-neighbour cells are allowed too, but each one must be paid for with an extra five.
The panel's sheet is an icosahedron subdivided three times: 642 cells, 630 with six neighbours and 12 with five. Tick the box under the panel to see where the twelve sit. They are the points where the sheet was forced to curve before any rule was applied.
Turing, with the surface allowed to move
Turing (1952) showed that two chemicals reacting with each other and spreading at different rates can turn a uniform mixture into a steady pattern of spots or stripes: the activator makes more of itself locally, the inhibitor it produces spreads faster and suppresses it further away. The pattern's spacing is set by the two spreading rates, not by anything outside. The panel's third rule runs a standard two-chemical system of this kind (the Schnakenberg reaction) along the sheet's own connections:
du/dt = a − u + u2v + Du Δu dv/dt = b − u2v + Dv Δv (a = 0.1, b = 0.9, Du = 0.4, Dv = 16)
where Δ sums the differences with a cell's neighbours. Wherever the activator u is high, the springs lengthen and the hinges curve outward. Coupling a pattern of this kind to the shape of the surface it sits on is an active research field; Mietke, Jülicher and Sbalzarini (2019) studied surfaces whose shape and chemistry organise each other. The panel is a far simpler cousin: the pattern sets the shape, but the shape does not change the chemistry.
The fourth rule closes that loop in the simplest way. A substance q, of fixed total amount, drifts toward cells that are curved more than the average, and wherever q is high the sheet prefers to curve more. Curvature gathers q, and q makes curvature:
a cell is curved more than average → q drifts in → its hinges prefer more curvature → the sheet bends further there → more q drifts in
The sheet, run here
642 cells joined by springs and hinges, enclosing a volume the panel holds fixed. Each cell carries a number q from 0 to 1, and a rule sets it. High q lengthens that cell's springs by up to a quarter and makes its hinges prefer to curve outward. Nothing else is told what shape to take.
How to read the picture. The pale sheet is the membrane. Coral marks where the rule's q is high. Drag to turn the body. "Settled" in the corner means the forces have balanced and the pattern has stopped changing. When a shape settles, the panel marks it; after a push, it waits for the sheet to settle again and reports whether it came back to the marked shape. The distance it reports is how far the cells ended from where they were marked, after turning and moving the body to line them up, as a fraction of the body's size. Below 0.015 counts as back.
Measured from the panel
| what was done | what happened |
|---|---|
| no rule, from the starting sphere (10 runs) | one inward dent, a cup, in all 10; axes 1 : 1 : 0.82 |
| no rule, squashed by 40% along a random direction (5 runs), or dented and bulged at random (5 runs) | back to the same cup in all 10, distance 0.001 or less |
| two caps, from the starting sphere (15 runs) | axes 1 : 0.94 : 0.75 in 13; in the other 2, a second shape, 1 : 0.93 : 0.71 |
| two caps, squashed (5 runs) or dented and bulged (5 runs) | back to the same shape in 8 of 10; the two runs that started in the second shape ended in the usual one, 0.16 away |
| two caps, rule switched off (5 runs) | the plain cup of the no-rule sheet in every run, 0.097 from the capped shape |
| two caps, rule switched back on | back to the capped shape in all 5, distance 0.0003 or less |
| Turing spots, from the starting sphere (20 runs) | 6 patches in 18 runs, settled after 8,000 to 25,000 steps; in the other 2, 7 patches still shifting at 40,000 steps |
| Turing spots, squashed (5 runs) | back to the same shape in all 5, distance 0.0015 or less |
| Turing spots, dented and bulged (5 runs) | still 6 patches, but in none of 5 the same shape: 4 hopped between two six-lobed arrangements (axes 1 : 0.97 : 0.94 and 1 : 0.97 : 0.97), 0.035 to 0.07 away; the seven-patch run moved 0.16 |
| Turing spots, pattern scrambled (5 runs) | a new pattern formed: 6 patches in 4 runs, 7 in one, in new places, 0.13 to 0.23 away |
| Turing spots, rule off, then back on (5 runs) | off: the plain cup every time. On: six lobes again, but never exactly the marked shape, 0.04 to 0.17 away |
| curvature sensing, from the starting sphere (20 runs) | 25 to 30 small patches, each a shallow bud; settled within 2,000 to 5,200 steps |
| curvature sensing, squashed (5 runs) | back in all 5, distance 0.011 or less |
| curvature sensing, dented and bulged (5 runs) | back in 3; in 2 the buds rearranged (29 to 26 patches, 28 to 27), about 0.06 away |
| curvature sensing, pattern scrambled (5 runs) | a new pattern every time, with fewer buds (20 to 25), 0.02 to 0.11 away |
| curvature sensing, rule off, then back on (5 runs) | off: the plain cup every time. On: back to the marked shape in 1 run; in 4, a different arrangement with fewer buds, 0.05 to 0.11 away |
Four results are worth reading twice. The same sheet, with the same volume and from the same sphere, took four different bodies under four rules, and the shape was never specified in any of them. Each rule's body survives a squash: 19 of the 20 squashes in the table came back, and the one that did not started in the capped sheet's rarer shape. Switching a rule off takes its body away within a few thousand steps and leaves the plain cup, so the shape lives in the rule, not in the sheet. And switching it back on brings the two caps back exactly, brings the Turing body back as six lobes in some arrangement, and gives the curvature-sensing sheet a new arrangement of buds in 4 runs of 5.
One row does not match the classical picture. Canham's shape is a biconcave disc. This sheet never made one: with no rule it dented into a cup, the other shape Canham mentions, at every volume setting from 0.84 down to 0.60 (one run at 0.60 had a second, smaller dent). The panel's springs rest at the lengths of the starting sphere, so the sheet remembers being a sphere, and a deflated shell that remembers a sphere dents inward on one side. A real red blood cell's membrane is a fluid that does not remember any rest shape.
Layer 2 of 2 · the framework's reading
The wrapped reading
The Shape of the Basin
The attractors here are settled shapes: a body whose forces have balanced and whose pattern has stopped changing. The basin of each is the set of starting shapes and patterns that settle into it. The counts below are return rates from the runs above, at a fullness setting of 0.80.
For a squash, the basins are wide. A 40% squash moves the cells about 0.14 to 0.19 of the body's size. At the 0.80 setting, 19 of 20 squashes came back; the exception was a capped run that started in its rarer shape. In the volume table below, one Turing sheet in three hopped to its other six-lobed arrangement instead. Dents and bulges of 0.09 to 0.16 were undone every time by the plain sheet, and by the capped sheet whenever it began in its usual shape.
Some rules have neighbouring basins. The capped sheet has a second shape it occasionally starts in, and a push moves it out of that one into the usual one. The Turing sheet has two six-lobed arrangements of nearly equal energy, and a dent-and-bulge push moved it from one to the other in 4 runs of 5. The curvature-sensing sheet has many nearby arrangements of its buds; a push moved it to another one in 2 runs of 5.
The pattern has a basin of its own, and it is the rule's. Scramble the Turing chemistry and six spots form again in 4 runs of 5, somewhere new: the number is held, the positions are free, much as the fly's heading bump is held but can sit anywhere on its ring. Scramble the curvature-sensing pattern and it re-forms with fewer buds, 20 to 25 instead of 25 to 30, so the number itself depends on the history.
Outside every basin is the plain cup. With any rule switched off, the body falls into the no-rule cup in every run. The rule is what keeps each body from falling there.
The Morph of the Basin
The fullness setting is the dial that reshapes these basins. It sets the volume against the volume of the sphere the springs would make at rest. Each row below is three runs from the starting sphere; "squash" gives how many came back to their own shape after a 40% squash.
| volume setting | no rule | two caps | Turing spots | curvature sensing |
|---|---|---|---|---|
| 1.00 | sphere; squash 3 of 3 | 1 : 0.91 : 0.91; 3 of 3 | 6 lobes; 3 of 3 | 20 buds; 3 of 3 |
| 0.90 | sphere, springs squeezed; 3 of 3 | 1 : 0.90 : 0.89; 3 of 3 | 6 lobes; 3 of 3 | 28 to 30 buds; 3 of 3 |
| 0.80 | cup, 1 : 1 : 0.82; 3 of 3 | 1 : 0.94 : 0.75; 3 of 3 | 6 lobes; 2 of 3 | 29 buds; 3 of 3 |
| 0.70 | cup, 1 : 1 : 0.73; 3 of 3 | 1 : 0.91 : 0.67, two dents; 3 of 3 | 6 lobes; 3 of 3 | 17 to 21 buds; 3 of 3 |
| 0.60 | deep cup, 1 : 0.93 to 0.97 : 0.64 to 0.66; 2 of 3 | 1 : 0.95 to 0.98 : 0.56 to 0.59, two dents; 1 of 3 | 6 lobes; 3 of 3 | 13 to 14 buds; 3 of 3 |
The plain sphere's basin shrinks, then vanishes. With no rule, the sheet stays a sphere down to a setting of 0.86 by squeezing its springs instead of denting. But the sphere's basin narrows as the volume falls. At 0.90 a squash sent it back to the sphere in 5 runs of 5; at 0.88 in 1 of 5; at 0.86 in none, and every squash left a permanent dent. At 0.84 and below it did not form in any run: the sheet dents on its way down from the start. Between 0.88 and 0.86 the sphere is still an attractor, but a shallow one, and the cup beside it has taken almost all the room.
Lower the volume and the neighbouring basins crowd in. At 0.60 a squash sent the plain sheet to a different shape in 1 run of 3 and the capped sheet in 2 of 3. There is more spare area, so there are more ways to fold it.
Only the loop's pattern depends on the body. The Turing chemistry made six spots at every setting, because its spacing is set by its own two spreading rates and the body does not feed back on it. The curvature-sensing pattern changed with the volume, from 20 buds at 1.00 to 13 or 14 at 0.60, because the body's curvature is what gathers it. In the one rule where the shape feeds back, the pattern is a product of the body it makes.
What the morph can and cannot say here. Every basin above is drawn inside a model whose sheet, rules, volume and running costs are supplied. No maintenance cost is represented anywhere in it: a bud costs nothing to keep, and the volume is held exactly by the code, not paid for. The first two rules settle into a minimum of the sheet's energy, which is an equilibrium; the Turing and curvature-sensing rules settle into steady states of equations written into them. No quantity that pays to keep its shape is running here, so recurcline is undefined for all four, not zero. The morph shows which settings make each body possible and how hard it is to knock out of. It says nothing about whether any of these bodies could hold its shape while paying for itself.
What would show this reading wrong
The shape staying put after its rule is switched off. That would mean the body was stored in the sheet, not held by the rule. In every run it fell to the plain cup.
The Turing spot count changing with the volume setting. That would mean the body feeds back on the chemistry, which the model does not allow. It was six at every setting tried.
The curvature-sensing pattern coming back exactly after a pause or a scramble, at the rate the two caps do. That would mean the loop carries no history of its own. It came back exactly in 1 run of 5 after a pause, and in none after a scramble.
Honest limits
The sheet has 642 cells and fixed connections, so it bends and stretches like a thin solid shell, and its springs rest at the lengths of the starting sphere. That is why it dents into a cup instead of forming Canham's disc. A fluid membrane, whose cells can slide past each other, would behave differently, and nothing here speaks for it.
Research models of membranes handle this with dynamically triangulated surfaces: the simulation keeps changing which points are joined, by flipping the shared edge of two triangles, so the surface flows like a two-dimensional liquid and forgets the shape it started from. One current example is OrganL (Allolio, Fábián and Dostalík, 2024), which also makes each triangle curved; its authors describe it as a method "for simulating biomembranes of arbitrary shape." A real red cell is a fluid lipid layer over a protein skeleton that does resist shearing, so a faithful red-cell model needs both layers. This panel has only a solid one.
The stiffnesses, how much q lengthens a spring and curves a hinge, and every rule's constants were chosen so the four rules give clearly different bodies, not measured from anything. Different choices would give different bodies and different counts.
The Turing chemistry spreads along the sheet's connections and counts neighbours; distances play no part. Stretching or bending the sheet therefore does nothing to the Turing pattern. Under the Turing rule the chemistry shapes the body and the body never shapes the chemistry back. The curvature-sensing rule is the only one on this page where the shape feeds back into the pattern. In the curvature-sensing rule, q stops moving once it reaches 0 or 1 in a patch, and the buds are then pinned where they formed; that pinning is a property of the rule as written, and it is part of why that rule's history matters.
Every count comes from a handful of runs: five per row at a setting of 0.80, three per cell in the volume table, five per setting in the sphere's basin. "Back" means within 0.015 of the marked shape after the best turn and shift. The volume setting is not the measured fullness: a sheet that squeezes its springs instead of denting keeps a measured fullness near 1.
Sources
Canham, P. B. (1970). The minimum energy of bending as a possible explanation of the biconcave shape of the human red blood cell. Journal of Theoretical Biology 26(1), 61–81. Source of the quoted passages on shapes of equal area and volume, and on cup shapes. Abstract read. doi.org/10.1016/S0022-5193(70)80032-7
Klein, Y., Efrati, E., & Sharon, E. (2007). Shaping of elastic sheets by prescription of non-Euclidean metrics. Science 315(5815), 1116. Identified by title and venue; not read. Cited for the general idea that local preferred lengths decide a sheet's shape.
Mietke, A., Jülicher, F., & Sbalzarini, I. F. (2019). Self-organized shape dynamics of active surfaces. Proceedings of the National Academy of Sciences 116, 29. Identified by title and venue; not read. Cited only to show that coupling surface chemistry and surface shape is an active field.
Allolio, C., Fábián, B., & Dostalík, M. (2024). OrganL: Dynamic triangulation of biomembranes using curved elements. Biophysical Journal. Abstract read, from the authors' manuscript. Cited as a current example of a dynamically triangulated membrane model. pmc.ncbi.nlm.nih.gov/articles/PMC11213972
Turing, A. M. (1952). The chemical basis of morphogenesis. Philosophical Transactions of the Royal Society of London B 237(641), 37–72. Not read for this page; the site's Turing morphogenesis page covers it.
Related on this site: Living Structures; the Self-Maintaining Membrane, a boundary that rebuilds itself; and Growth & Repair, a form that regrows toward a target.
T0-B). An authored design claim, not a measurement.
SOV §7.2), what this
panel supplies, not how it scored- exercised 1 · Recursion lock. The curvature-sensing rule closes a loop, shape gathers the pattern and the pattern bends the shape, and after a pause or a scramble you can watch whether the same body comes back; the other three rules close no loop.
- supplied 2 · Internal recurcline persistence. Each body comes back from a squash, but only while its rule is on: switch the rule off and every body falls to the same plain cup, so the return is held by the authored rule, not by the structure.
- supplied 3 · Boundary retention. The membrane is a modeled boundary, but it is never built or repaired: its cells, connections and enclosed volume are given by the code and cannot be lost.
- supplied 4 · Maintenance-bearing continuation. No maintenance cost is represented: a bud costs nothing to keep and the volume is held exactly by the code, so persistence is not paid for by the structure.
No sovereignty verdict is claimed here or anywhere in this gallery. The Sovereign column is empty, and that emptiness is the honest reading: no panel here has been shown to produce its own order.