building the icosahedron's plane arrangement…

Cells walkthrough

The Cells table is the control that actually builds a stellation. It looks like a wall of numbers. It is really a map of the shells of space around a polyhedron — and once one row of it can be read, the great icosahedron is two clicks away.

Every figure below is live and clickable. They run the same code as the app.

1 · The idea, in two dimensions

Take a pentagon and extend its five sides. Each side runs on until it meets another, and the crossings outline a five-pointed star — a pentagram. Nothing was added: it is the same five lines, followed further.

That is a stellation. The word is Kepler's, from stella, star; he used the construction in 1619 to obtain two star polyhedra that now carry his name.

In three dimensions you extend a polyhedron's face planes rather than a polygon's edges. The planes carry on past the solid, cut through one another, and slice the space around it into bounded pieces.

2 · Twenty planes, seen from one face

An icosahedron has twenty faces, and so twenty planes. Viewed perpendicular to one of them, the other nineteen cross it in a set of lines that divide it into regions. This figure is the stellation diagram, and it has been the working drawing of the subject since the 1870s. Edmund Hess was using such diagrams by 1876; Max Brückner systematised the icosahedron's in Vielecke und Vielflache (1900), crediting the method to Hess, and every later treatment descends from those two.

The little triangle at dead centre is the icosahedron's own face. Everything further out is space the extended planes have carved up. The ten-pointed star is characteristic of the icosahedron; every solid has its own signature diagram.

3 · Cells, and the shells they sit in

The twenty planes cut the space around the icosahedron into 473 bounded cells, of only twelve distinct shapes. They are not scattered at random: they fall into concentric shells, and the program numbers those shells layers, counting outward from the solid itself.

Layer 0 is the icosahedron. Layer 1 is the first shell of cells resting on its faces — twenty of them, one per face. Layer 2 is the next shell out, and so on to layer 7, where the last cells run off into the long spikes of the diagram. In the table the core is the bottom row and each shell sits on the one below it, so the rows stack the way the solid does.

Click any box. Click the row number to take a whole layer at once.

Walk outward:

4 · How to read one row

The table has a compact grammar. A row is a layer, and along it sit the cells of that layer — though not one box per cell, which would be unusable: layer 3 alone contains sixty.

Cells that the solid's symmetry carries onto one another are instead bundled into a single box. Rotating the icosahedron carries any one of layer 1's twenty cells onto each of the other nineteen in turn; they are one cell occupying twenty positions, and receive one box. The bundle is an orbit of the symmetry group, and is the smallest addition that preserves the symmetry.

5 0 101

5 the layer — the fifth shell out from the core
0 the first bundle of cells in that layer: 12 congruent cells, one on each vertex direction
1 the second bundle: 120 cells — shaded, because it splits
0 1 the two halves it splits into, 60 cells each. This is a chiral pair: mirror images of one another. Take one and you get a left-handed solid, take the other and you get its reflection.
▬ the bar at the foot of every box — its colour is the number of congruent pieces, explained just below

What the coloured bars mean

The bar beneath each box encodes neither selection state nor layer. It records how many congruent cells the box represents, one hue per count. Six counts occur in the icosahedron's table:

Boxes sharing a colour therefore contain the same number of pieces, and that number is determined: it is 120 divided by the symmetry a single cell retains, as the table below sets out. Following a colour through the table identifies cells of the same kind — the vertex-centred ones, the edge-centred ones — which is the structure needed when searching for a particular solid.

Of particular note is the rarest, , which occurs once, in layer 5. It is the only bundle whose cells retain no symmetry of their own, and the only one that separates into two hands.

The split arises from a deliberate two-stage construction. Cells are first grouped under the full symmetry of the polyhedron — for the icosahedron the group I_h, of order 120, which includes reflections. Each bundle is then subdivided under the symmetry required of the stellation, normally I, the 60 rotations alone. A bundle preserved by reflection is unaffected; a chiral bundle separates into two mirror halves, either of which may be taken independently.

Why the counts are what they are

Every number in the table is determined. A bundle's size is the order of the symmetry group divided by the symmetry a single cell retains — the orbit–stabiliser theorem. Since I_h has order 120:

layer	Du Val	cells	what one cell keeps	so it sits…
0	a	1	all 120	at the centre
1	b	20	6 — a 3-fold axis and mirrors	one per face
2	c	30	4 — a 2-fold axis and mirrors	one per edge
3	d	60	2 — a single mirror	—
4	e	20 + 60	6, then 2	—
5	f	12 + 120	10 — a 5-fold axis and mirrors; then 1	one per vertex; and the odd one out
6	g	30 + 60	4, then 2	—
7	h	60	2	—

These sum to 472 cells outside the core, the count given in the literature. One bundle is anomalous: the 120-cell bundle in layer 5 is the only one whose cells retain no symmetry, which is precisely why it is the only one to split. A cell whose stabiliser contains a mirror is its own reflection; a cell whose stabiliser is trivial has a distinct mirror twin elsewhere in the bundle, and the bundle divides in half.

Du Val expressed the same structure typographically: subscripts (e₁, e₂, f₁…) where a shell held more than one shape, and roman against italic for the right- and left-handed forms of a chiral one. Layer, bundle, sub-cell. The table reproduces his notation with the italics replaced by checkboxes.

5 · Support

A cell selected from layer 5 alone yields twelve spikes suspended in mid-air, touching nothing. It is a legitimate set of cells, but not a usable solid.

Compare:

The second selection adds every cell beneath the first, down to the core. In the application this is a single gesture: shift-click follows the chain of contacts downward and takes the entire supporting set.

The third selection illustrates a subtler failure. {0,2} — core plus shell c, omitting shell b — is externally indistinguishable from the compound of five octahedra, because the missing shell is buried. It is nonetheless hollow. The compound is {0,1,2}, and the two can be distinguished only by asking whether every cell rests on something. The support relation exists to catch precisely this; visual inspection does not.

Miller's rules, and where 59 comes from. When Coxeter, Du Val, Flather and Petrie published The Fifty-Nine Icosahedra in 1938, the number 59 was not a discovery about space — it was a consequence of a definition. The five criteria came from J. C. P. Miller, a Cambridge contemporary who was not one of the authors: the faces must lie in the twenty planes; the same pattern must appear in every plane; that pattern must have three-fold symmetry; every part must be reachable from outside, so no sealed cavities; and the figure must not fall into two pieces each with the full symmetry — with one exception, a mirror-image pair sharing nothing, which is what keeps the compound of ten tetrahedra on the list. Change the rules and you change the number.

One clause matters here more than the rest. Miller's third rule asks the parts in each face plane to have three-fold symmetry with or without reflection — and that "with or without" is precisely the I_h-versus-I dial in the program's symmetry panel. Of the 59, 32 are mirror-symmetric and 27 are chiral; count both hands of the chiral ones and you have 86 physically distinct solids. Only 18 of the 59 are fully supported.

The program does not enforce Miller's rules for you — it lets you build whatever you like, including the floating spikes above. What it gives you is the support relation, so that a supported solid is always one gesture away. Coxeter's own enumeration ran on a graph of which region-sets adjoin which, which makes that relation the direct ancestor of what shift-click does here.

6 · Six solids, constructed by hand

Each of the following is a well-known polyhedron, and each is a few clicks in the table.

The capital letters are Coxeter's mainline. A stellation that takes every cell out to some shell is named after that shell in capitals: A is the core, B is core plus shell b, and so on to H, everything. Those eight are the mainline stellations, and four of the six above are among them. Mixed selections concatenate — Ef₁ means "everything through shell e, plus the f₁ cells". That is the same information as {0,1,2,3,4,5(1)}, written by hand in 1938.

7 · Chirality

Layer 5's second bundle divides into the halves marked 0 and 1. Taking everything out to layer 4 and adding one of those halves gives:

The two hands:

The two are genuinely distinct solids: no rotation carries one onto the other, only a reflection. Taking both halves restores mirror symmetry.

This is why the program requires two symmetry groups rather than one. Under I_h the halves are inseparable; under I they may be chosen independently.

The textbook case is a pair of solids usually credited to Edmund Hess in 1876, though Inchbald reads Cauchy as having had them in 1812. Five regular tetrahedra can be inscribed in a dodecahedron, and the resulting compound of five tetrahedra has only the 60 rotations — it is chiral, and there are two of them, mirror images that no rotation can interchange. Put both together and you get the compound of ten tetrahedra, which is mirror-symmetric again. Both are stellations of the icosahedron, and in cell language they differ in exactly one place. Both take every cell out to shell e. Then the ten-tetrahedron compound takes the whole chiral bundle in shell f, while each five-tetrahedron compound takes one of its two halves. Choosing a sub-cell instead of its parent is literally the difference between building one and building the other.

The figure above is a well-known solid. The selection shown, {0,1,2,3,4,5(1[1])}, is Du Val's Ef₁ — the compound of five tetrahedra, one of the five regular compounds, and one of only two chiral stellations of the icosahedron that are fully supported. Both counts agree: 1+20+30+60+20+60 cells out to shell e, plus 60 for one hand of f₁, is 251, which is the figure in the literature.

It is also what the original program shipped as an example: sample_02.stel, one of four .stel files in the repository, contains exactly this string — and it is what a single shift-click on that half-bundle produces, its supporting set being everything beneath it.

8 · Exercises

The table below begins empty and is unconstrained.

Click 0 in row 0 to restore the icosahedron.
Click a box high in the table, well away from the core, then shift-click the same box, and compare the floating shell with the supported solid.
ctrl-click a cell in a low layer to remove it together with everything resting on it. (A right-click does nothing.)
Click a row number to select an entire layer.
Hover over any box: the readout gives the cell's identifier, its number of congruent pieces, and its volume.

The string underneath is the same notation the program saves in its files. {0,1,2} means layers 0, 1 and 2 entire; 5(1[1]) means layer 5, bundle 1, half 1.

9 · Where this comes from

Little in the Cells table is new. The shells are Du Val's, the selection rules Miller's, the diagram Hess's by way of Brückner, and the solids largely Hess's and Poinsot's. The program's contribution is that they can be manipulated directly.

The case for a table. Most stellation software selects cells geometrically: by clicking a region of the stellation diagram, or a face of the solid. That is intuitive, but it answers the question "which cell lies behind this facet?" only indirectly, and a single click may cascade into many cells. A table presents the whole state space at once, at the cost of abstraction.

Of the stellation programs written since 1980, a survey turns up essentially only Bulatov's that offers a layers × cells × sub-cells grid. The nearest relative is Great Stella's cell diagram, which holds the same information drawn as a graph rather than a grid — and its author cautions in his own manual that it is usually not an intuitive way to build stellations.

What justifies the grid here is its third column: the sub-cell split, which arises only when the stellation is permitted less symmetry than the polyhedron. Without it the table is a list; with it, the table is the only representation in which a solid's handedness is directly addressable.

H. S. M. Coxeter, P. Du Val, H. T. Flather and J. F. Petrie, The Fifty-Nine Icosahedra, University of Toronto Press, 1938. Second edition Springer-Verlag 1982; third edition Tarquin 1999, with new material and photographs by Kate and David Crennell. Flather made the card models, Petrie drew the figures, Du Val devised the shell notation, and the selection criteria are J. C. P. Miller's. The source of the shells a–h, the sub-cell handedness convention, and the number 59.
Max Brückner, Vielecke und Vielflache: Theorie und Geschichte, Teubner, 1900. Where the icosahedron's stellation diagram is worked out in print — §153, "the complete figure of the planes of the icosahedron" — and the source of the photographic plates that carried these solids into the twentieth century.
Edmund Hess, 1876 — the earliest documented use of stellation diagrams, and the compounds of five octahedra, five tetrahedra and ten tetrahedra. Three of the solids in section 6, and the chirality in section 7. The 1876 date rests on secondary sources; Inchbald's timeline puts the relevant work in 1883 and credits the compounds to Cauchy.
Louis Poinsot, 1809 — the great icosahedron and great stellated dodecahedron, completing the four Kepler–Poinsot star polyhedra begun by Kepler in 1619.
Magnus Wenninger, Polyhedron Models, Cambridge University Press. Tabulates the icosahedron's cell shapes and counts — 20, 30, 60, 20, 60, 120, 12, 30, 60, 60, which is 472 cells outside the core. The program computes the same multiset independently.
MathWorld, Icosahedron Stellations Of the 59: 32 reflexible, 27 chiral, 18 fully supported.
Guy Inchbald, Stellating the Icosahedron and Faceting the Dodecahedron A modern reading of Miller's rules and Du Val's notation, and of what the rules leave out.
Vladimir Bulatov, Stellation applet, 1998–2001. The program this one is a port of, and the origin of the Cells window.

The program is a JavaScript port of the Stellation applet written by Vladimir Bulatov between 1998 and 2001. The geometry here is a direct port of his algorithm, checked against the original Java: the same facet counts, the same cell structure, the same volumes to six decimal places. Source and notes are on GitHub.