The nested canon

These are the five regular solids: in each, every face is the same shape and every corner is the same. Here the five share one frame. From the outside in: dodecahedron, cube, icosahedron, octahedron, tetrahedron.

The cube's 8 corners are 8 of the dodecahedron's 20, so the two share one sphere.

The icosahedron's 12 corners rest on the cube's faces, two to a face. The octahedron's 6 corners sit at the middles of the icosahedron's edges.

The tetrahedron's 4 corners sit at the centres of the octahedron's faces. That makes 8 + 12 + 6 + 4 = 30 contacts, and the largest error among them is 0.00000000000000011, the limit of the arithmetic.

In the Timaeus, written around 360 BC, Plato gives four of the solids to the elements: the tetrahedron to fire, the octahedron to air, the icosahedron to water, the cube to earth. Then he builds their faces from two triangles.

He cuts every equilateral triangle into six right triangles with sides in the ratio 1 : √3 : 2, and every square into four with sides 1 : 1 : √2.

Count them. The tetrahedron has 4 faces, so 24 small triangles. The octahedron has 48, the icosahedron 120. The cube is built from the other triangle: 24 of them.

Fire, air and water share a triangle, so they can trade. One water, 120 triangles, breaks into one fire and two airs: 120 = 24 + 48 + 48. Earth never trades. The dodecahedron uses neither triangle; of the fifth figure Plato writes that “God used it up for the Universe in his decoration thereof.”

At every corner of a solid at least three faces meet. Unfold them flat around the point and there has to be a gap. With no gap they lie flat, and cannot fold up into a corner.

A triangle's corner is 60°. Three triangles leave a gap of 180° and fold into a tetrahedron; four make an octahedron, five an icosahedron. Six triangles fill all 360°, and the floor stays flat.

Squares, at 90°, allow three: the cube. Pentagons, at 108°, allow three: the dodecahedron. Hexagons already fill 360° with three, and nothing larger fits at all. That is the whole list.

With p sides to a face and q faces at a corner, the rule is (p − 2)(q − 2) < 4. The red line is where the product is exactly 4, and the three flat floors sit on it.

Around 1630 Descartes found one more thing. Add up the gaps at every corner, and all five give the same total, 720°.

Hypsicles proved it in the second century BC. His book travelled with Euclid's Elements as Book XIV, and in it he credits the result to an older geometer, Aristaeus.

The reason is duality. Put a point at the centre of every face of a dodecahedron and you get the corners of an icosahedron, and the other way round.

A solid and its dual have the same ratio between their inner and outer spheres. Inside one outer sphere they share the inner one too, so every face of either sits 0.79465 of the radius from the centre, and cuts the sphere in one circle of radius 0.60706.

So the theorem belongs to duality, not to pentagons. The cube and the octahedron share a circle the same way, at 0.81650. The tetrahedron is its own dual, at 0.94281.

Saturn's sphere holds a cube. The cube holds Jupiter's sphere, then come a tetrahedron, Mars, a dodecahedron, Earth, an icosahedron, Venus, an octahedron and, at the centre, Mercury. Each solid fixes the ratio of the sphere outside it to the sphere inside.

Kepler then checked the model against the best distances he had, from Copernicus, and printed the comparison. Mars and Venus, he wrote, came out the same; Earth and Mercury not very different; Jupiter alone was far off.

For Mercury he bent his own rule. Instead of the sphere inside the octahedron, 577 parts per thousand, he used the sphere through the middles of its edges, 707. In red, that bend.

Even his best match, Venus at 794 against the law's 795, rests on a slip in his arithmetic. Worked correctly from Copernicus's own figures, it is 772.

Luca Pacioli's book De divina proportione printed solids drawn by Leonardo da Vinci. Each name follows a small grammar: the solid, then plain, cut or raised, then solid or open.

Cut the icosahedron's corners at a third of each edge and you get 32 faces, 60 corners and 90 edges, the pattern of a football. Raise a three-sided pyramid on every face and you get 60 faces, 32 corners and 90 edges.

Every version keeps Euler's count: corners minus edges plus faces is 2. Leonardo's open frames show the front and the back at once.

Six plates, XXI to XXVI, each seen along four axes, make 24 figures: the number Wenzel Jamnitzer drew for each solid in 1568. In red, plate XXIIII's name as both manuscripts write it.

1. XXI YCOCEDRON PLANVS SOLIDVS
2. XXII YCOCEDRON PLANVS VACVVS
3. XXIII YCOCEDRON ABSCISVS SOLIDVS
4. XXIIII YCOCEDRON ABSCISVS VACVVS
5. XXV YCOCEDRON ELEVATVS SOLIDVS
6. XXVI YCOCEDRON ELEVATVS VACVVS

The nest's cube uses 8 of the dodecahedron's 20 corners. Turn the dodecahedron a fifth of a turn and the same 20 corners hold a different cube.

Five turns, five cubes, drawn here in five colours: 5 × 8 = 40 places on 20 corners, so every corner belongs to exactly two cubes.

The dodecahedron's 60 turns shuffle the five cubes in every even way, and there are exactly 60 even rearrangements of five things.

Inside each cube sit two tetrahedra, one of each hand: Kepler's stella octangula of 1619. They cross in an octahedron, and together fill exactly half the cube.

On a loom the long threads, the warp, hang from frames called shafts. Foot pedals, the treadles, lift the shafts, and the cross thread, the weft, passes under every lifted thread.

A weaver's draft has four parts. The threading says which shaft holds each warp thread. The tie-up says which shafts each treadle lifts. The treadling says which treadle goes down for each pass of the weft. Multiply the three and you get the cloth.

Around 1804 Joseph-Marie Jacquard replaced the treadles with a chain of punched cards: one card for each pass, with a hole wherever a thread must rise.

This draft repeats every 8 passes and runs back on itself, so passes 6, 7 and 8 reuse cards 4, 3 and 2. Five cards do the work of eight: 80 holes instead of 135.

“We may say most aptly that the Analytical Engine weaves algebraical patterns just as the Jacquard-loom weaves flowers and leaves.”

Six triangles fill 360° at a point, so they make a floor, not a solid. Alternate triangles and hexagons around every point, 60° + 120° + 60° + 120° = 360°, and the floor becomes a basket weave. Physicists call it kagome, a Japanese word for the eyes of a woven basket.

Weave three families of threads at 60° and their crossings are its points. X passes over Y, Y over Z, and Z over X: no thread lies on top of all the others, and that is why the cloth holds.

Count one repeat of the pattern: 3 points, 6 edges, 3 faces, and 3 − 6 + 3 = 0. On a sphere the same count gives 2 and the gaps add to 720°; on the flat weave it gives 0 and every gap is 0°. One rule covers both: the total gap is 360° times Euler's count.

In red, the eye: the point where the cloth parts.

On the folio's first plate the nest stands on this weave. Its threads run along the cube's three axes and cross in the same cycle: X over Y, Y over Z, Z over X.

Count every corner, edge and face of all five: 50 − 90 + 50 = 10, two for each solid. Corners and faces both come to 50, so the set is its own dual.

Every drawing on this page is computed in your browser from closed forms, none drawn by hand, and every number agrees with the nine plates of the folio.