From: dtd@world.std.com (Don Davis) Subject: 600-cell from two tori (long) Date: Wed, 8 Sep 1999 22:35:27 GMT Newsgroups: sci.math Keywords: construction of 120-cell and 600-cell (solids in R^4) In article <7r5bil$elq$1@mail.pl.unisys.com>, "Clive Tooth" wrote: > I wonder if the same approach could be used to illustrate the > construction of a 600-cell. My first reaction is that the tetra- > hedron is too "pointy" to make nice-looking diagrams, but maybe not... i have a more-or-less simple-to-visualize construction of the 600-cell, which i figured out last winter (it took me a while). i also found, in writing this up, that a similar construction also makes the 120-cell even easier to visualize. i have looked around on the web, and haven't seen anything as easy as my technique. i must admit i am very pleased with it. i don't have pictures yet, but here's a quickie description, without any proofs: the 600-cell construction has three parts: two solid tori of 150 cells each, and a hollow torus of 300 cells. build each solid torus as follows: using 100 tetrahedra, assemble 5 solid icosahedra (this is possible in R^4). daisy-chain five such icosahedra pole-to-pole. between every pair of adjacent icosahedra, surround the common vertex with 10 tetrahedra. each solid torus has a decagonal "axis" running through the centers and poles of the icosa- hedra. each solid torus contains 5*20 + 5*10 = 150 tetra- hedra, and its surface is tiled with 100 equilateral triangles. on this surface, six triangles meet at every vertex. we will link these solid tori, like two links of a chain. with the hollow torus acting as a glue layer between them. build the hollow torus as follows: lay out a 5x10 grid of unit edges. omit the left-hand and lower boundaries' edges, because we're going to roll this grid into a torus later. thus, the grid contains 100 edges: 50 running N-S, and 50 running E-W. attach one tetrahedron to each edge from above the grid. the opposite edges of these tetrahedra will form a new 5x10 grid, whose vertices overlie the centers of the squares in the lower grid. thus, these 100 tetrahedra now form an egg-carton shape, with 50 square-pyramid cups on each side. divide each cup into two non-unit tetrahedra, by erecting a right-triangular wall across the cup, corner-to-corner. make the upper cups' dividers run NE/SW, and make the upside-down lower cups' dividers run NW/SE. note that the egg-carton is now a solid flat layer, one tetrahedron deep, containing 100 unit tetra- hedra and 200 non-unit tetrahedra. when we shrink the right-triangular dividing walls into equilateral triangles, we distort each egg-cup into a pair of unit-tetrahedra. at the same time, the opening of each egg-cup changes from a square to a bent rhombus. as the square openings bend, the flat sheet of 300 tethrahedra is forced to wrap around into a hollow torus with a one-unit- thick shell. surprisingly, this bends each 5x10 grid into a toroidal sheet of 100 equilateral triangles. each grid's short edge is now a pentagon that threads through the donut hole. the grid's long edge is now a decagon that wraps around both holes in its donut. the two grids' long edges are now linked decagons. this wrapping cannot occur in R^3, but it works fine in R^4. i admit that this part of my presentation is not easy to visualize. perhaps a localized visualization image will help: as an upper egg-cup is squeezed in one direction, the edge-tetrahedra around it rotate, squeezing the nearby lower egg-cups in the other direction. this forces the flat sheet into a saddle-shape. in R^4, when this saddle-bending happens across the whole egg-carton at once, the carton's edges can meet to make the toroidal sheet. finally, put one solid torus inside the hollow toroidal sheet, attaching the 100 triangular faces of the solid to the 100 triangles of the sheet's inner surface. this gives us a fat solid torus, 10 units around and 4 units thick, containing 450 tetrahedral cells. nevertheless, its surface has only 100 triangular faces. thread the second 150-cell solid torus through this fat torus, and attach the two solids' triangular faces. this is the 600-cell polytope. symmetry: recall the decagonal "axes" of the original 150-cell solid tori. these two linked decagons are now the equators of a 3-sphere in R^4. the equators are 3 units apart. all of the grids' N/S unit edges are linked up into decagons, too. indeed, each edge in the 600 cell is part of a unique planar decagon that girdles the figure. there are (600*6)/5 = 720 edges in all, so 72 such decagons criscross the 600-cell. it's a beautiful fact that these decagons can be grouped into sets of 12, s.t. the 12 decagons trace the linked circles of a Hopf fibration of S^3! for each of the 6 vertex-to- vertex rotational axes of the icosahedron, there are two arrangements of these 12 decagons: * through each axis there is exactly one equatorial decagon; * linked with this equatorial decagon are 5 decagons, wrapped around the equator in a barber-pole pattern; * 5 more decagons are wrapped around the barber-pole in the same direction, but in a shallower spiral; * the second equatorial decagon girdles the lot, in a plane perpendicular to the first equator. there are two such wrappings for each axis, because the barber pole can carry a left-handed or a right-handed stripe. thus, the 600-cell presents 12 distinct Hopf fibrations of itself. -------==================------- in general, this "stitch 2 tori together" approach can be very natural for visualising the complicated polytopes of 3-d cells, since the tori display the S^3 symmetry enjoyed by these shapes. for example, it's nicer to con- sider the tesseract as two linked rings of 4 cubes, than as a russian doll of nestled cubes, or as a 3-D cross. note that the 600-cell has (600*4)/20 = 120 vertices, and it is the dual of the 120-cell. thus, it is straight- forward to assemble the 120-cell in an analogous but easier way, starting with two simple rings of 10 dodecahedra apiece. each ring has 10 neck-like indentations between pairs of dodecahedra. we cover each neck with 5 dodecahedra, making two bumpy tori of 60 cells apiece. the bumps and hollows form a simple square-grid arrangement on the surface of each torus. link these tori, and stitch them together, fitting the bumps of one torus into the hollows of the other. this bumpy interface between the two tori is a semi- regular toroidal surface, comprising 50*4 = 200 regular pentagons. three pentagons meet at each vertex in the concave and convex parts of the surface, but in the saddle- shaped parts, four pentagons meet at each vertex. when we project this tesselation onto the plane, we get a familiar tiling of irregular pentagons. in this projection, each pentagon has two right angles and three 120-degree angles, and four pentagons are arranged to form a squat hexagon. these hexagons tile the plane (and thus the toroidal surface) in the usual way (you'll need a fixed-width font here): :, ,: :, ,: :, ,: __/ "|" \___/ "|" \___/ "|" \___ \ ,|, / \ ,|, / \ ,|, / ,:" ":, ,:" ":, ,:" ":, |" \___/ "|" \___/ "|" \___/ "|" |, / \ ,|, / \ ,|, / \ ,|, ":, ,:" ":, ,:" ":, ,:" __/ "|" \___/ "|" \___/ "|" \___ \ ,|, / \ ,|, / \ ,|, / ,:" ":, ,:" ":, ,:" ":, |" \___/ "|" \___/ "|" \___/ "|" |, / \ ,|, / \ ,|, / \ ,|, ":, ,:" ":, ,:" ":, ,:" __/ "|" \___/ "|" \___/ "|" \___ \ ,|, / \ ,|, / \ ,|, / ,:" ":, ,:" ":, ,:" ":, |" \___/ "|" \___/ "|" \___/ "|" |, / \ ,|, / \ ,|, / \ ,|, ":, ,:" ":, ,:" ":, ,:" __/ "|" \___/ "|" \___/ "|" \___ in this diagram, suppose that the ring of 10 dodecahedra is running vertically inside its sheath of 50 dodecahedra, whose exposed facets appear as irregular pentagons. then the horizontal edges are at the tops of the bumps in the surface, while the vertical edges are at the bottoms of the hollows. thus, the fat torus of 60 dodecahedra is covered with 10 rows of squat hexagons, with 5 hexagons in each row. similarly, this pattern is overlaid with 10 columns of tall hexagons, with 5 hexagons in each column. together, these overlaid patterns show that the two fat 60-cell tori can indeed be fit together, facet-to-facet and bump-into-hollow. while every edge in the 600-cell is part of exactly one decagon, every dodecahedron in the 120-cell is part of 6 different 10-cell ring. the 120-cell can be parti- tioned into 12 linked rings, pairwise nearly parallel, so that the rings again trace the circles of the Hopf fibration. there 12 ways to perform this Hopf-like partition of the 120-cell. these 12 arrangements of rings correspond exactly to the 12 Hopf-like arrange- ments of decagons in the 600-cell. - don davis, boston -