From: dtd@world.std.com (Don Davis)
Subject: 600cell from two tori (long)
Date: Wed, 8 Sep 1999 22:35:27 GMT
Newsgroups: sci.math
Keywords: construction of 120cell and 600cell (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 600cell. My first reaction is that the tetra
> hedron is too "pointy" to make nicelooking diagrams, but maybe not...
i have a moreorless simpletovisualize construction
of the 600cell, 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 120cell 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 600cell 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).
daisychain five such icosahedra poletopole. 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 lefthand and lower boundaries'
edges, because we're going to roll this grid into a torus
later. thus, the grid contains 100 edges: 50 running NS,
and 50 running EW. 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 eggcarton shape,
with 50 squarepyramid cups on each side. divide each cup
into two nonunit tetrahedra, by erecting a righttriangular
wall across the cup, cornertocorner. make the upper cups'
dividers run NE/SW, and make the upsidedown lower cups'
dividers run NW/SE. note that the eggcarton is now a solid
flat layer, one tetrahedron deep, containing 100 unit tetra
hedra and 200 nonunit tetrahedra.
when we shrink the righttriangular dividing walls into
equilateral triangles, we distort each eggcup into a pair
of unittetrahedra. at the same time, the opening of each
eggcup 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 oneunit
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 eggcup is squeezed in one direction,
the edgetetrahedra around it rotate, squeezing the nearby
lower eggcups in the other direction. this forces the flat
sheet into a saddleshape. in R^4, when this saddlebending
happens across the whole eggcarton 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 150cell
solid torus through this fat torus, and attach the two solids'
triangular faces. this is the 600cell polytope.
symmetry: recall the decagonal "axes" of the original
150cell solid tori. these two linked decagons are now
the equators of a 3sphere 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 600cell.
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 vertexto
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 barberpole pattern;
* 5 more decagons are wrapped around the barberpole
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 lefthanded or a righthanded stripe.
thus, the 600cell 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 3d 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 3D cross.
note that the 600cell has (600*4)/20 = 120 vertices,
and it is the dual of the 120cell. thus, it is straight
forward to assemble the 120cell in an analogous but easier
way, starting with two simple rings of 10 dodecahedra apiece.
each ring has 10 necklike 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 squaregrid 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 120degree 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 fixedwidth 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 60cell tori can
indeed be fit together, facettofacet and bumpintohollow.
while every edge in the 600cell is part of exactly
one decagon, every dodecahedron in the 120cell is part
of 6 different 10cell ring. the 120cell 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 Hopflike
partition of the 120cell. these 12 arrangements of
rings correspond exactly to the 12 Hopflike arrange
ments of decagons in the 600cell.
 don davis, boston
