H(4)-Polychora with Zome

Introduction.

This page has some data on the construction of some Zome models. In particular, it gives a parts list for the Zome models of the convex uniform polychora with the same symmetry group as the 600-cell. In the context of Coxeter groups, this group is often denoted by H(4).

There are 15 convex uniform polychora with H(4) symmetry, classified by their Wythoff symbols. A Wythoff symbol is an extension of a Coxeter graph, obtained by circling a non-empty subset of its vertices. The Wythoff symbol designates where each vertex lies, in reference to a fundamental region of the Coxeter-group action on 4-space. Also, the Coxeter graph obtained by omitting the circled vertices yields the symmetry group of the vertex figure of the corresponding polychoron. Notice that this page does not have any instructions on how to construct the models once one has all the required pieces. Presumably, if one can figure out the technicalities of the Wythoff symbol, one can probably figure out how to put the pieces together. (But not necessarily vice-versa!)

The Table.

+----------------------------+-----------------------+-------+-------+------+------+------+-------+--------+
|            Name            |                       |       |       |      |      |      |       |        |
|     (Vorthmann Symbol)     |     Wythoff Symbol    | Balls | R1=R2 |  Y2  |  B2  | d(B) | d(B2) |  Link  |
+----------------------------+-----------------------+-------+-------+------+------+------+-------+--------+
|                            |                       |       |       |      |      |      |       |        |
|          600-cell          |  0-----0-----0----(0) |   75  |   72  |  120 |  120 |  30  |   60  |  Here  |
|            (1)             |     5                 |       |       |      |      |      |       |        |
+----------------------------+-----------------------+-------+-------+------+------+------+-------+--------+
|                            |                       |       |       |      |      |      |       |        |
|     rectified 600-cell     |  0-----0----(0)----0  |  396  |  360  |  600 |  480 |  72  |   60  |  Here  |
|            (2)             |     5                 |       |       |      |      |      |       |        |
+----------------------------+-----------------------+-------+-------+------+------+------+-------+--------+
|                            |                       |       |       |      |      |      |       |        |
|     rectified 120-cell     |  0----(0)----0-----0  |  640  |  360  |  600 |  480 |  80  |   60  |  Here  |
|            (4)             |     5                 |       |       |      |      |      |       |        |
+----------------------------+-----------------------+-------+-------+------+------+------+-------+--------+
|                            |                       |       |       |      |      |      |       |        |
|          120-cell          | (0)----0-----0-----0  |  330  |  120  |  200 |  180 |  60  |   60  |  Here  |
|            (8)             |     5                 |       |       |      |      |      |       |        |
+----------------------------+-----------------------+-------+-------+------+------+------+-------+--------+
|                            |                       |       |       |      |      |      |       |        |
|     truncated 600-cell     |  0-----0----(0)---(0) |  780  |  432  | 720  |  600 |  120 |  120  |  Here  |
|            (3)             |     5                 |       |       |      |      |      |       |        |
+----------------------------+-----------------------+-------+-------+------+------+------+-------+--------+
|                            |                       |       |       |      |      |      |       |        |
|    cantellated 600-cell    |  0----(0)----0----(0) |  1860 |  1080 | 1800 | 1380 |  120 |   60  |  Here  |
|            (5)             |     5                 |       |       |      |      |      |       |        |
+----------------------------+-----------------------+-------+-------+------+------+------+-------+--------+
|                            |                       |       |       |      |      |      |       |        |
|   expanded 120/600-cell    | (0)----0-----0----(0) |  1260 |  720  | 1200 |  960 |  120 |  120  |  Here  |
|            (9)             |     5                 |       |       |      |      |      |       |        |
+----------------------------+-----------------------+-------+-------+------+------+------+-------+--------+
|                            |                       |       |       |      |      |      |       |        |
|  bitruncated 120/600-cell  |  0----(0)---(0)----0  |  1860 |  720  | 1200 |  960 |  120 |  120  |        |
|            (6)             |     5                 |       |       |      |      |      |       |        |
+----------------------------+-----------------------+-------+-------+------+------+------+-------+--------+
|                            |                       |       |       |      |      |      |       |        |
|    cantellated 120-cell    | (0)----0----(0)----0  |  1860 |  1080 | 1800 | 1380 |  120 |   60  |  Here  |
|            (A)             |     5                 |       |       |      |      |      |       |        |
+----------------------------+-----------------------+-------+-------+------+------+------+-------+--------+
|                            |                       |       |       |      |      |      |       |        |
|     truncated 120-cell     | (0)---(0)----0-----0  |  1260 |  480  |  800 |  660 |  120 |  120  |  Here  |
|            (C)             |     5                 |       |       |      |      |      |       |        |
+----------------------------+-----------------------+-------+-------+------+------+------+-------+--------+
|   cantitruncated 120-cell  |                       |       |       |      |      |      |       |        |
|            grix            |  0----(0)---(0)---(0) |  3660 |  1440 | 2400 | 1860 |  120 |  120  |  Here  |
|            (7)             |     5                 |       |       |      |      |      |       |        |
+----------------------------+-----------------------+-------+-------+------+------+------+-------+--------+
|   runcitruncated 600-cell  |                       |       |       |      |      |      |       |        |
|           prahi            | (0)----0----(0)---(0) |  3660 |  1800 | 3000 | 2280 |  120 |   60  |  Here  |
|            (B)             |     5                 |       |       |      |      |      |       |        |
+----------------------------+-----------------------+-------+-------+------+------+------+-------+--------+
|   runcitruncated 120-cell  |                       |       |       |      |      |      |       |        |
|            prix            | (0)---(0)----0----(0) |  3660 |  1800 | 3000 | 2280 |  120 |   60  |  Here  |
|            (D)             |     5                 |       |       |      |      |      |       |        |
+----------------------------+-----------------------+-------+-------+------+------+------+-------+--------+
|   cantitruncated 120-cell  |                       |       |       |      |      |      |       |        |
|           grahi            | (0)---(0)---(0)----0  |  3660 |  1440 | 2400 | 1860 |  120 |  120  |  Here  |
|            (E)             |     5                 |       |       |      |      |      |       |        |
+----------------------------+-----------------------+-------+-------+------+------+------+-------+--------+
|                            |                       |       |       |      |      |      |       |        |
| omnitruncated 120/600-cell | (0)---(0)---(0)---(0) |  7200 |  2880 | 4800 | 3600 |   0  |   0   |  Here  |
|            (F)             |     5                 |       |       |      |      |      |       |        |
+----------------------------+-----------------------+-------+-------+------+------+------+-------+--------+

Explanation of the Table.

The first and second columns of the table are fairly self-explanatory. The names used here for these polychora are somewhat arbitrary; it is believed that these are the names most commonly used. Jonathan Bowers created the unusual names, "grix", etc, for four of these polychora; he has names for all of these (and many others), and they are all essentially clever abbreviations of very long terms. Norman Johnson coined the terms "rectified" and "cantellated". Not many people have seen models of the last 5 polychora in the table, and this is part of the reason why it is difficult to find suitable names for them. Remarkably, all 15 of these polychora have been built using Zome pieces. The last in the column of the table provides a link to one or more photos of these models.

(Appearing in parentheses below the common name is a hexadecimal digit indicating the symbol whimsically proposed by Scott Vorthmann. It is based on the fact that all fifteen of these polychora may be interpreted as F2-linear combinations of the first four appearing in the table. Looking at the Wythoff symbol, it is clear that each polychora may be expressed using four bits, with 1 meaning that the node is circled and 0 meaning that the node is uncircled.)

The third column gives the required number of connector balls for each model. The column marked "R1=R2" gives the required numbers of R1 (short red) and R2 (medium red) struts. These two numbers are always equal for any particular model. The next two columns give the required numbers of Y2 (medium yellow) and B2 (medium blue) struts. Thus, all 15 of the models may be constructed from the family {R1,R2,Y2,B2}, and using no other lengths of struts. The seventh and eighth columns "d(B)" and "d(B2)" require further explanation. Briefly, these are the numbers of balls and B2 struts "on the boundary".

The Details.

A Zome model of a polychoron typically represents the image of a linear map

p: R^4 ---> R^3,
(w,x,y,z) |---> (x,y,z).
Thus, generally, p maps vertices to points (balls) and edges to line segments (struts). These are often called "shadow" projections of polychora.

Every polychoron considered here corresponds to a cellular decomposition of the hypersphere S^3 in R^4, and the image of S^3 under p is a closed 3-dimensional ball. This ball can be partitioned into two manifolds, one being an open 3-dimensional ball, and the other being the boundary, which is an ordinary (2-dimensional) sphere in 3-space. The significance of this partition is as follows: Generally, if a cell lies interior to p(S^3), then the pre-image under p consists of two cells, while if a cell lies on the boundary of p(S^3), then the preimage consists of one cell. This observation is one of the keys to determining the required numbers of connector balls and struts.

The projections of these H(4)-polychora all share another property. Specifically, the pre-image of a connector ball (resp. strut) consists of two vertices (resp. edges) if and only if it lies interior to the closed ball p(S^3). One should note that this is an artifact of these polychora and the way they are mapped by p. This phenomenon does not occur for all uniform polychora, and, indeed, it is possible to construct Zome models of other uniform polychora for which each of the vertices is rendered uniquely by a Zome connector ball.

The number of balls. Choose a convex uniform polychoron X with H(4) symmetry, and let v denote the number of vertices. One regards these v vertices, all lying at the same distance to the origin, as lying on the hypersphere S^3. Thus one may apply the above observation concerning the properties of the projection p.

Let B denote the number balls required to make a Zome model of X and d(B) denote the number of balls which lie on the boundary. Then one evidently has the relationship

B = v/2 + d(B)/2
between these quantities. For example, the 600-cell has v=120 vertices, and its Zome model has B=75 balls with d(B)=30 balls lying on the boundary. Notice that 75=120/2+30/2. In order to determine the number of required balls, therefore, one needs to know v and d(B). The number v of vertices is well-known, and one can determind d(B) fairly quickly by constructing a part of the boundary of the Zome model of X. This explains the column marked "d(B)".

The numbers of struts. These computations are more complicated, as they involve information about the conjugacy classes of the rotation group of the regular icosahedron. Nevertheless, one still applies the topological property of the projection p. As above, let X be a convex uniform H(4) polychoron, and let e denote the number of edges.

There is a close relationship between the the ratio [R1:R2:Y2:B2] of the numbers of struts and the ratio [12:12:20:15]. For example, if X is the 120-cell, this ratio is [120:120:200:180], which, although not equal, is similar to [12:12:20:15]. The ratio [12:12:20:15] is significant because these numbers appear in the class equation for the rotation group of the regular icosahedron:

60=1+12+12+20+15.
One intereprets this equation as saying that the rotation group of the icosahedron has 1 identity element, 12 rotations by 72°, 12 rotations by 144°, 20 rotations by 120°, and 15 rotations by 180°. Thus, there appears to be a correspondence,
R1 <---> 72°-rotations,
R2 <---> 144°-rotations,
Y2 <---> 120°-rotations,
B2 <---> 180°-rotations.
This correspondence is strengthened by noticing that R1 and R2 struts have pentagonal symmetry, Y2 struts have trihedral symmetry, and B2 struts have dihedral symmetry.

One can clarify this correspondence as follows. First, mark all of the edges of X according to how they are mapped by p. Thus, most of the edges are marked by one of the symbols {R1,R2,Y2,B2}, while the rest of them, being collapsed to single points by p, are marked as such. Now there is an "enhanced" polychoron (X,E), where X is the polychoron and E is the set of marked edges. Call the set E the "virtual struts" of X. Let n, r1, r2, y2, and b2 respectively denote the numbers of collapsed edges and virtual struts marked by R1, R2, Y2, and B2. Using the correspondence described above, one quickly obtains the formulae for the numbers of virtual struts in E:

n = e/60,
r1 = r2 = e/5,
y2 = e/3,
b2 = e/4.
One obtains these merely by requiring that
[n : r1 : r2 : y2 : b2] = [1:12:12:20:15] with n + r1 + r2 + y2 + b2 = e.

Next, as with the computation for the number of balls, one must take into account the fact that most struts in the Zome model of X correspond to two virtual struts. Of course, the only struts which correspond to one virtual strut lie on the boundary. As above, one can construct Zome models of the boundaries of all of the 15 such polychora. After doing this, one discovers that only B2 struts occur on the boundary. (An explanation of this phenomenon this uses the binary icosahedral group, a double cover of the rotation group of the icosahedron.) Thus, the formulae for the R1, R2, and Y2 struts appear to be

R1 = R2 = r1/2 = e/10 and Y2 = y2/2 = e/6.
One needs a correction only for the B2 struts. With that, let d(B2) denote the number of B2 struts on the boundary. As above, one has a relationship,
B2 = b2/2 + d(B2)/2 = e/8 + d(B2)/2.
Consider the 120-cell, for example. This has e=1200 edges and d(B2)=60 struts lying on the boundary. Notice that 180=1200/8+60/2. This explains the final column of the table.

Conclusion. There is likely a way to obtain the last two columns of the table without resorting to building partial Zome models of these polychora, but instead by carefully studying the geometry of the Wythoff symbol. This is probably easiest to see with the "omnitruncated 120/600-cell", where all four of the vertices are circled. The vertices for this polychoron lie on the interior of each fundamental region of the H(4) action, and so no balls or B2 struts can lie on the boundary.

Here and there, one can find many people who have discovered the naturality of building these objects. (Of course, Zome was made into a toy for this very reason!) George Hart has led many "barn-raisings" in constructing these models. Half of the links above show these events. Here are some other examples:
http://www.cornerpockets.com/Zome/zome_cantelated_120_cell.htm
http://web.mit.edu/hawksley/Public/zome.html
http://mathforum.org/pcmi/hstp/sum2004/morning/120cell.html
http://alc.hopto.org/~aaron/toys.php
http://www-math.mit.edu/~drew/ZomePhoto.html
(If you are aware of other links to Zome models in this family, please let me know.)


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