Regular polychoron compounds

"I'm a Platonist - a follower of Plato - who believes that one didn't invent these sorts of things, that one discovers them. In a sense, all these mathematical facts are right there waiting to be discovered."
H. S. M. Coxeter.

As in the case of polygons and polyhedra, stellating a polychoron can result either in a star polychoron or a regular compound. There are, in total, 36 regular and 16 partially regular polychoron compounds. The fully regular compounds must have regular "cells" and vertex polyhedra; the difference is that either can be a regular polyhedron compound as well, in the discussion below we'll call attention to such cases. Like the regular star polyhedra, they must be stellations and facetings of regular convex forms because they must also be symmetric around each vertex and around each cell. For the same reason, the constituent polychora must also be regular.

The partially regular compounds also consist of regular polychora, but they are stellations or facetings of a regular convex polychoron, but not both. In Fig. 5.4, we have already seen two examples of the latter class:
  1. The Compound of two Tesseracts can be represented by the vertex-first model of the Tesseract. It is cell-regular, i.e., its 16 Cubic cells are in the same 3-D hyperplanes as those of the 16-cell, making it a stellation of that polychoron. It is not vertex regular: it has 24 vertices, 8 of each are common to both Tesseracts, eight touch only one of the Tesseracts, and another eight touch only the second Tesseract; it is therefore not a faceting of any regular convex polychoron.
  2. Its dual, the Compound of two 16-cells, can be represented by the cell-first model of the 16-cell. By the rules of duality, this compound is vertex-regular, i.e. its 16 vertices are those of the Tesseract, making it a faceting of that polychoron. It is not cell regular: only 8 of its 24 cell planes are common to both 16-cells; it is therefore not a stellation of a regular convex polychoron.
There are nine additional Zomable compounds, all of them are fully regular. Below we show all their Zomable projections.

Triality

 Before proceeding, we present a brief explanation of the concept of Triality. In the case of duality in 3 dimensions, one can, by the definition of duality, scale and rotate two concentric dual polyhedra A and B in such a way that each edge from A meets a perpendicular edge of B (see examples in Figs. 3.3a, b and c). If A and B are regular, their edges meet at their midpoints, the black balls in Figs. 3.3a, b and c. Those two edges form a plane that is perpendicular to a third line joining the midpoint to the centres of A and B. Rectifying A and B results in the same rectified polyhedron (Fig. 3.4).

In the case of Triality in 4 dimensions, the situation is very similar, except that the perpendicular edges of three concentric "trial" polychora, A, B and C meet at right angles. A set of three 16-cells is the only regular case. Because the three 16-cells are regular, 3 edges from each of the three 16-cells meet at their midpoint. These edges are in a 3-D "plane" that is itself perpendicular to a fourth direction, the line joining the midpoint to their centre. Rectifying all three polychora results in the same rectified polychoron, the regular case being the 24-cell, whose vertices coincide with the previous edge midpoints.

The same applies to projections: if we pick three "trial" projections of the 16-cell and rectify them, we obtain an invariant projection of the 24-cell. In this site, we show a triad of such trialities. The first we met already in Fig. 5.4: the vertex-first and two cell-first projections of the 16-cell have the same rectification (the model of the 24-cell in Fig. 5.5a). The other two are shown in Figs. 7.1a (also 5.5b) and b below.


Fig. 7.1a: Three edge-first projections of the 16-cell, with the central edges of the projections (very) loosely aligned along 3 perpendicular orthogonal axes in 3-d space. Rectifying these three projections, we obtain the vertex-first projection of the 24-cell in the middle. This is the "pyritohedric" projection we have seen in Fig. 5.5b.


Fig. 7.1b: Bottom: The three "prismatic" projections of the 16-cell (*). All are built with the same set of parts: 8 connectors, 6 long reds, 6 medium reds, 6 long yellows, and 6 long blues. Rectifying all three projections results in the same projection of the 24-cell (*), also known as its "prismatic" projection, this is shown by the larger model above. The red balls represent edge intersections that happen only in the 3-d projection. One of the beauties of these models is that they show all vertices, edges, faces and cells without superpositions.


Fig. 7.1c: The prismatic projection of the 24-cell has axes of apparent (6-fold rotational) symmetry. This is known as "ghost symmetry", a phenomenon that will be discussed in more detail below.


Ultimately, Triality results from the aforementioned fact that the 16-cell is a Demi-Hypercube, and is a direct consequence of a unique feature of the demi-hypercubic symmetry in 4 dimensions, the three-fold symmetry of its Coxeter-Dynkin graph. For a more detailed explanation, see David Richter's Triality with Zometool page.

Facetings of the 24-cell

 As mentioned in the derivation of the 24-cell, it has the vertices of three 16-cells. We can therefore make a regular compound of thee 16-cells with the same vertex arrangement of the 24-cell, which is therefore its faceting. In the following Figures (7.2a, b and c), we show on the right three projections of this compound, each of them a superposition of 3 "trial" projections of the 16-cell that appeared, respectively, in Figs. 5.4, 7.1a and b. The real vertices of the compound are represented by the white balls. The edge midpoints, where as we mentioned above they intersect by threes at right angles to each other, are by analogy with Figs. 3.3a, b and c represented by the black balls. These false vertices have the same arrangement as the real vertices of the rectifications of these projections, the projections of the 24-cell on the left.

Importantly, there is a stellation operation, greatening, that transforms a 24-cell into a Compound of three 16-cells. This operation stellates each Octahedral cell of the 24-cell into a "Stella octangula" (Fig. 4.4b) in the same 3-D ``plane", with each Tetrahedron in the pair belonging to a different 16-cell. Each Triangular face is replaced by a dual Triangular face in the same 2-D plane, with each edge being replaced by a parallel one that is twice as long (Fig. 4.4a). Thus, for each strut in a Zomable projection of the 24-cell on the left there is, within the corresponding greatening on the right, a parallel set of two struts of the same colour and size in a straight line connected by a black ball. Therefore, if a projection of the 24-cell is Zomable, then there is a Zomable projection of the Compound of three 16-cells with the same strut colours. This will have important consequences below.


Fig. 7.2a: Combining one vertex-first projection of the 16-cell in Fig. 5.4 with the projection of the Compound of two 16-cells in that Figure, we obtain, on the right, the Octahedral projection of the Compound of three 16-cells. Since the projection of the 24-cell on the left has a Cuboctahedral envelope; its stellation's envelope is the first stellation of the Cuboctahedron: the composite of a Cube and a Octahedron in Fig. 3.3b.


Fig. 7.2b: Superposing the 3 orthogonal 16-cells in Fig. 7.1a, we obtain, on the right, a second Octahedral projection of the Compound of three 16-cells, its "pyritohedric" projection. Since the 24-cell on the left has the Rhombic dodecahedron as its outer envelope; its stellation's envelope is Escher's solid (Fig. 4.8b).

This projection is centred on two superposed three-edge intersections, which are seen here without distortion, one near (which we call N) and the other far (F) from our 3-D space. At the ends of those edges, we see green balls. Each of these represents the superposition of two real vertices (one called N' at the end of an edge of N and the other F' at the end of an edge of F) with a 3-edge intersection, between the edge that goes from N' to F' (which is shortened to a point by the projection) with the two other blue edges we see crossing there.


Fig. 7.2c: Combining the three prismatic projections of the 16-cell in Fig. 7.1b we obtain, on the right, the prismatic projection of the Compound of three 16-cells (*). The red balls in both models are edge intersections that appear because of the projection; they are not real edge intersections in 4 dimensions. This projection shows all true and false vertices without superposition.

How to build Compound of three 16-cells: Build any of the prismatic projections of the 16-cell in Fig. 7.1b with either size 2 or size 3 struts as the largest, with two struts connected by a black ball for each single strut in that projection in Fig. 7.1b. The models in Fig. 7.2c use size 3 struts. Then build the other two 16-cells in prismatic projection using the same black balls as edge mid-points. In this way, you will be able to figure out all the strut intersections. Building the model on a φ smaller scale requires 6 ultra-short red struts (R00), which are currently sold by Zometool. If you can find size 3 struts, the larger scale is preferable, otherwise the model looks very dense.

Check also David Richter's dedicated page, which provides more details on this model.


At the centre of the projection of the Compound of three 16-cells in Fig. 7.2a are two of the vertices of one of the 16-cells; each of these lies "above" two undistorted Tetrahedral cells from each of the other two 16-cells, the greatening of the ``central'' undistorted Octahedral cells of the 24-cell on the left. Thus the vertices of this Compound of three 16-cells lie above the 3-D planes of the original 24-cell, which means that they are the vertices (and thus the compound is a faceting) of a dual 24-cell.

The projections reflect this: the arrangement of true vertices in the projection of the Compound of three 16-cells in Fig. 7.2a is the same as in the projection of the 24-cell in Fig. 7.2b (which is the dual of the projection of the 24-cell in Fig. 7.2a); conversely the arrangement of true vertices in the projection of the Compound of three 16-cells in Fig. 7.2b is the same as in the projection of the 24-cell in Fig. 7.2a. This distinction between a 24-cell and its dual is important: The existence of two dual sets of 24-cell vertices implies that the same is true for the 3-D cell planes of two dual 24-cells.

Since the Compound of three 16-cells is a faceting of the 24-cell, the vertices of the prismatic projection of the Compound of three 16-cells in Fig. 7.2c are also the vertices of the ``anti-prismatic'' projection of the 24-cell, the dual of its prismatic projection in Fig. 7.2c. However, if you try to build it, you'll find that it is not Zomable (see Fig. 8.2). Its greatening, the anti-prismatic projection of the Compound of three 16-cells, will have the same vertices as the prismatic projection of the 24-cell and will have the edge directions of the anti-prismatic projection of the 24-cell (from the rule mentioned above on greatenings), therefore it is not Zomable either. The fact that these projections are not Zomable will have important consequences below.

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It follows from the properties of the Compound of three 16-cells that its dual, the Compound of three tesseracts, is necessarily an edge faceting of a 24-cell and a stellation of its dual 24-cell. The first statement means that it shares all three Zometool representations of the 24-cell:
  1. As mentioned in the derivation of the 24-cell, the pyritohedric projection of the 24-cell (Figs. 5.5b, 7.1a and 7.2b) looks exactly like a superposition of the Compound of two vertex-first Tesseracts and the cell-first projection of the Tesseract in Fig. 5.4.
  2. Its dual, the Octahedral projection of the 24-cell, looks like three superposed face-first projections of Tesseracts (see Fig. 7.3).
  3. Challenge to the reader: Locate the three Tesseracts in the prismatic projection of the 24-cell (Figs. 7.1b, 7.2c), and build Zometool models for them. How many Zometool models do you need? Are these the dual projections of the 16-cells in Fig. 7.1b or not?
The second statement, that it stellates the dual 24-cell, can be seen directly from its pyritohedric projection (Fig. 7.2b), where the Cubic cells appear directly ``under'' the vertices of the 24-cell, i.e. the Cubic cells are in 3-D planes that are in dual positions to the cells of the 24-cell being faceted. This will be important for the discussion below.

A consequence of these dualities is that the vertex arrangement of the Compound of three 16-cells is exactly the same as for its dual Compound of three tesseracts, as we can see in Fig. 7.3. The same happens, therefore, for their cell planes.


Fig. 7.3: Top left: The pyritohedric projection of the Compound of three 16-cells (Fig. 7.2b). Its outer envelope is Escher's solid (Fig. 4.8b).
Bottom left: One of the components of the latter, the edge-first projection of the 16-cell. Its outer envelope is the Yellow dipyramid (Fig. 4.8b).
Bottom right: Its dual, the face-first projection of the Tesseract. That central face is perpendicular to the central edge of the previous projection of the 16-cell.
Top right: The dual of the Compound of three 16-cells, the Compound of three Tesseracts. This projection, the dual of the projection on top right, shares the Zometool representation of the cell-first projection of the 24-cell in Fig. 7.2a. With some attention, you can see the 3 face-first Tesseracts in that model. Its 3-D envelope is the first faceting of the Cuboctahedron (Fig. 4.8c).
Note that these projections of the Compound of three 16-cells and the Compound of three tesseracts have the same vertex arrangement, which means the same happens with the polychoron compounds themselves. The same happens, therefore, with their cell planes.


The Tesseracts in this compound have a total of 16 × 3 = 48 vertices, twice as much as the number of vertices of the compound. This implies that each vertex in the compound is shared by two Tesseracts. Thus, the vertex "polyhedron" of this compound consists of two vertex polyhedra of the Tesseract, which are Tetrahedra. Since the Compound of three tesseracts is an edge-faceting of the 24-cell, the two Tetrahedra of its vertex polyhedron must be a faceting of the Cubic vertex polyhedron of the 24-cell, i.e., they are arranged as a Stella octangula (Fig. 4.4c). This has to be, because the dual Compound of three 16-cells has 24 Stellae octangulae as "cells".

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No additional regular compounds are edge facetings of the 24-cell: because the 24-cell is radially equilateral, it can only be edge-faceted by other radially equilateral polychora, the only one being the Tesseract. By duality, we conclude that there are no additional greatenings of the 24-cell.

We've described the two regular facetings and stellations of the 24-cell in detail because they are important for understanding what follows. For each compound of n 24-cells below (with n = 5 and 25), we will see a projection of a compound of 3n Tesseracts, which is an edge faceting of the compound of n 24-cells and therefore represented by the same model (this will have the Stella octangula as its vertex "polyhedron"), and a projection of a Compound of 3n 16-cells, which is a greatening (with twice the edge length, but the same edge directions) of the compound of n 24-cells; this will have n × 24 "Stellae octangulae" as composite "cells".

These projections are made of multiples of the pyritohedric and prismatic projections of the 24-cell or the Compound of three 16-cells above, with the prismatic projections being 4 times more numerous than the pyritohedric projections: the single axis of 3-fold symmetry of a Prismatic projection coincides with each of the 4 axes of 3-fold symmetry of the pyritohedric projections. Because the duals of the prismatic projections are not Zomable, then none of the duals of these projections are Zomable either (see, for instance, Figs. 8.3 and 8.4), even if they represent a Zomable compound, or even of they represent exactly the same compound (as in the case of the self-dual Compound of five 24-cells). As in the case of the Compounds of three 16-cells and three tesseracts, the dual compounds of 3n 16-cells and 3n tesseracts have the same vertex arrangement (which is also valid for the projections) and the same arrangement of 3-D cell "planes".

All of these Zometool models suffer from the multiple blue intersections that we found with the models of the regular star polychora, the reason is that, as we will see later, they share their edge directions. Like the latter models, all these models are built on a scale φ times smaller than the corresponding models in David Richter's Zometool projects page, and for the same reasons.

Facetings of the 600-cell

 In the case of the Icosahedron, there are only three facetings: three of the regular star polyhedra. We have already seen above, the 600-cell also has several regular facetings: nine of the regular star polychora. However, and unlike the Icosahedron, it also has several facetings that are regular compounds, all of which share its full Hexacosichoric symmetry.

We start with the Compound of five 24-cells. If you followed attentively the construction of the 600-cell from the 24-cell, you can check that the last vertices to be added (the 24 pyramidal apices) have the same arrangement as the vertices of a 24-cell. Again, the symmetry of the 600-cell implies that there is nothing special about these apices: the 96 vertices of the Snub 24-cell can be decomposed into 4 groups of 24 vertices, each also with the same arrangement as the vertices of a 24-cell. This means that we can build a set of five 24-cells with the same vertex arrangement of the 600-cell. A projection, with the same vertex arrangement as the Icosahedral projection of the 600-cell in Figs. 5.10 and 6.1 is shown in Figs. 7.4a and b.


Fig. 7.4a: This model represents a projection of the Compound of five 24-cells, here seen through one of the 8 directions along the 4 genuine three-fold rotational symmetry axes.


Fig. 7.4b: The same projection as in the previous Figure, now seen through one of 12 special directions, along the 6 Pentagonal "Ghost symmetry" axes (see text). There are an additional 6 Triangular ghost symmetry axes.

See detailed explanation of this model in David Richter's dedicated page, which includes a part count.


Fig. 7.4c: The core of the Compound of five 24-cells.

How to build: The model in Fig. 7.4c is the core of the model in Figs. 7.4a and b. It is nearly identical with a Stellated dodecahedron, with three pais of opposite edges (in orthogonal directions) removed, the inner eight yellow struts connect the centre to the inner false vertices where three blue edges cross.
By examining the centre of the prismatic projection of the 24-cell (Figs. 7.1b and 7.2c), you can see that the struts nearer to the centre are blue, making two Triangles in dual positions. The four pairs of opposite blue triangles in this model (perpendicular to the yellow struts) are the starting point for building 4 replicas of the prismatic projection of the 24-cell; for this having a model of the latter is very handy.
The central yellow struts are the starting point for building a 24-cell in pyritohedric projection, this must be built at the same scale as that of the prismatic projections (the blue edges must have the same lengths).

Building the model at this scale has an interesting implication: If we can find some size 3 struts (12 R3 and 12 Y3), we can use each of them to replace a combination of two struts and one ball where the ball is not strictly necessary (i.e., a ball where there is no intersection with other edges). Doing this, we can save 48 regular-sized struts and 24 balls compared to the part count in David Richter's page on this model and improve its appearance.


This regular polychoron compound is one of the very few that are self-dual. Since it is a faceting of the 600-cell, its dual - in this case itself - is necessarily a stellation on the 120-cell.

Despite the Icosahedral vertex arrangement, this projection has the pyritohedral symmetry of the model of the irregular icosahedron and the Snub 24-cell in Figs. 5.7 and 5.8. This is unusual since the Compound of five 24-cells has full Hexacosichoric symmetry: for most other similarly symmetric objects, the only Zomable projections have Icosahedral symmetry. The reason for this is explained in the next page.

In this projection there are no axes of 5-fold symmetry, as we would naturally find in a model with Icosahedral symmetry. However, if we project this model into 2 dimensions along a set of 12 special directions - as in Figure 7.4b - we see again a semblance of 5-fold symmetry. This is a remnant of the fact that the original compound can be inscribed in the 600-cell. The original paper describing this (Richter and Vorthmann, 2006) calls this phenomenon "Tenacious Symmetry". However, the name "Ghost symmetry" has gained wider acceptance, since it is more evocative. It is a widespread phenomenon, caused by the fact that the 2-D symmetries are rather "small": one can choose the vertices of a polychoron to project symmetrically into two dimensions, and have an intermediate 3-D projection that is still highly asymmetric. An extreme example are these two projections of the 600-cell.

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This model also represents the Compound of fifteen tesseracts, which is obtained by edge-faceting the 24-cells of the previous compound. As discussed above, for the Compound of three tesseracts that results from each of those 5 24-cells, the 3-D planes of the Cubic cells are in the 3-D planes of the dual 24-cell. Therefore, the 15 × 8 = 120 Cubic cells of the compound of fifteen Tesseracts are in the same planes as the the cells of the dual of the Compound of five 24-cells, which is of course another Compound of five 24-cells. As we saw above, the latter is a stellation of the 120-cell, so the same applies to the Compound of fifteen tesseracts.

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Since the Compound of five 24-cells is self-dual, its edge faceting (the Compound of fifteen tesseracts in Figs. 7.4a and b) is the dual of its own greatening, the Compound of fifteen 16-cells. A projection of the latter is shown in Figs. 7.5a and b. Like its dual, it is a faceting of the 600-cell and a stellation of the 120-cell.


Fig. 7.5a: A projection of the Compound of fifteen 16-cells. Here we see the model through one of the 4 three-fold rotational symmetry axes.


Fig. 7.5b: The same model as in the previous Figure, here seen through one of the 6 Pentagonal "ghost symmetry" axes.

See also David Richter's dedicated page, which has a detailed description and part count.


Fig. 7.5c: The core of the Compound of fifteen 16-cells.

How to build: By examining the centre of the prismatic projection of the Compound of three 16-cells (Fig. 7.2c), you can see that the struts nearer to the centre are yellow, in particular two sets of three radiating from two opposite apexes. The model in Fig. 7.5c is the core of the model in Figs. 7.5a and b. The outer 4 pairs of opposite apexes where three yellow struts converge are the starting points for building 4 replicas of the prismatic projection of the Compound of three 16-cells; for this having a model of the latter is very handy. Note that at the same time, one has to start building, from the centre of this model, a projection of the Compound of three 16-cells in pyritohedric projection (Fig. 7.2b) using a central ball and blue struts (again, having that projection at hand is useful). Those blue struts must be oriented in a way that is consistent with the remainder of the model, and the pyritohedric projection must be made at the same scale as the prismatic projections. Building these projections slowly inside out you will be able to figure out all the strut intersections. Building this model is a bit tricky because the Compound of three 16-cells in pyritohedric projection remains disconnected from the remainder of the model for quite a while!

This model cannot be built on a smaller scale, because it uses the smallest available yellow struts (Y0). At this scale, we can use size 3 struts (in this case, 72 B3, 48 R3 and 120 Y3) to save 480 regular-sized struts and 240 balls from the part count in David Richter's page on this model and improve its appearance.


This particular projection is the greatening of the projection of the Compound of five 24-cells in Figs. 7.4a and b, therefore it has the same pyritohedric symmetry, despite the full Hexacosichoric symmetry of the Compound of fifteen 16-cells. The 24-cell in pyritohedric projection is greatened into a Compound of three 16-cells in pyritohedric projection (see Fig. 7.2b), and the four 24-cells in prismatic projection are greatened into four Compounds of three 16-cells in prismatic projection (see Fig. 7.2c).

However, and uniquely in this projection, its vertex arrangement is not that of the Icosahedral projection of the 600-cell in Fig. 5.10, but instead the vertex arrangement of its Tetrahedral projection, shown in Fig. 8.1. This is a 1/5 subset of the vertex arrangement of the Icosahedral projection of the 120-cell in Fig. 5.11. This reflects a fundamental fact, that the 120-cell can be faceted by 600-cells, either five or ten (see Fig. 8.5). We will see a few important consequences of this below.

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There are four regular compounds of twenty-five 24-cells. Among them, there is a dual pair with Hexacosichoric symmetry, where one of the compounds is a faceting of the 600-cell (see Fig. 7.6) and the other is a faceting of the 120-cell (Fig. 8.3). This implies that the former is also a stellation of the 600-cell and the latter is also a stellation of the 120-cell.


Fig. 7.6: This model of the Icosahedral projection of the Stellated 120-cell and Great grand 120-cell, also represents the Icosahedral projections of two regular polychoron compounds: one of the Compounds of twenty-five 24-cells and one of the Compounds of seventy-five Tesseracts.


We now discuss the Compound of twenty-five 24-cells that facets and stellates the 600-cell, in Fig. 7.6. Since the twenty-five 24-cells have a total of 600 vertices, they have to coincide by fives with the 120 vertices of the 600-cell. Since the vertex polyhedron of the 24-cell is a Cube, the vertex "polyhedron" of this compound is a Compound of five cubes. This is a faceting of the Dodecahedral vertex polyhedron of the Stellated 120-cell. Thus the edge arrangement, which is defined by the overall vertex arrangement (of the 600-cell) and the vertices of the vertex polyhedron (identical to those of the Stellated 120-cell) is that of the Stellated 120-cell. This means that the Icosahedral projections of this compound and the Stellated 120-cell are represented by the same Zometool model! The 24-cells "fit" because, like the Stellated 120-cell, they are radially equilateral! The cells of these 24-cells are the 600 Octahedral sections of the Stellated 120-cell.

Furthermore, since the Compound of five cubes shares the edge arrangement of the ditrigonal polyhedra, the Compound of twenty-five 24-cells shares the face arrangement of the ditrigonal polychora. Because the faces in this arrangement are Triangles, the edges of this compound will necessarily trace its vertex ``polyhedron", the Compound of five cubes; this is one of the aforementioned edge sections of the Stellated 120-cell.

The twenty-five 24-cells in the compound have a total of 25 × 96 = 2400 edges, which is twice the number of edges of the stellated 120-cell. This means that the edges of the 24-cells in this compound coincide in pairs. This is to be expected from the vertex "polyhedron": in the Compound of five cubes, each vertex is touched by two different Cubes.

The projection in this Figure can be built from the model of the Compound of five 24-cells in Figs. 7.4a and b by quintuplication (see Fig. 4.1b): To that projection, we add four more identical ones, each rotated around one of the "ghost symmetry" axes of the first compound by an additional 72 degrees. The resulting model has Icosahedral symmetry.

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This model also represents one of the regular Compounds of seventy-five Tesseracts, which is obtained by edge-faceting the 24-cells of the previous compound. The 16 × 75 = 1200 vertices of all these Tesseracts fall in sets of 10 on the 120 vertices of this compound. Since the vertex polyhedron of the Tesseract is a Tetrahedron, the vertex "polyhedron" of this compound is the Compound of ten tetrahedra*, which is also a faceting of the Dodecahedral vertex polyhedron of the Stellated 120-cell. As in the previous compound, the 75 × 32 = 2400 edges of the Tesseracts coincide in pairs. This is to be expected from the vertex "polyhedron": in the Compound of ten tetrahedra, each vertex is touched by two different Tetrahedra.

As discussed above, when we facet a 24-cell to obtain a Compound of 3 Tesseracts, the latter's cells are in the 3-D "planes" of a dual 24-cell. Thus, the 3-D cell "planes" of the Compound of seventy-five tesseracts in Fig. 7.6 are identical to those of the dual Compound of twenty-five 24-cells in Fig. 8.3, which as mentioned above, is a stellation of the 120-cell. Thus, its 75 × 8 = 600 Cubic cells appear in sets of five in the cell planes of the 120-cell, i.e., this compound can be seen as a polychoron with 120 Compounds of five cubes as composite cells, which edges coincident with the edge sections of the Stellated 120-cell and Compound of twenty-five 24-cells.

No additional regular compounds are edge facetings of the Stellated 120-cell: because the latter is radially equilateral, any such compounds can only consist of radially equilateral polychora - the 24-cell and Tesseract. Thus, the list of regular edge facetings of the Stellated 120-cell is complete.

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In the facetings of the 600-cell above, we used edge arrangements that result from connecting each vertex of the 600-cell to vertex section 1 in Table 2 (Fig. 6.1), to vertex section 2 in Table 2 (Figs. 6.2 a and b, Fig. 7.6) and section 5 in Table 2 (Figs. 6.3a and b). What happens if we try to connect the vertices with vertex sections 3, 4, 6 or 7 in Table 2? Do we get any additional facetings of the 600-cell?

The answer is exactly one: the Compound of seventy-five 16-cells in Fig. 8.4; which is the dual of the Compound of seventy-five tesseracts in Fig. 7.6, and therefore a stellation of the 120-cell. The edges of this compound connect each vertex to vertex section 4 in Table 2. This compound is shown in the next page because, as we'll see there, this is the only faceting of the 600-cell that is not Zomable.

The reason for this single additional object is the following: Given the central symmetry of their vertex polyhedra, any regular facetings of the 600-cell (polychora or compounds) must have rectifications (or compounds of rectifications) of regular polyhedra as their equatorial polyhedra, furthermore, these must facet the Icosidodecahedron, which is the equatorial polyhedron of the 600-cell. All such objects are depicted in Fig. 4.10. The only such faceting in that model that has not been used as an equatorial polyhedron in previous models is the Compound of five octahedra; this is the equatorial ``polyhedron" of the Compound of seventy-five 16-cells in Fig. 8.4. Since there are no additional such facetings, the list of facetings of the 600-cell is complete.

* When discussing the Compound of three tesseracts above, we said that other compounds of 3 n Tesseracts also have the Stella octangula as vertex figures. This is still true in this case, since the Compound of ten tetrahedra can be thought of as a Compound of five Stellae octangulae.

Facetings of the 120-cell

 As we've seen, the Dodecahedron has only one regular star polyhedron (the Great stellated dodecahedron) as a faceting, but three regular compounds: the Compounds of five and ten Tetrahedra, and the Compound of five Cubes. The situation is analogous for the 120-cell: only one regular faceting (the Great grand stellated 120-cell) but many regular compounds, which have its vertex sections (listed in Table V of Coxeter 1973) as cells. Its regular facetings can be divided in three main classes:
  1. The Great grand stellated 120-cell, the only regular polychoron in this list,
  2. The largest group (39 regular and partially regular compounds) results from the fact, mentioned after Fig. 7.5b, that we can facet a 120-cell with 5 or 10 600-cells (see Fig. 8.5). This has a major implication: all facetings of the 600-cell (either 9 of the 10 regular star polychora or the regular polychoron compounds just mentioned above) can then be used to form new regular compounds that are also facetings of the 120-cell. Some of these compounds, like the Compounds of five and ten 600-cells themselves, are partially regular facetings of the 120-cell; they and their duals (which are partially regular stellations of the 600-cell) are also included in this group,
  3. The last group (3 regular compounds) arises from the fact that, unlike the 600-cell, the 120-cell can be faceted by 5-cells. This is analogous to the fact that the Dodecahedron can be faceted with Tetrahedra.
This list implies that the vertex arrangement of the 120-cell includes the vertex arrangements of all other regular polychora as subsets! This is very different from the Dodecahedron, which cannot be faceted by the Icosahedron and its facetings, or Octahedra for that matter. This implies, by duality, that the 3-D cell planes of all regular polychora are subsets of the 3-D cell planes of the 600-cell.

In Figs. 6.4a and b, we showed a model of the single member of the first class. In Figs. 7.7a, b, c, d and 7.9a, b, we present the single regular compounds from the second and third classes that are Zomable (for an explanation on why this is the case, look here). It's nice that there are Zomable members of the three classes! However, as we see in those Figures, these models are extremely complex and require many parts; this is an unavoidable consequence of the large number of vertices of the 120-cell (600). The outer vertices have the same separations as the model in Figs. 6.4a and b.

The compounds represented by the Zometool models below have full Hexacosichoric symmetry; however, this is not true for all facetings of the 120-cell (like the Compound of five 600-cells, see Fig. 8.5).

***

We start with the single Zomable faceting of the second class, the regular Compound of seventy-five 16-cells with full Hexacosichoric symmetry.


Fig. 7.7a: The Icosahedral projection of the Compound of seventy-five 16-cells that facets the 120-cell and has full Hexacosichoric symmetry.


Fig. 7.7b: The model is seen here from a 3-fold symmetry axis, and without counter-illumination.


Fig. 7.7c: When the Sun is at a good elevation, these models cast really nice shadows! Note the small bit of light right at the centre of the shadow: sunlight shone through all the pentagonal holes of the nine balls in one of the six central axes of 5-fold symmetry of the model. This is proof that the parts in the model are aligned with very good precision.


Fig. 7.7d: Same as Fig. 7.7c, but at at different time of the day, with the Sun shining through a 3-fold symmetry axis.

Unlike the picture of the models, which are perspective projections, these shadows are almost orthographic projections of the 3-D model to 2-D, but also from the 4-D object to 2-D. They are not fully orthographic because the vectors used in the projection - the Sunbeams - are not perpendicular to the 2-D surface (the floor of my living room); however, like orthographic projections they are affine transformations. Apart from that, they are quite beautiful!

See also David Richter's page on this compound.


Fig. 7.7e: The core of the Compound of seventy-five 16-cells.

How to build: By examining the centre of the prismatic projection of the Compound of three 16-cells (Fig. 7.2c), you can see that the struts nearer to the centre are yellow, in particular two sets of three radiating from two opposite apexes. The model in Fig. 7.7e is the core of the model in Figs. 7.7a, b, c and d. The 20 pairs of outer apexes where three yellow struts converge are the starting points for building 20 replicas of the prismatic projection of the Compound of three 16-cells; for this having a model of the latter is very handy. The 30 blue struts radiating from the centre (from all the rectangular holes of the central node) are the starting points for building five Compounds of three 16-cells in pyritohedric projection (Fig. 7.2b), these must be built at the same scale as the prismatic projections. Building these projections slowly inside out you will be able to figure out all the strut intersections.

The model cannot be built on smaller scale because it uses many B0 and Y0 struts. At this scale no size 3 struts are necessary!


This compound is the greatening of the Compound of twenty-five 24-cells in Fig. 7.6. Since all stellations preserve the 3-D cell "planes", then this compound is also a stellation of the 600-cell. Because this Compound can be derived by greatening the Compound of twenty-five 24-cells in Fig. 7.6, its dual Compound of seventy-five tesseracts is an edge faceting of the dual Compound of twenty-five 24-cells in Fig. 8.3. Therefore both of those Compounds are represented by the same edge model.

Regarding this particular projection, it can be derived from the projection of the Compound of fifteen 16-cells in Figs. 7.5a and b by quintuplication. Each of those five projections facets a Tetrahedral projection of the 600-cell, five of which cover all the vertices of the Icosahedral projection of the 120-cell in Fig. 5.11.

***

Until now, all fully regular compounds we have built were Compounds of n 24-cells (with n = 1, 5, and 25) and compounds derived from them by edge-faceting (with 3n Tesseracts, which are represented by the same models) or greatening (with 3n 16-cells). That basically sums it up. However, the last model below represents something truly different: it is a compound of 5-cells.

In Fig. 5.3, we saw the vertex/cell first projection of the 5-cell. In the Figure below, I show three more Zomable projections.


Fig. 7.8: Three Zomable projections of the 5-cell.


In this page I name these from left to right, the "triangular", "prismatic" and "ghost-symmetric" projections. The latter was found by Scott Vorthmann, the other two were found later by David Richter. The name for the projection on the right comes from the fact that it has pentagonal ghost symmetry: from this perspective, we can see a Pentagram inscribed in a Pentagon. For a more complete description of this projection and its implications, see Scott's vZome page on this projection.

With these three projections, we can build a model of the Icosahedral projection of the Compound of 120 5-cells, this is shown below.


Fig. 7.9a: The Icosahedral projection of the regular Compound of 120 5-cells with Hexacosichoric symmetry, here seen from a 5-fold symmetry axis.


Fig. 7.9b: Same model as in the previous Figure, seen here from a 3-fold symmetry axis.

The model cannot be built at a smaller scale because it already uses Y0 struts. At this scale, we can use 180 B3, 120 R3 and 260 Y3 struts; this saves 560 balls and 1120 regular sized struts and greatly improves the appearance of the model.


This regular polychoron compound is special for several reasons. It is self-dual, a property it shares only with the Compound of five 24-cells (Figs. 7.4a and b) and the other two regular 5-cell compounds. Among the latter, it is the only Zomable compound.

Since it is a faceting of the 120-cell, and it is made of 5-cells, it has 600 vertices with Tetrahedral vertex polyhedra, like the Great grand stellated 120-cell and the 120-cell. This and the fact that it is self-dual implies that it is a stellation of the 600-cell: like the Grand 600-cell, it has 600 Tetrahedral cells located in the same 3-D "planes" of the 600 Tetrahedral cells of the 600-cell. All of this makes it a 4-D analogue of the self-dual Compound of five tetrahedra.

The story behind this model is interesting. It is not in David Richter's list of Zometool projects, so I did not know whether the Compound of 120 5-cells was Zomable. However, when I saw the "ghost symmetric" projection of the 5-cell in Scott Vorthmann's vZome page on the 5-cell, I wondered whether it can be inscribed in the Icosahedral projection of the 120-cell (Fig. 5.11). I asked Scott about this, and he found out that the answer is "yes". Not only that, but the same is true for the other two projections of the 5-cell in Fig. 7.8.

Even more interestingly, he found that with only those three projections (20 triangular, 40 prismatic and 60 ghost-symmetric) we can cover all the vertices of the Icosahedral projection of the 120-cell. Thus, there is a Zomable projection of the Compound of 120 5-cells, with the same vertex arrangement as the model in Fig. 5.11 and therefore with the same Icosahedral symmetry! To demonstrate this, Scott made a very nice vZome model of the Icosahedral projection of this Compound, and Nan Ma (see star polytope page) made a Observable notebook on the Compounds of 120 5-cells. One of the cool things about this is that it also shows the dual projection.

David Richter then made us realise that the 120 × 10 = 1200 edges of the 120 5-cells in this compound can be obtained by extending the 1200 edges of the 120-cell and Stellated 120-cell - if we extend these further, we obtain the edges of the Great grand stellated 120-cell. This is important because it shows that the compound is isotoxal in 4-D (and therefore, by duality, isohedral). This parallelism also applies necessarily to their projections, so that shows that this model really is a correct projection of the Compound of 120 5-cells. Knowing this and having Scott's vZome model made the construction trivial. However, it should be noted that the Compound of 120 5-cells is not technically an edge stellation of the Stellated 120-cell; as we saw above, it is instead a stellation of the 600-cell.

When studying this compound, Nan Ma found a new regular, self-dual Compound of 720 5-cells. This was missed earlier in Coxeter's list of regular polychoron compounds! He then found out that Peter McMullen had discovered that regular compound in 2018! Not only that, but McMullen also discovered five other new regular compounds, all facetings of the 120-cell: a second Compound of 120 5-cells, a third Compound of seventy-five 16-cells, a third Compound of seventy-five Tesseracts, and a third and fourth partially regular Compounds of twenty-five 24-cells! Finally, McMullen demonstrated that the list of regular compounds is now complete.

That was a lot of fun!

As far as I know, this is the first physical model of this compound ever built. One has to build something very similar during the process of building the model of the Great grand stellated 120-cell in Figs. 6.4a and b, but there are some subtle differences that make it very unlikely anyone has built this exact model during that process.

This ends the list of the three Zomable regular facetings of the 120-cell, and our list of Zometool models of the regular compounds of regular polychora.


Paulo's polytope site / Next: Zomability.