Zomability

"Without doubt the authentic type of these figures exists in the mind of God the Creator and shares His eternity"

"Nature uses as little as possible of anything."
Johannes Kepler


We will now elaborate on which projections are Zomable and which are not. We start with the more complex Zomable polychora. Then, we extend the analysis to the regular polychoron compounds, showing why eleven of these are Zomable. Finally, we show why the majority of the regular compounds are not Zomable.

For this, we start with some general definitions. Two polytopes A and B in any dimension have a parallel set of edges if there is a suitable re-orientation of B where, for each edge of A, there is one or more edges of B that are parallel to it.
- In that case, if for each set of parallel edges in A there is an equally numerous set of parallel edges in B, and vice-versa, then we will say that the edges of A and B are n-parallel.
- In that case, if for each edge of A there is a parallel edge of B that lies along the same line, then those edges are collinear.
- In that case, if for each edge of A the collinear edge of B is also coincident (starting and ending at the same points), then A and B share the same edge arrangement.

If any of these relations applies between the edges of A and B, the same will then be true for the edges of any lower-dimensional projections of A and B, as long as both projections are made along the same vector (this will be assumed from now on).

We now recall the meaning of the three basic colours in the Zometool system. First, in any Zometool model, all connectors are aligned exactly with each other. Second, we can make a model with Icosahedral symmetry that is centered on a connector. The red struts in the central connector are aligned along the 6 axes of 5-fold symmetry of that connector and therefore of the whole model, the identical orientation of all connectors then means that any red strut in the model - connecting to any pentagonal hole of any connector - will be parallel to the axes of 5-fold symmetry of the full model. Similarly, all yellow struts, which connect to the triangular holes of the connectors, are parallel to the 10 axes of 3-fold symmetry of such a model; the blue struts, which connect to the rectangular holes of the connectors, are parallel to the 15 axes of 2-fold symmetry of the model. The Green color will be discussed later. The lengths of the struts yield distances between the centres of the connectors that are powers of the golden ratio, φ.

If the directions of the edges of A or a projection of A in any dimension can be represented with this limited set of Zometool directions, the same is true for the edges of B or a projection of B if the edges of B are parallel to the edges of A and the ratio of their lengths is a combination of multiples of 1 and φ.

Zomable polychora

The fact that the Icosahedral projection of the 600-cell is Zomable implies immediately that all polychora shown after Fig. 21 must be Zomable. This happens because all of them can be oriented in a way that all their edges are parallel to those of the 600-cell. Furthermore, all of them have edge lengths related to those of the 600-cell by multiples of 1 and φ.

To establish this, we will study Diagrams IIa to c. We start with polychora with edges collinear with the 720 edges of the 600-cell. All these edges have either Pentagonal or Pentagrammic figures, i.e, each edge is shared by five cells:
The following polychora have edges collinear to the 1200 edges of the 120-cell. All these edges have Triangular figures, i.e, each of them is shared by 3 cells. The edges of the 120-cell are themselves parallel to those of the 600-cell, as we see from the first entry in the list: Thus, seven of the regular star polychora (those in the red and yellow circuits in Diagram IIc) have edges that are collinear to the 720 edges of the 600-cell with either Pentagonal or Pentagrammic figures; the remaining three have edges that are collinear to the 1200 edges of the 120-cell with Triangular figures. This is the number and type of faces of their duals.

Finally: As we will see next, this implies that all the regular convex polychora also have Zomable projections in blue, red and yellow.

Zomable regular compounds

As we have seen, 11 of the 52 regular polychoron compounds are Zomable. This happens for a simple reason: the Compound of twenty-five 24-cells that facets the 600-cell can be represented by the same model of the Stellated 120-cell (see Fig. 33), which as we have just seen, must be Zomable.

The Compound of seventy-five tesseracts that facets the 600-cell, being an edge faceting of the Compound of twenty-five 24-cells in Fig. 33, can also represented by the same model. Furthermore, the Compound of seventy-five 16-cells in Figs. 34a, b and c is a greatening of that Compound of twenty-five 24-cells, therefore, as explained above for the Compound of three 16-cells, it will also be Zomable, with the edge lengths of its Zomable projection being twice those of the projection of the Compound of twenty-five 24-cells in Fig. 33. Finally, the Compound of 120 5-cells has edges collinear and 2φ − 1 larger than those of the Stellated 120-cell, for that reason it is Zomable as well (see Figs. 36a and b).

Since the projections of the Compounds of five 24-cells / fifteen tesseracts and the Compound of fifteen 16-cells are subsets of the projections listed above, they must also be Zomable (see respectively Figs. 31a and b, 32a and b). Being 1/5 subsets of an Icosahedral projection is also the reason why they are not themselves icosahedral. The Compounds of two and three 16-cells and two and three tesseracts are also subsets of those Zomable compounds, so they must all also have projections Zomable in blue, red and yellow (See Figs. 29b and c).

All these Zomable projections of polychoron compounds imply that the constituent regular convex polychora have themselves projections that are Zomable in blue, red and yellow. These are the projections of the 16-cell shown in Figs. 28a and b, of the 24-cell and Tesseract shown in Figs. 29b and c and of the 5-cell in Fig. 35.

Non-Zomable projections

We now discuss the non-Zomable projections, with the help of virtual vZome models of regular compounds made by Nan Ma, and used here with his permission.

Apart from the Icosahedral projection in Fig. 21, the 600-cell has other symmetric orthographic projections. The most symmetric is its cell-centered projection, which necessarily has the Tetrahedral symmetry of the central cell. However, such a projection is not Zomable. When we try to model it in vZome (see Fig. 37), we see that some of the edge directions (blue and yellow) are clearly Zomable, however, of the new colors that appear here (represented by vZome as green, lavender, purple and olive green), only green struts are made by Zometool. This means that this projection is not Zomable. Note that there are no red struts in this projection: indeed, in a model with Tetrahedral symmetry, there are no real axes of 5-fold symmetry. However, and as we've seen for some models above (Figs. 31b and 32b), this projection has axes of 5-fold ``Ghost symmetry''.


Fig. 37: A vZome model of a Tetrahedral projection of the 600-cell, here seen from an axis of 5-fold ``Ghost symmetry''.

Model by Nan Ma. In this and the following Figures, you can click on the image to view it in the online version of vZome. It might take a few tens of seconds to download. The scroll wheel will move it closer/farther, clicking and dragging will rotate it.

Note that the "near" and "far" Tetrahedral cells of this projection are not coincident. As in the projection of the Rectified tesseract in Fig. 16a, these cells superpose in a Stella Octangula configuration. A consequence of this is that the projection has a larger symmetry than the Tetrahedral symmetry: again, the pyritohedric symmetry.

Since all the Zomable polychora and polychoron compounds discussed above have edges parallel to those of the 600-cell, we can then immediately say that all their Tetrahedral projections have vZome representations with the same strut colors as the Tetrahedral projection of the 600-cell (like, for instance, the vertex-first projection of the 120-cell). For that reason those projections are not Zomable either - unless you build the parts yourself. The important exceptions are the Tetrahedral projections of the 5-cell and Rectified 5-cell in Fig. 15, which use only green and yellow struts.

***

We now look at the duals of the projections in the previous page. As discussed there, the dual projections to the prismatic projections (of the 24-cell, 16-cell and Tesseract) are not Zomable. The fact that all Zomable projections of regular compounds that facet the 600-cell and 120-cell include polychora in such prismatic projections implies that none of their dual projections is Zomable.

Given the central role of the Compound of twenty-five 24-cells in Fig. 33, we exemplify in Fig. 38 with a vZome model of its dual projection, which is an Icosahedral projection of the Compound of twenty-five 24-cells that facets the 120-cell. The duals of the 24-cells that appear in pyritohedric projection in Fig. 33 appear here in Octahedral projection, those are clearly Zomable with green and blue parts (see Fig. 17). This is not the case for the duals of the 20 24-cells that appear in prismatic projection in Fig. 33: those have lavender, purple and olive green struts. These are the same colors vZome uses for the Tetrahedral projection of the 600-cell, the reason for this will be explained below.


Fig. 38: A vZome virtual model of an Icosahedral projection of the Compound of twenty-five 24-cells that is a faceting (and a stellation) of the 120-cell.

Model made by Nan Ma.

The dual to the Icosahedral projection of the Compound of seventy-five 16-cells that facets the 120-cell (Figs. 34a, b and c) is the Icosahedral projection of the Compound of seventy-five tesseracts that facets the 120-cell, which is also represented by the model in Fig. 38.

Another example of a non-Zomable dual of a Zomable projection is presented in Fig. 39. This is the Icosahedral projection of the Compound of seventy-five 16-cells that facets the 600-cell, the dual of the Compound of seventy-five tesseracts that facets the 600-cell (also represented in Fig. 33), this means that the Compound in this Figure is a stellation of the 120-cell. This is the greatening of the projection in Fig. 38, so it has the same strut colors as well, therefore it is not Zomable either.


Fig. 39: A vZome virtual model of the Icosahedral projection of the Compound of seventy-five 16-cells that is a faceting of the 600-cell.
Model by Nan Ma.

This and the two compounds in Fig. 38 have no Zomable projections. The Compound of seventy-five 16-cells in Fig. 39 is the only regular faceting of the 600-cell that is not Zomable.

All other duals to the projections of the regular compounds and regular polychora in the previous page (which include, of course, all the duals of the different prismatic projections) are sub-sets of these three compounds. Although they are not Zomable, they are projections of compounds and polychora that have Zomable projections.

***

We now look more systematically at the facetings of the 120-cell. As mentioned above, these can be grouped in three classes. The first one includes the single regular polychoron among such facetings, the Great grand stellated 120-cell, which as we've seen, is Zomable.

All other regular facetings of the 120-cell are regular compounds. Those in class 2 result, by our definition, from the fact that the 120-cell can be faceted by five 600-cells. For this reason, we will now study a vZome model of the Icosahedral projection of the Compound of five 600-cells. This model also represents the Compound of ten 600-cells.


Fig. 40: A vZome virtual model of an Icosahedral projection of the Compound of five 600-cells that facets the 120-cell.
Model by Nan Ma.

The central blue Dodecahedron of the projection of the 120-cell in Fig. 22 is here faceted by five Tetrahedral cells (in green) from each of the 600-cells, forming a central Compound of five tetrahedra (see Fig. 8a). Therefore, like the Compound of five tetrahedra, the Compound of five 600-cells lacks mirror symmetry. However, the projection does have mirror symmetry: the "near" and "far" Compounds of five Tetrahedra do not coincide and superpose in the projection as a Compound of ten tetrahedra. It is for this reason that the model appears identical to that of the Compound of ten 600-cells, where each Dodecahedron of the 120-cell is faceted by ten Tetrahedra. The latter has full mirror symmetry, and the same applies to the Compound of ten 600-cells.

The important point about this model is that it merely contains five identical Tetrahedral projections of the 600-cell, i.e., it is the quintuplication of the projection in Fig. 37 around its ghost symmetry axis!

We are now in a good position to understand why most of the facetings of the 120-cell in class 2 are not Zomable. Each of these facetings is made either of five or ten 600-cells or their facetings. As we have shown above, with the important exception of the Compound of fifteen 16-cells, all Zomable projections of regular facetings of the 600-cell have not only the same strut colors as the Icosahedral projection of the 600-cell in Fig. 21, but also the same vertex arrangement. Therefore, the Tetrahedral projections of those facetings have the same colours and vertex arrangement as the Tetrahedral projection of the 600-cell in Fig. 37. Quintuplicating (or decuplicating) these projections we obtain the Icosahedral projections of most of the facetings of the 120-cell in class 2, which will therefore not be Zomable either.

An example of this is the Compound of twenty-five 24-cells that facets the 120-cell in Fig. 38. Each of the five 600-cells in Fig. 40 is there replaced by one of its facetings, the Compound of five 24-cells, here in its non-Zomable Tetrahedral projection. This is the reason why the models in Figs. 38, 39 and 40 have the same strut colors.

The important exception - the Compound of fifteen 16-cells - is the only faceting of the 600-cell with a Zomable projection that shares the vertex arrangement of its Tetrahedral projection. Quintuplicating it results in the only Zomable faceting of the 120-cell in class 2, the Compound of seventy-five 16-cells in Figs. 34a, b and c.

***

Some partially regular compounds in class 2 are not facetings of the 120-cell, but they are duals of such facetings. Let us now verify whether they are Zomable.

To do this, we first remark that the Compounds of five and ten 600-cells are only partially regular. Although they are facetings of the 120-cell, they are not stellations of the 600-cell; they have many more 3-D cell "planes". These properties will be shared by any facetings of the 120-cell in class 2 where the faceting of the 600-cell being quintuplicated (or decuplicated) is also its stellation. There are two such facetings: the Grand 600-cell and the Compound of twenty-five 24-cells in Fig. 33.

The duals of such facetings of the 120-cell are the objects we want to investigate: they are are stellations of the 600-cell, but are not facetings of the 120-cell. Their Icosahedral projections are made of five or ten regular stellations and facetings of the 120-cell (120-cell, Compound of twenty-five 24-cells that facets the 120-cell, Great grand Stellated 120-cell) in Tetrahedral projection. As we've seen, the 120-cell in Tetrahedral projection has the same colors as the Tetrahedral projection of the 600-cell, so the same is true for projection of the Great grand stellated 120-cell. The compounds of 24-cells have new edge directions, most of which are not even defined in vZome.

***

We now discuss the third class of facetings of the 120-cell, the regular 5-cell compounds, all fully regular. Apart from the Compound of 120 5-cells in Figs. 36a and b, none has Zomable projections with the same vertex arrangement of the model in Fig. 22. We can see this from the model of the Compound of 720 5-cells: most edges don't even have assigned colors in the vZome system. The same must apply to the second, self-dual Compound of 120-cells (see second model in Na Ma's Observable Notebook on the Compounds of 120 5-cells) because it is a sub-set of the Compound of 720 5-cells.

One important aspect of these notebooks is that they also show projections with the vertex arrangement of a Tetrahedral (i.e., vertex-first) projection of the 120-cell. The Tetrahedral projection of the Zomable Compound of 120 5-cells in Figs. 36a and b has the same strut colours of the Tetrahedral projection of the 600-cell in Fig. 37. This must be the case since, as we've seen, their edges are parallel. As for the Compound of 720 5-cells, we can see that its Tetrahedral projection has many fewer struts with Zometool colors than its Icosahedral projection, the same applies to the second Compound of 120 5-cells, which is a sub-set. Thus, we won't find any additional non-Icosahedral projections that are Zomable.

Therefore, our list of Zomable projections of the regular compounds of regular polychora is complete!



Since there are no more Zomable projections of regular compounds in four dimensions, it is time to go beyond.

Main polytope page.