"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 affine tranformation of A
and B, like an orthographic projection to a lower-dimensional space, as long as the
projections of A and B are done along the same vector.
We now derive some basic properties of the Zometool system. As mentioned previously, in any model built with the
Zometool, all connectors are aligned exactly with each other. Because these have
Icosahedral symmetry, we can make a model with Icosahedral symmetry that is centred 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 colour will be discussed later.
If the directions of the edges of A or any parallel projection of A in any dimension can
be represented with this limited set of directions, the same is true for the edges of B or
any parallel 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.
The fact that the Icosahedral projection of the 600-cell is Zomable with blue, red and
yellow struts reflects the fact that the angles between the 3-D planes where the edges
exist and the projection vector have only a few values. This implies immediately that
all polychora shown after Fig. 5.10 must be Zomable with the same colours, 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 Diagram IId.
Diagram IId. In this diagram, the solid lines indicate an identical edge arrangement (see
Diagram IIb). The dashed lines represent edge stellation (see Diagram IIc).
The seven star polychora in the red circuit have 720 edges collinear with the edges of the
600-cell. All these edges have either Pentagonal or Pentagrammic figures, i.e, each
edge is shared by five cells.
As we've seen, the three polychora built with the sections of the 600-cell
(Icosahedral 120-cell, Grand 120-cell and Great 120-cell) share its edge arrangement (top
solid red circuit).
As discussed before, the Great 120-cell and the Grand 120-cell edge-stellate into the
Great stellated 120-cell and the Grand stellated 120-cell. Those edges are therefore
collinear with those of the 600-cell but are φ3 times larger (see Fig.
4.6a).
The latter polychora are in the lower solid red circuit, sharing the edge arrangement
of the Grand 600-cell (see model in Figs. 6.3a and b).
The three star polychora in the blue path have 1200 edges colinear withe the edges of the 120-cell.
All these edges have Triangular figures, i.e, each of them is shared by 3 cells:
As indicated by its name, the Grand 120-cell is an aggrandization of the 120-cell.
This means that it has 120 Dodecahedral cells in the same 3-D "planes", and with the same
orientations, as those of the 120-cell; thus its edges are parallel to those of the
120-cell. However, as we've seen above, it also shares the edge arrangement of the
600-cell. Therefore, the edges of the 120-cell and 600-cell are parallel. They are not
n-parallel: The reason is that each edge of the Grand 120-cell is shared by 5 Dodecahedra (the edge
figure is a Pentagram), while each edge of the 120-cell is shared by only three Dodecahedra (the edge
figure is a Triangle). Thus, the number of edges of the 120-cell is 720 × 5 / 3 =
1200.
The Stellated 120-cell is an edge stellation of the 120-cell, thus its edges are
collinear with those of the 120-cell and φ3 larger.
The Great grand 120-cell is an edge faceting of the stellated 120-cell.
The Great grand stellated 120-cell can be obtained by edge-stellating the Great grand
120-cell. Thus, the edges of the Great grand stellated 120-cell are also collinear to
those of the 120-cell, and φ6 larger.
Apart from explaining the false vertices of the Stellated 120-cell and Great grand
stellated 120-cell, this justifies the method of construction of the latter's model:
extend the edges of the model of the Stellated 120-cell/Great grand 120-cell until they
meet at a larger set of vertices of the (Icosahedral model of the) 120-cell, those of the
Great grand stellated 120-cell (see Diagram IIb). Finally:
Just as diagram IId characterises the number of edges and their figures for the star
polychora, a dual diagram (a mirror reflection) characterises the number and type of
faces.
As we have seen, 11 of the 52 regular polychoron compounds are Zomable. This happens for a
simple reason: as we've seen, 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. 7.6),
which is one of the consequences of the latter's Dodecahedral vertex polyhedron. This
implies that:
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. 7.6, is also represented by the
same model.
The Compound of seventy-five 16-cells in Figs. 7.7a, b, c and d 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. 7.6.
The Compound of 120 5-cells has edges collinear and 2φ − 1 = √5 larger
than those of the Stellated 120-cell, for that reason it is Zomable as well (see Figs.
7.9a 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. 7.3a and b, 7.5a 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. 7.2b and c).
As we've seen, the Stellated 120-cell is Zomable because its edges are collinear with
those of the 120-cell. That is also the case for the Compound of 120 5-cells, which is
therefore Zomable as well (see Figs. 7.9a and b).
Finally, 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. 7.1a and b, of the
24-cell and Tesseract shown in Figs. 5.5b, 7.2b and c and of the 5-cell in Fig. 7.8.
We now discuss the non-Zomable projections, with the help of virtual vZome models of regular compounds made by Nan Ma, and, in the
case of Fig. 8.1, by Scott Vorthmann, and used here with their permission.
Apart from the Icosahedral projection in Fig. 5.10, the 600-cell has other symmetric
orthographic projections. The most symmetric is its cell-centred 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. 8.1), we see that some of the
edge directions (blue and yellow) are clearly Zomable, however, of the new colours 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. 7.4b and 7.5b), this projection has axes of 5-fold ``Ghost symmetry''.
Fig. 8.1: A vZome model of a Tetrahedral projection of the 600-cell. In the initial
orientation, it is seen from an axis of 5-fold ``Ghost symmetry''(*).
Note that the "near" and "far" Tetrahedral cells of this projection are not coincident,
because the central element (the Tetrahedron) lacks central symmetry. In the projection
they are superposed in a ``Stella Octangula'' configuration. A consequence of this is that
the projection has a symmetry larger than that of the Tetrahedron: 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 colours 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. 5.3, which use only green and yellow struts.
***
We now look at the duals of the projections in the previous
page. As discussed there, the
``anti-prismatic'' projections of the 24-cell, 16-cell and Tesseract, which are the duals
of their prismatic projections, are not Zomable. Indeed, when we render the anti-prismatic
projection of the 24-cell, vZome renders it with yellow struts (which are in the Zometool
system), but also lavender, purple and olive green struts. These are the same colours
vZome uses for the Tetrahedral projection of the 600-cell, the reason for this will be
explained below.
Fig. 8.2: A vZome virtual model of the
anti-prismatic projection of the 24-cell (*). The vertices here have the same
arrangement as the Prismatic projection of the Compound of three 16-cells in Fig. 7.2c.
Model by Scott Vorthmann, see also his
page on this projection.
Challenge to the reader: As discussed before, any edge frame projection of the
24-cell is also a projection of the Compound of three tesseracts. Can you see the
projections of three Tesseracts in this model? What are their colours? Are these the duals
of the Prismatic projections of the 16-cell in Fig. 7.2c or not?
Challenge to the reader: Like its dual projection, there is also an orientation of
this projection that shows ghost symmetry. Can you see it? Is it identical to that shown
in Fig. 7.8c or not?
The fact that all Zomable projections of regular compounds that facet the 600-cell and
120-cell include polychora in prismatic projections implies that none of their dual
projections is Zomable. In what follows, when we refer to the compounds, we also refer to
their Icosahedral projections.
An example, in Fig. 8.3 we show the dual of the Compound of twenty-five 24-cells that
facets and stellates the 600-cell in Fig. 7.5: the Compound of twenty-five 24-cells that
facets and stellates the 120-cell. The duals of the 5 24-cells that appear in in Fig. 7.5
in pyritohedric projection appear here in Octahedral projection, those are clearly Zomable
with green and blue parts (see Fig. 17a), however, the duals of the 20 24-cells that
appear in Fig. 7.5 in prismatic projection appear here in the anti-prismatic projection of
Fig. 8.1. For that reason, this projection is clearly not Zomable.
Fig. 8.3: A vZome virtual model of the
Icosahedral projection of the Compound of twenty-five 24-cells that is a faceting (and
a stellation) of the 120-cell.
The 24 × 25 = 600 Octahedra appear in fives in the 120 3-D planes of the compound.
Thus, it can be thought of as having Compounds of five Octahedra as cells. One such
compound appears undistorted (in green) at the centre of this model. This has to be
because its dual has the Compound of five cubes as a vertex polyhedron.
The dual of the Compound of seventy-five 16-cells that facets the 120-cell and stellates
the 600-cell (Figs. 7.7a, b, c and d) is the Compound of seventy-five tesseracts that
stellates the 600-cell and facets the 120-cell. Because the former is a greatening of the
Compound of twenty-five 24-cells in Fig. 7.6, the latter is an edge-faceting of the dual
Compound in Fig. 8.3, therefore represented by the same model.
Another example of a non-Zomable object is the dual of the Compound of seventy-five
tesseracts that facets the 600-cell and stellates the 120-cell (also represented in Fig.
7.6), the Compound of seventy-five 16-cells that facets the 600-cell and stellates the
120-cell (see Fig. 8.4). Since the former is an edge faceting of the Compound of
twenty-five 24-cells in Fig. 7.6, the latter is the greatening of the compound in Fig.
8.3. Being a greatening, its projection has the same strut colours as the model in Fig.
8.3, therefore it is not Zomable either.
Fig. 8.4: A vZome virtual model of the
Icosahedral projection of the Compound of seventy-five 16-cells that is a faceting of the
600-cell.
The 75 × 16 = 1200 Tetrahedra of the 16-cells appear here in tens in the 120 3-d
planes of the 120-cell. Thus, it can be thought of as having Compounds of ten tetrahedra
as cells. One such compound appears undistorted (in green) at the centre of this model.
This has to be because its dual has the Compound of ten tetrahedra as a vertex polyhedron.
As mentioned above, the edge arrangement results from connecting each vertex of the
600-cell with the vertices of its equatorial section (number 4 in Table 2). Because vertex
section 4 is an Icosidodecahedron, the vertex polyhedron must be its faceting, which it
is: The 16-cells have Octahedra as vertex polyhedra, and their 75 × 8 = 600 vertices
coincide here by fives with the 120 vertices of the 600-cell; this implies that the vertex
figure is the Compound of five octahedra (see Fig. 4.12). This has to be because, as we
have seen, the dual figure (the Compound of seventy-five tesseracts that facets the
600-cell in Fig. 7.6) has the Compound of five cubes as a composite "cell".
The edges of the Compound of fifteen 16-cells (Figs. 7.5a and b) are a subset of edges of
this compound, this shows that the projection of that compound that has the vertex
arrangement in this figure (the same as for the projection of the 600-cell in Fig. 5.10)
is also not Zomable. This and the two compounds in Fig. 8.2 have no Zomable projections;
this implies that the Compound of seventy-five 16-cells in Fig. 8.4 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 the projections of these three compounds. Although they are
not Zomable, they are projections of compounds and polychora that have Zomable
projections: the Compound of five 24-cells, the Compound of fifteen tesseracts and the
Compound of fifteen 16-cells.
***
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. 8.5: A vZome virtual model of an
Icosahedral projection of the Compound of five 600-cells that facets the 120-cell (*).
The central blue Dodecahedron of the projection of the 120-cell in Fig. 5.11 is here
faceted by five Tetrahedral cells (in green) from each of the 600-cells, forming a central
Compound of five tetrahedra (see Fig. 3.8a). 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. 8.1 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, apart from the Compound of fifteen 16-cells,
all Zomable projections of regular facetings of the 600-cell have not only the same vertex
arrangement as the Icosahedral projection of the 600-cell in Fig. 5.10, but also the same
strut colours - the reason is that, as we've seen, their edges are parallel. Therefore,
and for the same reason, the Tetrahedral projections of those facetings must have, apart
from the same vertex arrangement as the Tetrahedral projection of the 600-cell in Fig.
8.1, also the same strut colours. Therefore they are not Zomable either.
An example of this is the Compound of twenty-five 24-cells that facets the 120-cell in
Fig. 8.3. Each of the five 600-cells in Fig. 8.5 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. 8.3, 8.4 and 8.5 have the same strut colours.
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. 7.7a, b, c and d. As we've seen,
both must be zomable because they are greatenings of zomable compounds, the Compounds of
five and twenty-five 24-cells in Figs. 7.4a, b and 7.6.
***
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. 7.6.
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 colours 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. 7.9a and b, none has
Zomable projections with the same vertex arrangement of the model of the 120-cell in Fig.
5.11. We can see this from the model of the Compound of
720 5-cells: most edges don't even have assigned colours 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. 7.9a and b has the
same strut colours of the Tetrahedral projection of the 600-cell in Fig. 8.1. 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 the
Zometool colours 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!
Paulo's polytope site / Since there are no more
Zomable projections of regular compounds in four dimensions, it is time to go beyond.
