"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 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 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. 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 any parallel 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 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 when the projection vector is parallel to a line from a vertex of the
600-cell to its centre, then there is a small number (5) of values of sin α, where α
is the angle between the edges and the projection vector. This implies immediately that all
polychora shown after Fig. 21 must be Zomable with the same colours. 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 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 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.
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).
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. 11).
The latter polychora are in the lower solid red circuit, sharing the edge arrangement of the
Grand 600-cell (see model in Figs. 25a and b).
The edge stellations, which transform Pentagonal faces into Pentagrammic faces, explain why the
false vertices of the polychora in the lower circuit have the same arrangement as the vertices of
the 600-cell. They also justify the suggested method of construction for the model of the Grand
600-cell: extend the edges of the model of the 600-cell until they meet again in a larger set of
identical vertices - those of the Icosahedral projection of the Grand 600-cell (see Diagram IIb).
The three star polychora in the blue path 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:
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:
The Icosahedral 120-cell has the same edge arrangement as the 600-cell. It has 120 Icosahedral
cells, which appear in the same 120 3-D "planes", and with the same orientations as the 120
Icosahedral cells of a suitably oriented Rectified 600-cell. The latter polychoron is isotoxal: all
its edges are identical, they touch two Octahedra and a single Icosahedron. This means that all its
edges are parallel to those of the Icosahedral 120-cell and thus to those of the 600-cell. However,
they are not n-parallel: the edges of the Rectified 600-cell belong to a single Icosahedron, while
each edge of the Icosahedral 120-cell is shared by five Icosahedra (they have Pentagrammic edge
figures). Therefore, the Rectified 600-cell has 5 times more edges than the 600-cel, i.e., 3600.
The edges of the Snub 24-cell (and Grand antiprism) are a subset of the edges of the 600-cell.
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 will see next, this implies that all the regular convex polychora also have Zomable
projections in blue, red and yellow.
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. 33), 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. 33, is also represented by the same model.
The Compound of seventy-five 16-cells in Figs. 34a, 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. 33.
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).
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. 36a 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. 28a and b, of the 24-cell and Tesseract shown in
Figs. 17b, 29b and c and of the 5-cell in Fig. 35.
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. 38, by Scott Vorthmann, and used here with their permission.
Apart from the Icosahedral projection in Fig. 21, 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. 37), 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. 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. 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. 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 ``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. 38: 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. 29c.
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. 29c 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. 28c 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. 39 we show the dual of the Compound of twenty-five 24-cells
that facets and stellates the 600-cell in Fig. 33: 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. 33 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. 33 in prismatic projection appear here in
the anti-prismatic projection of Fig. 38. For that reason, this projection is clearly not Zomable.
Fig. 39: 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 has 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. 34a, 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. 33, the latter is an
edge-faceting of the dual Compound in Fig. 39, 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. 33),
the Compound of seventy-five 16-cells that facets the 600-cell and stellates the 120-cell
(see Fig. 40). Since the former is an edge faceting of the Compound of twenty-five 24-cells in
Fig. 33, the latter is the greatening of the compound in Fig. 39. Being a greatening, its projection
has the same strut colours as the model in Fig. 39, therefore it is not Zomable either.
Fig. 40: 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 has 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.
This edge arrangement results from connecting each vertex of the 600-cell with the vertices of its
equatorial section (number 4 in Table 2, the Icosidodecahedron). This happens because the vertex
polyhedron is a faceting of the Icosidodecahedron: the 16-cells have Octahedra as vertex figures,
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, which as we've seen is a
faceting of the Icosidodecahedron. 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. 33, has the Compound of five
cubes as a composite "cell". The edges of the Compound of fifteen 16-cells (Figs. 32a 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. 21) is
also not Zomable.
This and the two compounds in Fig. 39 have no Zomable projections. The Compound of seventy-five
16-cells in Fig. 40 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. 41: 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. 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, 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. 21, 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. 37, 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. 39. Each
of the five 600-cells in Fig. 41 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 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. 34a, b, c and d.
***
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 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. 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 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. 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 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!
Since there are no more Zomable projections of regular compounds in four dimensions, it is time to
go beyond.
