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 reorientation 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 viceversa, then we will say that the edges of A and B are nparallel.
 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 lowerdimensional 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 5fold 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 5fold 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 3fold symmetry of such a model; the blue struts, which connect to the
rectangular holes of the connectors, are parallel to the 15 axes of 2fold 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 600cell 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 600cell. Furthermore, all
of them have edge lengths related to those of the 600cell 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 600cell. All these edges have either Pentagonal or Pentagrammic
figures, i.e, each edge is shared by five cells:

As we have seen in Diagram IIc, there are three polychora (Icosahedral 120cell, Grand 120cell and
Great 120cell) with the same edge arrangement of the 600cell (the red circuit).

The Great 120cell and the Grand 120cell edgestellate into the Great stellated 120cell and the
Grand stellated 120cell. Those edges are therefore collinear with those of the 600cell but are
φ^{3} times larger (see Fig. 11). The latter polychora are in the yellow circuit,
sharing the edge arrangement of the Grand 600cell (see model in Figs. 25a and b).
* This justifies the suggested method of construction for the latter model: extend the edges of the
600cell until they meet again in a larger set of vertices of the 600cell  which are also the
vertices of the Grand 600cell (see Diagram IIc).
The following polychora have edges collinear to the 1200 edges of the 120cell. All these edges
have Triangular figures, i.e, each of them is shared by 3 cells. The edges of the 120cell are
themselves parallel to those of the 600cell, as we see from the first entry in the list:

The Grand 120cell, which has the same edge arrangement as the 600cell, has 120 Dodecahedra as
cells, which appear in the same 120 3D "planes", and with the same orientations, as the Dodecahedra
of a suitably oriented 120cell. This means that the edges of that 120cell are parallel to the
edges of the 600cell. They are not nparallel: The reason is that each edge of the Grand 600cell is
shared by 5 Dodecahedra (the edge figure is a Pentagram), while each edge of the 120cell is shared by
only three Dodecahedra (the edge figure is a Triangle). Thus, the number of edges of the
120cell is 720 × 5 / 3 = 1200.

The Stellated 120cell is an edge stellation of the 120cell, thus its edges are collinear with
those of the 120cell and φ^{3} larger. The Great grand 120cell has the same edge
arrangement as the stellated 120cell (Diagram IIc).

The Great grand stellated 120cell can be obtained by edgestellating the Great grand 120cell.
Thus, the edges of the Great grand stellated 120cell are also collinear to those of the 120cell,
and φ^{6} larger.
* This justifies the method of construction suggested for the model in Figs. 26a and b: extend the
edges of the Great grand 120cell until they meet at the vertices of a larger 120cell, which are
also the vertices of the Great grand stellated 120cell (see Diagram IIc).
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 600cell with either Pentagonal or Pentagrammic
figures; the remaining three have edges that are collinear to the 1200 edges of the 120cell with
Triangular figures. This is the number and type of faces of their duals.
Finally:

The Icosahedral 120cell has 120 Icosahedral cells, which appear in the same 120 3D "planes", and
with the same orientations as the 120 Icosahedral cells of a suitably oriented Rectified 600cell.
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 600cell. However,
they are not nparallel: the edges of the Rectified 600cell belong to a single Icosahedron, while
each edge of the Icosahedral 120cell is shared by five Icosahedra (they have Pentagrammic edge
figures). Therefore, the Rectified 600cell has 5 times more edges than the 600cel, i.e., 3600.

The edges of the Snub 24cell are a subset of the edges of the 600cell.
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 twentyfive 24cells that facets the 600cell can be
represented by the same model of the Stellated 120cell (see Fig. 33), which as we have just seen,
must be Zomable.
The Compound of seventyfive tesseracts that facets the 600cell, being an edge faceting of the
Compound of twentyfive 24cells in Fig. 33, can also represented by the same model. Furthermore,
the Compound of seventyfive 16cells in Figs. 34a, b and c is a
greatening of that Compound of twentyfive 24cells, therefore, as explained above for the Compound of three 16cells, it will also be Zomable,
with the edge lengths of its Zomable projection being twice those of the projection of the
Compound of twentyfive 24cells in Fig. 33. Finally, the Compound
of 120 5cells has edges collinear and 2φ − 1 larger than those of the Stellated 120cell,
for that reason it is Zomable as well (see Figs. 36a and b).
Since the projections of the Compounds of five 24cells / fifteen tesseracts and the Compound of
fifteen 16cells 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 16cells 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 16cell shown in Figs. 28a and b, of the 24cell and Tesseract shown in Figs.
29b and c and of the 5cell in Fig. 35.
NonZomable projections
We now discuss the nonZomable 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 600cell has other
symmetric orthographic projections. The most symmetric is its cellcentered 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 5fold symmetry.
However, and as we've seen for some models above (Figs. 31b and 32b), this projection has
axes of 5fold ``Ghost symmetry''.
Fig. 37: A vZome model of a Tetrahedral projection of the 600cell, here seen from an axis of
5fold ``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 600cell,
we can then immediately say that all their Tetrahedral projections have vZome representations with
the same strut colors as the Tetrahedral projection of the 600cell (like, for instance, the vertexfirst projection of the
120cell). For that reason those projections are not
Zomable either  unless you
build the parts yourself. The important exceptions are the Tetrahedral projections of the 5cell
and Rectified 5cell 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 24cell, 16cell and Tesseract) are not Zomable. The fact that all
Zomable projections of regular compounds that facet the 600cell and 120cell include polychora in
such prismatic projections implies that none of their dual projections is Zomable.
Given the central role of the Compound of twentyfive 24cells 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 twentyfive
24cells that facets the 120cell. The duals of the 24cells 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 24cells 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 600cell, the reason for
this will be explained below.
Fig. 38: A vZome virtual model of an Icosahedral
projection of the Compound of twentyfive 24cells that is a faceting (and a stellation) of the
120cell.
Model made by Nan Ma.
The dual to the Icosahedral projection of the Compound of seventyfive 16cells that facets the 120cell
(Figs. 34a, b and c) is the Icosahedral projection of the Compound of seventyfive tesseracts that
facets the 120cell, which is also represented by the model in Fig. 38.
Another example of a nonZomable dual of a Zomable projection is presented in Fig. 39. This is the
Icosahedral projection of the Compound of seventyfive 16cells that facets the 600cell, the dual
of the Compound of seventyfive tesseracts that facets the 600cell (also represented in Fig. 33),
this means that the Compound in this Figure is a stellation of the 120cell. 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 seventyfive 16cells that is a faceting of the 600cell.
Model by Nan Ma.
This and the two compounds in Fig. 38 have no Zomable projections. The Compound of seventyfive 16cells
in Fig. 39 is the only regular faceting of the 600cell 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 subsets 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 120cell. 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
120cell, which as we've seen, is Zomable.
All other regular facetings of the 120cell are regular compounds. Those in class 2 result, by our
definition, from the fact that the 120cell can be faceted by five 600cells. For this reason, we
will now study a vZome model of the Icosahedral projection of the Compound of five 600cells. This
model also represents the Compound of ten 600cells.
Fig. 40: A vZome virtual model of an Icosahedral
projection of the Compound of five 600cells that facets the 120cell.
Model by Nan Ma.
The central blue Dodecahedron of the projection of the
120cell in Fig. 22 is here faceted by five Tetrahedral cells (in green) from each of the 600cells,
forming a central Compound of five tetrahedra (see Fig. 8a). Therefore, like the Compound of five
tetrahedra, the Compound of five 600cells 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 600cells,
where each Dodecahedron of the 120cell is faceted by ten Tetrahedra. The latter has full mirror
symmetry, and the same applies to the Compound of ten 600cells.
The important point about this model is that it merely contains five identical
Tetrahedral projections of the 600cell, 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 120cell in class 2
are not Zomable. Each of these facetings is made either of five or ten 600cells or
their facetings. As we have shown above, with the important exception of the Compound of fifteen 16cells, all
Zomable projections of regular facetings of the 600cell have not only the same strut colors as the
Icosahedral projection of the 600cell 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 600cell in Fig. 37. Quintuplicating (or decuplicating) these projections we
obtain the Icosahedral projections of most of the facetings of the 120cell in class 2, which will
therefore not be Zomable either.
An example of this is the Compound of twentyfive 24cells that facets the 120cell in Fig. 38. Each
of the five 600cells in Fig. 40 is there replaced by one of its facetings, the Compound of five
24cells, here in its nonZomable 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 16cells  is the only faceting of
the 600cell with a Zomable projection that shares the vertex arrangement of its Tetrahedral
projection. Quintuplicating it results in the only Zomable faceting of the 120cell in class 2, the
Compound of seventyfive 16cells in Figs. 34a, b and c.
***
Some partially regular compounds in class 2 are not facetings of the 120cell, 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 600cells are only partially regular.
Although they are facetings of the 120cell, they are not stellations of the 600cell; they have
many more 3D cell "planes". These properties will be shared by any facetings of the 120cell in
class 2 where the faceting of the 600cell being
quintuplicated (or decuplicated) is also its stellation. There are two such facetings: the Grand
600cell and the Compound of twentyfive 24cells in Fig. 33.
The duals of such facetings of the 120cell are the objects we want to investigate: they are
are stellations of the 600cell, but are not facetings of the 120cell. Their Icosahedral projections are
made of five or ten regular stellations and facetings of the 120cell (120cell, Compound of twentyfive 24cells
that facets the 120cell, Great grand Stellated 120cell) in Tetrahedral projection.
As we've seen, the 120cell in Tetrahedral projection has the same colors as the
Tetrahedral projection of the 600cell, so the same is true for projection of the Great grand stellated
120cell. The compounds of 24cells have new edge directions, most of which are not even defined in vZome.
***
We now discuss the third class of facetings of the 120cell, the regular 5cell compounds, all fully
regular. Apart from the Compound of 120 5cells 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
5cells: most edges don't even have assigned colors in the vZome system. The same must apply to
the second, selfdual Compound of 120cells (see second model in Na Ma's Observable Notebook on the
Compounds of 120 5cells) because it is a subset of the Compound of 720 5cells.
One important aspect of these notebooks is that they also show projections with the vertex
arrangement of a Tetrahedral (i.e., vertexfirst) projection of the 120cell. The Tetrahedral
projection of the Zomable Compound of 120 5cells in Figs. 36a and b has the same strut colours of
the Tetrahedral projection of the 600cell in Fig. 37. This must be the case since, as we've seen,
their edges are parallel. As for the Compound of 720 5cells, 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 5cells, which is a subset. Thus, we won't find any
additional nonIcosahedral 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.