"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
In the previous pages, we payed special attention to the vertex and cell arrangements,
this is important for keeping track of which regular convex polychora are being faceted
and stellated to obtain a particular star polychoron or compound. We now discuss in more
detail the edge and face arrangements of polytopes. One of the advantages of this is that
it will allow us to figure out which polygons, polyhedra and projections of polytopes are
Zomable and which are not.
What was mentioned above regarding polygons and polyhedra applies equally to polychora and
higher-dimensional uniform polytopes. The fact that orthographic projections are affine
transformations means that if a set of edges is parallel and have equal lengths in the
polychoron, their projections are also parallel and have equal lengths in the projection.
This means that, if a regular face lacks central symmetry, then when its is rectified or
truncated in a higher-dimensional space, the new sides are parallel to previous existing
sides. Therefore, the same happens in their projections to 3-D space: even though the
polygons might appear distorted by the projection, a rectification or truncation will
produce new sides that are parallel to the previous existing sides, as shown in Fig. 8.1
for one of the Triangular faces of a projection of the 5-cell or 600-cell, or one of the
Pentagonal faces in the model of the 120-cell. Thus, the Zometool rule 1 also applies to
orthographic projections of polychora, or any higher-dimensional polytopes!
Fig. 8.1: Truncating a projection of a Triangle or Pentagon, where they appear distorted
by the projection, will result in distorted projections of the Hexagon and Decagon.
However, all of the sides of the projected Hexagon and Decagon are parallel to sides of
the projected Triangle or Pentagon, so the former are also necessarily Zomable.
Thus all uniform polychora with the symmetry
of the 5-cell are Zomable - and in multiple
ways, one for each projection of the 5-cell in Figs. 5.3 and 7.8, and with the same
colours as those projections. Furthermore, since the 600-cell is Zomable, all the uniform
polychora with the symmetry of the 600-cell and 120-cell are Zomable. All Zomable
projections of these polychora have Icosahedral symmetry and are Zomable with B, R and Y
struts, as the projections in Figs. 5.10 and 5.11.
However, Zometool rule 2 still implies that some of the polychora with the symmetries of
the 16-cell/Tesseract and of the 24-cell are not Zomable: those that have Octagons as
faces or sections. We discuss in more detail here which of those polychora are
Zomable.
Regular star polychora:
The fact that the Icosahedral projections of the 600-cell and 120-cell are Zomable with B,
R and Y 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. 5.11 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 and 120-cell.
Furthermore, all of them have edge lengths related to those of the 600-cell by multiples
of 1 and \(\phi\).
To establish this, we will study Diagram 8.1.
Diagram 8.1. In this diagram, the solid lines indicate an identical edge arrangement (see
Diagram 6.2). The dashed lines represent edge stellation (see Diagram 6.3c).
The seven star polychora in the red circuit have 720 edges that are parallel to 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
parallel to those of the 600-cell but are \(\phi^3\) 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 with the edges of the
120-cell. That the 120-cell is Zomable can be deduced from the fact that the Grand
120-cell is an aggrandisation 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. Their numbers are not the same: 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 \(720 \times 5 / 3 = 1200\):
The Stellated 120-cell is an edge stellation of the 120-cell, thus its edges are
parallel to the edges of the 120-cell and \(\phi^3\) 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 parallel to
those of the 120-cell, and \(\phi^6\) 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 6.2).
Just as Diagram 8.1 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.
Regular compounds and their components:
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 figure. 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, by Zometool rule 1, 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. This is a particular
case of the general principle on the Zomability of polygons mentioned above.
The Compound of 120 5-cells has edges parallel and \(2 \phi - 1 = \sqrt{5}\) 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 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).
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. 8.2, 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. 8.2), we see that some of the
edge directions (G, B and Y) are clearly Zomable, however, none of the new colours that
appear here (represented by vZome in lavender, purple and olive green) is 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. 8.2: 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" central 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.
Duals of the Zomable regular compounds
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. 8.3: 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, the dual of the Compound of twenty-five 24-cells that facets and stellates the
600-cell in Fig. 7.5 is the Compound of twenty-five 24-cells that facets and stellates the
120-cell, see Fig. 8.4. The duals of the five 24-cells that appear in Fig. 7.5 in
pyritohedric projection appear here in Octahedral projection, those are clearly Zomable
with green and blue parts (see Fig. 5.5a). However, the duals of the twenty 24-cells that
appear in Fig. 7.5 in prismatic projection appear here in the anti-prismatic projection of
Fig. 8.2. For that reason, this projection is clearly not Zomable.
Fig. 8.4: 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 \(24 \times 25 = 600\) Octahedral cells of the compound appear in fives in its 120 3-D
planes. 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 figure.
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. 8.4, 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. 8.5). 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.
8.4. Being a greatening, its projection has the same strut colours as the model in Fig.
8.4, therefore it is not Zomable either.
Fig. 8.5: 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 \(75 \times 16 = 1200\) 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 figure.
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 figure must be its faceting, which it
is: The 16-cells have Octahedra as vertex figure, and their \(75 \times 8 = 600\) 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. 8.3 have no Zomable projections;
this implies that the Compound of seventy-five 16-cells in Fig. 8.5 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.
Facetings of the 120-cell
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. 8.6: 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. 8.2 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.
8.2, 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. 8.4. Each of the five 600-cells in Fig. 8.6 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. 8.4, 8.5 and 8.6 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") shares its 3-D planes. 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 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. 8.2. 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!
