Star polychora
The mathematician does not study pure mathematics because it is useful; he studies it because he
delights in it and he delights in it because it is beautiful.
J. H. Poincaré
Like polyhedra, polychora are not necessarily convex. In what follows, we will show models of highly
symmetrical non-convex polychora, which are known as ``star polychora'' by analogy with the star
polygons and star polyhedra.
To understand many of these polychora, we must clarity the meaning of stellations and facetings
among 4-D objects. Stellating a polychoron extends its cells along their 3-D "planes" until they
meet identically extended cells, while faceting remove parts of the polychoron while preserving its
vertex arrangement.
As for polyhedra, the larger the number of cells/vertices of a polychoron, the
larger the number of its stellations/facetings. All such stellations/facetings are necessarily
non-convex.
Also, as for polyhedra, and for the same reasons, if polychora A and B are duals, then
the dual of the stellation of A is a faceting of B. It is very important to keep this in mind for
understanding what follows.
Regular star polychora
There are ten regular
star polychora, the 4-D analogues of the regular star polyhedra. Below and in Nan Ma's page on star polytopes you can see models of all of
them. The cool exploding polychoron animations in the latter site are from this Observable notebook.
All ten are stellations and/or facetings of the 600-cell and 120-cell; all share their Hexacosichoric
symmetry. As we'll see below, there are no regular star polychora with other symmetries. We depict
their geometric relations in Diagrams IIa, b and c below. Some preliminary conclusions from these
diagrams:
-
In Diagram IIb, we see that nine of the regular star polychora are facetings of the 600-cell and
that one of them - the Great grand stellated 120-cell - is a faceting of the 120-cell, with 600
vertices. This makes it a 4-D analog of the Great stellated dodecahedron, which it has as cells.
-
The dual of each regular star polychoron is also a regular star polychoron; this implies that nine
of them are stellations of the 120-cell and that the dual of the Great grand stellated 120-cell -
the Grand 600-cell - is a stellation of the 600-cell (see Diagram IIc). This makes it a 4-D analog
of the Great icosahedron, which is its vertex polyhedron.
Thus, eight regular star polychora are facetings of the 600-cell
and stellations of the 120-cell, i.e., they have 120 vertices and 120 cells. Both types of
elements are necessarily aligned with the 60 axes of Icosahedral symmetry, with the cells and
vertex polyhedra sharing that symmetry.
Thus, their cells are necessarily located under their vertices. These polychora are
analogous to the Stellated dodecahedron and Great dodecahedron, which have 12 faces with 5-fold
symmetry located under their 12 vertices; the vertex figures also have 5-fold symmetry.
***
Three of the regular star polyhedra are built from the 600-cell by using its three sets of regular
edge and face sections as cells. Because this faceting operation preserves the edge arrangement, the
Zometool representations of their Icosahedral projections are identical to that in Figs. 21 and 23.
Because the latter is a vertex-first projection, the same is the case for these three projections.
Since each set of regular edge or face sections has 120 instances each, the polychora built with
these are necessarily stellations of the 120-cell, with the cells appearing (in 4D) under the 120
vertices.
Fig. 23: The Zometool Icosahedral projection of the 600-cell, this time seen from a 3-fold symmetry
axis. This model also represents the Icosahedral projections of three regular star polychora: the
Icosahedral 120-cell, the Great 120-cell and the Grand 120-cell.
The shared edge arrangement of these polychora also implies that their vertex polyhedra must share
the vertex arrangement of the vertex polyhedron of the 600-cell: the Icosahedron.
In cases where the polychora share the same face arrangement, the vertex polyhedra must share the
same edge arrangement. Their faces are determined by the vertex figures of the cells, their
vertex figures are the edge figures of the polyhedron. We can
therefore fully characterise the star polychora built with the sections of the 600-cell:
- The Icosahedral face sections of the 600-cell are the cells of the Icosahedral 120-cell (see virtual model), which
shares the Triangular faces of the 600-cell. The vertex polyhedron is the Great dodecahedron,
the edge figure is the Pentagram.
- Since the Icosahedral 120-cell has Triangular faces, its vertex polyhedron (the Great
dodecahedron) must be highlighted by its edges. These are the Great dodecahedral edge
sections of the 600-cell and
the cells of the Great
120-cell (see virtual model).
The vertex polyhedron is the Stellated dodecahedron, the edge figure is the Pentagon. Since the
cells and vertex polyhedron are duals, this polychoron is self-dual.
- The Dodecahedral edge sections of the 600-cell are the cells of the Grand 120-cell (see virtual model). This polychoron shares the
arrangement of Pentagonal faces of the Great 120-cell. Its vertex polyhedron is the Great
icosahedron, the edge figure is the Pentagram.
The vertex polyhedra are the facetings of the Icosahedron depicted in Fig. 11c.
***
We now present Icosahedral projections of the 7 remaining regular star polychora. Like the model of
the 600-cell above, these models are made on a φ smaller scale than the corresponding models in
David Richter's Zometool
projects page, the reason for this is the fact that size 3 struts are no longer widely
available, but also because the space in my office is limited! Also, they all have points where
several blue struts intersect. Because I wanted to make these models a permanent addition to my
office, I cut the necessary blue parts with pliers.
Figs. 24a, b, 25a and b show models of Icosahedral projections of the 6 remaining regular facetings
of the 600-cell. Given their symmetry, they necessarily have the vertex arrangement of the
Icosahedral projection of the 600-cell (Figs. 21 and 23). Therefore, as for the three previous regular star
polychora, these are vertex-first projections and all vertex polyhedra of the polychora being
projected have Icosahedral symmetry. The Icosahedral symmetry of these projections allows the
appearance of undistorted cells and sections (in blue, which we list) that share that symmetry and
are located (in 4D) under the two central superposed vertices, at increasing depths. Since the
vertex polyhedra have Icosahedral symmetry, they also have central symmetry; thus the list of
sections necessarily includes equatorial polyhedra and Polygons.
The first model (Figs. 24a and b) represents projections of two regular star polychora. The
edge arrangement results from connecting each vertex of the 600-cell to the 20 vertices of the
Dodecahedral section closest to it, i.e., section 2 in Table 2. Interestingly,
the resulting edges highlight the large Icosahedral vertex sections of the 600-cell,
numbers 3 and 5 in Table 2. Here is a list of the undistorted polyhedra
visible in the model (which you meet as you build the
model) which give us a(n incomplete) list of its edge sections:
- At the centre, and closest to the vertices, are Great icosahedra, which have the
vertices of vertex sections 1 and 7 in Table 2 (more on this below),
- The latter's edges are shared with Stellated dodecahedra. These are the cells of the Small stellated
120-cell (henceforth "Stellated 120-cell" for short, see virtual model),
- The three partially regular polyhedra in Fig. 7, which have the vertices of vertex
sections 2 and 6 in Table 2 (more on this below),
- The latter's edges coincide with Compounds of five cubes (more on this in the discussion on the regular polychoron compounds),
- Great dodecahedra. These have the edges of vertex sections 3 and 5 in Table 2, they
are the cells of the Great grand 120-cell (see virtual model),
- The latter's edges coincide with large Icosahedra, which have the faces of vertex
sections 3 and 5 in Table 2. See more on these sections below.
- On the outside, enveloping the projection, is an undistorted Dodecadodecahedron,
the equatorial polyhedron of the Stellated 120-cell. This shares the vertices of
vertex section 4 in Table 2.
There are 120 identical instances of sections 1 to 6, and 60 instances of the equatorial
polyhedron. In addition to the 600 Cubes of the Compounds of five cubes, there are also
600 Tetrahedral and 600 Octahedral sections. Thus, the list of sections includes seven of
the nine regular polyhedra plus the three partially regular polyhedra. The model shows
projections of all instances of these cells, all 600 Tetrahedra and 600 Octahedra are
projected with distortion.
Note that Sections 1 and 6 are isomorphic, as are Sections 2 and 5. As we've seen, the partially regular
polyhedra represented listed in 3) are either isomorphic to each other, or their two sets of faces
are isomorphic to each other, as is also the case for Section 7, the equatorial polyhedron.
Fig. 24a: Icosahedral projection of the Stellated 120-cell and Great grand 120-cell.
Fig. 24b: This is how it looks in my office.
Picture by Jason Wu.
How to build: see instructions in David Richter's
dedicated page.
The stellation of the 120-cell into the Stellated 120-cell provides an example of stellation in
polychora:
- The edges of each Pentagonal face are extended to form Pentagrammic faces, with a new
vertex above each of the previous edges that is touched by the 2 extended neighbouring edges.
- Each Dodecahedral cell was stellated into a Stellated dodecahedral cell, with a new
vertex above each previous Pentagonal face that is touched by the 5 stellated neighbouring faces.
- The 120-cell was stellated into the Stellated 120-cell, with a new vertex above each previous
Dodecahedral cell that is touched by the 12 stellated neighbouring cells.
This stellation process explains why the false vertices of the Stellated 120-cell are the true
vertices of the 120-cell: the false vertices of the Stellated dodecahedral cells are the true
vertices of the Dodecahedral cells of the 120-cell. In the projection in Figs. 24a and b, these have
the same arrangement (and at the same scale) as the true vertices of the Icosahedral projection of
the 120-cell in Fig. 22.
Since the vertex figure of the Stellated dodecahedron is a Pentagon and 12 non-overlapping
Stellated dodecahedra touch each vertex, the vertex polyhedron of the Stellated 120-cell has 12
non-overlapping Pentagons as faces - a Dodecahedron. This is the reason why for these
polychora each vertex of the 600-cell connects to the vertices of a nearest Dodecahedral section
(number 2 in Table 2) via 20 edges. The edge figure is the Triangle.
Since the Great grand 120-cell has the edge arrangement of the Stellated 120-cell, its vertex polyhedron
must have the same vertex arrangement of the Dodecahedron and be regular. As we've seen in Diagram
Ib, the only regular faceting of the Dodecahedron is the Great stellated dodecahedron (this means
that the edge figure of the Great grand 120-cell is also the Triangle). However, as we've seen, the
Dodecahedron has many more facetings; one of the consequences of this is the profusion of
polychora and polychoron compounds that are edge facetings of the Stellated 120-cell, which we
will discuss below.
***
The model in Figs. 25a and b represents the Icosahedral projection of the Grand 600-cell (see virtual model). The
edge arrangement results from connecting each vertex of the 600-cell to the 12 vertices of
section 5 in Table 2.
As mentioned above, the Grand 600-cell is a faceting of the 600-cell and its only stellation;
like the latter it has 600 Tetrahedral cells, which have the same orientations of the Tetrahedral
cells of the 600-cell; as for the latter polychoron, 20 cells and 12 edges meet at each vertex.
Therefore, the projections of the 600 Tetrahedral cells in the projection in Fig. 25a and b are
identical to those of the Icosahedral projection of the 600-cell in Figs. 21, 23
except that they are larger in the former model; all appear distorted in these projections given
their Icosahedral symmetry. Since the cells are Tetrahedral and the Grand 600-cell is not convex,
its vertex polyhedron has to be non-convex and have 20 Triangular faces - it is therefore the Great
icosahedron, which has 12 vertices like section 5 of Table 2. The edge figure is the Pentagram.
Fig. 25a: This model represents the Icosahedral projections of the Grand 600-cell and three other
regular star polychora that share its edge arrangement.
See also David
Richter's page on this projection.
Fig. 25b: This is how it looks in my office.
Picture by Jason Wu.
How to build: Make a model of the 600-cell, but extend all internal edges: there can be no
broken lines. Then build a new, outer identical set of vertices, with their spacing increased by the
Golden ratio (only a few of these will suffice). For each new outer vertex, see what edges of the
internal model point towards it, then extend those edges to the outer vertex. Do this for a set of
neighboring vertices, which will allow you to work out the intersections of those edges. Once this
is done, apply the same pattern to all outer vertices using Icosahedral symmetry, while removing all
the edges of the outer 600-cell that were holding the outer vertices in place. Look here for the justification of this method of
construction.
Like the 600-cell, the Grand 600-cell has three types of regular edge and face sections, with 120
instances each, that result from its Tetrahedral cells, which we list below. In the model in
Figs. 25a and b two undistorted and superposed instances of each of these sets of sections appear
under the two superposed central vertices.
As in the case of the 600-cell, these sections are the cells of three other regular star polychora
that share its edge arrangement, also listed below. Since they have 120 cells, they must be, unlike
the Grand 600-cell, stellations of the 120-cell. The shared edge arrangement implies that their
vertex polyhedra must share the vertex arrangement of the Great icosahedron (the vertex polyhedron
of the Grand 600-cell); this means that they are facetings of the Icosahedron. In cases where the
polychora share the same face arrangement, the vertex polyhedra must share the same edge
arrangement; their faces are determined by the vertex figures of the cells:
- The Tetrahedral cells of the Grand 600-cell imply that its vertex polyhedron (the Great
icosahedron) is outlined by its faces. These Great icosahedral face sections,
which share their vertices with vertex sections 3 and 5 in Table 2, are the cells of the Great icosahedral
120-cell (virtual model),
which therefore shares the Triangular faces of the Grand 600-cell. The vertex polyhedron is the
Stellated dodecahedron, which means that the edge figure is the Pentagon.
- Since the Great icosahedral 120-cell has Triangular faces, its vertex polyhedron (the Stellated
dodecahedron) must be highlighted by its edges. These Stellated dodecahedral sections,
which share the edges and vertices of the Great icosahedral sections mentioned above, are the cells
of the Grand stellated
120-cell (virtual model).
The vertex polyhedron is the Great dodecahedron, thus the edge figure is the Pentagram. Since the
cells and vertex polyhedra are duals, this is a self-dual polychoron.
- The Pentagrammic faces of the previous sections are shared by Great stellated dodecahedral
edge sections. Their vertices are those of vertex sections 2 and 6 in Table 2. These are the cells
of the Great stellated 120-cell (virtual model), which therefore
shares the faces of the Grand stellated 120-cell. The vertex polyhedron is the Icosahedron, the
edge figure is therefore the Pentagon.
- On the outside, enveloping the projection, is an undistorted Great icosidodecahedron, the
projection of one of the 60 instances of the equatorial polyhedron of the Grand 600-cell.
This shares the vertices of vertex section 4 in Table 2.
Because these polychora share the edge arrangement of the Grand 600-cell, the Zometool
representations of their Icosahedral projections are identical to that in Figs. 25a and b. In these
figures, the false vertices have the same arrangement (and at the same scale) as the true vertices
of the Icosahedral projection of the 600-cell in Figs. 21 and 23, the reason for this is explained later.
The pattern of faceting of the vertex polyhedra is, as in the case of the facetings of the 600-cell,
depicted in Fig. 11c, but starting from the Great stellated 120-cell. Note that the regular
sections of the 600-cell are isomorphic to the ``sections'' of the Grand 600-cell discussed above:
1) Icosahedron - Great icosahedron (these are their vertex polyhedra), 2) Great dodecahedron -
Stellated dodecahedron, 3) Dodecahedron - Great stellated dodecahedron. The same applies to their
equatorial polyhedra - the Icosidodecahedron and Great icosidodecahedron, both of which share the
Triangular faces of their respective polychora.
***
As mentioned above, the equatorial polyhedron of the 600-cell is the Icosidodecahedron, therefore
the equatorial polyhedra of its facetings must be facetings of the Icosidodecahedron, which as we've
seen is the case. These equatorial polyhedra are useful for calculating the metric
properties of their polychora:
- As discussed above, for the Icosidodecahedron the distance of a vertex to the centre (0R)
is φ ℓ, as for its own equatorial Polygon, the Decagon. The same applies therefore
to the 600-cell and the 3 star polychora that share its edge arrangement.
- As discussed above, the equatorial polyhedron of the Stellated 120-cell, the Dodecadodecahedron,
is radially equilateral, because it has the Hexagon as its equatorial Polygon. This implies that the
Stellated 120-cell - and the Great grand 120-cell, which shares its edge arrangement - must also be
radially equilateral! This concludes the list of regular polytopes in n dimensions with this
characteristic (which already included three convex members, the Hexagon, Tesseract and 24-cell).
- For the equatorial polyhedron of the Grand 600-cell (the Great icosidodecahedron),
0R = Φ ℓ (where Φ = 1 / φ), as for its own equatorial
Polygon, the Decagram. The same applies therefore to the Grand 600-cell and the other 3 star
polychora that share its edge arrangement.
It is an interesting fact that all rectifications of regular polyhedra
occur as equatorial polyhedra of regular polychora!
***
In Figs. 26a and b, we present the Icosahedral projection of the last of the regular star polychora,
the most complex of the 16 regular polychora, the final regular stellation of the 120-cell and its
only regular faceting: The Great grand stellated
120-cell (virtual model).
Being an Icosahedral projection of a faceting of the 120-cell, its vertex arrangement is
necessarily the same as for the Icosahedral projection of the 120-cell in Fig. 22. This also means
that, like that projection, this is necessarily a cell-first projection: Indeed, there are two
superposed, undistorted Great stellated dodecahedra concentric with the model.
Fig. 26a: Icosahedral projection of the Great grand stellated 120-cell, here seen from a 5-fold symmetry axis.
See also David Richter's page on this
projection.
Fig. 26b: Same model as in the Figure above, here seen from a 3-fold symmetry axis.
How to build: Make a smaller version of the Stellated 120-cell (in Fig. 24a), but with all
internal edges extended - there can be no broken lines. Then identify the equivalent vertices of the
largest possible Dodecahedron in the model, i.e., the outermost balls along the 3-fold symmetry
axes. These are the vertices of the central Dodecahedron of a (large) Icosahedral projection of the
120-cell. Now, based on this, build a few of the remaining "outer" vertices of that projection of
the 120-cell at an appropriate scale. Then, from each of those vertices, see what edges in the inner
model point towards it, and extend those edges until they reach that vertex. Doing this for a set of
neighboring vertices will then allow you to figure out the network of edge intersections. Once this
is done, repeat for all vertices of the 120-cell using Icosahedral symmetry, while removing all
edges of the large projection of the 120-cell that were holding the first set of outer vertices in
place. Look here for the justification of
this method of construction.
This model can use 20 Y3 struts. These will be directly aligned with the 3-fold symmetry axis shown
in this Figure. This saves 40 regular-sized struts (Y2 and Y1) and 20 balls and slightly improves
the appearance of the model. In this model, I omitted intersecting blue struts.
As in the 120-cell, apart from the cells we see no edge sections because there
are no Triangular faces; there are no equatorial polyhedra because the vertex polyhedron (the
Tetrahedron) has no central symmetry. However, as in the 120-cell, there are equatorial rings with
10 cells!
Apart from the large number of vertices (600), this polychoron has a great number of false vertices:
an inner set with the vertex arrangement of a small 120-cell, a slightly larger one with the
arrangement of the 600-cell (together these are the false and true vertices of the Stellated
120-cell), and a second, larger set that also has the arrangement of the 120-cell. As we will see later, these are the vertices of the Compound of 120 5-cells. As shown by
the construction method, and discussed in more detail later, these represent successive stages of
extension of the edges of the 120-cell.
Geometric relations of the star polychora
We will now summarise the properties of the regular polychora. In this summary, we will also
show that there are no additional regular polychora, despite the fact that the 600-cell
and 120-cell have many more stellations and facetings. We will do this with the help of Table 3,
where we list the possible cells and vertex polyhedra.
Unlike Table 1, this Table is finite, since the cells or vertex polyhedra can only be one of the 9
regular polyhedra. However, of the 81 entries, only 25 are logically possible. The reason for
this is that the vertex figure of the cell must match the face of the vertex polyhedron. To give
an example, if the cell is a Cube (which has a Triangular vertex figure), then the vertex
polyhedron must have Triangular faces. These possibilities appear in 4 groups, depending on the
vertex figure of the cells: 16 in the Triangular group (yellow), 1 in the Square group (red), 4 in
the Pentagonal group (green) and 4 in the Pentagrammic group (Gold). These numbers are the squares
of the number of polyhedra with Triangular, Square, Pentagonal and Pentagrammic faces in Table 1.
The convex regular polychora have their names in boldface.
The order of the types of cells and vertex polyhedra is such that
for the objects along the diagonal the cells and vertex polyhedra are duals, these objects are
therefore self-dual. Flipping the table around this diagonal, we find dual
polychora: for instance, the Great grand stellated 120-cell is the dual of the Grand 600-cell.
Table 3: Regular polychora have, by definition, regular cells and vertex polyhedra.
Thus, if we list all possible cells in columns and all possible vertex polyhedra in rows we
can enumerate all regular polychora (colored according to the vertex
figures of the cells: yellow - Triangles, red - Squares, green - Pentagons, gold - Pentagrams) and
all regular honeycombs of the Euclidean (dark gray) and hyperbolic (gray) 3-D spaces.
The objects in light gray are not honeycombs, they are degenerate polychora with an infinite number
of elements, the reason for this is that they are isomorphic to the honeycombs of 3-D hyperbolic
space indicated with the same letters.
Note the similarity of the top left 4 × 4 square with Table 1.
If the vertex polyhedron is convex and the
Solid angles of all
cells meeting at a particular vertex equal or exceed 4π
steradians, then we don't
have a finite polychoron, but a regular division of a 3-D space: a honeycomb. In Table 3, we
find all five compact regular honeycombs: the self-dual Cubic honeycomb of the Euclidean 3-D space and the four regular
honeycombs of 3-D hyperbolic
space: the Order-5 cubic honeycomb, its
dual, the Order-4 dodecahedral
honeycomb and the self-dual Order-5 dodecahedral
and Icosahedral honeycombs.
In this process, we see that the closest analogues of polychoral surfaces are honeycombs of
3-Dimensional ``surfaces'', Euclidean and hyperbolic. There is a close correspondence between
polychoral ``surfaces" and the finite honeycombs of the
3-D
spherical surface, but as for polyhedra, this correspondence is not one-to-one, because of
objects like the regular
multi-dimensional beach balls.
Without these five honeycombs, we still have 20 objects. To proceed, we rely on the concept of
isomorphism. Two isomorphic polychora are topologically identical, sharing the
same configuration
matrix. This implies that the cells and vertex polyhedra and all other elements
are also isomorphic. The four objects indicated with letters in
parentheses are not honeycombs (see detailed treatment in Coxeter 1973), however they are
isomorphic to the four hyperbolic honeycombs indicated with the same letters. The resulting infinite
number of elements implies that, although we might attempt to build polychora with these types of
cells and vertex polyhedra, they never close on themselves.
The remaining 16 objects were already mentioned above. There are therefore no additional regular
polychora, and therefore no star polychora with symmetries other than that of the 600-cell.
***
We now discuss the geometric relations between the 120-cell, 600-cell and the regular star
polychora. Instead of 6 polyhedra shown in Diagrams Ia and b, we now have 12 polychora, in
Diagrams IIa, b and c we follow Conway, Burgiel and Goodman-Strauss (1991, see references) and arrange them as the vertices of a Cuboctahedron, which
has 12 vertices and is the only Archimedean polyhedron that is radially equilateral.
Diagram IIa
As in those earlier diagrams, the vertical scale indicates the density. The thick black lines
represent sets of 4 polychora with identical edge arrangements (of the 600-cell and Grand
600-cell, as discussed above). The aforementioned reciprocal geometric relations between
polytopes are indicated by reflections:
-
Duality. This operation is represented by the yellow
reflections across the central vertical line. Their horizontal direction means that
this operation preserves the density. The aforementioned self-dual star polychora, the Great
120-cell and the Grand stellated 120-cell, are necessarily represented in that central vertical
line. Together with the 24-cell, they are the only regular polytopes in n-dimensional spaces with
this property, other than Polygons and Simplices.
-
Isomorphism. This operation is represented by the red reflections through the centre.
As for polyhedra, it changes all Pentagonal elements (either the cell's faces, vertex figures, or
edge figures) into Pentagrammic and all Pentagrammic elements into Pentagonal, as necessary
in a reciprocal operation.
As an example of Isomorphism, the Grand 600-cell is isomorphic to the 600-cell. Both have 600
Tetrahedral cells in the same arrangement of 3-D planes, 20 meeting at each of the 120 vertices,
which have an identical arrangement. As we've seen above, the same applies to their sections.
This has a major implication: when we use the sections of the 600-cell to make the regular star
polychora in the top black circuit (represented by Fig. 23) the results are isomorphic to using the sections of
the Grand 600-cell to make the regular star polychora in the lower black circuit (represented
by Figs. 25a and b). We have thus established the geometric reason for four of the isomorphisms.
As for polyhedra, if two polychora are isomorphic, so are their duals. This is enough to establish
the two remaining isomorphisms (120-cell - Great grand stellated 120-cell, Stellated 120-cell -
Great grand 120-cell). This can be quickly verified by checking that their cells and vertex
polyhedra are indeed isomorphic to each other. The isomorphism between the Stellated 120-cell and
the Great grand 120-cell can be verified independently from the fact that both share the same
vertices and edges and that their cells (and vertex figures) are isomorphic to each other, as
mentioned in the discussion on Figs. 24a and b.
Diagram IIb
What we have learned above about the the edge sections of the 600-cell, Grand 600-cell and
Stellated 120-cell and the polychora derived from them using these sections as cells is
systematized in Diagram IIb. The four faceting operations that occur among regular polychora are
represented by parallel
translations within
the same Cuboctahedral arrangement (represented by arrows), the reason for this is the central
symmetry of the isomorphism operation in Diagrams Ia and IIa:
- Face faceting. As we've seen from the study of the sections, several polychora share the
same face
arrangements. These polychora are linked by the dark blue arrows. This operation
changes Pentagonal edge figures into Pentagrammic ones
- Edge faceting. Equally, we have seen that several polychora share the same edge
arrangement, but without sharing face arrangements. These are indicated by the light blue arrows.
This transformation preserves the edge figures, but being a faceting operation, it decreases their
angular size.
As we've seen in the study of the sections, because these two operations preserve the polychoron's
edge arrangement, they necessarily facet the vertex polyhedra. These facetings work in the same
sense (top right to lower left) and with the same vertex polyhedra in the upper and lower blue
circuits. This leads us to the next faceting operation.
- Vertex faceting. The light green arrows represent a faceting that preserves the type of
vertex polyhedron; however, since this is a faceting, it necessarily reduces the vertex polyhedron's
angular size.
- Dual vertex faceting. This operation, represented by the dark green arrows, is necessary
to completely account for all facetings. It transforms the vertex polyhedron into its dual, while
reducing its angular size. Thus, the aforementioned Dodecahedral vertex polyhedron of the Stellated
120-cell is the dual of the Icosahedral vertex polyhedron of the 600-cell.
Diagram IIc
As we've seen, for two dual polychora A and B, a stellation of A (sA) is the dual of a faceting of B
(fB). We can therefore say that the stellation operation from A to sA is the dual of the faceting
operation from B to fB. Being the duals of the faceting operations, these operations are, in this
diagram, a reflection around the central vertical line of the faceting operations in Diagram IIb;
for this reason they are also represented by sets of parallel translations, in this case arrows with
the same colours as those of their dual transformations. These stellation operations give the regular
star polychora their names.
- Edge stellation. This operation is represented by the dark blue arrows. Since the dual
operation (face faceting) preserves the face arrangement, this operation preserves the dual
(perpendicular) edge directions, but extends them, stellating Pentagonal faces into Pentagrammic
ones. This has to be since the dual operation transforms Pentagonal edge figures into Pentagrammic ones.
- Greatening. This operation is represented by the light blue arrows. Since the dual
operation (edge faceting) preserves the edge arrangement, this operation preserves the perpendicular
2-D facial planes. Since the dual operation preserves the type of edge figure but makes it smaller,
this operation replaces a face by a larger one of the same type (see Fig. 11a).
Since the dual operations facet the vertex polyhedra, these two operations stellate the cells.
Indeed, the pattern of cell stellation in the top blue circuit is shown in Fig. 11b.
- Aggrandizing. This operation is depicted by the light green arrows. Since the dual
operation replaces vertex polyhedra by smaller ones of the same kind, this operation replaces cells by
larger ones of the same kind. From this, we see that the cell types in the lower blue circuit are a
repeat of the cell types in the top blue circuit.
- Dual aggrandizing. The second operation is depicted by the dark green arrows. Since the
dual operation changes a vertex polyhedron into a smaller dual, this operation changes a cell into a
larger dual.
Challenge to the reader: As mentioned above, the Cuboctahedron has 4 Hexagons as equatorial
Polygons. In Diagrams IIa, b and c, which Hexagons touch polychora where the cells match
the polyhedra touched by the Hexagon in Diagrams Ia and b? Do the geometric operations along
them match those in Diagrams Ia and b? If not, what are they replaced with?
In the Figure below, we show a Zometool representation of Diagrams IIa, b and c.
Fig. 27: Zometool representation of diagrams IIa, b and c.
The black balls - the vertices of the outer Cuboctahedron - represent the 120-cell, 600-cell and the
10 regular star polychora, the struts represent the geometrical operations discussed above.
Note how the edge stellation and its dual, the face faceting, are special, being the only
operations represented by the blue struts. All others, including duality, can be represented by
green struts and include special double-length cases going through the centre of the figure: the
isomorphisms.
Partially regular polychora
Apart from the convex Archimedean polychora and the regular star polychora,
we know many more uniform
polychora. It is not known if the list is complete or not, but it likely isn't, because
hundreds of them have been found since 2020. However, apart from some of the regular star polychora,
only four are edge facetings of the regular polychora, and thus represented by the models above. We now list them.
There is one faceting of the 16-cell, the Tesseractihemioctachoron; it
is an edge faceting, so it shares the Zometool representations of the 16-cell. This polychoron is
constructed with the 4 equatorial Octahedra of the 16 cell plus 8 of its 16 Tetrahedra. Since all
cells are regular, this is a semi-regular non-convex polychoron. It is the 4-dimensional analog of a
partially regular polyhedron, the Tetrahemihexahedron, which it has as
vertex polyhedron; the latter is necessarily a faceting of the Octahedral vertex polyhedron of the
16-cell.
The Stellated 120-cell is special because it is the only regular polychoron where the vertex
polyhedron is a Dodecahedron. As we've seen, Dodecahedra can be faceted in 22 different ways, this
results in 22 edge facetings of
the Stellated 120-cell, which necessarily share its Zometool representation and are radially
equilateral. One of these is another regular polychoron, the Great grand 120-cell, which has the only
regular faceting of the Dodecahedron, the Great stellated dodecahedron, as a vertex polyhedron. If
we use instead the ditrigonal polyhedra in Fig. 7 as vertex polyhedra, we obtain three
additional semi-regular polychora:
- With the Small ditrigonal
icosidodecahedron we obtain the Small
ditrigonal 120-600-cell, which has 120 Great icosahedra and 600 Tetrahedra as cells.
- With the Ditrigonal dodecadodecahedron
we obtain the Ditrigonal
120-120-cell, which has 120 Great icosahedra and 120 Icosahedra as cells.
- Finally, with the Great ditrigonal
icosidodecahedron we obtain the Great
ditrigonal 120-600-cell, which has 120 Icosahedra and 600 Tetrahedra as cells.
These Icosahedral, Great icosahedral and Tetrahedral cells are the aforementioned Icosahedral,
Great icosahedral and Tetrahedral sections of the Stellated 120-cell.
The names of these three polychora are consistent with those of the ditrigonal polyhedra,
which indicate the number of the two types of faces separately. In addition to sharing the vertex
arrangement of the 600-cell and edge arrangement of the Stellated 120-cell, they share an identical
face arrangement, a consequence of the fact that their vertex polyhedra, the partially regular
polyhedra in Fig. 7, share the same edge arrangement. Furthermore, since all faces are Triangular,
their vertex polyhedra are necessarily among their edge sections. These are the partially regular
polyhedral sections of the Stellated 120-cell mentioned above.
The isomorphisms between these polytopes are the same as for their vertex polyhedra: the Small
ditrigonal 120-600-cell is isomorphic to the Great ditrigonal 120-600-cell and the two sets of
cells of the Ditrigonal 120-120 cell are isomorphic to each other.
Being edge facetings of regular convex polychora, the Tesseractihemioctachoron and the three
ditrigonal polychora are isogonal and isotoxal. Furthermore, their vertex polyhedra are the
partially regular polyhedra, all isotoxal and which each edge separating the two types of faces;
this implies that these four polychora are also isohedral, with each face (Triangles in all four
cases) separating the two types of cells of each object, all of which are regular. No other
semi-regular polychora come this close to full regularity. For this reason I call them, by
analogy with their vertex polyhedra, the ``partially regular'' polychora.
Additionally, two of the facetings of this Dodecahedral vertex polyhedron are regular compounds: the
Compound of five cubes and the Compound of ten tetrahedra. This implies that two of the facetings of
the Stellated 120-cell are Zomable regular polychoron compounds.