Polychora

"As for the analogous figures in four or more dimensions, we can never fully comprehend them by direct observation. In attempting to do so, however, we seem to peep though a chink in the wall of our physical limitations, into a new world of dazzling beauty."
H. S. M. Coxeter, in the preface of "Regular Polytopes".

The generalization of "polygon" in 2 dimensions and "polyhedron" in 3 dimensions to any number of dimensions is a polytope. A 4-dimensional polytope is a polychoron (plural "polychora"). This is a finite region of 4-D Euclidean space* bound by at least 5 polyhedral 3-faces, or cells; it must also have at least 5 vertices, 10 edges and 10 faces. These surface elements are, together with the inner 4-D ``volume", part of the polychoron. In a regular polychoron all cells and vertex polyhedra are regular. This implies that it is isochoral: the cells are not only identical, but the polychoron looks the same seen from all of them; this implies, for instance, that the polychoron has a well defined centre and that all cells are equidistant from it. The same is true for the other types of elements, i.e., the polychoron is also isogonal, isotoxal and isohedral. As we'll see, there are sixteen regular polychora: six are convex and ten are non-convex.

If polychoron A is regular, its dual B is also regular. To each vertex of A corresponds a cell in B, which is the dual of the vertex polyhedron of A, and vice-versa. Each edge of A goes through a perpendicular face of B, which is identical to that edge's figure, and vice-versa. Two dual polychora must necessarily have the same symmetry.

* This is, again, a traditional definition that excludes honeycombs and apeirotopes.

Projections and sections

Projections:

Of course, we cannot build models of the polychora - the 3-D space we inhabit does not allow it. The models below represent projections of polychora to our 3-dimensional Euclidean space. These cannot represent the inner 4-D volume of the polychoron, only its ``surface'' elements.

The best-known projections of polychora are perspective projections, where vectors emanate from a single point at a finite distance. Photographs and digital pictures are of this type, where nearby 3-D objects are projected into the focal plane of the camera. This type of projection can be used to portray 4-D objects into 3 dimensions, an example of which are Schlegel diagrams. The most common representation of any polychoron, the Tesseract represented as a Cube within a Cube (see Fig. 5.1) is of this type.


Fig. 5.1: A perspective projection of the Tesseract in 3-D space, a 3-dimensional Schlegel diagram, represented with the Zometool.


The projections below are not of this type. They are orthographic projections; where the projection vectors are parallel to each other and perpendicular to the ``plane'' of the projection. Fig. 5.2 shows three orthographic projections of three-dimensional figures (the Octahedron and the Cube) to 2-D space. This will be used to illustrate many properties of these projections.


Fig. 5.2: Projections of the Octahedron (A) and Cube (B and C) into 2-dimensional space represented with the Zometool. Projections A and B are "vertex-first", projection "C" is face-first.


This type of projection is an example of an affine transformation, meaning that a set of parallel lines in the polychoron are still parallel to each other in its projection. This is not the case for the perspective projection in Fig. 5.1, where the edges represented in yellow, which are parallel to each other in the Tesseract, are not parallel in the projection.

There are infinite orthographic projections for any polytope, corresponding to an infinite number of orientations of the polytope relative to the projection plane. In Fig. 5.2 and below we show a few projections that can be represented with the Zometool system, i.e., that are "Zomable". If a polychoron has at least one such projection, we will likewise say it is Zomable. All regular polychora are Zomable, below we show all their Zomable projections. In addition, we show Zomable projections of closely related forms, the Archimedean polychora, the 4-D analogues of the Archimedean solids.

The Zometool has two apparent disadvantages: First, it can only represent a small number of projections. This is not much a problem, though, because the projections that can be represented are almost always the most symmetric and beautiful. Second, it can represent only the vertices (balls) and edges (struts) of the projection, the faces and cells must be deduced from their vertices and edges. However, this feature is actually useful: since all the projections below have a lot of important internal structure, representing the faces (as in a paper model) would hide the internal structure of the projection. If a set of polychora have different face and/or cell arrangements, but share an edge arrangement, then their projections will share the same Zometool representation.

As we can see in Fig. 5.2, the central elements of a projection represent the elements of the polytope that are ``closest'' and ``farthest'' from the plane of the projection. In cases A and B, these are vertices, in case C the faces. In the projections below, these are generally cells or vertices, although occasionally we show other types of projections.

The symmetry of these central elements reflects the symmetry of the axis along which the polytope is being projected and determines the symmetry of the projection: as an example, the projections of the Octahedron and Cube with Square symmetry (Fig. 5.2, A and C) result from projecting the Octahedron and Cube along one of their axes of Square symmetry. An important consequence is that, if the polytope and the central element have central symmetry, then the ``near'' and ``far'' sides of the polytope coincide in the projection, as in A and C. This is not the case for the vertex-first projection of the Cube in Fig. 5.2 (B): the central element (a vertex) has a Triangular vertex figure, which lacks central symmetry. This implies that the "near" and "far" sides of the projection do not coincide, with the nearest and most distant edges in different colours. The projections of polychora below where this happens are marked by an asterisk (*); these are generally centered on Tetrahedral cells or vertices with Tetrahedral vertex polyhedra.

In any projections, most or all elements are necessarily distorted. One of the main reasons why the Zometool is a superb way to study the geometrical properties of the polychora shown below is that it provides an easy way of seeing how distorted an element is: their colour. Indeed, because orthographic projections are affine, in a polytope with edges of identical lengths ℓ, a set S of parallel edges in the polytope are still parallel and have identical length ℓ' in the projection, where ℓ' = ℓ sin(ζ) and ζ is the angle between the edges of S and the projection vector. A set of edges with identical lengths and directions can only be represented, in a Zometool model, by struts of the same colour (and length and direction, although the length can be divided between more than one strut because of intersections with other edges).

This implies that each strut colour in the models below represents a specific value of sin ζ that occurs in the projection. To calculate this, we use the Zometool strut lengths. For most models, the longest struts are blue, in that case the values of ζ are:

Colour sin ζ ζ
Blue 1 90
Red √ (5 + √5) / 8) 72 or 108
Yellow √ 3 / 2 60 or 120
Smaller red √(5 + √5) / 8 ) / φ = √2 √(5 − √5) / 4 36 or 144
Points 0 0 or 180

Note that these are the angles appeared in the discussion of the metric properties of the polygons. If the model includes Tetrahedra, Octahedra and Cuboctahedra, the longest struts are green, in that case the angles are:

Colour sin ζ ζ
Green 1 90
Blue √ 2 / 2 45 or 135
Yellow √ 6 / 4 37.761 243... or 142.238 756...
Points 0 0 or 180

It is therefore clear, in a model made with the Zometool, which elements are not distorted. If one of these undistorted polyhedra is a cell (it can also be a section, see below) then in 4-D it is the element closest or most distant from us, or lies directly ``under'' it; this corresponds in the projection to the central element or an object concentric with it. For each instance of these undistorted elements, there will be many more instances of the same element that are not in 3-D planes parallel to ours that will appear distorted by the projection.

Sections:

Another way to understand a polytope is by cutting it with successive "planes". The resulting sections are polytopes of a lower dimension. The succession of sections of a polychoron gives us an idea of its 4-dimensional shape, in the same way a set of two-dimensional sections describes a polyhedron, as in a 3-D printer.

For any polytope, there is an infinity of sections: as for the projections, we can orient the polytope relative to relevant plane in an infinite number of ways. Below we will only refer to symmetric sections starting either from a vertex or a cell (which will therefore have the symmetry of the cell or vertex polyhedron) that share vertices, edges or faces with the polychoron. These we call vertex, edge or face sections.

If a polychoron has only Triangular faces, all vertex polyhedra are necessarily represented by the polychoron's edges; if these edges are in the same 3-D "plane" they represent edge sections of the polychoron. If it has only Tetrahedral cells, all vertex polyhedra are necessarily represented by its faces; if these faces are in the same 3-D plane they represent face sections of the polychoron.

If the cells of a regular polychoron A have central symmetry, then each cell has opposite pairs of faces, which communicate with other identical cells. This means that those cells form equatorial ``rings'' linked by a set of parallel faces. This implies, by duality, that the vertex polyhedra of the dual regular polychoron B also have central symmetry, and that B's edges form equatorial Polygons, with vertices and edges corresponding to the cells and faces of the equatorial cell ring of A. B's equatorial Polygons must form equatorial polyhedra. The only polyhedra where all edges are part of equatorial Polygons are the rectifications of regular polyhedra: even the Octahedron, the only Platonic solid with equatorial Polygons, is a rectification (of the Tetrahedron).

Thus, all regular polychora with centrally symmetric vertex polyhedra - and only those - have equatorial polyhedra, and the latter are always rectifications of regular polyhedra. Such equatorial polyhedra are edge sections of the polychoron by 3-D planes that passes through its centre and are located either between opposite pairs of cells or vertices. The duals of such polychora have equatorial cell rings.

Sections in projections:

Any edge or face section, having edges that belong to the polychoron, will be represented in any projection of the polychoron that shows its edges, like Zomable projections.

Any instances of the edge or face sections projected without distortion in the models below will appear, as mentioned above, as green or blue polyhedra (according to the model). In 4-D, these instances occur "under" the element that is being projected to the center, in 3-D their projections will be concentric with the projection of the central element. If an instance of the equatorial sections is projected without distortion, then it is located (in 4-D) between the two diametrically opposed elements that appear, in 3-D, superposed at the centre of the projection, its projection envelops the projection of the whole polychoron.

Simple regular polychora

 Before we move on, we need a clarification on the names. Some regular polychora have many names - one is known as the 120-cell, Dodecaplex, Hyperdodecahedron, Polydodecahedron, Hecatonicosachoron, Dodecacontachoron and Hecatonicosahedroid. The convention adopted here is the same adopted by Wikipedia, which is to use the simpler names, for this example "120-cell". This is consistent with the names of the regular polyhedra, which refer only to the number of faces, not their type. This simple system works because the resulting names are unambiguous. However, unlike the case of polyhedra and polygons, we don't use Greek names: "120-cell" is much easier to understand, and shorter to write, than "Hecatonicosachoron". For the 8-cell we will use the widely known term "Tesseract". This is also consistent with the polyhedral and polygonal names, where the terms "Cube" and "Square" are at the same time technically precise (they imply regularity) and do not obviously indicate the number of faces or edges.

***

The simplest regular polychoron, the 5-cell, is the 4-D analogue of the Tetrahedron. It is bound by 5 Tetrahedral cells and has 5 vertices, with Tetrahedral vertex polyhedra. These cells have 5 × 4 = 20 Triangular faces, however, since each face is shared by two cells the total number of faces is 10. The cells also have 5 × 6 = 30 edges, but since the edges have Triangular figures (i.e., since each edge is shared by three cells), there are only 10 edges. As mentioned above, this is the smallest number of elements of these types any polychoron can have.

The symmetry of the numbers of elements (5-10-10-5) and the duality of the Tetrahedral cells and Tetrahedral vertex polyhedra imply that the 5-cell is, like the Tetrahedron, self-dual. This property is shared by all their n-dimensional analogues, the Simplices.

The Tetrahedral projection of the 5-cell is shown in Fig. 5.3. Note how the outer Tetrahedron, which is an undistorted cell, also represents the undistorted vertex polyhedron of the central vertex (this is also the case for any apex-first projections of 4-D pyramids, like the Cubic or Icosahedral pyramids, which we'll meet later). To its left is a projection of the Rectified 5-cell.


Fig. 5.3: Projections of the 5-cell and its rectification, a semi-regular polychoron.

Left: A "vertex-first" and "cell-first" Tetrahedral projection of simplest regular polychoron, the 5-cell (*).
Right: A Tetrahedral (thus, cell-centred) projection of the Rectified 5-cell (*).


We now use the 5-cell to illustrate the extension of the concept of rectification (discussed previously for polygons and polyhedra) to polychora.

When rectifying the 5-cell, each of its 5 Tetrahedral cells is rectified into an Octahedron. Each vertex of the 5-cell is replaced by its vertex polyhedron, a Tetrahedron. The number of vertices of the rectification is the same as the number of edges of the 5-cell, 10. In the 5-cell, each edge is shared by 3 Tetrahedral cells, this means that in the Rectified 5-cell each vertex (located in the middle of the edge of the 5-cell) is touched by three Octahedral rectifications of the previous cells, and by the two new Tetrahedral cells, which relative to this vertex are opposite to each other. This means that the vertex polyhedron of the Rectified 5-cell is the Triangular prism. Since all faces of the Rectified 5-cell are Triangular, these Triangular prisms are edge sections lying under each of its 10 vertices, it is fun to find them in the model. This polychoron is isotoxal, with each edge shared by one Tetrahedron and two Octahedra.

Since all its cells are regular, the Rectified 5-cell is one of the three Semi-regular convex polychora. Together with its vertex polyhedron, the Triangular prism, it is a member of the finite multi-dimensional k21 family of Semi-regular polytopes, where each n-dimensional polytope is the vertex polytope of the n+1 dimensional polytope in the family.

Both polytopes in Fig. 5.3 have Pentachoric symmetry. For more Zomable projections of the 5-cell, see Fig. 7.8.

***

The 16-cell has four pairs of vertices located along four orthogonal directions at the same distance from the centre. It is the 4-D analogue of the Octahedron. It is bound by 16 Tetrahedral cells and has Octahedral vertex polyhedra. It has a Square edge figure. Following the logic used above to calculate the number of elements of the 5-cell, we find it has 24 edges, and 32 Triangular faces. Its dual, the Tesseract, is the 4-D analogue of the Cube. This is bound by 8 Cubic cells and has 16 vertices with Tetrahedral vertex polyhedra. It has 32 edges with Triangular figures and 24 Square faces. These polychora share the Hexadecachoric symmetry.

In Fig. 5.4, we show different Octahedral projections of the 16-cell and Tesseract. Three of these are analogs of the projections of the Octahedron and Cube in Fig. 5.2. For more Zomable projections of the 16-cell, see Figs. 7.1a and b. For another projection of the Tesseract, see Fig. 7.3.


Fig. 5.4: Octahedral projections of the 16-cell and Tesseract.

Top left: the vertex-first projection of the 16-cell. Top right: The vertex-first projection of the Tesseract (*). This projection represents two different projections of the Tesseract, which have different sets of four edges closer to our 3-D ``plane''.

Bottom left: The cell-first projection of the 16-cell (*). This projection is especially good for highlighting its 8 vertices separately. Two opposite Tetrahedral cells superpose at the centre, in a Stella Octangula configuration. This projection represents two different projections of the 16-cell, which have different central Tetrahedra closer to our 3-D ``plane''. Bottom right: The cell-first projection of the Tesseract.


We described above the meaning of duality between polychora. We now describe the meaning of duality between projections. In Fig. 5.4, the vertex-first projection of the 16-cell is the dual of the cell-first projection of the Tesseract; each cell of the former is centred exactly on a vertex of the latter, and vice-versa. Equally, the cell-first projection of the 16-cell is the dual of the vertex-first projection of the Tesseract. Two dual projections necessarily have the same symmetry.

Like all regular polychora with Tetrahedral cells, the edges and faces of the 16-cell trace its vertex polyhedron, the Octahedron, which in this case is an equatorial polyhedron between pairs of opposite vertices. In the vertex-first projection in Fig. 5.4 we see one undistorted Octahedron enveloping the two superposed opposite vertices at the centre, and three flattened projections of Octahedra (Fig. 5.2, A) in three perpendicular planes passing through the centre of the projection lying halfway between the outer vertices. These equatorial Octahedra have themselves equatorial Polygons, Squares, which correspond, in its dual (the Tesseract), to equatorial rings of 4 Cubes joined with each other by 4 Square faces. As for these equatorial figures, in the 16-cell the distance of a vertex to the centre is √ 2 / 2 times the edge length (ℓ).

Like the 5-cell, these regular polychora have analogues not only in 3-D, but also in any n-dimensional Euclidean space. The Tesseract is the 4-D Hypercube, and the 16-cell is a 4-D cross polytope. However, they have unique properties: This means that we can make a Compound of two 16-cells with the vertex arrangement of the Tesseract; interestingly, the cell-first projection of the 16-cell in Fig. 5.4 can represent such a compound! This implies, by duality, that its dual projection (the vertex-first projection of the Tesseract) also represents a Compound of two Tesseracts. In the projections of these compounds, the ``near'' and ``far'' sides are identical.

One of the consequences of the 16-cell being a Demi-tesseract is that if we start with a Tesseract honeycomb and do an alternating deletion of half of its vertices, we end up with a new regular honeycomb made of 16-cells: this is unique to 4 dimensions and the 16-cell is the only cross polytope that can fill space! One of its projections would look like a filling of 3-D space with the cell-first projections of the 16-cell, which would represent the 16-cells concentric with the previous Tesseracts. The "new" 16-cells that fill the gaps between the previous ones would be represented by vertex-first projections, which appear automatically around the vertices where 6 blue struts meet.

How these properties of the 16-cell and Tesseract are related is discussed in more detail below.

The 24-cell

 Ultimately, the Rectified 5-cell owes its semi-regular status to a fact we mentioned when discussing polyhedra: the rectification of a Tetrahedron is another regular polyhedron, the Octahedron. This means that, when we rectify any regular polychoron with Tetrahedral cells, we obtain a semi-regular polychoron bound by Octahedral cells and cells identical to the former's (regular) vertex polyhedron. The new vertex polyhedra are equilateral prisms, with a "base" given by the edge figure of the regular polychoron that was rectified.

Things are even more interesting when we rectify the 16-cell. Doing this, we create a polychoron where 16 Octahedral cells result from rectifying the Tetrahedra of the 16-cell and where 8 new Octahedral cells are the vertex polyhedra of the 16-cell. The edge figure of the 16-cell is the Square, this implies that the vertex polyhedron of the Rectified 16-cell is the Square prism, i.e., the Cube. The regularity of cells and vertex polyhedra means that this rectification is regular: the 24-cell. The 24-cell has its own unique Icositetrachoric symmetry.

In Fig. 5.5a, we show the cell-first projection of the 24-cell. This results from rectifying either the cell-first or the vertex-first projections of the 16-cell in Fig. 5.4.


Fig. 5.5a: The cell-first projection of the 24-cell.


Because it is a rectification of the 16-cell, the number of its vertices has to be the same as the number of edges of the 16-cell, also 24. The number of edges (which have Triangular figures) and Triangular faces is 96. This and the fact that the cells and vertex polyhedra are duals implies that the 24-cell is self-dual. In all Euclidean n-dimensional spaces, this is the only regular convex polytope with this characteristic that is not a Simplex or a Polygon. Furthermore, the 24-cell and its symmetry have no analogs in any other n-dimensional space, i.e., the 24-cell and its symmetry are clearly exceptional objects! We will now find why.

The previous way of building the 24-cell (by rectifying the 16-cell) is known as Cesàro's construction (Coxeter, 1973); this is the 4-D equivalent of obtaining a Cuboctahedron from an Octahedron by rectification. The "dual" construction of the 24-cell (Gosset's construction) is equivalent to the construction of a Rhombic dodecahedron from two Cubes mentioned above. This construction illustrates especially well why the 24-cell is unique to 4 dimensions.

Step 1: Divide Tesseract A into 8 Cubic pyramids (these can be represented by the model in Fig. 4.3a). Their 4-D volume is limited by a Cubic "base" (one of the previous cells of the Tesseract) and by 6 Square pyramids, the "sides", the apex lies 1/2 ℓ above the base, the distance of a Cubic cell to the centre of its Tesseract.

Step 2: Separate these Cubic pyramids and "glue" their Cubic bases to the Cubic cells of a second identical Tesseract, B, making those cells - and all Square faces - disappear. Square pyramids from two neighbouring Cubic pyramids now contact each other through what were their Square bases, thus forming 24 Octahedra - the cells of the 24-cell.

As shown by the construction of the Rhombic dodecahedron from Cubes, similar constructions can be made in any dimensional space. The difference in 4 dimensions is that Tesseracts are, as mentioned above, radially equilateral. Therefore, the Cubic pyramids and their Square pyramid "sides" used in the previous steps are also equilateral, a necessary pre-condition for building Octahedra.

To the 16 vertices of Tesseract B, all ℓ from the centre, 8 new ones were added: the apices of the Cubic pyramids, which lie "above" each Cubic cell of B, also ℓ from its centre (see Fig. 5.5b, where Tesseract B appears in cell-first projection, in blue). Thus, the 24-cell is, like the Tesseract, radially equilateral.


Fig. 5.5b: The vertex-first projection of the 24-cell, known as its "pyritohedric" projection. This is the dual of the projection in Fig. 5.5a. The two superposed central vertices appear ``above'' two superposed undistorted Cubic edge sections, in blue. The outermost vertices appear above Cubes flattened by the projection.


Since each of the 8 new pyramidal apices (touched only by yellow struts in Fig. 5.5b) hangs over a cell of Tesseract B, they have the vertex arrangement of the dual of the Tesseract, the 16-cell. Thus, the 24-cell has the vertices of a dual compound, the Tesseract and the 16-cell. Furthermore, as we've seen above, we can divide the vertices of any Tesseract into those of another two 16-cells, therefore the vertices of the 24-cell can be decomposed into three sets of vertices of the 16-cell. As we will see, a set of three 16-cells with the vertices of the 24-cell is a regular polychoron compound (see Figs. 7.2a, b and c).

Since the 24-cell has Triangular faces, its vertex polyhedron (the Cube) is necessarily an edge section, with one such Cube located under each vertex of the 24-cell (see Fig. 5.5b). Since these vertices have the same arrangement as the vertices of the Compound of three 16-cells, those Cubic sections must be arranged as the cells of the dual compound, which necessarily consists of three Tesseracts. This includes Tesseract B plus two others that appear in Fig. 5.5b in vertex-first projection (see Fig. 5.4).

As suggested by their derivations, the Cuboctahedron and Rhombic dodecahedron are the closest polyhedral analogues to the 24-cell: The existence of equatorial polyhedra and Polygons is a consequence of the central symmetry of the vertex polyhedron of the 24-cell. The self-duality of the 24-cell means that the cells must also have central symmetry, which means that the 24-cell also has equatorial cell rings: it is the only regular convex polychoron with both, and they are dual to each other. In Fig. 5.5a, we see one of these rings as a small ``tunnel'' at the centre of the projection, we also see an outer Hexagon that is perpendicular to that ring. In the projection in Fig. 5.5b of the dual 24-cell, the central cell rings are replaced by their duals, radial line-like Hexagons (distorted in this way by the projection) and the outer perpendicular Hexagons are replaced by their duals, rings of 6 Octahedral cells, all completely flattened by the projection.

Like the Rhombic dodecahedron, the 24-cell can fill space. To see this, we apply the process used above to transform a Cubic honeycomb into a Rhombic dodecahedron honeycomb to a Tesseractic honeycomb: First, we divide the latter into two alternating sets of Tesseracts. Second, we divide all Tesseracts in one of the sets into Cubic pyramids. Third, we attach their Cubic bases to the Cubic cells of their neighbouring Tesseracts, thus making 24-cells via Gosset's construction above. This is the regular 24-cell honeycomb.

The dual of the latter honeycomb must also be regular; it is in fact the 16-cell honeycomb mentioned above! The reason for this is simple: if we start with two dual Tesseractic honeycombs A and B, then the alternate removal of half of all Tesseracts in A (to make 24-cells) will cause the alternate removal of half of all vertices in B. The addition of vertices in A (at the centres of alternating sets of Tesseracts) marks the centres of "new" 16-cells in B. B thus becomes the 16-cell honeycomb mentioned above.

Thus, the fact that the Tesseract is radially equilateral leads to the existence of the of the 24-cell and is the ultimate reason why 4-D space is, together with the 2-D Euclidean space, the only n-dimensional Euclidean space with three regular honeycombs. All others have only their respective Hypercubic honeycombs!

These unique honeycombs have an interesting property demonstrated only recently: if we replace the 24-cells of the 24-cell honeycomb (or the vertices of the 16-cell honeycomb) with concentric 3-spheres, the resulting arrangement is the densest possible regular sphere packing in 4 dimensions, where each sphere kisses 24 others.

The 600-cell and the 120-cell

 We will now describe a construction of the more complex polychora based on the 24-cell, also known as Gosset's construction (see Coxeter 1973).

Step 1: Rectify the 24-cell, obtaining the Rectified 24-cell (Fig. 5.6). For each of the 24 Octahedra of the 24-cell the latter has one Cuboctahedron (the rectification of the Octahedron). For each of the 24 vertices of the 24-cell, the Rectified 24-cell has a Cube (the vertex polyhedron of the 24-cell). This polychoron has 96 vertices, one for each edge of the 24-cell.


Fig. 5.6: A cuboctahedral-cell-first projection of an Archimedean polychoron, the Rectified 24-cell.
All projections of the Cube and Cuboctahedron in this model are already present in the model of the 24-cell in Fig. 5.5a.

How to build: Have a look at the Eusebeia page on the Rectified 24-cell. For this projection, see the part on the "Cantellated 16-cell".


Step 2: Diagonally bisect all of its Squares. If the diagonals are chosen correctly, then each of the 12 vertices of the Cuboctahedral cells has five edges and five triangular faces converging on it, i.e., it has been transformed into an irregular icosahedron (Fig. 5.7). The Cubic cells of the Rectified 24-cell have been transformed into sets of five Tetrahedra, with only the central one being regular.


Fig. 5.7: Being a rectification of the Octahedron or Cube, the Cuboctahedron (in green, see also Fig. 3.4) has 12 vertices, the number of edges of the Cube and Octahedron. This is the same number of vertices as the Icosahedron. That is why their duals are both dodecahedra, respectively the Rhombic dodecahedron (Fig. 3.4) and the Dodecahedron (Fig. 3.1).

The reason for the identical number of vertices is that if we add new edges along the diagonals of the Square faces of the Cuboctahedron as shown here (in blue), it becomes an irregular Icosahedron, with 5 edges and 5 triangles converging on each of its vertices. By shortening those diagonals to the size of the other edges, we transform the irregular icosahedron into a regular one. None of these transformations changed the number of vertices.

The blue diagonals break the Octahedral symmetry of the Cuboctahedron, this model has pyritohedric symmetry.


Step 3: Shorten these Square diagonals until they have the same length as the other edges. This is possible because of the very peculiar kinematics of the Cuboctahedron. This generates one regular Icosahedron for each of the previous 24 irregular icosahedra, and 24 clusters of five Tetrahedra, 120 in total. The resultant polychoron is the Snub 24-cell (Fig. 5.8).


Fig. 5.8: This is an icosahedral-cell-first projection of an Archimedean polychoron, the Snub 24-cell.
Like the model in Fig. 5.7, this model has pyritohedric symmetry.

How to build: Have a look at the Eusebeia page on the Snub 24-cell.
Also recommended is David Richter's page on the 24-cell and its snub.


The second and third steps preserved the number of vertices of the Rectified 24-cell, 96, all of them are still identical to each other, the vertex polyhedron is a Johnson solid, the tri-diminished icosahedron. Because all faces are Triangles, this vertex polyhedron is an edge section of the Snub 24-cell, with one tri-diminished Icosahedron appearing (in 4-D) under each vertex.

Because it is isohedral and its cells are Uniform, the Snub 24-cell is a Uniform polychoron. Because all its cells are regular, it is also semi-regular. However, and unlike all other polychora in this page, this is neither regular nor derived from a regular polychoron by rectification; it is also not isotoxal. Although it is derived from the symmetry of the 24-cell, it is less symmetric, having the ionic diminished Icositetrachoric symmetry.

Step 4: Glue an equilateral Icosahedral pyramid (consisting of an Icosahedral "base" and 20 regular Tetrahedra as "sides", see Fig. 5.9) by its base to each of the 24 Icosahedral cells of the Snub 24-cell.


Fig. 5.9: Icosahedral projection of the Icosahedral pyramid, a non-uniform polychoron. This can be seen as an "apex-first" projection, but also as a "base-first" projection, where the base is the outer Icosahedron. This pyramid can be equilateral (in 4-D) because the Icosahedron's circumradius (red struts) is slightly smaller than its edge length (blue struts).


The Icosahedral cells disappear. The 120 Tetrahedra of the Snub 24-cell are now joined by the 24 × 20 Tetrahedra = 480 Tetrahedra of the new 4-D "Pyramids", making a total of 600 Tetrahedral cells. Together with the previous 96 vertices of the Snub 24-cell, the new 24 pyramidal apices add to a total of 120 vertices. The former's vertex polyhedra are still identical to each other; in fact they have all become regular: 20 Tetrahedra now meet at each vertex, which means that the vertex polyhedron is an Icosahedron. This is also the case for the 24 new pyramidal apices!

Since all cells and vertex polyhedra are regular, we have built a new regular polychoron, the 600-cell! It has 1200 Triangular faces and 720 edges with Pentagonal figures. This is the 4-D analogue of its vertex polyhedron, the Icosahedron; its symmetry, known as the Hexacosichoric symmetry, is the 4-D analogue of the Icosahedral symmetry. Its Icosahedral projection is shown in Fig. 5.10; this is necessarily a "vertex-first" projection.


Fig. 5.10: The Icosahedral projection of the 600-cell.

How to build: see instructions in David Richter's list of projects with the Zometool.


By studying the model, we can make a Table with all the vertices starting from the central one. This also lists all the vertex sections of the 600-cell, starting from the central vertex. This list will be important for understanding much of what follows. For lists of vertex sections centered on vertices and cells for all convex regular polychora see Table V of Coxeter (1973).

Section N. of vertices Shape of the sectionNote
0 1 Point The vertex closest to us, at the centre of the projection.
1 12 Icosahedron This matches the vertex polyhedron of the 600-cell.
2 20 Dodecahedron These vertices are a reflection of the central vertex through each of the 20 faces of the previous Icosahedron.
3 12 Icosahedron These vertices are a central reflection of the central vertex through each vertex of the previous Icosahedron. They appear in the projection as a φ times larger arrangement than the inner Icosahedron. They are the outermost vertices with that arrangement.
4 30 Icosidodecahedron This is the outermost set of vertices of the model, representing the equatorial polyhedron, the section of the 600-cell through its middle between opposite vertices.
Its Triangular faces are a central reflection (via the vertices of the previous Dodecahedral section) of the Triangular faces of the inner Icosahedral section.
After this, the sections are a repeat of the sections above in reverse order, they are closer to the most distant vertex.
5 12 Icosahedron These 12 vertices are superposed to those of section 3.
6 20 Dodecahedron These 20 vertices are superposed to those of section 2.
7 12 Icosahedron These 12 vertices are superposed to those of section 1.
8 1 Point Most distant vertex from us. Appears superposed to that of section 0.
Total: 120 - -
Table 2: Vertex sections of the 600-cell, starting from a vertex. All have the symmetry of the vertex polyhedron of the 600-cell.

Since there are 120 vertices, there are 120 instances of the three regular sections, each under its vertex. There are also 60 instances of the equatorial Icosidodecahedra, each lying halfway between each pair of opposed vertices.

In the model of the 600-cell in Fig. 5.10 all 600 Tetrahedral cells appear distorted; however, we can see undistorted instances of most of the sections listed in Table 2 as blue polyhedra, which means that those vertex sections are also edge sections of the 600-cell. These necessarily have the Icosahedral symmetry of the model and are thus concentric with the central vertex:
  1. As mentioned above, for regular polychora with Tetrahedral cells, their vertex polyhedra - in this case Icosahedra - are inevitably highlighted by their faces. This means that sections 1 and 7 in Table 2 are face sections of the 600-cell.
  2. The edges of the Icosahedral sections are shared by Great dodecahedral sections, which have the Pentagonal sections of the Icosahedron as faces. These are edge sections of the 600-cell.
  3. We also see that the vertices of the Dodecahedral sections of the 600-cell - numbers 2 and 6 in Table 2 - are also linked by edges of the 600-cell. This means that they are edge sections of the 600-cell.
  4. The vertices of the equatorial Icosidodecahedral sections (section 4 in table 1) are also linked by the edges of the 600-cell; its Triangular faces are also those of the 600-cell, i.e., this is partly a face section. The equatorial polygon of the 600-cell is thus the same as for the Icosidodecahedron, the Decagon.
The fact that the edges of the 600-cell highlight equatorial Icosidodecahedra and Decagons implies that, as for those two figures, R0 = φ ℓ.

Thus, the edges of the 600-cell - where each vertex is connected to the closest vertex in the arrangement (section 1) - also represent sections 1,2,4,6 and 7 in Table 2, with 0 and 8 being represented by the connectors themselves. In the model, we can see projections of all 120 instances of the three regular edge and face sections, and of all 60 equatorial Icosidodecahedra.

Challenge to the reader: After building the model of the 600-cell, locate the perfectly flat projections of the Icosahedron, Dodecahedron and Icosidodecahedron under each vertex of the outer blue Icosidodecahedron.

The Icosahedral face sections share the Triangular faces of the Tetrahedral cells, the Great dodecahedral edge sections share the Pentagonal faces of the Dodecahedral edge sections. As we'll see next, these regular edge and face sections are cells of three regular facetings of the 600-cell that have the same edges, and in some cases the same faces. They are analogous to the Great dodecahedron, which has the edges of the Icosahedron and the latter's Pentagonal edge sections as faces.

***

The dual of the 600-cell is necessarily a regular polychoron sharing its symmetry, 600 vertices with Tetrahedral vertex polyhedra, 1200 edges with Triangular figures, 720 Pentagonal faces and 120 Dodecahedral cells: the 120-cell. This is the 4-D equivalent of its cell, the Dodecahedron. Fig. 5.11 shows a model of its Icosahedral projection. Two of its Dodecahedral cells appear undistorted at the centre of projection: an Icosahedral projection of the 120-cell is necessarily a cell-first projection.


Fig. 5.11: This is the Icosahedral projection of the 120-cell. This projection is the dual of the projection of the 600-cell in Fig. 5.10.

How to build: see instructions here. The model in the figure is φ times larger than the set sold by the Zometool company as the hyperdo.


If you pay attention to the model, you will see that all of these projections of Dodecahedral cells can already be seen as the Dodecahedral edge sections in the model of the 600-cell! This situation is similar to that of the Icosahedron, which contains all faces of the Dodecahedron as Pentagonal sections.

Apart from the cells, we see no additional edge sections, as there are no Triangular faces, however, as we will see, there are plenty of vertex sections, either centered on the vertices or the cells. There are no equatorial polyhedra because the vertex polyhedron of the 120-cell (the Tetrahedron) has no central symmetry. However, for each equatorial Decagon of the 600-cell, we see in the 120-cell a dual ring of 10 Dodecahedra joined with each other by 10 Pentagonal faces. Some of these rings are the central tunnels we see in this model; these are the duals of the Decagons projected as red lines going through the centre of the model of the 600-cell in Fig. 5.10.

With the 120-cell, the list of regular convex polychora is complete. This will be shown in detail next in our enumeration of all regular polychora in the next page.

Paulo's polytope site / Next: The regular star polychora.