Polyhedra with Zometool

"Thus, the chief reason for studying regular polyhedra is still the same as in the time of the Pythagoreans, namely, that their symmetrical shapes appeal to one's artistic sense."
H. S. M. Coxeter, in the preface to "Regular Polytopes".

In Figs. 3a, b and c, we have seen three Zometool models. Such models display only the vertex and edge arrangements of a geometrical object; unlike paper models they cannot display face arrangements. A consequence of this is that the Icosahedron and the Great dodecahedron, which share the same edge arrangements, are represented by the same Zometool model. The same would happen for the Stellated dodecahedron and the Great icosahedron. The ditrigonal polyhedra in Fig. 7 and the Compound of five cubes in Fig. 8b are all represented by the same Zometool model.

However, this can also be an advantage: because they don't display the faces, they allow the display of internal structure. For the models that follow that is absolutely necessary.

As an example, the first two Zometool models below represent the five Platonic solids. However, they are not represented in isolation (something that would merely repeat the paper models above), but instead in an arrangement that illustrates many interesting geometric relations between them, in a very elegant and economic way. Clearly, such a model cannot be made with paper. The first model in particular was the first Zometool kit I purchased; it was especially useful for me to familiarise myself with the interesting properties of that system, something that I suggest the reader does before attempting to build any of the models that follow:

Vertices are represented by connectors (balls), all are identical: they have 12 Pentagonal holes, 20 Triangular holes and 30 rectangular holes, in a pattern that is similar, but not identical, to the faces of the Rhombicosidodecahedron. They have, therefore, an overall Icosahedral symmetry. They are normally provided in white, but Zometool sells them in a variety of colours.
Edges are represented by the struts, they are always identical at both ends. Those that connect to the rectangular holes of the balls are in blue (B), they align with the 15 axes of 2-fold symmetry of the connector. Triangular holes - yellow (Y), which align with its 10 axes of 3-fold symmetry, Pentagonal holes - red (R), which align with its 6 axes of 5-fold symmetry. Green struts (G) also connect to the Pentagonal holes, but they have a twist that allows them to represent objects with Tetrahedral and Octahedral symmetry. They also come in half-sizes (HG). We will keep this B, R, Y, G terminology to represent the shapes of the struts.
All of these struts come in three regular sizes (n = 0, 1, 2), with distances between the centres of balls attached at both ends multiplied by powers of φ relative to the lower size. Red struts also have a very short (00) size. Size 3 struts were made by Zometool in the early days, but they have been discontinued, and for that reason are a bit harder to find. I buy them mostly on eBay.
The R1 and Y1 struts are the distances from the centres to the vertices of, respectively, a Icosahedron built with B1 edges and a Dodecahedron built with B0 edges. The Y1 struts also represent the distance from the centre of the vertex of a Cube built with B1 edges, and the distance from the centre of a Tetrahedron built with G1 struts. A G1 strut is the diagonal of a Square built with B1 struts.
All the struts have a twist in the middle, which mean that the connectors at both ends are aligned with each other. This means that, in any Zometool model, all connectors are aligned exactly with each other!

The Tetrahedron and Octahedron are built with G struts; all other regular polyhedra are built with B struts. These colours are the same for their rectifications: for instance, the Octahedron is built with G struts like the Tetrahedron, the Cuboctahedron is built with G struts like the Octahedron (not the Cube); the Icosidodecahedron is built with B struts like the Icosahedron or Dodecahedron (there is a deep reason for this, the Wythoff construction).

Most models in this page (but not the following pages on polychora) have G or B struts with non-native colours. These can be purchased from Zometool on demand, but they are slightly more expensive than struts in native colours. They are used to help distinguish the regular polyhedra from each other, which would otherwise only be represented in green or blue.

Kepler's Cosmos

With that said, let's have a look at the models. The model in Fig. 9a includes models of the five Platonic solids. This is a kit sold by Zometool with the name ``Kepler's Kosmos''. The arrangement was designed by John H. Conway. Fig. 9b shows a larger version I have built.

Fig. 9a: Myself holding a beautiful geometrical model sold by Zometool as Kepler's Kosmos. Photo by Aris Noutsos.

Fig. 9b: This is a larger version of the model above, which I have made using some of the specially coloured struts from the model above and some other rare Zometool parts*.

The innermost polyhedron, the Icosahedron (represented in the model in Fig. 9b by the red B1 struts) is the dual of the outermost polyhedron, the Dodecahedron (dark blue B2 struts). The second-innermost polyhedron, the Octahedron (light blue HG1 and HG2 struts) is the dual of the second outermost one, the Cube (yellow B3 struts). The middle polyhedron, the Tetrahedron (green HG3 struts) is self-dual. The model is put together in such a way that each vertex of a polyhedron lines up with the face of its dual, and vice-versa; furthermore, the edges of dual polyhedra are perpendicular to each other. An interesting implication of this is that the whole model is, in a way, also self-dual! Also,

The 12 vertices of the Icosahedron touch the 12 edges of the Octahedron.
The 6 vertices of the Octahedron touch, in their midpoints, the 6 edges of the Tetrahedron.
The 6 edges of the Tetrahedron lie in the 6 facial planes of the Cube.
The 12 edges of the Cube lie in the 12 facial planes of the Dodecahedron.

Thus, when the vertex of an inner polyhedron A touches and edge of inner polyhedron B, the edge of the dual of B lies in the face of the dual of A. The construction is possible because the number of vertices of the Octahedron (6) and Icosahedron (12) are the same as the number of edges of the Tetrahedron (6) and Octahedron (12) - by duality, the 6 and 12 edges of the Tetrahedron and Cube correspond to the 6 and 12 faces of the Cube and Dodecahedron.

Challenge to the reader: The numerical coincidence of vertices of the Octahedron and edges of the Tetrahedron is easily explained: the Octahedron is the rectification of the Tetrahedron (see previous page). However, is there any special reason why the number of vertices of the Icosahedron (12) has to be the same as the number of edges of the Octahedron? (An answer is provided in Fig. 18b).

We can also see that:

The 8 faces of the Octahedron are in the same planes as 8 of the 20 faces of the Icosahedron.
The 4 facial planes of the Tetrahedron coincide with 4 of the 8 facial planes of the Octahedron.
The 4 vertices of this Tetrahedron coincide with 4 of the 8 vertices of the Cube (i.e., the Tetrahedron is "inscribed" in the Cube).
The 8 vertices of the Cube coincide with 8 of the 20 vertices of an outer Dodecahedron (i.e., the Cube is inscribed in the Dodecahedron).

Thus, the facial plane coincidences of the inner polyhedra - all of them with Triangular faces - are the duals of the vertex coincidences of the outer polyhedra - all of them with Triangular vertex figures. These vertex coincidences and their figures explain why the Y struts, which connect to Triangular holes, represent the distance of a vertex to the centre in a Dodecahedron, Cube and Tetrahedron built with the Zometool system.

Many other things can be seen in the model. For instance, if we extend the edges of the inner Icosahedron, they meet two vertices of the outer Dodecahedron. The edge arrangement would then be that of the Great stellated dodecahedron. Also, all the edges of the inner Octahedron do this as well, but only in one direction! It is not clear to me why this happens!

The models above have no mirror symmetry; only a type of symmetry called chiral Tetrahedral symmetry. To achieve mirror symmetry, we would need to add a second Tetrahedron, the dual to the one in the model, making a Stella Octangula configuration (Fig. 8a and 10a, b below). Then, we'd have a complete stellation of the Octahedron and a complete faceting of the Cube in this model. The resulting model would then have an overall pyritohedral symmetry.

Now, if we rotate that model four times around an axis going through the centre of two opposite Pentagonal faces of the Dodecahedron, and superpose these to the original - a process we will call here "Quintuplication" - we find that the inner Icosahedron stays unchanged, the Octahedron becomes a Compound of five octahedra (illustrating how it results from a stellation of the Icosahedron), the Stella Octangula becomes a Compound of ten tetrahedra (illustrating how the latter compound results from a stellation of the Icosahedron, being self-dual it is also a faceting of the Dodecahedron), the Cube becomes a Compound of 5 Cubes (illustrating how it results from the faceting of a Dodecahedron, this is the dual of the inner Compound of five octahedra) and the outermost Dodecahedron (the dual of the innermost Icosahedron) stays the same. All of these facetings and stellations are shown in the models below.

Thus, a 5-fold rotation of an object with pyritohedral symmetry around a carefully chosen axis transforms it into an object with full Icosahedral symmetry! This is a fact we will be using often for what follows.

* Some words on this. The model in 9b represents an improvement compared to the model in Fig. 9a because:
a) each polyhedron is now represented by a different colour: in Fig. 9a, the Octahedron and the Tetrahedron are in the same colour, blue-green. That error (later fixed by Zometool) was useful: I used the 12 extremely rare light blue HG2 struts of the Tetrahedron in 9a to build the larger Octahedron in 9b.
b) The edges of this Octahedron are no longer dominated by the twists of the HG parts.
The inner Icosahedron in 9b is made with the red B1 struts of the outer Dodecahedron in 9a. The outer Dodecahedron in 9b is made using commonplace B2 struts. For the Cube in 9b, I've had for a long time the rare yellow B3 equivalents of the yellow B2 struts in 9a. What finally made the model in 9b possible was the surprise inclusion of a set of extremely rare green HG3 struts in a large lot of Zometool parts I have purchased in 2019, these are used to make the Tetrahedron.
I like the colour scheme of the model in Fig. 9b very much. The colours of the dual pairs are opposed: If we associate these colours with a sequence in the Rainbow (red, yellow, green, blue-green, dark blue/violet), then the inner/outer solids have the extremes of the sequence, the middle solid has the middle colour (green) and the others have the remaining, not-so-extreme colours. Relative to Fig. 9a, the colours of the inner/outer pair were exchanged, something that increases the contrast between polyhedra and makes it easier to distinguish them.

Stellations and facetings

In what follows, we depict how many of the polyhedra depicted by the paper models above - especially the star polyhedra - relate to the convex polyhedra by stellation and faceting. For this, we will use stellation diagrams; also represented in Zometool. In the stellations, the real vertices are indicated by the white balls; false vertices (where edges intersect, but don't end) appear in other colours. For the facetings, the only real vertices (of the outer polyhedron, which is the convex hull of all objects further in) will be depicted in black, false vertices appear in other colours. The facial plane coincidences in the models showing stellations correspond to the vertex coincidences in the models showing facetings. Most models with icosahedral symmetry use size 3 struts.

Fig. 10a: This is the stellation diagram of the Octahedron. The inner Triangle is the face of the Octahedron. The outer dual Triangle, with an edge that is twice as long as that of the inner Triangle, is the face of its only stellation, the Compound of two tetrahedra (Fig. 8a). This type of stellation is called a "greatening".

Fig. 10b: Expanding all faces of the Octahedron as in the previous Figure, we obtain on the left a model of the Octahedron (light blue) and the Compound of two tetrahedra (green). On the right is the dual of the previous model, showing the Cube (in yellow) and further in, with the same vertex arrangement, the Compound of two tetrahedra (green); this also appears here because it is self-dual.

The models in Fig. 10b follow the colour scheme of Fig. 9b, and represent the three medial polyhedra in that model. We cannot invert the relations because there are no stellations of the Cube, therefore no facetings of the Octahedron.

The more complex models below continue the same theme of stellations and facetings. One common characteristic is that the inner and outer polyhedra are isomorphic to each other.

We start with the stellations of the Dodecahedron and the facetings of the Icosahedron. Here we can represent all stellations and facetings in relatively simple models!

Fig. 11a: This is the stellation diagram of the Dodecahedron; its face is the inner Pentagon. The inner Pentagram, with edge length 2φ +1 = φ³ times that of the inner Pentagon, is the face of the Stellated dodecahedron. The outer Pentagon - a φ + 1 = φ² larger dual of the inner Pentagon - is the face of the Great dodecahedron, and the outer Pentagram - a φ² larger dual of the inner Pentagram - is the face of the Great stellated dodecahedron.

Fig. 11b: Extending the 12 faces of the Dodecahedron as in Fig. 11a, we obtain the four stellations of the Dodecahedron (see Fig. 5a, blue rectangle in Diagram Ib).

Fig. 11c: In this model, the dual of the model in Fig. 11b, we represent the four facetings of the Icosahedron (see Fig. 5a and green rectangle in Diagram Ib), which is polyhedron in yellow B2 struts. Also represented, with the same edge arrangement but pentagonal faces, is the Great dodecahedron. Further in, in blue, is the Stellated dodecahedron. Finally, the innermost polyhedron, with the same edge and vertex arrangement of the Stellated dodecahedron but with triangular faces is the Great icosahedron. The model has the same edges as the model in Fig. 11b, except for the absence of the outer layer.

We now continue with the stellations of the Icosahedron and the facetings of the Dodecahedron. We represent only a few of the 22 cases in each class, mostly objects that we have represented as paper models.

Fig. 12a: An incomplete stellation diagram of the Icosahedron, representing four of the figures that can be built with a small Zometool model. The inner blue Triangle is the face of the Icosahedron, the green Triangle is a face of the Compound of five tetrahedra (see Fig. 8a) and the outer φ⁴ larger dual Triangle is a face of the Great icosahedron (see Fig. 5a). An intermediate figure where each face is composed of three disjointed (blue) triangles, touching the face of the Icosahedron at its vertex, is the aforementioned Excavated dodecahedron. A few other stellations were presented already in Fig 7, but those cannot be represented with Zometool.

Fig. 12b: Extending the 20 faces of the Icosahedron as shown above, we have the four stellations of the Icosahedron mentioned above.

Fig. 12c: This model represents facetings of the Dodecahedron (the outer polyhedron represented by the white B3 struts) that were represented by paper models.

Fig. 12d: At the centre of the previous model, in yellow B1 and B2 struts, is the Great stellated dodecahedron (Fig. 5a).

Fig. 12e: The green struts represent the Compound of five tetrahedra (Fig. 8a). The Compound of ten tetrahedra (Fig. 8a) is also a faceting of the Dodecahedron, but it cannot be represented in Zometool, as it would require two green struts to connect to the same holes.

Fig. 12f: The blue B1/B2 struts represent the ditrigonal polyhedra (Fig. 7) and the Compound of five cubes (Fig. 8b), all of which share the same edge arrangement.

Below, we show a model of the four isotoxal stellations of the rhombic triacontahedron. For more on them and instructions on how to build them, and many more stellations of the rhombic triacontahedron, look here. We also show the dual model, with the four isotoxal facetings of the Icosidodecahedron.

Fig. 13a: Within the inner region of the stellation diagram of the Rhombic triacontahedron (Fig. 4), we can find its four isotoxal stellations, for which we have built paper models above. The inner Golden rhombus (in red) is one of the 30 faces of the Rhombic triacontahedron itself. The yellow Rhombus is one of the 30 faces of the Medial rhombic triacontahedron (see Fig. 6a). The blue Square is one of the 30 faces of the Compound of 5 cubes (Fig. 8b). The outer Golden rhombus, which is 2φ + 1 = φ³ times larger than the inner red Rhombus, is one of the 30 faces of the Great rhombic triacontahedron (Fig. 6a).

Fig. 13b: Extending the 30 faces of the Rhombic triacontahedron as in Fig. 13a, we obtain this model. The four stellations can be found very easily since they are represented by different colours.

Fig. 13c: This model, the dual of the model in Fig. 13b, shows the four isotoxal facetings of the Icosidodecahedron, which is the outer shape in yellow B3 struts.

Fig. 13d: At the centre of the model is the Great icosidodecahedron, in white B1 and B2 struts.

Fig. 13e: Just outside the Great icosidodecahedron, in green, is the Compound of five octahedra. Note the 3-intersections of green struts in the 3-fold symmetry axes, there they have to bend a bit.

Fig. 13f: Just above the Compound of five octahedra is the Dodecadodecahedron in blue B1 and B2 struts. Its Pentagrammic faces are inscribed in the Pentagonal faces of the Icosidodecahedron, its Pentagonal faces circumscribe the Pentagrammic faces of the Great icosidodecahedron.

One could also make models with the stellations of the Icosahedron and the facetings of the Rhombic triacontahedron, but none of those stellations is isotoxal.

One last word about the models in this page. They are the closest in the whole site to artistic objects: one can choose which combinations of polyhedra to show, and their sizes and orientations relative to each other, furthermore we have some freedom in the choice of colours. The models in Figs. 9a and b, by John Conway, are in my opinion elegant works of art. Regarding the other models, although they merely display stellations and facetings, they can also be very pleasing. I am particularly happy with those in Figs. 12b and 13b.

Next: Polychora