Extending the faces of a particular polyhedron until they intersect other
faces creates stellations. In the
case of the Rhombic triacontahedron, this process can generate many new, beautifully symmetric
polyhedra: according to George Heart's page on this topic, the
number is an incredible 358 833 072. Of these, 226 are fully supported, i.e.,
with no hollows between any point and the centre.
As you might have seen in the page on Zometool models of polyhedra,
Zometool
is a great tool for representing such stellations.
One can, of course, represent them individually, as was done in
this page in
David Richter's
list of Zometool projects. However, because Zometool depicts
only the edge arrangement, we can build a stellation around a smaller one
and still see both. This allows one to depict several stellations
with the same model; this in turn allows a direct visualization of how the
different stellations relate to each other. This process can be extended to
represent many stellations at the same time, as in the cases shown below.
Stellation diagrams for the Rhombic triacontahedron:
Another tool that helps understand the geometric relations between
the different stellations is the
stellation diagram.
Building it in Zometool is extremely instructive, because apart from
the geometric insight it provides, it also gives us
an idea of whether the model is feasible within the Zometool system, provides
a count of the number of parts necessary to build the model
and furthermore provides a simple almost step-by-step guide on how to build it.
We exemplify all of this below.
We start with the full stellation diagram of the Rhombic triacontahedron:
Fig. 1: Full stellation diagram of the Rhombic triacontahedron,
made in Zometool. Only a few ultra-short yellow struts are missing,
this issue can be solved by making the whole diagram larger by the
golden ratio. Only one quarter of the diagram was actually built, the
rest of it was built by flipping the images.
Amazingly, as Fig. 1 shows,
the Zometool system can depict the whole diagram (the many super-long
struts needed can be built from shorter ones). This does not happen with other objects,
like for instance, the stellation diagram of the Icosahedron, where many lines
cannot be represented in the Zometool system.
Expanding the 30 faces of the Rhombic triacontahedron as in Fig. 1,
we obtain a simultaneous representation of all 358 833 072 stellations!
See virtual vZome model
at the end of the page.
However, that takes a few thousand Zometool parts, and many of those would be
extra long, for that reason I have not built an actual
Zometool model of the full set of stellations.
The Zometool models below represent different sub-sets of the
full stellation diagram.
In what follows, we will focus on the inner part of the diagram that is covered by the
Great rhombic triacontahedron.
This includes the well-known stellations of the Rhombic triacontahedron mentioned
in the polyhedron page.
We later move farther out.
The Great rhombic triacontahedron, with face intersections:
In our first model, we will stellate the Rhombic triacontahedron until
we reach the Great rhombic triacontahedron. We will only represent, for now, the
mutual intersections of the finite rhombic faces.
Fig. 2a: Inner part of the stellation diagram of the Rhombic triacontahedron,
the scale has been multiplied by the golden ratio relative to Fig. 1.
The outer Rhomb depicts one of the 30 faces of the Great rhombic triacontahedron,
all other lines depict the mutual intersections of those faces.
The inner red Rhombus is one of the 30 faces of the
Rhombic triacontahedron. The inner yellow Rhombus (without yellow side extensions) is one of the 30 faces of the
Medial rhombic triacontahedron. The blue square is one of the 30 faces of the Compound of five cubes.
Fig. 2b: Same as Fig. 2a, but now with numbering of the different parts of the stellation diagram.
This will be useful for understanding the construction stages below.
The model in Fig. 2a shows a) the relative positions of those
important stellations, b)
provides a count of the parts needed (it is the number of struts in the
model times 15) and c) in Fig. 2b, we highlight how it shows us how to build the stellation
models.
Like most complex Zometool models, we start from the centre,
in this case by building first the Rhombic triacontahedron itself, this is region
1 in Fig. 2b. As we build each layer
on top of the previous, we meet successive stellations of the Rhombic triacontahedron.
Below we follow the order shown in Fig. 2b. This is not the
only way - there are many other models and stellations that can be obtained
by merely switching the order in which we build the different layers!
The larger finished model is the
Great rhombic triacontahedron itself, with all intersections of its
Rhombic faces visible.
Fig. 3: First step: the Rhombic triacontahedron itself. Each face is represented by region 1 in figure 2b.
Fig. 4: Adding red and yellow struts, we cover region 2, obtaining the first stellation of the Rhombic triacontahedron.
Fig. 5: Adding shorter red and yellow struts, we cover region 3, obtaining the second stellation of the Rhombic triacontahedron.
Fig. 6: Adding some more red and yellow struts, we cover region 4, obtaining another stellation of the Rhombic triacontahedron.
Fig. 7: Adding some short blue struts, we cover region 5. This is yet another stellation of the Rhombic triacontahedron.
Fig. 8: With pentagonal pyramids of yellow struts on top of the blue pentagons, we cover region 6.
The outer shape is now the Medial rhombic triacontahedron.
Fig. 9: Adding some red/yellow pyramids on top of the deepest holes in the previous model, we now cover region 7, obtaining
a beautiful stellation of the Rhombic triacontahedron.
Fig. 10: With some blue struts, we now cover region 8. The outer shape is now a combination of the medial rhombic triacontahedron
and the Compound of five cubes.
Fig. 11: As a side excursion, we can remove the yellow struts from region 6 and obtain the
Compound of five cubes.
The intersection of those 5 Cubes is the Rhombic triacontahedron itself.
The convex hull is
the Dodecahedron, which has the same vertex arrangement.
Fig. 12: Continuing from Fig. 10, and adding a few red struts, we now cover also regions 9 and 10. More stellations of the Rhombic triacontahedron.
Fig. 13: Adding some long red struts, we cover region 11 and obtain a very beautiful stellation of the Rhombic triacontahedron.
Fig. 14: With 20 sets of three R3 struts we cover region 12, and the model of the Great rhombic triacontahedron is completed!
The Great rhombic triacontahedron, with facial plane intersections:
The representation of the Great rhombic triacontahedron we have just shown
is not the only possibility. We can instead make a model
where, instead of the intersections of its
30 finite Rhombic faces, we represent the intersections of the 30 (infinite) facial
planes.
In this case, we get a larger portion of the stellation diagram:
Fig. 15: Inner part of the stellation diagram of the Rhombic triacontahedron,
this time including all intersections of its 30 facial planes that are found within the
volume and surface of the Great rhombic triacontahedron. The yellow
connections that were problematic in Fig. 1 (too short) can be done at this scale.
In this case, the sequence of stellations continues from the
penultimate step in the sequence above.
Fig. 13, repeat: OK, this is where we were...
Fig. 16: Another stellation of the Rhombic triacontahedron.
Fig. 17: Another stellation of the Rhombic triacontahedron.
Fig. 18: With 20 sets of three long red struts the new, more complex model of the Great rhombic triacontahedron is completed!
The isotoxal stellations of the Rhombic triacontahedron:
The models above might look too complex for some - they certainly use a lot of parts! A
simpler alternative is presented below (also in the page on Zometool models
of polyhedra), which depicts only the four isotoxal stellations, which were mentioned specifically in the
caption of Fig. 2a above. This is presented in two sizes.
Fig. 19: Excluding most struts and connectors from the stellation diagrams
in Figs. 2 or 15, we see the faces of the four isotoxal stellations of the Rhombic triacontahedron much more clearly.
Fig. 20: Extending the 30 faces of the Rhombic triacontahedron as in Fig. 19,
we obtain this model, where the four stellations are easy to spot since they are
in different colors. This model uses 120 R3 struts.
Fig. 21: The latter model can also be built in a smaller scale, which is cute and avoids the use of size 3 struts.
All stellations of the Rhombic triacontahedron:
Making a model with the full stellation diagram of the Rhombic Triacontahedron
represented in Fig. 1 is very expensive, time-consuming and
takes a lot of space. It would also require, to be built properly, many size 4 and 5 struts,
which do not exist (although they can be built by adding smaller struts).
For this reason, I spend about 30 minutes on vZome reproducing 1/4 of the stellation
diagram in Fig. 1, and then applied full Icosahedral symmetry. The result is shown
below.
Fig. 24: vZome model of the full set of stellations of the Rhombic triacontahedron.
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