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. C.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. C.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. C.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. C.2a: Inner part of the stellation diagram of the Rhombic triacontahedron, the scale
has been multiplied by the golden ratio relative to Fig. C.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. C.2b: Same as Fig. C.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. C.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. C.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. C.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. C.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. C.3: First step: the Rhombic triacontahedron itself. Each face is represented by region 1 in figure C.2b.
Fig. C.4: Adding red and yellow struts, we cover region 2, obtaining the first stellation of the Rhombic triacontahedron.
Fig. C.5: Adding shorter red and yellow struts, we cover region 3, obtaining the second stellation of the Rhombic triacontahedron.
Fig. C.6: Adding some more red and yellow struts, we cover region 4, obtaining another stellation of the Rhombic triacontahedron.
Fig. C.7: Adding some short blue struts, we cover region 5. This is yet another stellation of the Rhombic triacontahedron.
Fig. C.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. C.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. C.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. C.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. C.12: Continuing from Fig. C.10, and adding a few red struts, we now cover also regions 9 and 10. More stellations of the Rhombic triacontahedron.
Fig. C.13: Adding some long red struts, we cover region 11 and obtain a very beautiful stellation of the Rhombic triacontahedron.
Fig. C.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. C.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. C.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. C.13, repeat: OK, this is where we were...
Fig. C.16: Another stellation of the Rhombic triacontahedron.
Fig. C.17: Another stellation of the Rhombic triacontahedron.
Fig. C.18: With 20 sets of three long red struts the new, more complex model of the Great
rhombic triacontahedron is completed!
The models above might look too complex for some - they certainly use a lot of parts! A
simpler alternative is presented in Figs. 4.8a and b, which depict only the four isotoxal
stellations that were mentioned specifically in the caption of Fig. C.2a above. Below
we present a smaller version of this model.
Fig. C.19: The model in Fig. 4.8b can 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. C.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. C.1, and then applied full Icosahedral symmetry. The result is shown
below.
Fig. C.20: vZome model of the full set of stellations of the Rhombic triacontahedron.
Paulo's polytope site.
