Simple City

[This is a sequel to my previous post Magforming the Johnson Solids. Please see that for a disclaimer and (if you want to understand the words in this post) the geometrical background. If you only want to see some pretty pictures, then ignore all this stuff and just scroll down…] 

Last time, we investigated the shapes that can be built out of regular polygons, focusing on convex shapes – roughly speaking those with no holes, dents, or spikes. There are exactly 98 theoretically magformable convex polyhedra: all 5 of the Platonic solids, 11 Archimedean solids (out of 13 in total), 4 prisms (of an infinite family), 4 antiprisms (ditto), and 74 Johnson solids (out of 92).

Once you drop the requirement for convexity, the only limits are your imagination and the size of your magformer collection. In principle there are inifinitely many magformable polyhedra, because you can always stick more bits on. See robopenguin here, for example.

So what to do? Let’s return to the starting point of every discussion of polyhedra: the Platonic Solids.

 

 

 

 

Here they are again:

These are the only 5 polyhedra satisfying all of the following wonderful properties:

1. Every face is a regular polygon
2. Every corner (or vertex) is identical to every other (vertex-transitivity)
3. Every face is identical to every other (face-transitivity)
4. Convexity (roughly speaking, no holes, dents, or spikes)

The main goal of this post is to explore the geometrical landscape that opens up when we drop requirement 4. But first…

Magforming the 4th Dimension

We’ve focused on 3-dimensional polyhedra. Last time we also looked at 2-dimensional structures – what about 4-dimensional structures? Just as in 3-dimensions there are 5 regular convex polyhedra (the Platonic solids), so in 4 dimensions, it turns out there are 6 regular convex polychora. You might object that we don’t have much chance of actually building them… but, just as a 3d shape such as a tetrahedron can be unfolded into 2-dimensions…

…so a 4-dimensional shape can be unfolded into 3-dimensions, where their nets form interesting non-convex polyhedra:

I don’t have enough pieces to create nets of the remaining Platonic polychora – in fact you might notice some squares missing in the hypercube. The hyperoctahedron is within range if I had a few more triangles but the remaining three – yes six in total! – Platonic polychora have impractically many faces. But do check out Hyrodium’s amazing models of them.

However, we needn’t give up on 4D just yet. There are always the Archimedean polychora, such as:

And exciting new constructions such as the duoprisms:

 

Stewart’s G3

There are several other avenues to explore up in 4D, but for now let’s come back down to Earth. What other non-convex polyhedra might we try? The mathematician Bonnie Stewart^[2] discovered several interesting ones, such as this compact little beauty known only as G3:

 

We’ll come back to G3 later. Stewart also found a few others, including this one, X:

 

You might wonder how Stewart found these – we will come to the answer in a little while.

Non-Convex Uniformity

Let’s be more systematic. Conditions 1, 2, & 3 above together give the definition of a regular polyhedron, the most symmetric type of all. If the Platonic solids are the only convex regular polyhedra, an obvious question arises: are there any non-convex regular polyhedra? The answer is yes! There are the four Kepler-Poinsot polyhedra. Great! Let’s grab the magformers and set to work….

But no. The Kepler-Poinsot polyhedra are non-magformable. Here’s why: suppose a solid is non-convex because it has a spike. At the end of that spike will be a vertex. And if one vertex is at the end of a spike, then by vertex-transitivity every corner must be at the end of a spike. And that is impossible, unless we relax one other hitherto unspoken condition, and allow the shape’s faces to cut through each other along “false edges”. The great dodecahedron, for instance, is composed of 12 regular pentagons, just like the ordinary dodecahedron, but this time they are configured so that they intersect:

(Image credits: [1])

Meanwhile the small stellated dodecahedron is made from 12 regular pentagrams – because if we’re going to allow faces to cut through each other, then why not also permit polygons whose edges cut through each other?

That’s all very well, but there’s no hope of making these without taking a hacksaw to my precious magformers. Still, there is one nice shape we can make along these lines. The polyhedra just discussed share one other property that we have taken for granted: they come in one piece. That is, if you start at any corner, travelling only along the shape’s edges, you can reach any other corner. That sounds obvious, but once we allow faces to slice through each other, it is no longer automatic. There are five regular polyhedral compounds which satisfy 1-3 above, have faces which cut through each other, and don’t come in one piece. One of them is the compound of two tetrahedra, also known as…

You can think of this shape in two different ways: as a stellated octahedron (which matches how one builds it – first make an octahedron and then construct a tetrahedron on every face); or as a polyhedral compound: two big tetrahedra (the yellowy one and the blueish one) pushed through each other until their centres coincide.

So much for non-convex regular polyhedra and compounds. So much, in fact, for this entire line of thought. There is a beautiful theory of non-convex uniform polyhedra & polyhedral compounds, those satisfying conditions 1 & 2 above. Besides everything we have met so far, there are the non-convex (e.g. pentagrammic) prisms and antiprisms, and analogues of the Archimedean solids: 53 uniform star polyhedra such as the dodecadodecahedron build from 12 intersecting pentagons and 12 non-intersecting pentagrams:

There are also 70 uniform polyhedral compounds, on top of the 5 regular compounds already discussed. But, alas, none of these are of any use to the aspiring magformist.

Stewart’s idea: quasiconvexity

Actually, there is door hiding here, spotted by Bonnie Stewart^[2], that opens to reveal an Aladdin’s cave of beautiful, magformable, non-convex polyhedra. It comes from deconstructing the meaning of convex. Remember that a solid is convex, roughly, if it doesn’t have any holes, dents or spikes. Stewart coined the term quasiconvex for polyhedra which (again roughly) have no spikes, but may have holes or dents.

More precisely: given any polyhedron, one can form its convex hull by filling in all its holes and the gaps between its spikes. Technically, whenever you draw an imaginary line connecting two points within the shape, then all the points on that line are incorporated into the convex hull. As expected, the convex hull is always convex. Stewart said that initial shape is quasiconvex if this procedure doesn’t create any new edges, that is to say if every edge of the convex hull is already present in the original. Looking at the stella octangula above, for example, its convex hull will have a new edge joining the tips of the two nearest spikes. So that shape is not quasiconvex. (In fact, it is not too hard to see that the convex hull of the stella octangula is a cube.)

A method for discovering quasiconvex shapes is to begin with a convex polyhedron, such as an Archimedean solid, and then start gouging out, or excavating its faces. Excavation is the opposite of augmentation which we me last time. To augment a truncated cube, for instance, we replace one of its octagonal faces with an outward-facing square cupola. To excavate it, one replaces an octagonal faces with an inward-facing square cupola. Doing this with an opposite pair of faces, and then removing a final cube to connect the two cupolas, brings us our first Stewart toroid:

The word toroid here indicates that these shapes are like a torus: they have a hole (or more than one) in the same sense that a donut has. Another can be formed by removing two triangular cupolas and an octahedron from a truncated octahedron:

You can perform augmentation and excavation on the same face. Starting again with a truncated cube, and augmenting two opposite faces with square cupolas produces a biaugmented cube (one of the Johnson solids). Removing two cuboctahedra and a central cube from this shape gets us:

 

A couple of further examples. Extracting a triangular cupola and trinagular prism from an augmented triangular cupola produces:

 

Here’s a good one. Taking a pentagonal cupola and pentagonal antisprism out of a rotunda produces:

 

A different approach, also investigated by Bonnie Stewart, instead of creating toroids by gouging out faces from uniform solids is to glue them together:

Toroids of this form are not quasiconvex. Still, some amazing things are possible – see for example RobertLovesPi’s arrangement of 92 dodecahedra into a single rhomic triacontahedron.

Back to quasiconvex solids, here’s one of an interesting and rare type, again discovered by Bonnie Stewart:

On first glance it looks unremarkable: a Johnson solid with four hexagonal faces excavated. But which Johnson solid? On closer inspection, there is something unexpected: the excavated faces are irregular hexagons. This establishes, surprisingly, that a quasiconvex polyhedron built entirely from regular polygons may have a convex hull with irregular faces. K4′ (pronounced “K4 prime”) is built by starting with the truncated cuboctahedron (also known as K4) and excavating four of its octagonal faces with square cupolas. That gives polyhedron with a a sort of belt of squares around it. Deleting that belt gives K4′, with its slightly squashed appearance.

 

Not many polyhedra like K4′ are known. But the good news is that K4′ can itself be drilled out with two more square cupolas and a cuboctahedron:

 

So how many Stewart toroids are there? If we allow shapes like the ring of hexagonal prisms above, then there are infinitely many. What if we limit ourselves to quasiconvex polyhedra whose convex hull (unlike K4′) above is an Archimedean solid? So far as I am aware, the exact number isn’t known, but is at least guaranteed to be finite. Stewart estimated, in February 1980, that there are 6031 ways of drilling through the truncated cuboctahedron alone, and several quadrillion ways of tunneling through the (decagon-requiring and therefore non-magformable) truncated icosidodecahedron. This is how Stewart discovered the unusual non-convex solids G3 and X above: by carving smaller polyhedra out of a bigger one, those were the parts which were left behind.

Let’s end with a couple I am rather pleased with. Last time, I said that my favourite Johnson solid was the hebesphenerotunda. The back of this is built from pentagons and triangles, like a patch of icosidodecahedron. So you can use a hebesphenerotunda to excavate an icosidodecahedron, and if you do that twice on opposite sides, happily their hexagonal faces match up exactly:

 

(Strictly speaking the hexagonal face should be removed from the right hand picture to give a proper “tunnel”, but aesthetically and structurally it’s nicer to to leave it in. This is of course the elephant in the room in building with magformers – faces are only there in skeletal form anyway.)

Likewise, Stewart’s G3 (pictured above) has on one side three pentagons, like a patch of dodecahedron. So you can use G3 to excavate a dodecahedron:

 

You can then do the same again…

 

What this gives is a decomposition of the dodecahedron into two G3s, four pentagonal pyramids, and one of… whatever this is:

 

Having got this far myself, I had wondered whether anyone else had followed the same line of thought, an perhaps even given this shape a name. The answer is that Alex Doskey and a few others have, but the only written description of this shape I can find is Alex’s “another figure with fewer faces”.

Anyway, putting all those back together gives a very pleasing decomposition of the dodecahedron, which is where we leave this post.

 

References

[1] All the pictures in this post are either my own photos of magformers, or are created using Robert Webb’s Stella 4d, the best software I know for easily manipulating a wide variety of polyhedra.

[2] The ideas in this post are based on Bonnie Stewart’s wonderful, but sadly largely unavailable, self-published, hand-lettered, and hand-illustrated book Adventures Among The Toroids (1980, 2nd edition).