—And He Built a Crooked House

One of the great science fiction short classics is Heinlein’s “—And He Built a Crooked House,” in which a character builds a house in the shape of an unfolded four-dimensional hypercube or tesseract (I’ve always wanted one like it):


Just as a cube folds out into six squares, a tesseract folds out into eight cubes. Now this makes for an interesting house design, but even more interesting is that, during an earthquake, the house “folds up” in the fourth dimension. Which is hard to picture. This is one view, which gets the connections between the eight cubes right, but ignores the fact that they all have the same size and shape:


Here is another view (on second thoughts, maybe living in a house like that would make me sea-sick):


Apparently you can actually build such a house in Second Life.

Tilings in the real world

Following on from my last post about tiling, here are some nice real-world examples of the 11 vertex-transitive tilings:


Hexagons (Rome): photo by David Shay

Triangles (as a quilt): photo by Oops-Lah

Octagons and squares

Hexagons and triangles (Seville): photo by Anneke Bart

Ditto (as a quilt): photo by M. L. Henneman

Essentially the same thing (merging the triangles with alternate hexagons to form stars): photo by Yvette

Hexagons, squares, triangles (Seville): photo by Anneke Bart
Please comment if you’ve seen others. We’ve all seen square tilings, but there are a few types of vertex-transitive tiling that I’ve not seen in real life (although if you zoom in on Dr. Henneman’s wonderful quilt, you will see that all the other tilings are actually present).

Eleven vertex-transitive tilings

Somebody asked me about my last post, which mentioned the eleven regular and Archimedean tilings:


One way of looking at these is as infinitely large (and hence totally flat) versions of the five Platonic solids and thirteen Archimedean solids.

We can also think of this as a game with bathroom tiles. However, there are two rules. First the tiles must be laid edge-to-edge, so that corners (vertices) meet up with corners. The following tiling is ruled out, because some corners meet halfway along the edges of the larger (yellow) tiles:


The second condition (“vertex transitivity”) is that all corners must be equivalent. This tiling is also ruled out, because corners where three tiles meet differ from corners where four tiles meet:


More specifically, it should be possible to map any corner in the tiling onto any other corner by a rotation, reflection, translation, or glide reflection, leaving the tiling looking identical. For this example pair of corners, either a rotation or a glide reflection does the job:


In particular, equivalence of corners on the same tile means that all the tiles must be regular convex polygons.

Exploring all possibilities for tiles meeting at a corner (we need only consider one corner, since all corners are equivalent) shows that only eleven tilings satisfy the conditions. The first three tilings use only a single kind of tile; the other eight mix different tiles. One of these eight also comes in a mirror-image form.

M. C. Escher produced some interesting non-vertex-transitive tilings, as well as some beautiful variations on the basic square and hexagonal themes:

Counting

I love to count: There is one God.

(and one side on a Möbius strip)


There are two sides to a coin.


There are three dimensions in ordinary space.


There are four colours needed to colour a map.


There are five Platonic solids.


There are six sides on a cube.


(there are also six ways to arrange three items)


There are seven days in a week.


There are eight notes in an octave.


There are nine numbers in the smallest magic square.


There are ten digits.


There are eleven regular and Archimedean tilings.


There are twelve different pentominos.


There are thirteen Archimedean solids.

How mathematics is done: a conversation

Alice: Hi! Let’s think about symmetrical polygons.

Bob: You mean polygons with identical sides? Like this?


Alice: Hmmm. Not quite what I meant. I think “symmetrical” probably should mean that the corners are all the same too, so that middle one should be a square. In fact, I think the regular convex polygons are what I had in mind.

Bob: What about in three dimensions? I guess the sides should be identical regular polygons?


Alice: Yes, but that’s got the same problem as before – the corners aren’t all the same. The cube and dodecahedron are OK, but the shape on the left is two pentagonal pyramids stuck together.

Bob: Oh, I see what you mean. Two of the corners have five triangles meeting, and the others have only four. The tetrahedron, octahedron, and icosahedron should be OK, though:


Alice: Is that all?

Plato: Without a doubt. There are only five. What’s more, they correspond to the five elements:


Alice: I think there are a few more elements than that, but that isn’t important right now. How can we be certain there are only five?

Bob: Well, look at the shapes that have to meet at a corner. If four squares met, we would have a flat surface, and ditto for six triangles or three hexagons. That means the only possible corners are three triangles meeting, four triangles, five triangles, three squares, or three pentagons. Which are the five cases we’ve got.

Alice: I think you’re right. But we can actually get identical corners with more more than one kind of regular polygon. Look at the prisms – rotate any one whichever way you like, and you’ll see the corners are all identical:


Bob: That middle one is actually just a cube.

Alice: So it is.

Kepler: There’s also something called an antiprism, which has triangles along the side, instead of squares:


Alice: That first one is just an octahedron. And I’ve just realised that the soccer-ball shape works too – the sides are regular pentagons or hexagons, and all the corners are identical:


Bob: is that all?

Archimedes: Actually, I made a list of thirteen that included that last one, but I seem to have mislaid the paper...

Kepler: And with those, that really is all.

Rudy: Unless you’d like to try for four dimensions?