Tag: Polyhedra

In a previous post I showed some examples of irregular polyhedra that I’ve been making out of paper. These polyhedra were all based on points distributed quasi-randomly over a sphere. At each point, my program placed a plane tangent to the sphere. The result of the intersection of all of those planes was an irregular polyhedron.

The 47-hedron in the picture above was made in a rather different way. It’s certainly irregular, but the process that created it was not at all random. As in the previous polyhedra, the starting point was a sphere with a number of points distributed over it. But this time, the points were placed according to a simple and perfectly regular rule.

Imagine you you draw a line from one pole of a sphere to the opposite pole, winding sort-of-helically around the sphere. Then place a number of points at exactly equal spacings along this line. In the examples below, there are 43 points.

If your quasi-helix had 2.5 turns, the result would look like the diagram on the right. The numbers are where the points are placed on the imaginary sphere; the resulting polyhedron is also drawn. The helical structure is very clear, but notice the irregularity of the faces: for example, the lower edges of faces 26, 27, and 28 are all different. This is because the radius of the quasi-helix is not constant, so turns of the helix near the poles will have fewer faces on them than turns near the equator, which means that the way the numbers on one turn line up with the numbers on the previous and subsequent turns keeps changing.

If we increase the number of turns in the quasi-helix to 5, the helical structure is still clear:With 7.5 turns in the quasi-helix (below), adjacent-numbered faces are now so far apart that the faces on the turns above and below them are starting to intrude into the spaces between them. See, for example, how faces 27 and 28 are almost pushed apart by faces 19 and 35.By the time we have 10 turns (below), consecutively-numbered faces are so far apart that faces from the turns above and below often meet between them, making the helical structure hard to discern (see, for example, how faces 34 and 22 squeeze in between faces 28 and 29). The polyhedron is beginning to look quite irregular, but note that consecutively-numbered faces are still more similar to each other than they are to the other faces.

The 47-hedron on the right (same as at the top of the post) was based on a quasi-helix with about 12.5 turns. At first sight, it looks quite random, and it’s certainly irregular, but there are still visible similarities between faces if you look hard enough. Compare, for example, the mid-blue and light-blue faces at roughly 10 o’clock and 2 o’clock respectively. And what about the dark-blue and very dark-blue faces just below the centre? One looks suspiciously like a rotated version of the other – and it is! In fact the whole polyhedron has an axis of 180° rotational symmetry. This suprised me when I spotted it, but it shouldn’t have. A helix has 180° rotational symmetry (turn a corkscrew upside down and it looks the same) so it’s inevitable that shapes based on a helix will have that symmetry also.

 

 

 

 

 

It’s a remarkable fact that there are only five regular convex polyhedra, that is, solid shapes whose flat faces are all identical regular polygons. The most familiar example is the cube, with 6 identical square faces. There are other, less regular but still orderly polyhedra, such as the truncated icosahedron, seen in a bloated form in some footballs.

But what, I wondered, about truly irregular polyhedra, where every face is a different irregular polygon? What would they look like? Last January I set out to make some and find out.

My irregular polyhedra are all based on spheres. Imagine placing a number of dots on a sphere. At each dot, let there be a plane just touching the sphere. These planes will all intersect each other, and if we remove the parts of each plane cut off by the neighbouring planes, we’re left with a polyhedron, with one face for each dot that we placed on the sphere. The nature of this polyhedron depends upon how the points are distributed over the sphere. In the polyhedra shown here, the placement of dots was random but subject to certain constraints.

I wrote some software that allowed me to choose how many faces I wanted, and to regulate how evenly spread over the sphere the dots were. I could preview the resulting polyhedron, and when I saw one that I liked, the program produced a set of images of the faces of the polyhedron, with numbered tabs on them. Then all I had to do was print them, cut them out, fold the tabs, and glue them all together. This is not a task for the impatient: the 83-hedron at the top took about 3 days.

The polyhedra shown here differ principally in how evenly the dots were spread over the imaginary internal sphere. After randomly placing the dots, the program simulated repulsion between them (as if they were electric charges). The longer this repulsion process went on, the more evenly distributed the dots became. For the 29-hedron, no repulsion process was done, and for the 43-hedron, the process was allowed to continue until the dots stopped moving; presumably this is now (in some undefined sense) as regular as a 43-hedron can get.

For the 29-hedron (right), I took advantage of the four-colour map theorem, which tells me that if I want to colour the faces such that no two neighbouring faces have the same shade, 4 shades of card are all I need. I leave it to you to convince yourself that if this theorem is true for flat maps, it must be true for maps on balls too. (If you don’t see where maps come into it, think of each face of the polyhedron as a country on a political map.)

None of this would have happened had it not been for a visit to Jenny Dockett to talk about her Illuminating Geometry project. It was while talking to Jenny that the idea for making irregular polyhedra came to me.

In 2015 I exhibited these polyhedra, and others, at Mini Maker Faire and Porty Art Walk in Edinburgh.

I wrote the software in Python. The internally-illuminated shapes are made out of layout paper (very thin paper), and the 29-hedron is made of thin card. I used UHU Office Pen glue (not to be confused with UHU Pen glue). This glue doesn’t wrinkle the thin paper, and dries at about the right speed, but has proved to be somewhat unreliable in humid weather.