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John Derbyshire

Math Corner     Comments on last month’s Math Corner are here.

For this month, since Egypt is in the news, consider the humble pyramid: a square base with four sides all equilateral triangles.

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That makes the pyramid a polyhedron — a solid figure with plane faces. Each face is a polygon — a plane figure with straight-line sides. For the Egyptian-style pyramid the polygons are four equilateral triangles and a square.

(The sides of the Egyptian pyramids are not in fact equilateral triangles: the base angles are a tad more than 58 degrees, not 60 degrees. I’m going to ignore this inconvenient fact. This is math. The heck with reality.)

I’m only going to talk about solids whose faces are regular polygons: equilateral triangles, perfect squares, perfect pentagons, perfect hexagons, and so on. Furthermore, all the faces are convex: no star-shapes.

The best-known polyhedra are the five Platonic solids: tetrahedron, cube, octahedron, dodecahedron, icosahedron. These obey the very strict conditions that not only are all faces regular convex polygons, they are all the same polygon, with the same number meeting at every vertex.

By loosening those conditions a little, you get interesting new families of polyhedra. If you drop that “same number meeting at every vertex” condition, for example, you get the deltahedra. If you drop the “convex” condition, you get various kinds of stellated polyhedra.

The first condition dropped by the ancients was in fact the one insisting that all the faces have to be the same regular polygon. What if you allow a mix — some triangles, some squares, for instance — while holding on to the condition that all the vertices must be the same (same numbers of each polygon meeting at each vertex, in the same order)?

Well, you then get the so-called Archimedean polyhedra, the simplest of which is the truncated tetrahedron, which has as its faces four equilateral triangles and four regular hexagons.

Here’s a formal definition:

Archimedean polyhedron:  All faces are convex regular polygons, though not all the same polygon; all vertices are congruent; prisms and antiprisms don’t count (because they’re boring).

If you now additionally drop the condition that all the vertices must be the same, you get the Johnson polyhedra. That gets us back to the Egyptian-style square-based pyramid, which is the simplest Johnson polyhedron. The four base vertices are the same (square meets two triangles) but the summit vertex is different (four triangles).

There are 92 Johnson polyhedra, a fact not proved until 1969 — incredibly, it seems to me, when you consider that polyhedra have interested mathematicians since antiquity.

Here’s something even more incredible. The number of Archimedean polyhedra as defined above was given as 13 by Archimedes himself (according to Pappus), and this number was repeated down through the succeeding 22 centuries into my own schooldays. Cundy and Rollett’s classic Mathematical Models, for example (my edition 1961), repeats it on page 100. Yet it is wrong! And in all those centuries nobody (with the possible exception of Kepler) noticed!

The whole amazing story is told by Branko Grünbaum in an essay titled “An Enduring Error,” which you can find in this splendid anthology.

Okay, okay, a brainteaser. The true number of Archimedean polyhedra, on the definition above, is 14, not 13. The 14th one, though none of the other 13, is included among Johnson’s 92. Can you spot it in the Mathworld list? What is its Johnson number?

Oh, and those boys in the back row sniggering over “Johnson number” can report to the principal directly after class. You know who you are.



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