Archimedean solids

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Archimedean solids

Postby Joseph Malkevitch » Mon Nov 27, 2017 7:11 pm

Dear Colleagues,

I am doing some writing that made think about the so called Archimedean polyhedra. How many Archimedean Solids (polyhedra) are there? Though Archimedean solids/polyhedra are often talked about when discussing convex polyhedra, we have no manuscript due to Archimedes that discusses them. Our knowledge of them comes from Pappus, who lived many years after Archimedes, who discusses them, in Greek, not surprisingly. Pappus attributes the list of polyhedra and descriptions he provides to Archimedes. Here is Pappus's account and a translation appears there of what is said in Greek:

https://www.math.nyu.edu/~crorres/Archi ... appus.html

Note that the solid often referred to as the Buckyball, or truncated iscosahedron appears in the list.

Why is there a problem? From a modern mathematical perspective it seems strange to give a name, without a "mathematical definition" to a collection of things, because someone in ancient times credited them to Archimedes. The issue comes down to what Archimedes thought he was counting - probably convex 3-dimensional solids that had the same pattern of regular polygons around each vertex. With this definition, Archimedes and Pappus missed one. (There are also the prisms and antiprisms that are not mentioned.) Using a more modern approach, involving groups theory, probably not what Archimedes had in mind, there are indeed only 13 such solids. (The idea being that the isometry group of the solid acts transitively on the vertices, that is, can move any vertex to any other vertex with a symmetry of the solid.) The 14th solid, seemingly noticed first by Kepler, does not have a symmetry group that is transitive on the vertices.

For a more detailed discussion look at Branko Grunbaum's paper:

https://digital.lib.washington.edu/rese ... sequence=1

Definitions do matter, and when counting things it is often possible to make a mistake!

Regards,

Joe
Joseph Malkevitch
Department of Mathematics
York College (CUNY)
Jamaica, New York 11451

email:

malkevitch@york.cuny.edu

web page:

http://york.cuny.edu/~malk

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