Prefixes for Naming Polyhedra

Date: 12/07/2000 at 20:53:46
From: David
Subject: Polygons

I am having trouble understanding this paragraph that Prof. Conway 
wrote. Could you please explain it to me?

Thank you,
David

Prof. Conway writes:

  Antreas Hatzipolakis and I worked out a complete system up to the 
  millions from which this is taken, and which has also been "vetted" 
  by several other scholars. The most important of the reasons which 
  make me prefer the "kai" forms is that they permit these prefixes to 
  be unambiguously parsed even when concatenated, as they are in 
  Kepler's names for certain polyhedra; for example, the 
  icosidodecahedron or (20,12)-hedron, so called because it has 20 
  faces of one type and 12 of another. Kepler said "this particular 
  triacontakaidihedron I call the icosidodecahedron", a remark showing 
  that he also preferred the kai forms.

   John Conway

See Naming Polygons and Polyhedra from the Dr. Math FAQ:

  http://mathforum.org/dr.math/faq/faq.polygon.names.html   

Date: 12/07/2000 at 23:16:47
From: Doctor Peterson
Subject: Re: Polygons

Hi, David.

Prof. Conway, like me, is interested not only in math but also in 
etymology, the study of the origin of words, and is concerned that we 
form our words consistently, rather than, for instance, accidentally 
combining Latin and Greek roots in the same word. 

In this case, he is telling us that he tried to make a consistent 
system of numeric prefixes with the help of a Greek math historian. He 
had to make some choices that have not been agreed upon in the past, 
when it came to putting together prefixes for bigger numbers like 23, 
where there is no standard prefix. He chose to use the Greek word 
"kai," meaning "and," rather than, for example, just sticking the two 
prefixes together (concatenating them). In his example, he points out 
that his system agrees with the astronomer Kepler, who used these two 
names for the same thing:

     icosi dodeca hedron
       20    12    faces = 20 faces of one kind, 12 of another

     triaconta kai di hedron
         30    and 2   faces = 32 faces in all

By using the "kai" to combine digits of a number (such as 32), his 
system distinguishes between these two forms; otherwise you might use 
"triacontadihedron," which could be taken as "30 of one kind and 2 of 
another." You use "kai" when you mean one number, and leave it out 
when you mean two. I might have preferred to do it the other way 
around, but the icosidodecahedron has a long history, and we don't 
want to change established names.

Does that help? Write back if I've left anything out.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/