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Naming Polygons and Polyhedra

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#### What do you call an 11-sided polygon? A 14-sided polygon? Can you give me a list of names of polygons?

This answer represents excerpts from contributions made by Prof. John Conway of Princeton University to the geometry.college and geometry.pre-college newsgroups. For an etymological discussion in the Dr. Math archive, see "Naming Polygons."

When naming polygons, for the "numerical" part of the name, we use the Greek prefixes:

``` mono, di, tri, tetra, penta, hexa, hepta, octa, ennea,
1    2   3     4      5      6     7      8     9

deca, hendeca, dodeca, triskaideca, tetrakaideca, ..., enneakaideca,
10      11      12        13            14                  19

icosa, icosikaihena, icosikaidi, icosikaitri, ..., icosikaiennea,
20       21            22          23                  29

triaconta, triacontakaihena, ..., triacontakaiennea, tetraconta, ...,
30             31                     39             40

pentaconta, hexaconta, heptaconta, octaconta, enneaconta, hecta
50         60          70          80          90      100
```

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

Names of Polygons
```    1  monogon                  (Monogon and digon can only
2  digon                     be used in rather special
3  trigon, triangle          circumstances. Trigon and
4  tetragon, quadrilateral   tetragon are alternatives to
5  pentagon                  triangle and quadrilateral;
6  hexagon                   the adjectival forms trigonal
7  heptagon                  and tetragonal are more common.)
8  octagon
9  enneagon

10  decagon
11  hendecagon
12  dodecagon
13  triskaidecagon
14  tetrakaidecagon, tetradecagon
15  pentakaidecagon, pentadecagon
16  hexakaidecagon, hexadecagon
17  heptakaidecagon
18  octakaidecagon
19  enneakaidecagon

20  icosagon
21  icosikaihenagon, icosihenagon
22  icosikaidigon
23  icosikaitrigon
24  icosikaitetragon
25  icosikaipentagon
26  icosikaihexagon
27  icosikaiheptagon
28  icosikaioctagon
29  icosikaienneagon

30  triacontagon
31  triacontakaihenagon
32  triacontakaidigon
33  triacontakaitrigon
34  triacontakaitetragon
35  triacontakaipentagon
36  triacontakaihexagon
37  triacontakaiheptagon
38  triacontakaioctagon
39  triacontakaienneagon

40  tetracontagon
41  tetracontakaihenagon
42  tetracontakaidigon
43  tetracontakaitrigon
44  tetracontakaitetragon
45  tetracontakaipentagon
46  tetracontakaihexagon
47  tetracontakaiheptagon
48  tetracontakaioctagon
49  tetracontakaienneagon

50  pentacontagon ...
60  hexacontagon ...
70  heptacontagon ...
80  octacontagon ...
90  enneacontagon ...
100  hectogon, hecatontagon
1000  chiliagon
10000  myriagon

```
The "gon" has an interesting etymology: it is ultimately derived from the Greek word "gonu" for "knee", which they transferred to "angle". This word goes straight back to the Indo-European, and is essentially the same in lots of languages:
```     gonu    (Greek)
genu    (Latin)
k nee   (English)

```
French is similar to Latin here, and German to English (except that the "K" is still pronounced.

Naming Polyhedra

The "hedron" in "polyhedron" is also an Indo-European word, meaning "seat". A "cathedral" church is one that contains a seat from which a bishop hands down authoritative opinions (cata = down, hedron = seat).

In accordance with Grimm's law, the "h" in Greek corresponds to "s" in English, while "d" may soften to "t" and "p" or "b" to "f" or "v". So look:

```    penta
five     hexa
six     hepta
seven    hedr
seat    ped
foot

```
In summary, a "polygon" is a thing with many knees, and a "polyhedron" a thing with many seats.

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