Dozenal Nomenclature of Geometrical Figures

A proposal for replacing decimal based names of geometrical figures by dozenal names of Greek derivation.

Table of Names for Polygons

Number of Vertices or Edges	Polygon Name	Notes
one	henagon	point
two	digon	segment of a line
three	trigon	triangle
four	tetragon	quadrilateral
five	pentagon
six	hexagon
seven	heptagon
eight	octagon
nine	enneagon	nonagon
ten	decagon
eleven	enkomigon
twelve	henzigon
a dozen plus one	henkaizigon
a dozen plus two	dokaizigon
a dozen plus three	trikaizigon	includes a constructible polygon
a dozen plus four	tetrakaizigon
a dozen plus five	pentakaizigon	includes a constructible regular polygon
a dozen plus six	hexakaizigon
a dozen plus seven	heptakaizigon
a dozen plus eight	octakaizigon
a dozen plus nine	enneakaizigon
a dozen plus ten	decakaizigon
a dozen plus eleven	enkomikaizigon
two dozen	dozigon
two dozen plus one	henkaidozigon
three dozen	trizigon
four dozen	tetrazigon
five dozen	pentazigon
six dozen	hexazigon
seven dozen	heptazigon
eight dozen	octazigon
nine dozen	enneazigon
ten dozen	decazigon
eleven dozen	enkomizigon
square dozen	zardogon
square dozen plus one	henkaizardogon
square dozen plus twelve	henzikaizardogon
square dozen plus twelve plus one	henkaihenzikaizardogon
square dozen plus nine dozen plus five	pentakaienneazikaizardogon	includes a constructible regular polygon
cubic dozen	zartrigon
cubic dozen plus one	henkaizartrigon
quartic dozen	zartetragon
three quartic dozen + a cubic dozen + eleven square dozen + a dozen + five	pentakaihenzikaienkomizardokaizartrikaitrizartetragon	includes a constructible regular polygon
quintic dozen	zarpentagon
sextic dozen	zarhexagon

After Gauss, the number of edges of constructible regular polygons by ungraduated straight-edged rule and unfixed compass have prime factorisations of the form \(2^{n}\) times the zeroth or first powers of primes of the form \(2^{2^m}+1\).

Table of Polyhedral Names

Number of faces	Polyhedron Name	Notes
one	henahedron
two	dohedron	includes polygons
three	trihedron
four	tetrahedron	includes triangular-based pyramids
five	pentahedron	includes square-based pyramids, triangular prisms
six	hexahedron	includes the cube
seven	heptahedron
eight	octahedron	includes a Platonic solid
nine	enneahedron
ten	decahedron
eleven	enkomihedron
twelve	henzihedron	includes a Platonic solid
twelve plus one	henkaizihedron
twelve plus two	dokaizihedron
twelve plus three	trikaizihedron
twelve plus five	pentakaizihedron
twelve plus eight	octakaizihedron	includes the Platonic solid icosahedron
two dozen	dozihedron
three dozen	trizihedron

I'm not really fond of naming polytopes after their face-count. 

Twelftychoron for example, impressed Norman Johnson, and I have been using it in one form or another since 2002, maybe earlier.

I've used 'fifhundchoron' for {3,3,5}, which has 420z tetrahedral faces, but it's used interchangably with sixhundchoron.  I tend to write one of my notations following names, this notation is the mainstay at hi.gher.space forum.

A selection of six vertices of the icosahedron gives rise to Weimholt's hexahedron, which i call Weimholt's cube.  There is a polytope in 4d that is bounded by 72 Weimholt's cubes, and 48 vertices.  It's dual has 72 vertices and 48 tri-diminished icosahedra (teddi).

The names by Bowers, who discovered all of the 'starry' uniform polychora, is to heavily truncate the names, so you need to use Klitzing's conversion between Bowers and my names.  For example, the polytope Coxeter describes as \( 2_{21}\) is fy in Bower's names, and /4B in my name, it inverts to give a 4/B.

John Kodegadulo tried a dozenal name system with his SDN, but never figured out how to get the apposition correctly.  I had to demonstrate how to do this to Jonathan Bowers, my names form #4 to #8 in his system.

Names for figures in higher dimensions were not mentioned, but the plan is that the names for numbers of bounding figures would be applied instead of decimal in every instance where Greek names are used. I am not decreeing that figures have to be named for the count of some feature rather than another, but recommend that the count be in dozenal.

wendy.krieger wrote:I'm not really fond of naming polytopes after their face-count. 

Twelftychoron for example, impressed Norman Johnson, and I have been using it in one form or another since 2002, maybe earlier.

The pentatope has ten triangular faces or 'εδρών. The pattern was that the bounding non-curved figure in dimension n is called a -tope, whatever its dimension be. Etymologically, this may be incorrect. I would rather not recycle terms from lower dimensions, so a face would be a two-dimensional figure always, and it would not be right to call the bounding solid figures faces. My reflex would be to call the polyhedrons στερεών, so there would be the pentastereon. However, pentachoron from Greek χώρων meaning "spaces" would seem apt if it conveys that the solids are all connected into a larger building.

wendy.krieger wrote:my names form #4 to #8 in his system.

It was taking up a bit of time trying to track this down. Perhaps you could provide a link or type a few examples?

I normally use http://hi.gher.space/forum/ as the main chat room, and http://www.os2fan2.com/gloss/index.html is where the dictionary is maintained.  The bulk of the conversation is on a private mail list. 

The two systems are discussed on http://www.os2fan2.com/gloss/pglosstu.html .