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A board for discussion of the number twelve as the base of numeration in mathematics and physics.

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2 posters

# Dozenal Nomenclature of Geometrical Figures

Phaethon

Posts : 118
Points : 195
Join date : 2019-08-05
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

wendy.krieger

Posts : 47
Points : 67
Join date : 2019-08-07
Age : 65
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.

Phaethon

Posts : 118
Points : 195
Join date : 2019-08-05
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?

wendy.krieger

Posts : 47
Points : 67
Join date : 2019-08-07
Age : 65
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 .