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Dozenal Nomenclature of Geometrical Figures

Phaethon
Phaethon
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Dozenal Nomenclature of Geometrical Figures Empty Dozenal Nomenclature of Geometrical Figures

Post by Phaethon Sat Sep 07, 2019 8:04 pm

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 enneagonnonagon
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 NameNotes
one henahedron
two dohedronincludes polygons
three trihedron
four tetrahedron includes triangular-based pyramids
five pentahedronincludes 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 henzihedronincludes 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
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Dozenal Nomenclature of Geometrical Figures Empty Re: Dozenal Nomenclature of Geometrical Figures

Post by wendy.krieger Sun Sep 08, 2019 12:10 pm

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
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Post by Phaethon Sun Sep 08, 2019 1:50 pm

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
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Post by wendy.krieger Mon Sep 09, 2019 7:52 am

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 .

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