Answering njas: definition of POLYPON

[Jonathan Post]

Awkward Graeco-Latin coinage, to be sure, but that is not uncommon in
Math terminology.

Perhaps your attempt to improve the missing definition still needs an
amendment to deal with holes in these equivalence classes of
side-connected, non-overlapping assemblies of congruent 120-30-30
triangles. Or maybe that's a just new sequence waiting to be
enumerated.

On 12/10/07, Richard Mathar <mathar at strw.leidenuniv.nl> wrote:
>
> jp> From seqfan-owner at ext.jussieu.fr  Mon Dec 10 00:46:54 2007
> jp> Date: Sun, 9 Dec 2007 15:46:39 -0800
> jp> From: "Jonathan Post" <jvospost3 at gmail.com>
> jp> To: seqfan <seqfan at ext.jussieu.fr>, "jonathan post" <jvospost2 at yahoo.com>
> jp> Subject: Answering njas: definition of POLYPON
> jp>
> jp> A057785  Blunt polypons (no 30 deg. angles) with n cells.
> jp> COMMENT
> jp> It would be nice to have a definition of "polypon"! - njas, May 09 2007
> jp>
> jp> After looking at over 170 web pages, I have one likely positive hit, a
> jp> vast number of typographical errors, and two weird false positive.
> jp>
> jp> The probable correct coinage:
> jp>
> jp> "If we use the grid below then we obtain sets of shapes which we could
> jp> call polypons from the isosceles triangle (pons asinorum) which is the
> jp> basis of the tessellation."
> jp> http://geocities.com/alclarke0/PolyPages/PolyX/PolyX.htm
> jp>
> jp> The two weirdities:
> jp>
> jp> polypon: J 1 [where J = Liber Juratus (Sworn Book of Honorius)]
> jp> Index of Angel names, magical words, and names of God
> jp> http://www.esotericarchives.com/solomon/nameindx.htm
> jp> Copyright 1998 by Joseph H. Peterson. All rights reserved.
> jp>
> jp> and, stranger still, as I took courses from this professor:
> jp>
> jp> Collected Plays, Volumes I and II by Oscar Mandel
> jp> Professor Snaffle's Polypon, and Of Angels and Eskimos
> jp> Santa Barbara, California:
> jp> Unicorn Press
> jp> Distributed by Spectrum Productions
> jp> 979 Casiano Road
> jp> Los Angeles CA 90049
> jp> Volume One ISBN 0-87775-000-0
> jp> Volume Two ISBN 0-87775-001-7
> jp> 211 + 204 pages
> jp> The hard cover set: $24.00
> jp>
> jp> I disregard the Swedish Jazz group: "Helges Polypon-Boys" because we
> jp> don't know the etymology: angel, typo, or variant of "polyphone"?
>
> This looks like a racial mix of the Latin pons (=bridge) and the Greek
> poly = many/multi. The unit element is a isosceles triangle with the
> basis the longest side which spans a river or anything else of this kind:
>
>       /\
>      /  \      <= bird and flying fish this way
>     /    \
>    /      \    <= walk, jog, drive (everything subject to grativation) here
>    --------
>    ^      ^  points of support
>       \/ abyss here
>
> By looking at the Clarke pictures, I guess, the unit element is a triangle with
> internal angles of 120 degrees and two times 30 degrees. The polypons are
> connected, non-overlapping assemblies of these, where connectivity is defined
> via common sides; a common point is not enough. Only non-congruential
> assemblies are counted, those which cannot be mapped onto each other by
> rotations, translations or mirrors along a line or point. However, the
> polypons are not all of these, because some of the free-form assemblies of
> this kind would need placement of the unit that violates the format by the
> grid. (The first case where this happens is with assemblies of 3 units: the
> picture shows 2 examples with assemblies of 3 units, but I can imagine at
> least 1 more where the unit would need to hide/cover one of the grid's edges.)
>
> Richard
>



Possible duplicates:
--------------------

http://www.research.att.com/~njas/sequences/?q=id:A071683|id:A001076&fmt=1
(aside from initial term in A001076)

http://www.research.att.com/~njas/sequences/?q=id:A041299|id:A001112&fmt=1
(aside from initial zero in A001112)

http://www.research.att.com/~njas/sequences/?q=id:A118579|id:A001297&fmt=1
(aside from initial zero in A118579)

http://www.research.att.com/~njas/sequences/?q=id:A125121|id:A001481&fmt=1
(duplicates as listed, will they ever differ?)


Misc:
------

http://www.research.att.com/~njas/sequences/?q=id:A087327|id:A000982&fmt=1
(equation for A000982 already noted in comments to A087327, ought to have a
reference to A000982)

http://www.research.att.com/~njas/sequences/?q=id:A004729|id:A001317&fmt=1
(A004729 is a subset of A001317)