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RE: Hexagonal indexing?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg67740] RE: [mg67675] Hexagonal indexing?
  • From: "Erickson Paul-CPTP18" <Paul.Erickson at Motorola.com>
  • Date: Thu, 6 Jul 2006 06:52:43 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

I don't know how  standard, but I've commonly seen the indexing axes at
60 degs where each axes is oriented with the intersection in the center
of a hexagonal and each axes goes through the center of one of the
edges. It's complete and unique, but not orthogonal. That's not hard, as
there is a straight forward mapping to / from a Euclidian space.

As to nearest neighbors, {{i+1,j},{i, j+i}, {i-1, j+1}, {i-1, j}, {i,
j-1}, {i+1, j-1}} are the six neighbor indeces. These can also be from
selecting -1,0,+1 for each i and j, but not allowing {-1,-1}, {0,0}, and
{+1,+1}.

Paul 

-----Original Message-----
From: AES [mailto:siegman at stanford.edu] 
To: mathgroup at smc.vnet.net
Subject: [mg67740] [mg67675] Hexagonal indexing?

Partly out of a practical problem, partly out of curiousity, are there
any standard conventions for "hexagonal indexing"? -- that is, for
attaching a single index  k  or a double index  [m, n]  to the points in
a planar hexagonal array so that one can conveniently do things like

--FInd the 6 nearest neighbors {m', n'] to a point [n, m] ?

--Find the coordinates of an arbitrary array point relative to an
optimally chosen origin or center?

--Find the distances between two array points [m', n] and [m", n"] ?

--Find all the array points on the outer rim of a finite hexagonal array
having 6 identical flat faces, or a finite hexagonal array maximally
filling a spherical shell?

--Efficiently convert the double indices (if used) to a single
Mathematica style array index?

I can surely work out some of these answers for myself, but I've never
encountered a "standard method" for doing these, and wonder if there is
an optimal or conventional approach?


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