In search of the perfect digital hexagon

[coreolyn]

In all honesty this is where the music major in me regrets having such problems with math. I was reading reordering segments to form a polygon this morning and it scratched an old itch that I have tried many times in the past to itch. What I'd like to find is the most elegant way to express perfect hexagons from data. I have always marveled at the hex since the old Avalon Hill Bookshelf games and held a facination with them. Looking at the quick elagent responses to that node I was wondering if some 'big brain' around here could do that in a snap as it would be most appreciated.

As this expresses laziness not of the Perlish kind downvotes would be understood.

[japhy]

I'm not sure I understand the question... but, if you're dealing with a "perfect" hexagon (I believe you mean one with equal sides and equal angles (120 degrees)), then I think the simplest way to describe that hexagon in 2-d space would be with four values (x_c, y_c, l_s, θ):
• x_c is the x-coordinate of the center of the hexagon
• y_c is the y-coordinate of the center of the hexagon
• l_s is the length of a side of the hexagon
• θ is the angle (measured from the x-axis) of the tilt of the hexagon
A tilt of 0 degrees would mean the hexagon looks like:

 _
/ \
\_/

where a tilt of 90 degrees would mean the hexagon looks like:

/\
||
\/

[BrowserUk]

Perhaps you could clarify what you mean by "express perfect hexagons from data"?

A quick google for "Perfect hexagons" doesn't turn up anything that immediately leaps off the page as a "well know definition".

[coreolyn]

Thanks for the responses. I'll attempt to clarify.

All of the responses are relevant and have given me things to play with and research. What I keep trying to express in code is 3 dimensional space. Much like a 10 sided dice in D&D hexagons fill space perfectly. Additionally smaller hexes fill larger hexes perfectly. So what I'm looking to do is figure a coordinate system for defining what is in a space by getting code to scale space via hexagonal scaling. The crux of this is figuring the best algorithm for representing and storing a hex ( as japhy pointed out ) with 120 degree angles.

Update -> After finding the most efficient hex comes translating this type of 2D thinking into 3D. :)

[VSarkiss]

It sounds like what you're looking for is regular solid tilings. There's actually a large body of data on the subject. Try googling for regular tiling, or take a look at some math reference sites.

[coreolyn]

Ouch, but yeah.. thanks for the leads.

[pbeckingham]

I am shamelessly ripping off fletch's work here, but the following (hexagonal) grid:

 -8        . . . - -             -8
 -7       . . . . - ^            -7
 -6    . . . . - - - -           -6
 -5   . . - . a a - ^ ? ? -      -5
 -4  . . . . k o ! - ^ ? ? ?     -4
 -3 . - . a h j a ^ ^ ^ ^ ^ -    -3
 -2  . . - . a a a ^ a a - - -   -2
 -1   . . . a a a b + + ^ ^ -    -1
  0  . . . m c u a a - ^ - ^     0
  1   . - . a a . a a - - ^      1
  2    - ^ ^ a a a ^ a - - ^     2
  3     - - - - - - - . . -      3
  4      ^ - - - - - ^ - -       4
  5       - ^ - - ^ - - -        5

Is actually the same grid as this rectangular grid:

 -8    ...--      -8
 -7   ....-^      -7
 -6  ....----     -6
 -5 ..-.aa-^??-   -5
 -4 ....ko!-^???  -4
 -3.-.ahja^^^^^-  -3
 -2 ..-.aaa^aa--- -2
 -1 ...aaab++^^-  -1
  0 ...mcuaa-^-^  0
  1 .-.aa.aa--^   1
  2  -^^aaa^a--^  2
  3  -------..-   3
  4   ^-----^--   4
  5   -^--^---    5

It is just a question of
1. How the grid is rendered.
2. how a cell's neighboring cells are determined.
Once the data structure is reduced to rectangular, the mystery of the hexagon disappears. It is this kind of thinking that reduces a checkerboard to an 8 x 4 board, for certain games.

[water]

Also not sure what you mean by "express perfect hexagons", but the reference to board games makes me think of a using a graph (vertices V_1... V_n, adjacency matrix A_ij=1 when edge between V_i and V_j) to represent the 6 neighbors of each cell.

[Fletch]

Another way is to use a two dimensional array (matrix), but only some locations are valid. Empire uses a scheme sort of like this:

    ---------0000000000111111111
    9876543210123456789012345678
 -8        . . . - -             -8
 -7       . . . . - ^            -7
 -6    . . . . - - - -           -6
 -5   . . - . a a - ^ ? ? -      -5
 -4  . . . . k o ! - ^ ? ? ?     -4
 -3 . - . a h j a ^ ^ ^ ^ ^ -    -3
 -2  . . - . a a a ^ a a - - -   -2
 -1   . . . a a a b + + ^ ^ -    -1
  0  . . . m c u a a - ^ - ^     0
  1   . - . a a . a a - - ^      1
  2    - ^ ^ a a a ^ a - - ^     2
  3     - - - - - - - . . -      3
  4      ^ - - - - - ^ - -       4
  5       - ^ - - ^ - - -        5
    ---------0000000000111111111
    9876543210123456789012345678

(From the docs for the map command)

To move east or west you use an increment of two to the x index (i.e. west would be ( x - 2, y )); north / south is forbidden in this scheme since the faces are lined up W/NW/NE/E/SE/SW, although that's by convention and there's nothing saying you couldn't rule out east / west by adding to y instead (and in effect rotating the hex grid 90 degrees). To move any of the other four directions you add/subtract one to both coordinates (northeast would be ( x + 1, y - 1)). You can easily verify if a coordinate pair's valid as both x and y have to be both odd or both even.

If this is anywhere near what you were asking about you might take a gander at PEI, the Perl Empire Interface, which has code for dealing with maps of this type.