Newsgroups: rec.puzzles
From: hoey@AIC.NRL.Navy.Mil (Dan Hoey)
Date: Wed, 7 Jul 1993 16:30:32 GMT
Subject: Re: Floor Tiling

aa...@eecs.nwu.edu (Aaron Feigelson) writes:

> Of the twenty four distinct (up to symmetry) heptadiamonds
                                               ^^^^^^^^^^^^^
heptiamonds

> (shapes composed of seven equilateral triangles with adjacent
> edges) EXACTLY ONE cannot be used as the prototile of a monohedral
> planar tiling.

Well, it's been nearly two months since this problem was last posted
in rec.puzzles, so I guess it's time again.

Of course, I turned immediately to George E. Martin's Polyominoes_
_A_Guide_to_Puzzles_and_Problems_in_Tiling.  He tells us it's the V
heptiamond, but unfortunately his figure 10.8 was printed upside-
down, so it's not immediate what he means.  But it's pretty easy.
Twenty-one of the heptiamonds satisfy Conway's criterion, and two form
easy satisfying constellations, so the odd man out is

  *---*   *---*
   \   \ /   /
    *   *   *
     \     /
      *---*

which looks like it should be called the V heptiamond, anyway.  I
wonder how many other of them have names?  It's easy to show the V
doesn't tile: The concavity must be filled, and there's only one way
up to symmetry (A) then the right concavity must be filled, in the
only way doesn't fail immediately (B).  Then the top concavity must be
filled in one of two ways, each of which has an unfillable concavity
(C1,C2).

   (A)       (B)          *---*---*                 *---*---*
                           \       \               /       /
*---*---*   *---*           *---*   *             *   *---*
 \       \ /     \        (C1) /   /               \   \ (C2)
  *---*   *   *   *   *---*---*   *---*     *---*---*   *---*
     /   /   / \   \   \       \ /     \     \       \ /     \
*---*   *---*   *---*   *---*   *   *   *     *---*   *   *   *
 \   \ /   /               /   /   / \   \       /   /   / \   \
  *   *   *               *   *---*   *---*     *   *---*   *---*
   \     /                 \ /                   \ /
    *---*                   *                     *

Aaron's use of `heptadiamonds' brings up some other questions.
Obviously, the heptadiamond should refer to a special kind of
14-iamond, one that can have its triangles paired into seven diamonds.
Since all the diamonds, tetriamonds, hexiamonds, and octiamonds tile
the plane, the first n-diamond candidate for non-tiling is the
pentadiamond.  Can anyone find a non-tiling pentadiamond?  The two
with holes doesn't count!

Dan Hoey
Hoey@AIC.NRL.Navy.Mil