One can fill the surface R^2 with triangles, squares and hexagons as shown below.
geometry/tiling/triangles.squares.hexagons New
New entry, edited from material collected for the archive and not yet in a released version.
Problem
One can fill the surface R^2 with triangles, squares and hexagons as
shown below. What is the ratio between the 3, 4, and 6-gons if the
surface is filled with this pattern?
____ ____ ____ ____
,-'| |`-. ,-'| |`-. ,-'| |`-. ,-'| |`-.
/`-.|__|,-'\ /`-.|__|,-'\ /`-.|__|,-'\ /`-.|__|,-'\
/ / \ \____/ / \ \____/ / \ \____/ / \ \
|`-./ \,-'| |`-./ \,-'| |`-./ \,-'| |`-./ \,-'|
|,-'\ /`-.|__|,-'\ /`-.|__|,-'\ /`-.|__|,-'\ /`-.|
\ \____/ / \ \____/ / \ \____/ / \ \____/ /
\,-'| |`-./ \,-'| |`-./ \,-'| |`-./ \,-'| |`-./
/`-.|__|,-'\ /`-.|__|,-'\ /`-.|__|,-'\ /`-.|__|,-'\
/ / \ \____/ / \ \____/ / \ \____/ / \ \
|`-./ \,-'| |`-./ \,-'| |`-./ \,-'| |`-./ \,-'|
|,-'\ /`-.|__|,-'\ /`-.|__|,-'\ /`-.|__|,-'\ /`-.|
\ \____/ / \ \____/ / \ \____/ / \ \____/ /
\,-'| |`-./ \,-'| |`-./ \,-'| |`-./ \,-'| |`-./
`-.|__|,-'\ /`-.|__|,-'\ /`-.|__|,-'\ /`-.|__|,-'
\ \____/ / \ \____/ / \ \____/ /
\,-'| |`-./ \,-'| |`-./ \,-'| |`-./
`-.|__|,-' `-.|__|,-' `-.|__|,-'
-- Mark van Hoeij <hoeij@sci.kun.nl>
[ Editor: van Hoeij's figure was PostScript ("shown in the postscript
file included below" in the original); the ASCII drawing is Dan
Hoey's, shortened here by one band. ]
Show solution
The easy way is to erase some of the lines, to form a monohedral tiling
by seashells.
____ ____ ____ ____
,-' `-. ,-' `-. ,-' `-. ,-' `-.
/ \ / \ / \ / \
/ \____/ \____/ \____/ \
`-. ,-' `-. ,-' `-. ,-' `-. ,-'
\ / \ / \ / \ /
\____/ \____/ \____/ \____/
,-' `-. ,-' `-. ,-' `-. ,-' `-.
/ \ / \ / \ / \
/ \____/ \____/ \____/ \
`-. ,-' `-. ,-'| |`-. ,-' `-. ,-'
\ / \ /`-.|__|,-'\ / \ /
\____/ \____/ / \ \____/ \____/
,-' `-. ,-' `-./ \,-' `-. ,-' `-.
/ \ / \ / \ / \
/ \____/ \____/ \____/ \
`-. ,-' `-. ,-' `-. ,-' `-. ,-'
\ / \ / \ / \ /
\____/ \____/ \____/ \____/
,-' `-. ,-' `-. ,-' `-. ,-' `-.
/ \ / \ / \ / \
/ \____/ \____/ \____/ \
`-. ,-' `-. ,-' `-. ,-' `-. ,-'
\ / \ / \ / \ /
\____/ \____/ \____/ \____/
As the one in the center shows, each shell is composed of one hexagon,
two triangles, and three squares.
ObPuzzle: Is there a tiling in which every two congruent tiles are
equivalent, and which cannot be converted into a monohedral tiling?
By "equivalent", I mean that there is a symmetry of the tiling taking
one tile to the other.
-- Dan Hoey <Hoey@AIC.NRL.Navy.Mil>
[ Editor: so the ratio of triangles to squares to hexagons is 2:3:1.
No answer to Hoey's ObPuzzle was found in the collected postings. ]