Heesch 28 Interlocking Tiles

Dutch artist M.C. Escher is famous for the images of a bird or a lizard that fits together with adjacent tiles without gaps. These tiles can extend to infinity on a surface. The essence of "Escher Tile" is that these are enclosures with one or two motifs.

I feel that the best mathematics to describe Escher tile is the set of 28 interlocking tiles published by Heinrich Heesch (1906-1995). Most people who studied this topic tends to use 17 Wallpaper groups to analyze the symmetry of the aggregate. There may be more than one way to look at this tiling. My friend Dr. Robert Fathauer said all tilings are patterns. But not all patterns are tilings. The 17 Wallpaper groups is best suited to study symmetry and pattern. The Escher Tile and the Heesch tiles are more related to tiles (finite enclosures). I hope the following two tables will show some of the differences.

Compare the two systems:
ICU Wallpaper Heesch Tile
Heinrich Heesch (1906-1995)
Properties of the aggregate Dissection of tile boundary into
lines and vertices
Each pattern has 1 to 4 properties Each pair of line has one and
only one property
All patterns have translation symmetry. Only TT lines have translation symmetry.
A pattern may have no boundary
and no limit on size.
A tile is composed of minimum 2 sets
of lines. And a maximum of 4 sets
of lines.
A set is normally two separate lines
from a symmetry movement.
C line with half turn symmetry
at center is a pair by itself.
Half the boundary of the tile is a
copy of the other half.
Patterns and images can fill the
inside of a silhouette.
Inside of the tile is not part of
the tile definition.
All p4 and p6 patterns have p2
property. Only smallest angle
rotation is counted.
All line properties are counted.
C lines (half turn) are counted
with C3, C4 & C6.
Mirror and glide mirror lines have
angle properties (parallel, intersect,
45 degree) and coincide (or not)
with rotation centers
Lines on tile boundary cannot be
mirrored. Glide mirror axes may
be parallel or perpendicular.


When the pattern or tile has no mirror and no glide mirror property,
these are the different classifications:
ICU
17 Wallpaper Groups
(Symmetry of Pattern)
Heesch
28 Basic Tiles
(Symmetry of Line Components)
Escher
example
p1 TTTT
TTTTTT
Pegasus TTTT
Sky and Water
p2 CCC
CCCC
TCCTCC
TCTCC
TCTC
Lizard TCTC
Sea horse CCCC
p3 C3C3C3C3
C3C3C3C3C3C3
Escher Reptiles
p4 CC4C4
C4C4C4C4
CC4C4C4C4
Lizard C4C4C4C4
p6 CC6C6
CC3C3
C3C3C6C6
CC3C3C6C6
Dolphins
CC3C3C6C6




Heesch’s table of the 28 types of asymmetric isohedral tiles

Heesch's diagrams of the 28 types of interlocking shapes

Computer generated Escher tiles by William Chow 1979

Penrose tile in Architecture

Beautiful designs with mathematics ideas

Escher's lizards everywhere

Animated tiles by David Chow

Screen shots of a CD to teach Tessellations

by William Chow October 17, 2007