Steven Dutch, Natural and Applied Sciences, University
of Wisconsin - Green Bay
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The terms below are mostly from Grunbaum and Shephard, Tilings and Patterns: an
Introduction (Freeman, 1989).
- Any geometrical operation that preserves all distances, including translation, rotation
- Any isometry that copies an object or pattern onto itself. Rotating a square by 90
degrees about its center rotates the square onto itself and is a symmetry; rotating the
square by 60 degrees, or rotating it about an arbitrary point do not copy the square onto
itself. These operations are isometries, but not symmetries.
- Transitivity Class
- A group of tiles that are all related to each other by symmetry operations. For example,
in a square tiling all the tiles belong to a single transitivity class. However, if we
stagger the rows of squares randomly, none of the squares in adjacent rows are related by
symmetry, and each row consists of tiles of a different transitivity class. Obviously
tiles of differing shapes cannot belong to the same transitivity class.
- A tiling that consists of a single transitivity class; all the tiles are related by
symmetry. A tiling with k transitivity classes is said to be k-isohedral. For
example, the tiling consisting of squares and octagons covering the plane is 2-isohedral.
- A tiling with tiles all the same shape. An isohedral tiling is monohedral but not vice
versa. A tiling with randomly staggered rows of squares is monohedral but not isohedral.
- A tiling in which all vertices belong to a single transitivity class. The tiling
consisting of squares and octagons covering the plane is isogonal, as is the tiling
consisting of alternating rows of triangles and squares. However, neither tiling is
- A tiling in which all edges belong to a single transitivity class. The 6-3-6-3 tiling
consisting of alternating triangles and hexagons covering the plane is isotoxal. The
tiling consisting of alternating rows of triangles and squares is not.
- A tiling in which the edges of all adjacent tiles coincide; that is, there are no
vertices in the middle of an edge. A regular square tiling is edge-to-edge, a tiling with
staggered rows of squares is not.
- Tilings in which all mutually congruent tiles belong to a single transitivity class. The
tiling consisting of alternating rows of triangles and squares is equitransitive because
all the squares are related by symmetry and so are all the triangles. But a tiling
consisting of two rows of squares alternating with rows of triangles is not.
There is no way to predict from the symmetry of the tiling that there should be two rows
of squares, and each row of squares is a separate transitivity class.
Normality and Balance
Grunbaum and Shephard introduced the concepts of normality and balance to allow
distinctions between what we intuitively consider "well-behaved" tilings and
others that can be downright paradoxical.
Consider the following tiling: draw a regular 7-gon (heptagon), then surround it by
other (non-regular) heptagons. Around that ring of heptagons draw another, always having
three heptagons meet at a vertex. You can keep adding heptagons indefinitely, but they
become narrower with each successive ring. Now, since all the figures are heptagons, the
average of all the angles in the tiling should be 5*180/7 = 128.57 degrees. Just as
certainly, since three edges meet at every vertex, the average of all the angles should be
360/3 = 120 degrees. What gives?
This tiling is not normal (the tiles get arbitrarily thin) and is also not balanced,
and it was precisely to eliminate paradoxes like this that Grunbaum and Shephard found it
necessary to introduce the concepts of normality and balance.
- A normal tiling obeys the following conditions:
- Every tile is a topological disk (no holes or disconnected pieces)
- All tiles intersect in a connected set of points (no disconnected edges, for example two
tiles wrapping around a third on all sides)
- All tiles are uniformly bounded (they are of finite size and greater than zero width;
they all have a clearly defined finite upper and lower size limit.)
- A tiling is balanced if the ratio of vertices to tiles and edges to tiles tends to a
stable value as we look at larger and larger regions. This definition excludes tilings
where tiles get progressively larger or smaller with distance from a center. Balanced
tilings need not be normal, and vice versa. A tiling consisting of squares each enclosing
a smaller square in the center is clearly balanced, but it is not normal.
- Strongly Balanced
- A tiling which is both normal and balanced.
- Prototile Balanced
- A tiling where the ratios of different types of tiles tends to a stable limit as we
examine larger and larger regions. This definition excludes tilings where the shapes and
relative numbers of tiles vary systematically from one part of the plane to another. All
normal periodic tilings are both strongly and prototile balanced.
- Metrically Balanced
- A tiling where the numbers of tiles, vertices and edges per unit area tend toward a
stable limit as we examine larger and larger areas. All normal periodic tilings are
- A transformation that maps points (x,y) in the plane into other points (X,Y) in a
one-to-one fashion. That is, every point (x,y) is mapped onto a single point (X,Y) and
there is no other point that also maps onto (X,Y).
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Created 2 July 1999, Last Update 2 July 1999