Steven Dutch, Professor Emeritus, Natural and Applied Sciences,
Universityof Wisconsin - Green Bay
The terms below are mostly from Grunbaum and Shephard, tilingsand Patterns:
anIntroduction (Freeman, 1989).
- Any geometrical operation that preserves all distances, including translation,
rotation and reflection.
- 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 isohedral.
- 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.
- tilingsin 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
allowdistinctions between what we intuitively consider "well-behaved"
tilings andothers that can be downright paradoxical.
Consider the following tiling: draw a regular 7-gon (heptagon), then surround
it byother (non-regular) heptagons. Around that ring of heptagons draw another,
always havingthree heptagons meet at a vertex. You can keep adding heptagons
indefinitely, but theybecome narrower with each successive ring. Now, since
all the figures are heptagons, theaverage of all the angles in the tiling should
be 5*180/7 = 128.57 degrees. Just ascertainly, since three edges meet at every
vertex, the average of all the angles should be360/3 = 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 itnecessary 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
- 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 metrically balanced.
- 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