11.3 TILINGS AND ESCHER-LIKE DESIGNS
JESSICA WILLIAMS
First we need to know what a tiling even is.
Definition 0.1. A tile is a simple, closed curve together with its interior.
Examples drawn on board of tiles vs. non-tiles, recalling what a
simple, closed curve means.
Definition 0.2. A set of tiles forms a tiling of a figure if the figure is
completely covered by the tiles without overlapping any interior points
of the tiles.
• Tiles are “edge-to-edge.”
• We say a “tiling of the plane” when we mean the tiling can
repeat forever.
• Also sometimes called a “tesselation.”
• There is a lot of artwork and design using tilings. (Show picture
slide of various tilings.)
• There are both simple and complex tilings.
ACTIVITY # 1 : Try to make a tiling with each of the shapes
given to you (equilateral triangle, square, regular pentagon, regular
hexagon). Only use one shape at a time.
The tiles given in this activity were all regular polygons (equilateral and equiangular). You should have been able to tile with the
triangle, square, and hexagon, but not the pentagon.
Since we only used one type of shape at a time and they were regular
polygons, this is a specific type of tiling:
Definition 0.3. A regular tiling is a tiling consisting of congruent
regular polygons joined edge to edge.
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JESSICA WILLIAMS
Theorem 0.4. There are exactly three regular tilings of the plane:
(1) by equilateral triangles,
(2) by squares,
(3) by regular hexagons.
WHY is this true? To help us answer this question, let’s first define
what a vertex figure is. It will help us understand.
Definition 0.5. A vertex figure is any arrangement of nonoverlapping polygonal tiles surrounding a common vertex.
• They are again “edge-to-edge.”
• The vertex is the “center point.”
• Need at least three shapes for a vertex. Two only gives you an
edge.
• (Show regular tiling vertex figure examples.)
Now, back to the question. Why are these the only three regular
tilings?
Question 1: Why are there six polygons in the triangle vertex
figure, four in the square one, and 3 in the hexagon one?
• Leading question: What do the integerior angles at the vertex
have to sum to? – 360◦ !
• So equilateral triangle have interior angles of 60 degrees, squares
have 90, and regular hexagons have 120. So how many interior
angles meet at the vertex? 6 for the triangle, 4 for the square,
3 for the hexagon.
Question 2: So why couldn’t we possibly have a regular tiling by a
polygon of more than six sides?
• The interior angles would be to large to have more than two
polygons meeting – not a vertex figure.
• So this means the polygons to form a regular tiling can only
have 3,4,5, or 6 sides. NOTE: This does NOT mean that these
all work! It just means they are the only possibilities. We’ve
already demonstrated that 3,4, and 6 work, but not 5.
11.3 TILINGS AND ESCHER-LIKE DESIGNS
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Question 3: Why doesn’t the pentagon work?
• What’s the interior angle measure?
• Does 108 divide 360? – No. Would have overlap.
Present summary of the proof.
ACTIVITY #2: Try to form vertex figures with the same shapes,
but this time you can use more than one shape at a time.
You should have found some possibilities. Tilings with vertex figures
made out of regular polygons, but with more than one kind of shape
at a time, are a different classification of tiling:
Definition 0.6. An edge-to-edge tiling of the plane with more than
one type of regular polygon and with identical vertex figures is called
a semiregular tiling.
• The same types of polygons must surround each vertex, and
they must occur in the same order.
Theorem 0.7. There are only 18 ways to form a vertex figure with
regular polygons of two or more types. Only 8 of these vertex figures
can be made into semiregular tilings.
• Show picture of all 18 vertex figures.
• Show picture of 8 semiregular tilings.
• This means 21 possible vertex figures of regular polygons, in
total, and 11 possible regular or semi-regular tilings, in total.
• The fact that some can’t be extended to semiregular tilings is
surprising and apparently a little difficult to show. (gruze.org Imperfect Congruence)
It is possible to make tilings with irregular polygons, too. In fact,
the plane can be tiled by:
• any triangular tile (because you can form a parallellogram, and
then tile with that)
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JESSICA WILLIAMS
• any quadrilateral tile (Rotate 180 degrees about the midpoint
of a side to form a hexagon. Each vertex figure has the four
angles of the quadrilateral.)
• certain pentagonal tiles (always possible if one set of sides are
parallel)
• certain hexagonal tiles
If time permits, more on irregular tilings:
• Pentagonal tiles can always tile when they have two parallel
sides.
• Hexagonal tiles can always tile if they have two opposite parallel
sides of the same length.
• Matching opposite sides is the main technique.
• The plane can NOT be tiled with any convex polygon of seven
or more sides.
• Nonconvex polygons of seven or more sides CAN tile the plane.
(example of nonagon in the text)
Escher-like Designs
Okay, finally, what is the “Escher-like” designs part all about?
(Show Escher pictures)
Maurits Cornelis Escher (1898-1972) is one of the world’s most
famous graphic artists.
M.C. Escher, during his lifetime, made 448 lithographs, woodcuts and
wood engravings and over 2000 drawings and sketches.
• Escher discovered new principles of tiling formation based on
symmetries that mathematicians overlooked.
• He used a technique called translation. (cutting out pieces
from tiles and shifting them to opposite sides) (show pictures
again to point out translation)
• Rotations can also be used to create Escher-like tiles.
11.3 TILINGS AND ESCHER-LIKE DESIGNS
Resources:
morphingtiling.wordpress.com - Morphing Tilings
gruze.org - Imperfect Congruence
euler.slu.edu/escher - Math & the Art of MC Escher
Department of Mathematics, University of Iowa, Iowa City, IA
E-mail address: [email protected]
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