ILS 4430 Symmetry in Geometry Mathematics Problems

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Topic 17 Classwork Assignment
Problem 1: Construct the dual tiling of the given tiling
(a) Regular Tiling: 6.6.6
(b) Semiregular Tiling: 3.3.3.4.4
Problem 2: Show that the polygon is a reptile.
(a)
(b)
Problem 3: Determine if the Escher tiling is constructed from a translation, a glide reflection, a midpoint
rotation, or a side rotation. Sketch an underlying polygonal grid.
(a)
(b)
Modification:
(c)
Modification:
(d)
Modification:
Modification:
Topic 17
Symmetry in Geometry
Tilings – Part II
Constructing Dual Tilings
To each polygonal tiling we can associate a dual tiling.
A dual tiling is a type of tiling that is superimposed a polygon tiling.
A dual tiling may or may not be regular or semiregular.
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Constructing Dual Tilings
Example:
The dual tiling of the semiregular tiling 4.8.8 (grey) is a tiling of 45°- 45°- 90° right
triangles (blue).
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Constructing Dual Tilings
To construct the dual tiling of a polygon tiling:
Step 1: Place a vertex at the center of each polygon in the original tiling.
Step 2: Whenever two polygons share an edge in the original tiling, draw a dual
edge connecting the new vertices at the centers of those polygons.
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Example: Dual Tilings
Construct the dual tiling of the semiregular tiling 3.4.6.4.
Solution:
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Irregular Tilings
An irregular tiling is a tiling of polygons that are not regular. Any quadrilateral can
tile the plane if the copies are put together right.
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Irregular Tilings
Any convex quadrilateral can tile the plane.
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Irregular Tilings
Similarly, any non-convex quadrilateral can tile the plane.
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Example: Irregular Tilings
Show that any parallelogram can tile the plane.
Solution:
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Reptiles
A reptile (short for repeating tile) is a tile that can be arranged to form a larger copy
of itself. The larger copy must be an exact scaled replica of the original.
Example:
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Example: Reptiles
Show that the following shape is a reptile.
Solution:
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Escher Tilings
Maurits Cornelis Escher was a mathematical artist.
Escher developed a own system of cataloging patterns, not only in terms of 17
wallpaper patterns, but also in terms of the relationship between motif and underlying
grid, and in terms of coloring.
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Escher Tilings
To construct an Escher tiling,
Step 1: Choose a grid.

The grid my be a regular or semiregular tiling, or a tiling by irregular polygons.
Step 2: Modify a tile in the grid using one or more or the following transformations:

Translation
Glide Reflection
Midpoint Rotation
Side Rotation
Step 3: Use the modified tile to tile the plane.
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Escher Tilings: Modifying a Tile Using Translation
To modify a tile using translation, the original tile must have at least one pair of
parallel sides of equal length.
Grid: squares, rectangles, parallelograms, hexagons.
Tile Modification: Cut out a section of the tile that leaves the vertices of the tile intact.
Translate this section to the parallel side of the tile. The modification may be repeated
for the other pair of parallel sides of the tile.
Example:
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Escher Tilings: Modifying a Tile Using Translation
Escher Example:
Grid: squares
Modification: translation
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Escher Tilings: Modifying a Tile Using Translation
Escher Example:
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Escher Tilings: Modifying a Tiling Using Glide Reflection
To modify a tile using glide reflection, the original tile must have at least one pair of
parallel sides of equal length.
Grid: squares, rectangles, parallelograms, hexagons
Tile Modification: Take the original tile and cut something off of one of the parallel
sides. Flip the cut piece over and move it to the parallel side.
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Escher Tilings: Modifying a Tiling Using Glide Reflection
Escher Like Example:
Grid: parallelograms
Modification: glide reflection and translation
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Escher Tilings: Modifying a Tiling Using Glide Reflection
Escher Like Example:
Grid: parallelograms
Modification: glide reflection and translation
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Escher Tilings: Modifying a Tiling Using Glide Reflection
Escher Like Example:
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Escher Tilings: Modifying a Tiling Using Midpoint Rotation
To modify a tile using midpoint rotation, any original tiling may be used.
Grid: any tiling
Tile Modification: Cut out a section of the tile that begins at one endpoint and ends at
the midpoint of the same side. Rotate the section by 180° about the midpoint.
O
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Escher Tilings: Modifying a Tile Using Midpoint Rotation
Escher Like Example:
Grid: parallelograms
O
Modification: midpoint rotation
O
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Escher Tilings: Modifying a Tile Using Midpoint Rotation
Escher Like Example:
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Escher Tilings: Modifying a Tiling Using Side Rotation
To modify a tile using side rotation, the original tile must have a pair of adjacent sides of
equal length
Grid: squares, equilateral triangles, rhombi, or regular hexagons.
Tile Modification: Cut out a section of the tile that that begins at one vertex of one of
the equal edges of the tile and ends at the other vertex. Rotate the section about the
vertex connecting the two equal sides and fix it to the adjacent edge. The modification
may be repeated for the other sides of the tile.
O
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Escher Tilings: Modifying a Tile Using Side Rotation
Escher Example:
Modification: side rotation
Modification: side rotation
O
O
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Escher Tilings: Modifying a Tile Using Side Rotation
Escher Example:
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Symmetry in Geometry

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