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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.

6

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:

10

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:

14

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

21

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

24

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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Explanation & Answer:

3 Problems

Tags:

polygon

Regular Tiling

side rotation

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