1.	Tile the plane with a scalene triangle.	

2.	Tile the plane with a non-square rhombus.	

3.	Tile the plane with a (tileable) pentagon.	

4.	Tile the plane with a (tileable) hexagon.	

5.	Create a non-regular tiling with a right triangle tile.	

6.	Create a non-regular tiling with pentagon.	

7.	Create a non-regular tiling with octagons and squares.	

8.	Create a tiling with a parallelogram.  Is it regular?	

9.	Create a tiling with a non-convex quadrilateral.  Is it regular?	

10.	Use the figure below to tile the plane, if possible.

11.	Use the figure below to tile the plane, if possible.

12.	Use the figure below to tile the plane, if possible.

13.	Choose one of Marjorie Rice's pentagon tiles, and use it to tile the plane. 	

14.	Why are regular pentagons not usable to tile the plane? 	

15.	Why are regular octagons not useable to tile the plane? 	

16.	Create a tiling of the plane using squares and hexagons, if possible.	

17.	Create a tiling of the plane using triangles and octagons, if possible.	

18.	Create an Escher-like tiling that uses only translations. 	

19.	Create an Escher-like tiling that uses only translations and half-turns. 	

20.	Create an Escher-like tiling that uses only translations and reflections. 	

21.	Why are Penrose tilings of interest to chemists? 	

22.	If a tiling has only translations, how could it be classified in wallpaper pattern notation? 	

23.	How can you "tile" three dimensions using cubes? 	

24.	How can you "tile" three dimensions using non-cube solid "tiles"? 	

25.	Give a numerical explanation of why a regular decagon (ten sides) cannot tile the plane by itself. 	

26.	At the vertex of a tiling of the plane there is a regular octagon, an equilateral triangle, and one other regular polygon. Which regular polygon is it? 	

27.	At the vertex of a tiling of the plane there is a regular pentagon, a square, and one other regular polygon. Which regular polygon is it? 	

28.	Explain why the tiling below does not form a semiregular tiling of the plane.

29.	Explain why the tiling below does not form a semiregular tiling of the plane.

30.	At the vertex of a tiling of the plane there is a regular octagon, a square, and one other regular polygon. Which regular polygon is it? 	

Answer Key

1.	
	
2.	
	
3.	
	
4.	
	
5.	
	
6.	
	
7.	
	
8.	

Yes.	
9.	

Yes.	
10.	
	
11.	
	
12.	
	
13.	See illustration in Spotlight 20.2, for example.	
14.	Angles do not evenly divide 360 degrees.	
15.	Angles do not evenly divide 360 degrees.	
16.	
	
17.	
	
18.	See Figure 20.10, for example.	
19.	See Figure 20.15, for example.	
20.	See Figure 20.11(a), for example.	
21.	For example, they describe the structure of quasicrystals.	
22.	p1	
23.	For example, stack them as alphabet building blocks.	
24.	For example, stack them as cinder blocks.	
25.	A regular decagon has interior angles measuring 144 degrees each. Since 144 does not evenly divide 360 degrees, decagons cannot tile the plane by themselves.	
26.	A regular 24-gon	
27.	A regular 20-gon	
28.	Not every vertex figure is the same; for example, at one vertex there are two squares and two hexagons, while a different vertex contains two squares, a triangle, and a hexagon.	
29.	The tiles are not edge-to-edge.	
30.	Another regular octagon.