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2 hexagons, 2 triangles<br />
1 hexagon, 4 triangles<br />
2 octagons, 1 square<br />
2 dodecagons, 1 triangle<br />
2 squares, 3 triangles<br />
1 decagon, 2 pentagons<br />
1 hexagon, 2 squares, 1 triangle<br />
1 dodecagon, 1 hexagon, 1 square<br />
1 dodecagon, 1 square, 2 triangles<br />
5. Answers will vary.<br />
6. Answers will vary. Example: If you start a tiling<br />
by surrounding a vertex with two pentagons<br />
and a decagon, you find that a third pentagon<br />
is necessary next to the first two. On the other<br />
side of the pentagons, you are forced to use<br />
two decagons, but those must overlap each<br />
other, which ruins the tiling.<br />
C.<br />
1<br />
The tiles are placed in<br />
order from 1 to 6; each<br />
step is forced by the<br />
available angles.<br />
2<br />
4<br />
3<br />
6<br />
5<br />
Overlap<br />
Discussion Answers<br />
A. This can be done with equilateral triangles;<br />
squares; and hexagons with triangles.<br />
B. 15-gon: triangle, decagon<br />
18-gon: triangle, nonagon (9-gon)<br />
20-gon: square, pentagon<br />
24-gon: triangle, octagon<br />
42-gon: triangle, heptagon<br />
D. Answers will vary. Most tilings will have<br />
parallel and nonparallel lines of symmetry.<br />
Nonparallel lines will form 30°, 45°, 60°, or<br />
90° angles. Centers of rotation symmetry will<br />
be found at centers of polygons, at midpoints<br />
of edges, and at vertices.They may have the<br />
same or different n-fold symmetries: twofold,<br />
three-fold, four-fold, or six-fold.<br />
Geometry Labs Notes and Answers 207<br />
© 1999 Henri Picciotto, www.MathEducationPage.org
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