| About the Author |
|
xi | |
| Preface |
|
xiii | |
|
1 Introduction to Tessellations |
|
|
1 | (16) |
|
Historical examples of tessellations |
|
|
2 | (2) |
|
Tessellations in the world around us |
|
|
4 | (1) |
|
Escheresque tessellations |
|
|
5 | (4) |
|
Tessellations and recreational mathematics |
|
|
9 | (2) |
|
Tessellations and mathematics education |
|
|
11 | (2) |
|
Activity 1.1 Recognizing tessellations |
|
|
13 | (1) |
|
Worksheet 1.1 Recognizing tessellations |
|
|
14 | (1) |
|
Activity 1.2 Historical tessellations |
|
|
15 | (1) |
|
Worksheet 1.2 Historical tessellations |
|
|
16 | (1) |
|
2 Geometric Tessellations |
|
|
17 | (34) |
|
|
|
17 | (1) |
|
|
|
18 | (1) |
|
Vertices and edge-to-edge tessellations |
|
|
19 | (1) |
|
Regular polygons and regular tessellations |
|
|
20 | (1) |
|
|
|
21 | (2) |
|
|
|
23 | (1) |
|
Semi-regular tessellations |
|
|
23 | (1) |
|
|
|
24 | (1) |
|
General triangle and quadrilateral tessellations |
|
|
25 | (2) |
|
Dual tessellations and Laves tessellations |
|
|
27 | (2) |
|
Pentagon and hexagon tessellations |
|
|
29 | (2) |
|
Stellation of regular polygons and star polygons |
|
|
31 | (2) |
|
Star polygon tessellations |
|
|
33 | (3) |
|
Regular polygon tessellations that are not edge-to-edge |
|
|
36 | (1) |
|
|
|
37 | (1) |
|
Modifying tessellations to create new tessellations |
|
|
38 | (4) |
|
Circle packings and tessellations |
|
|
42 | (3) |
|
Activity 2.1 Basic properties of tiles |
|
|
45 | (1) |
|
Worksheet 2.1 Basic properties of tiles |
|
|
46 | (1) |
|
Activity 2.2 Edge-to-edge tessellations |
|
|
47 | (1) |
|
Worksheet 2.2 Edge-to-edge tessellations |
|
|
48 | (1) |
|
Activity 2.3 Classifying tessellations by their vertices |
|
|
49 | (1) |
|
Worksheet 2.3 Classifying tessellations by their vertices |
|
|
50 | (1) |
|
3 Symmetry and Transformations in Tessellations |
|
|
51 | (26) |
|
|
|
51 | (3) |
|
|
|
54 | (2) |
|
Symmetry in tessellations |
|
|
56 | (2) |
|
|
|
58 | (2) |
|
|
|
60 | (5) |
|
|
|
60 | (1) |
|
|
|
60 | (1) |
|
|
|
60 | (1) |
|
|
|
60 | (2) |
|
|
|
62 | (1) |
|
|
|
62 | (1) |
|
|
|
63 | (1) |
|
|
|
63 | (1) |
|
|
|
64 | (1) |
|
|
|
64 | (1) |
|
|
|
64 | (1) |
|
|
|
64 | (1) |
|
|
|
64 | (1) |
|
|
|
65 | (1) |
|
|
|
65 | (1) |
|
|
|
65 | (1) |
|
|
|
65 | (1) |
|
Heesch types and orbifold notation |
|
|
65 | (1) |
|
Coloring of tessellations and symmetry |
|
|
66 | (3) |
|
Activity 3.1 Symmetry in objects |
|
|
69 | (1) |
|
Worksheet 3.1 Identifying symmetry in objects |
|
|
70 | (1) |
|
Activity 3.2 Transformations |
|
|
71 | (1) |
|
Worksheet 3.2 Identifying transformations |
|
|
72 | (1) |
|
Activity 3.3 Translational symmetry in tessellations |
|
|
73 | (1) |
|
Worksheet 3.3 Identifying symmetry in tessellations |
|
|
74 | (1) |
|
Activity 3.4 Rotational symmetry in tessellations |
|
|
75 | (1) |
|
Activity 3.5 Glide reflection symmetry in tessellations |
|
|
76 | (1) |
|
4 Tessellations in Nature |
|
|
77 | (12) |
|
Modeling of natural tessellations |
|
|
78 | (1) |
|
|
|
78 | (1) |
|
|
|
79 | (1) |
|
|
|
79 | (1) |
|
Divisions in plants and animals |
|
|
80 | (3) |
|
|
|
83 | (1) |
|
|
|
83 | (2) |
|
Activity 4.1 Modeling natural tessellations using geometric tessellations |
|
|
85 | (1) |
|
Worksheet 4.1 Modeling natural tessellations using geometric tessellations |
|
|
86 | (1) |
|
Activity 4.2 Quantitative analysis of natural tessellations |
|
|
87 | (1) |
|
Worksheet 4.2 Modeling natural tessellations using geometric tessellations |
|
|
88 | (1) |
|
5 Decorative and Utilitarian Tessellations |
|
|
89 | (14) |
|
|
|
89 | (1) |
|
Building blocks and coverings |
|
|
90 | (3) |
|
|
|
93 | (4) |
|
|
|
97 | (1) |
|
|
|
97 | (1) |
|
|
|
98 | (1) |
|
Islamic art and architecture |
|
|
99 | (1) |
|
|
|
99 | (2) |
|
Activity 5.1 Building with tessellations |
|
|
101 | (1) |
|
Worksheet 5.1 Building with tessellations |
|
|
102 | (1) |
|
|
|
103 | (12) |
|
|
|
104 | (2) |
|
Tessellations of polyforms |
|
|
106 | (2) |
|
The translation and Conway criteria |
|
|
108 | (1) |
|
Other recreations using polyforms |
|
|
108 | (2) |
|
|
|
110 | (1) |
|
|
|
111 | (1) |
|
Activity 6.1 Discovering and classifying polyforms |
|
|
112 | (1) |
|
Worksheet 6.1 Discovering and classifying polyforms |
|
|
113 | (2) |
|
|
|
115 | (20) |
|
|
|
115 | (3) |
|
|
|
118 | (4) |
|
Logarithmic spiral tessellations |
|
|
122 | (6) |
|
Archimedean spiral tessellations |
|
|
128 | (5) |
|
Activity 7.1 Exploring spiral tessellations |
|
|
133 | (1) |
|
Worksheet 7.1 Exploring spiral tessellations |
|
|
134 | (1) |
|
8 Matching Rules, Aperiodic Tiles, and Substitution Tilings |
|
|
135 | (14) |
|
Matching rules and tiling |
|
|
136 | (1) |
|
Periodicity in tessellations |
|
|
137 | (1) |
|
|
|
138 | (4) |
|
Other aperiodic sets and substitution tilings |
|
|
142 | (3) |
|
Socolar-Taylor aperiodic monotile |
|
|
145 | (1) |
|
Escheresque tessellations based on aperiodic tiles |
|
|
145 | (2) |
|
Activity 8.1 Penrose tiles and the golden number |
|
|
147 | (1) |
|
Worksheet 8.1 Penrose tiles and the golden number |
|
|
148 | (1) |
|
9 Fractal Tiles and Fractal Tilings |
|
|
149 | (24) |
|
Tessellations of fractal tiles |
|
|
150 | (2) |
|
|
|
152 | (2) |
|
Two-fold f-tilings based on segments of regular polygons |
|
|
154 | (4) |
|
f-tilings based on kite-, dart-, and v-shaped prototiles |
|
|
158 | (5) |
|
f-tilings based on polyforms |
|
|
163 | (5) |
|
|
|
168 | (3) |
|
Activity 9.1 Prototiles for fractal tilings |
|
|
171 | (1) |
|
Worksheet 9.1 Prototiles for fractal tilings |
|
|
172 | (1) |
|
10 Non-Euclidean Tessellations |
|
|
173 | (10) |
|
|
|
174 | (4) |
|
|
|
178 | (3) |
|
Activity 10.1 Non-Euclidean tessellations of regular polygons |
|
|
181 | (1) |
|
Worksheet 10.1 Non-Euclidean tessellations of regular polygons |
|
|
182 | (1) |
|
11 Tips on Designing and Drawing Escheresque Tessellations |
|
|
183 | (20) |
|
Drawing tessellations by hand |
|
|
184 | (1) |
|
Using general computer graphics programs |
|
|
184 | (1) |
|
Using a tessellations computer program |
|
|
185 | (1) |
|
|
|
185 | (1) |
|
Tip 1 The outline of the tile should suggest the motif |
|
|
185 | (1) |
|
Tip 2 The tiles should make orientational sense |
|
|
186 | (1) |
|
Tip 3 Choose motifs that go together |
|
|
187 | (2) |
|
Tip 4 Different motifs should be commensurately scaled |
|
|
189 | (1) |
|
Tip 5 Use source material to get the details right |
|
|
190 | (1) |
|
|
|
191 | (1) |
|
Tip 7 Choose a style that fits your taste and abilities |
|
|
192 | (1) |
|
Tip 8 Choose colors that suit your taste and bring out the tiles |
|
|
193 | (2) |
|
Activity 11.1 Finding motifs for a tile shape |
|
|
195 | (1) |
|
Worksheet 11.1 Finding motifs for a tile shape |
|
|
196 | (1) |
|
Activity 11.2 Refining a tile shape using translation |
|
|
197 | (1) |
|
Worksheet 11.2 Refining a tile shape using translation |
|
|
198 | (1) |
|
Activity 11.3 Refining a tile shape using glide reflection |
|
|
199 | (1) |
|
Worksheet 11.3 Refining a tile shape using glide reflection |
|
|
200 | (1) |
|
Activity 11.4 Locating and using source material for real-life motifs |
|
|
201 | (1) |
|
Worksheet 11.4 Locating and using source material for real-life motifs |
|
|
202 | (1) |
|
12 Special Techniques to Solve Design Problems |
|
|
203 | (10) |
|
Technique 1 Distorting the entire tessellation |
|
|
204 | (1) |
|
Technique 2 Breaking symmetries |
|
|
205 | (2) |
|
Technique 3 Splitting a tile into smaller tiles |
|
|
207 | (1) |
|
Technique 4 Splitting and moving vertices |
|
|
207 | (3) |
|
Activity 12.1 Reshaping a tile by splitting and moving vertices |
|
|
210 | (1) |
|
Worksheet 12.1 Reshaping a tile by splitting and moving vertices |
|
|
211 | (2) |
|
13 Escheresque Tessellations Based on Squares |
|
|
213 | (24) |
|
Creating a tessellation by hand |
|
|
214 | (5) |
|
Template 13.1 Tessellation with translational symmetry only |
|
|
219 | (1) |
|
Template 13.2 Tessellation with two-and four-fold rotational symmetry |
|
|
220 | (2) |
|
Template 13.3 Tessellation with glide reflection symmetry |
|
|
222 | (1) |
|
Template 13.4 Tessellation with a simple reflection and glide reflection symmetry |
|
|
223 | (2) |
|
Template 13.5 Tessellation with glide reflection symmetry in two orthogonal directions |
|
|
225 | (1) |
|
Template 13.6 Tessellation with twofold rotational and glide reflection symmetry |
|
|
226 | (1) |
|
Template 13.7 Tessellation with two motifs and glide reflection symmetry |
|
|
227 | (2) |
|
Template 13.8 Tessellation with two different tiles and reflection symmetry in one direction |
|
|
229 | (1) |
|
Template 13.9 Tessellation with two different tiles and reflection symmetry in two orthogonal directions |
|
|
230 | (3) |
|
Activity 13.1 Creating an Escheresque tessellation with translational symmetry |
|
|
233 | (1) |
|
Activity 13.2 Creating an Escheresque tessellation with rotational symmetry |
|
|
234 | (1) |
|
Activity 13.3 Creating an Escheresque tessellation with glide reflection symmetry |
|
|
235 | (2) |
|
14 Escheresque Tessellations Based on Isosceles Right Triangle and Kite-Shaped Tiles |
|
|
237 | (12) |
|
Template 14.1 Right-triangle tessellation with two-fold rotational symmetry |
|
|
239 | (1) |
|
Template 14.2 Right-triangle tessellation with two- and four-fold rotational symmetry |
|
|
240 | (1) |
|
Template 14.3 Right-triangle tessellation with two- and four-fold rotational and reflection symmetry |
|
|
241 | (2) |
|
Template 14.4 Kite tessellation with glide reflection symmetry |
|
|
243 | (1) |
|
Template 14.5 Kite tessellation with two motifs and glide reflection symmetry |
|
|
244 | (2) |
|
Activity 14.1 Creating a tessellation based on right-triangle tiles |
|
|
246 | (1) |
|
Activity 14.2 Creating a tessellation based on kite-shaped tiles |
|
|
247 | (2) |
|
15 Escheresque Tessellations Based on Equilateral Triangle Tiles |
|
|
249 | (12) |
|
Template 15.1 Tessellation with sixfold rotational symmetry |
|
|
252 | (2) |
|
Template 15.2 Tessellation with rotational and glide reflection symmetry |
|
|
254 | (2) |
|
Template 15.3 Tessellation with twofold rotational symmetry only |
|
|
256 | (1) |
|
Template 15.4 Tessellation with translational symmetry only |
|
|
257 | (2) |
|
Activity 15.1 Creating an equilateral triangle-based tessellation with rotational symmetry |
|
|
259 | (1) |
|
Activity 15.2 Creating an equilateral triangle-based tessellation with glide reflection symmetry |
|
|
260 | (1) |
|
16 Escheresque Tessellations Based on 60°-120° Rhombus Tiles |
|
|
261 | (16) |
|
Template 16.1 Tessellation with translational symmetry only |
|
|
264 | (1) |
|
Template 16.2 Tessellation with reflection symmetry |
|
|
265 | (2) |
|
Template 16.3 Tessellation with three-fold rotational symmetry |
|
|
267 | (2) |
|
Template 16.4 Tessellation with glide reflection symmetry |
|
|
269 | (2) |
|
Template 16.5 Rhombus tessellation with rotational and glide reflection symmetry |
|
|
271 | (1) |
|
Template 16.6 Tessellation with kaleidoscopic symmetry |
|
|
272 | (2) |
|
Activity 16.1 Creating a tessellation with bilaterally symmetry tiles |
|
|
274 | (1) |
|
Activity 16.2 Creating a tessellation with kaleidoscopic symmetry |
|
|
275 | (2) |
|
17 Escheresque Tessellations Based on Hexagonal Tiles |
|
|
277 | (10) |
|
Template 17.1 Tessellation with threefold rotational symmetry |
|
|
279 | (1) |
|
Template 17.2 Tessellation with sixfold rotational symmetry |
|
|
280 | (2) |
|
Template 17.3 Tessellation based on hexagons and hexagrams |
|
|
282 | (3) |
|
Activity 17.1 Creating a tessellation based on hexagonal tiles |
|
|
285 | (1) |
|
Activity 17.2 Creating a hexagon-based tessellation with two-, three-, and six-fold rotational symmetry |
|
|
286 | (1) |
|
18 Decorating Tiles to Create Knots and Other Designs |
|
|
287 | (14) |
|
The role of combinatorics |
|
|
287 | (2) |
|
Using tessellations to create knots and links |
|
|
289 | (2) |
|
Creating iterated and fractal knots and links with fractal tilings |
|
|
291 | (5) |
|
Other types of decorative graphics |
|
|
296 | (3) |
|
Activity 18.1 Creating symmetrical designs by decorating tessellations |
|
|
299 | (1) |
|
Worksheet 18.1 Creating symmetrical designs by decorating tessellations |
|
|
300 | (1) |
|
19 Tessellation Metamorphoses and Dissections |
|
|
301 | (10) |
|
|
|
301 | (2) |
|
Positive and negative space |
|
|
303 | (1) |
|
Techniques for transitioning between Escheresque tessellation motifs |
|
|
304 | (3) |
|
|
|
307 | (3) |
|
Activity 19.1 Learning to draw a tessellation metamorphosis |
|
|
310 | (1) |
|
20 Introduction to Polyhedra |
|
|
311 | (16) |
|
Basic properties of polyhedra |
|
|
311 | (3) |
|
Polyhedra in art and architecture |
|
|
314 | (3) |
|
|
|
317 | (1) |
|
Tiling three-dimensional space |
|
|
318 | (1) |
|
Slicing 3-honeycombs to reveal plane tessellations |
|
|
319 | (2) |
|
Activity 20.1 Identifying and characterizing polyhedra in nature |
|
|
321 | (1) |
|
Worksheet 20.1 Identifying and characterizing polyhedra in nature |
|
|
322 | (2) |
|
Activity 20.2 Identifying polyhedral in art and architecture |
|
|
324 | (1) |
|
Worksheet 20.2 Identifying polyhedra in art and architecture |
|
|
325 | (2) |
|
21 Adapting Plane Tessellations to Polyhedra |
|
|
327 | (14) |
|
|
|
328 | (1) |
|
Restrictions on plane tessellations for use on polyhedra |
|
|
328 | (1) |
|
Distorting plane tessellations to fit polyhedra |
|
|
329 | (3) |
|
Designing and drawing tessellations for polyhedra using the templates |
|
|
332 | (1) |
|
Coloring of tessellations on polyhedra |
|
|
333 | (1) |
|
Tips on building the models |
|
|
334 | (2) |
|
Activity 21.1 Representing solids using nets |
|
|
336 | (1) |
|
Worksheet 21.1 Representing solids using nets |
|
|
337 | (1) |
|
Activity 21.2 Using transformations to apply a tessellation motif to a net |
|
|
338 | (1) |
|
Worksheet 21.2 Using transformations to apply a tessellation motif to a net |
|
|
339 | (2) |
|
22 Tessellating the Platonic Solids |
|
|
341 | (18) |
|
Background on the Platonic solids |
|
|
341 | (12) |
|
Tessellation templates for the Platonic solids |
|
|
353 | (2) |
|
Activity 22.1 Attributes of the Platonic solids and Euler's formula |
|
|
355 | (1) |
|
Worksheet 22.1 Attributes of the Platonic solids and Euler's formula |
|
|
356 | (1) |
|
Activity 22.2 Drawing the Platonic solids |
|
|
357 | (1) |
|
Worksheet 22.2 Drawing the Platonic solids |
|
|
358 | (1) |
|
23 Tessellating the Archimedean Solids |
|
|
359 | (42) |
|
Background on the Archimedean solids |
|
|
359 | (33) |
|
Tessellation templates for the Archimedean solids |
|
|
392 | (5) |
|
Activity 23.1 Surface area of Archimedean solids |
|
|
397 | (1) |
|
Worksheet 23.1 Surface area of Archimedean solids |
|
|
398 | (1) |
|
Activity 23.2 Volume of a truncated cube |
|
|
399 | (1) |
|
Worksheet 23.2 Volume of a truncated cube |
|
|
400 | (1) |
|
24 Tessellating Other Polyhedra |
|
|
401 | (24) |
|
|
|
401 | (16) |
|
|
|
417 | (5) |
|
Activity 24.1 Cross-sections of polyhedra |
|
|
422 | (1) |
|
Worksheet 24.1 Cross-sections of polyhedra |
|
|
423 | (2) |
|
25 Tessellating Other Surfaces |
|
|
425 | (12) |
|
Other surfaces to tessellate |
|
|
425 | (9) |
|
Tessellation templates for other surfaces |
|
|
434 | (1) |
|
Activity 25.1 Surface area and volume of cylinders and cones |
|
|
435 | (1) |
|
Worksheet 25.1 Surface area and volume of cylinders and cones |
|
|
436 | (1) |
| References |
|
437 | (4) |
| Glossary of Terms |
|
441 | (8) |
| Index |
|
449 | |
More info about Taylor & Francis e-books