Tessellations: Mathematics, Art and Recreation aims to present a comprehensive introduction to tessellations (tiling) at a level accessible to non-specialists. Additionally, it covers techniques, tips, and templates to facilitate the creation of mathematical art based on tessellations. Inclusion of special topics like spiral tilings and tessellation metamorphoses allows the reader to explore beautiful and entertaining math and art.
The book has a particular focus on 'Escheresque' designs, in which the individual tiles are recognizable real-world motifs. These are extremely popular with students and math hobbyists but are typically very challenging to execute. Techniques demonstrated in the book are aimed at making these designs more achievable. Going beyond planar designs, the book contains numerous nets of polyhedra and templates for applying Escheresque designs to them.
Activities and worksheets are spread throughout the book, and examples of real-world tessellations are also provided.
Introduces the mathematics of tessellations, including symmetry
Covers polygonal, aperiodic, and non-Euclidean tilings
Contains tutorial content on designing and drawing Escheresque tessellations
Highlights numerous examples of tessellations in the real world
Activities for individuals or classes
Filled with templates to aid in creating Escheresque tessellations
Treats special topics like tiling rosettes, fractal tessellations, and decoration of tiles
weitere Ausgaben werden ermittelt
Robert Fathauer has had a life-long interest in art, but studied Physics and Mathematics in college, going on to earn a Ph.D. from Cornell University in Electrical Engineering. For several years he was a researcher at the Jet Propulsion Laboratory in Pasadena, California. Long a fan of M.C. Escher, he began designing his own tessellations with lifelike motifs in the late 1980's. In 1993, he founded a business, Tessellations, to produce puzzles based on his designs. Over time, Tessellations has grown to include mathematics manipulatives, polyhedral dice, and books.
Dr. Fathauer's mathematical art has always been coupled with recreational math explorations. These include Escheresque tessellations, fractal tilings, and iterated knots. After many years of creating two-dimensional art, he has recently been building ceramic sculptures inspired by both mathematics and biological forms. Another interest of his is photographing mathematics in natural and synthetic objects, particularly tessellations. In addition to creating mathematical art, he's strongly committed to promoting it through group exhibitions at both the Bridges Conference and the Joint Mathematics Meetings.
Chapter 1. Introduction to Tessellations
Historical examples of tessellations. Tessellations in the world around us. Escheresque tessellations. Tessellations and recreational mathematics. Tessellations and mathematics education. Activity 1.1. Recognizing tessellations. Activity 1.2. Historical tessellations.
Chapter 2. Geometric Tessellations
Tiles. Angles. Vertices and edge-to-edge tessellations. Regular polygons and regular tessellations. Regular-polygon vertices. Prototiles. Semi-regular tessellations. Other types of polygons. General triangle and quadrilateral tessellations. Dual tessellations and Laves tessellations. Pentagon and hexagon tessellations. Stellation of regular polygons and star polygons. Star polygon tessellations. Regular polygon tessellations that are not edge to edge. Squared Squares. Modifying tessellations to create new tessellations. Circle packings and tessellations. Activity 2-1. Basic properties of tiles. Activity 2-2. Edge-to-edge tessellations. Activity 2-3. Classifying tessellations by their vertices.
Chapter 3. Symmetry and Transformations in Tessellations
Symmetry in objects. Transformations. Symmetry in tessellations. Frieze groups. Wallpaper groups. Heesch types and orbifold notation. Coloring of tessellations and symmetry. Activity 3-1. Symmetries in Objects. Activity 3-2. Transformations. Activity 3-3. Translational symmetry in tessellations. Activity 3-4. Rotational symmetry in tessellations. Activity 3-5. Glide reflection symmetry in tessellations.
Chapter 4. Tessellations in Nature
Modeling of natural tessellations. Crystals. Lattices. Cracking and crazing. Divisions in plants and animals. Coloration in animals. Voronoi tessellations. Activity 4-1. Modeling natural tessellations using geometric tessellations. Activity 4-2. Quantitative analysis of natural tessellations.
Chapter 5. Decorative and Utilitarian Tessellations
Tiling. Building blocks and coverings. Permeable barriers. Other divisions. Fiber arts. Games and puzzles. Islamic art and architecture. Spherical tessellations. Activity 5.1. Building with tessellations.
Chapter 6. Polyforms and Reptiles
Properties of polyforms. Tessellations of polyforms. The translation and Conway criteria. Other recreations using polyforms. Heesch number. Reptiles. Activity 6.1. Discovering and classifying polyforms.
Chapter 7. Rosettes and Spirals
Rhombus rosettes. Other rosettes. Logarithmic spiral tessellations. Archimedean spiral tessellations. Activity 7.1. Exploring spiral tessellations.
Chapter 8. Matching Rules, Aperiodic Tiles, and Substitution Tilings
Matching rules and tiling. Periodicity in tessellations. Penrose tiles. Other aperiodic sets and substitution tilings. Socolar-Taylor aperiodic monotile. Eschereque tessellations based on aperiodic tiles. Activity 8-1. Penrose tiles and the golden number.
Chapter 9. Fractal Tiles and Fractal Tilings
Tessellations of fractal tiles.Fractal Tessellations. Two-fold f-tilings based on segments of regular polygons. f-tilings based on kite-, dart-, and v-shaped prototiles. f-tilings based on polyforms. Miscellaneous f-tilings. Activity 9-1. Prototiles for fractal tilings.
Chapter 10. Non-Euclidean Tessellations
Hyperbolic tessellations. Spherical tessellations. Spherical symmetry groups. Activity 10-1. Non-Euclidean tessellations of regular polygons.
Chapter 11. Tips on Designing and Drawing Escheresque Tessellations
Drawing tessellations by hand. Using general computer graphics programs. Using a tessellations computer program. Mixing techniques. Tip 1: The outline of the tile should suggest the motif. Tip 2: The tiles should make orientational sense. Tip 3: Choose motifs that go together. Tip 4: Different motifs should be commensurately scaled. Tip 5: Use source material to get the details right. Tip 6: Stylize the design. Tip 7: Choose a style that fits your taste and abilities. Tip 8: Choose colors that suit your taste and bring out the tiles. Activity 11-1. Finding motifs for a tile shape. Activity 11-2. Refining a tile shape using translation. Activity 11-3. Refining a tile shape using glide reflection. Activity 11-4. Locating and using source material for real-life motifs.
Chapter 12. Special Techniques for Solving Design Problems
Technique 1: Distorting the entire tessellation. Technique 2: Breaking symmetries. Technique 3: Splitting a tile into smaller tiles. Technique 4: Splitting and moving vertices. Activity 12-1. Reshaping a tile by splitting and moving vertices.
Chapter 13. Escheresque Tessellations Based on Square Tiles
Creating a tessellation by hand. Square tile grids. Square tessellation grid. Templates 13.1 - 13.9. Activity 13-1. Creating an Eschersque tessellation with translational symmetry. Activity 13-2. Creating an Eschesque tessellation with rotational symmetry. Activity 13-3. Creating an Eschereque tessellation with glide reflection symmetry
Chapter 14. Escheresque Tessellations Based on Isosceles Right Triangle and Kite-shaped Tiles
Templates 14.1 - 14.5. Activity 14.1. Creating a tessellation based on right-triangle tiles. Activity 8-2. Creating a tessellation based on kite-shaped tiles.
Chapter 15. Escheresque Tessellations Based on Equilateral Triangle Tiles
Templates 15.1 - 15.4. Activity 15.1. Creating an equilateral-triangle-based tessellation with rotational symmetry. Activity 15.2. Creating an equilateral-triangle-based tessellation with glide reflection symmetry.
Chapter 16. Escheresque Tessellations Based on 60°-120° Rhombus Tiles
Templates 16.1 - 16.6. Activity 16.1. Creating a tessellation with bilaterally-symmetry tiles. Activity 16.2. Creating a tessellations with kaleidoscopic symmetry.
Chapter 17. Escheresque Tessellations Based on Hexagonal Tiles
Templates 17.1 - 17.3. Template 17.1. Tessellation with three-fold rotational symmetry. Template 17.2. Tessellation with six-fold rotational symmetry.
Chapter 18. Decorating Tiles to Create Knots and Other Designs
The role of combinatorics. Using tessellations to create knots and links. Creating iterated and fractal knots and links with fractal tilings. Other types of decorative. Activity 18.1. Creating symmetrical designs by decorating tessellations.
Chapter 19. Tessellation Metamorphoses and Dissections
Geometric Metamorphoses. Positive and negative space. Techniques for transitioning between Escheresque tessellation motifs. Tessellation dissections. Activity 19.1. Learning to draw a tessellation metamorphosis.
Chapter 20. Introduction to Polyhedra
Basic properties of polyhedra. Polyhedra in art and architecture. Polyhedra in nature. Tiling three dimensional space. Slicing 3-honeycombs to reveal plane tessellations. Activity 20.1. Identifying and characterizing polyhedra in nature. Activity 20.2. Identifying polyhedra in art and architecture.
Chapter 21. Adapting Plane Tessellations to Polyhedra
Nets of polyhedra. Restrictions on plane tessellations for use on polyhedra. Distorting plane tessellations to fit polyhedra. Designing and drawing tessellations for polyhedra using the templates. Coloring of tessellations on polyhedra. Tips on building the models. Activity 21.1. Representing solids using nets. Activity 21.2. Using transformations to apply a tessellation motif to a net.
Chapter 22. Tessellating the Platonic Solids
Background on the Platonic solids. Tetrahedron (1). Cube (2). Octahedron (3). Dodecahedron (4). Icosahedron (5). Tessellation templates for the Platonic solids. Activity 22.1. Attributes of the Platonic solids and Euler's formula. Activity 22.2. Drawing the Platonic solids.
Chapter 23. Tessellating the Archimedean Solids
Background on the Archimedean solids. Truncated tetrahedron (6). Cuboctahedron (7). Truncated Octahedron (8). Icosidodecahedron (9). Small rhombicuboctahedron (10). Great rhombicuboctahedron (11). Small rhomicosidodecahedron (12). Great rhomicosidodecahedron (13). Truncated Cube (14). Snub cube (15). Truncated dodecahedron (16). Snub dodecahedron (17). Truncated icosahedron (18). Tessellation templates for the Archimedean solids. Activity 23.1. Surface area of Archimedean solids. Activity 23.2. Volume of a truncated cube.
Chapter 24. Tessellating other Polyhedra
Other popular polyhedra. 14-gon prism (19). Hexagonal antiprism (20). Heptagonal pyramid (21). Rhombic dodecahedron (22). Rhombic triacontahedron (23). Stella octangula (24). Tessellation templates. Activity 24.1. Cross sections of polyhedra.
Chapter 25. Tessellating other Surfaces
Other surfaces to tessellate. Cylinder (25). Cone (26). Möbius strip (27), Tessellation templates. Activity 25.1. Surface area and volume of cylinders and cones.