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.
Key features:
- 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
Author(s): Robert Fathauer
Publisher: CRC
Year: 2021
Contents
Preface
Intro to Tessellations
Historical Examples of Tessellations
Tessellations in the World around us
Escheresque Tessellations
Tessellations & recreational Mathematics
Tessellations & Mathematics Education
Recognizing Tessellations
Historical Tessellations
Geometric Tessellations
Tiles
Angles
Vertices & Edge-to-Edge Tessellations
Regular Polygons & regular Tessellations
Regular-Polygon Vertices
Prototiles
Semi-regular Tessellations
Other Types of Polygons
General triangle & quadrilateral Tessellations
Dual & Laves Tessellations
Pentagon & Hexagon Tessellations
Stellation of regular Polygons & Star Polygons
Star Polygon Tessellations
Regular Polygon Tessellations that are not Edge-to-Edge
Squared Squares
Modifying Tessellations to create new Tessellations
Circle Packings & Tessellations
Basic Properties of Tiles
Edge-to-Edge Tessellations
Classifying Tessellations by their Vertices
Symmetry & Transformations in Tessellations
Symmetry in Objects
Transformations
Symmetry in Tessellations
Frieze Groups
Wallpaper Groups
Heesch Types & Orbifold Notation
Coloring of Tessellations & Symmetry
Symmetry in Objects
Transformations
Translational Symmetry in Tessellations
Rotational Symmetry in Tessellations
Glide Refection Symmetry in Tessellations
Tessellations in Nature
Modeling of natural Tessellations
Crystals
Lattices
Cracking & Crazing
Divisions in Plants & Animals
Coloration in Animals
Voronoi Tessellations
Modeling natural Tessellations using geometric Tessellations
Quantitative Analysis of natural Tessellations
Decorative & Utilitarian Tessellations
Tiling
Building Blocks & Coverings
Permeable Barriers
Other Divisions
Fiber Arts
Games & Puzzles
Islamic Art & Architecture
Spherical Tessellations
Building with Tessellations
Polyforms & Reptiles
Properties of Polyforms
Tessellations of Polyforms
The Translation & Conway Criteria
Other Recreations using Polyforms
Heesch Number
Reptiles
Discovering & classifying Polyforms
Rosettes & Spirals
Rhombus Rosettes
Other Rosettes
Logarithmic spiral Tessellations
Archimedean spiral Tessellations
Exploring spiral Tessellations
Matching Rules, Aperiodic Tiles & Substitution Tilings
Matching Rules & Tiling
Periodicity in Tessellations
Penrose Tiles
Other aperiodic Sets & Substitution Tilings
Socolar-Taylor aperiodic monotile Escheresque Tessellations based on aperiodic Tiles
Penrose Tiles & the Golden Number
Fractal Tiles & Fractal Tilings
Tessellations of fractal Tiles
Fractal Tessellations
2-fold f-Tilings based on Segments of regular Polygons
f-Tilings based on kite-, dart- & v-shaped Prototiles
f-Tilings based on Polyforms
Miscellaneous f-Tilings
Prototiles for fractal Tilings
Non-Euclidean Tessellations
Hyperbolic Tessellations
Spherical Tessellations
Non- Euclidean Tessellations of regular Polygons
Escheresque Tessellations
Drawing Tessellations by Hand
Using general computer Graphics Programs
Using 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 style that fts your taste & abilities
Tip 8 Choose colors that suit your taste & bring out the tiles
Finding Motifs for Tile Shape
Refining Tile Shape using Translation
Refining Tile Shape using glide Refection
Locating & using Source Material for Real-Life Motifs
Special Techniques to solve Design Problems
Distorting the entire Tessellations
Breaking Symmetries
Splitting Tile into smaller Tiles
Splitting & moving Vertices
Reshaping Tile by splitting & moving Vertices
Escheresque Tessellations based on Squares
Creating Tessellation by Hand
Tessellation with translational Symmetry only
Tessellation with 2- & 4-fold rotational Symmetry
Tessellation with glide Refection Symmetry
Tessellation with simple Refection & glide Refection Symmetry
Tessellation with glide Refection Symmetry in 2 orthogonal Directions
Tessellation with 2-fold rotational & glide Refection Symmetry
Tessellation with 2 Motifs & glide Refection Symmetry
Tessellation with 2 different Tiles & Refection Symmetry in 1 Direction
Tessellation with 2 different Tiles & Refection Symmetry in 2 orthogonal Directions
Escheresque Tessellation with translational Symmetry
Escheresque Tessellation with rotational Symmetry
Escheresque Tessellation with glide Refection Symmetry
Escheresque Tessellations based on Isosceles Right Triangle & Kite-shaped Tiles
Right-Triangle Tessellation with 2-fold rotational Symmetry
Right-Triangle Tessellation with 2- & 4-fold rotational Symmetry
Right-Triangle Tessellation with 2- & 4-fold rotational & Refection Symmetry
Kite Tessellation with glide Refection Symmetry
Kite Tessellation with 2 Motifs & glide Refection Symmetry
Tessellation based on Right-Triangle Tiles
Tessellation based on Kite-shaped Tiles
Escheresque Tessellations based on Equilateral Triangle Tiles
Tessellation with 6-fold rotational Symmetry
Tessellation with rotational & glide Refection Symmetry
Tessellation with 2-fold rotational Symmetry only
Tessellation with translational Symmetry only
Equilateral Triangle-based Tessellation with rotational Symmetry
Equilateral Triangle-based Tessellation with glide Refection Symmetry
Escheresque Tessellations based on 60°-120° Rhombus Tiles
Tessellation with translational Symmetry only
Tessellation with refection Symmetry
Tessellation with 3-fold rotational Symmetry
Tessellation with glide Refection Symmetry
Rhombus Tessellation with rotational & glide Refection Symmetry
Tessellation with kaleidoscopic Symmetry
Tessellation with bilaterally Symmetry Tiles
Tessellation with kaleidoscopic Symmetry
Escheresque Tessellations based on Hexagonal Tiles
Tessellation with 3-fold rotational Symmetry
Tessellation with 6-fold rotational Symmetry
Tessellation based on Hexagons & Hexagrams
Tessellation based on hexagonal Tiles
Hexagon-based Tessellation with 2-, 3- & 6-fold rotational Symmetry
Decorating Tiles to create Knots & other Designs
Role of Combinatorics
Tessellations to create Knots & Links
Iterated & fractal Knots & Links with fractal Tilings
Other Types of decorative Graphics
Symmetrical Designs by decorating Tessellations
Tessellation Metamorphoses & Dissections
Geometric Metamorphoses
Positive & negative Space
Techniques for Transitioning btw Escheresque Tessellation Motifs
Tessellation Dissections
Draw Tessellation Metamorphosis
Intro to Polyhedra
Basic Properties of Polyhedra
Polyhedra in Art & Architecture
Polyhedra in Nature
Tiling 3D Space
Slicing 3-Honeycombs to reveal Plane Tessellations
Identifying & characterizing Polyhedra in Nature
Identifying Polyhedra in Art & Architecture
Adapting Plane Tessellations to Polyhedra
Nets of Polyhedra
Restrictions on Plane Tessellations for Use on Polyhedra
Distorting Plane Tessellations to fit Polyhedra
Designing & drawing Tessellations for Polyhedra using the Templates
Coloring of Tessellations on Polyhedra
Tips on building the Models
Representing Solids using Nets
Transformations to apply Tessellation Motif to Net
Background on the Platonic Solids
Tessellating Platonic Solids
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
Tessellating Archimedean Solids
Background on Archimedean Solids
Tessellation Templates for Archimedean Solids
Surface Area of Archimedean Solids
Volume of truncated Cube
Tessellating other Polyhedra
Other popular Polyhedra
Tessellation Templates
Cross-Sections of Polyhedra
Tessellating other Surfaces
Other Surfaces to tessellate
Tessellation Templates for other Surfaces
Surface Area & Volume of Cylinders & Cones
Refs
Glossary
Index
