Tessellations. Mathematics, Art and Recreation

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