Artifacts in black and white

or in any coloured light,

patterns knit by maddest hatters

defy the use of words and letters.

Design or game or maths as art ? -

I really don't know where to start

and how to tell you what I see;

it's mystic, mantra-like to me,

it's puzzling, difficult and vexing,

complex, astounding and perplexing.

And still - it could be after all

just your neighbour's bathroom wall.

Introduction

Chapter 1 Basic Notions

Chapter 2 Size-Alternating Tilings

Alternating tilings with different sizes of one shape.

Chapter 3 Squares of All Sizes - a Tricky Problem and a Tricky Solution

Chapter 4 Shape-Alternating Tilings of the Plane

Chapter 5 Alternating Tilings of the
Plane Using N-gons and M-gons

An introduction to mosaics made from m-sided and n-sided polygons

Chapter 6 The Hall of Fame - A Gallery of
Mosaics

Neat alternating tilings of the plane using n-gons and m-gons.

We try to find those with the smallest number of prototiles.

We give all solutions for pairs (n,m) where n, m < 11.

Chapter 7 The Ones That Did Not Quite Make
It.

Some interesting mosaics with a larger than minimum number of prototiles.

Chapter 8 The Toolbox

General construction methods for neat alternating tilings.

Upper bounds for the minimum number of prototiles for certain (n,m).

Chapter 9 The Existence Theorem of Neat Alternating
Tilings of the Plane

Using N-gons and M-gons.

We find upper bounds for the minimum number of prototiles.

Chapter 10 Symmetric Tiles

Neat alternating tesselations whose tiles are symmetric.

Chapter 11 Neat Alternating Tilings of the Plane
Using N-gons, M-gons and P-gons.

We find solutions for all triples (n,m,p) where n, m, p < 11.

Chapter 12 Neat Alternating Tilings of the Plane
Using N-gons, M-gons, ...., and Q-gons.

We prove a general existence theorem for neat alternating

tilings of the plane of type (n,m,p,...,q).

We find upper bounds for the minimum number of prototiles.

Chapter 13 Nowhere-Neat Tilings of the Plane

We try to find all mosaics with one or two prototiles.

Chapter 14 The Existence Theorem and the Extension
Theorem for Nowhere-Neat Tilings.

We find upper bounds for the minimum number of prototiles.

Chapter 15 Nowhere-Neat Alternating Tilings of the Plane

Literature

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