NEW MOSAICS



Geomantra

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.


Contents:

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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