Doris Schattschneider

Fascinating and unique book. Large format, with two column text and numerous black and white illustrations. The nets for the models in the back are all in color. The text is fairly mathematical and not directed to children.

Detailed notes:

Part I: In Three Dimensions: Extensions of M.C. Escher's Art

Chapter 1: The Geometric Solids
Platonic and Archimedean solids. Special focus on the dodecahedron and the cuboctahedron. The kaleidocycle, a ring of joined tetrahedra, in introduced. The IsoAxis is explained, but for some reason we don't get a net for that, perhaps because it is trademarked. I should make one for myself. The author, a mathematician, chose to study the possible objects that could be constructed by modifications to the IsoAxis net. Modifying the net so that the triangles are equilateral, rather than right-angled isosceles yields the pattern of the hexagonal kaleidocycles in the book. A slightly less extreme modification, and 8 instead of six tetrahedra yields the square kaleidocycle. At the time the author was studying the variations of the IsoAxis she was also studying the mathematics of repeating patterns. It occurred to her that these things could be combined.

Chapter 2: The Repeating Designs
Discussions of repeating designs. Translation, rotation, reflection. We call the study of all these things transformation geometry. Essentially, you can think of Escher repeating designs as unusually shaped interlocking tiles.

Chapter 3: Decorating the Solids
The platonic solids only have three types of faces: triangle, square, pentagon. It is possible to tile the plane w/ triangles and squares, but this does not entirely solve the problem of tiling the solids with those faces, as the patterns do not match up in the same way. The cubeoctahedron has both square and triangular faces, however Escher's "Circle Limit III" gives an example of a tiling on a hyperbolic surface with squares and triangles, so that gave the author a good start. It is not possible to tile the plane w/ pentagons.

Part II: Notes on the Models
Chapter 1: The tetrahedron
This one was super easy because the reptile design that was chosen had six-fold rotational symmetry. Thus no alteration had to be made in the pattern to make sure that it was not disrupted when wrapped around the tetrahedron.

Chapter 2: The octahedron
This one uses Escher's "Three Elements" and seems like it must have been remarkably straightforward to construct.… (plus d'informations)

Indeholder "In drei Dimensionen", "Die geometrischen Körper", "Die Kaleidozyklen", "Zyklische Flächenaufteilungen", "Oberflächengestaltung bei den geometrischen Körpern", "Oberflächengestaltung bei den Kaleidozyklen", "Die farbliche Gestaltung der Entwürfe", "Einzelheiten zu den Modellen", "Geometrische Körper", "Sechseckige Kaleidozyklen", "Quadratische Kaleidozyklen", "Verdrillte Kaleidozyklen", "Weiterführende Literatur", "Bauanleitungen für die Modelle".

En kombination af en bog om Eschers designs og nogle skabeloner af karton som kan foldes til polyedre og kaleidocykler, som er lange kæder af polyedre, som kan sættes sammen til en cirkel.

Ganske underholdende! Doris Schattschneider er omtalt i "Mathematical Recreations: A Collection in Honor of Martin Gardner". Og har skrevet "In Praise of Amateurs" i "The Mathematical Gardner".… (plus d'informations)