Algebraic Geometry - Bowdoin 1985, Part 2 by Bloch S. (ed.)

By Bloch S. (ed.)

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Example text

Is that up to isotopic reshaping there are just 17 plane crystallographic groups. As we said, you'll have to wait to see why the Magic Theorem is t rue. The next two chapters will discuss the versions of it that apply to patterns on t he sphere and to planar frieze patterns. Exercises We've told you how to find the signature of a pattern, but most people need some practice to get it right. Follow the steps on page 31 to identify the types of the patterns on pages 42-49. zj~ \>O 42 3. The Magic Theorem l.

The new patterns are somewhat simpler but have all the symmetries of the original; for our purposes all three patterns are identical. 1. A simpler pattern. 2. Another simple pattern. Repeating patterns like the ones studied in this book are made up of many symmetric copies of a motif. What we are studying here are the symmetries relating each motif to each other motif in the pattern. Describing Kaleidoscopes 17 Describing Kaleidoscopes Patterns whose symmetries are defined by reflections are called kaleidoscopic because of their similarity to the patterns seen in kaleidoscopes.

RO · ~ · ti. o~ ' ·""-·''·''·'-1"-w-- ~;.. (el Spiral bond has type 2222 (f) Zigzag running bond has type 22 * (g) Old Frist bond has type 22* (h) New Frist bond also has type 22* 44 3. The Magic Theorem 2. The placement of the dots changes the symmetry types of these patterns. Identify them. (el (f) Exercises 45 Check your answers. I ~~ 2 *22 (cl *X (d) 0 (el 2222 (f) 22 * 3. The Magic Theorem 46 3. Find the signatures of these patterns. Cal (b) Ccl Ce) (fl 47 Exercises Check your answers.