1 / 9

Exploring Symmetries in Islamic Patterns: A Study of Geometrical Foundations

This paper delves into the intricate symmetries of Islamic geometrical patterns, grounded in Euclidean principles. It discusses the definition of points and distances in the Euclidean plane and categorizes isometries into translations, rotations, reflections, and glide reflections. The work elaborates on how symmetries can be represented as groups, specifically focusing on the dihedral group and its implications in repeating patterns. Additionally, it highlights the crystallographic theorem, revealing the finite number of symmetries prevalent in Islamic art—specifically the 17 distinct groups that define these patterns.

dora-sellers
Download Presentation

Exploring Symmetries in Islamic Patterns: A Study of Geometrical Foundations

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Spain 2007 • Ralph Abraham Talk #4

  2. Islamic Patterns • Ref: Syed Jan Abas & Amer Shaker Salman • Symmetries of Islamic Geometrical Patterns • Singapore: World Scientific, 1995 • pp. 57-66

  3. The Euclidean Plane • Descartes, Geometry + Algebra, ca 1630 AD • A point in E2 is defined by coordinates (x, y) • Distance from (xa, ya) to (xb, yb) = • Square root of sum of squares

  4. Isometries of E2 • An isometry is a function from E2 to itself • preserving distances • Theorem: there are only four types: • translation, rotation, reflection, and • glide (translation plus reflection)

  5. Symmetries • A symmetry of a pattern (subset) P of E2: • an isometry of E2 • that maps P exactly onto itself

  6. Symmetry Groups • The symmetry group of a pattern is the set of all symmetries of the pattern • It as a group: • closed under composition • composition is associative • Each symmetry has an inverse • There is an identity

  7. The Dihedral Group • I = Identity, R1 = rotate 90 degrees CCW • M1 = flip 42, M2 = flip DB, etc • D8 = {I, R1, R2, R3, M1, M2, M3, M4} • combination T2.T1 means apply T2 after T1

  8. Repeating Patterns • Crystallographic Theorem: • the only rotational symmetries are • 2, 3, 4, or 6-fold

  9. Crystallographic Groups • Theorem: • There are only 17. • p6m, p4m, cmm, pmm, and p6 (later ...) • are the most common symmetries • of Islamic patterns

More Related