Spain 2007Ralph Abraham Talk #4

Islamic Patterns

•Ref: Syed Jan Abas & Amer Shaker Salman

•Symmetries of Islamic Geometrical Patterns

•Singapore: World Scientific, 1995

•pp. 57-66

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

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)

Symmetries

•A symmetry of a pattern (subset) P of E2:

•an isometry of E2

•that maps P exactly onto itself

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

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

Repeating Patterns

•Crystallographic Theorem:

•the only rotational symmetries are

•2, 3, 4, or 6-fold

Crystallographic Groups

•Theorem:

•There are only 17.

•p6m, p4m, cmm, pmm, and p6 (later ...)

•are the most common symmetries

•of Islamic patterns