Spain 2007
Ralph Abraham Talk #4
Islamic Patterns
(This is the text of the PPT.)
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Text: Syed Jan Abas  & Amer Shaker Salman,
	Symmetries of Islamic Geometrical Patterns,
	Singapore: World Scientific, 1995; pp. 57-66.
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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
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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)
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Symmetries
A symmetry of a pattern (subset) P of E2:
an isometry of E2  that maps P exactly onto itself.
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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
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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
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Repeating Patterns
Crystallographic Theorem: 
The only rotational symmetries are 2, 3, 4, or 6-fold.
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Crystallographic Groups
Theorem:  There are only 17.
p6m, p4m, cmm, pmm, and p6 are the most common symmetries of Islamic patterns.
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