The Escher Problem Frieze Groups Frieze embroidery from
The Escher Problem
Frieze Groups • Frieze = embroidery from Friez, horizotal ornamented band (architecture). • We are interested in symmetry groups of such bends. There are 7 frieze groups. • We start with a rectangular stamp.
Transformations • • Translation Halfturn Vertical Reflection Glide-reflection
Seven Frieze Types • Groups (notation): • • • 11 (translations only) 12 (translations and halfturns) m 1 (translations and vertical reflections) 1 g (translations and glidereflections) mg (translations, halfturns, vertical reflections and glide reflections) • 1 m (translations and horizontal reflections) • mm (translations, halfturns, vertical reflections, glide reflections, horizontal reflections)
11
12
m 1
1 g
mg
1 m
mm
The Groups • Group Elements • • Identity I Translation T Halfturn R Glidereflection G Vertical mirror V Horizontal mirror H. Some relations: R 2 = V 2 = H 2 = I, RV = VR = H, . .
Subgroups • • (1) T (2) G (3) T, R (4) T, V (5) T, H (6) G, V (7) all(T, G, R, V, H) P P, R(P) B A S A, R(A) H C 1 C 1 £ D 1 D 1 D 1 £ D 1
Exercise b d q • Explain how the Frieze groups can be described by the four letters (aspects of the pattern): p • b, p, q, d.
Discrete Isometries • Each metric space M determines the group of distance preserving maps, isometries Iso(M). • A subgroup of Iso(M) is discrete, if any isometry in it either fixes an element of M or moves is far enough. • Discrete subgroups of I(R 2) fall into three classes: • 17 crystallographic groups • 7 frieze groups • finite groups (grups of rozettes).
Theorem of Leonardo da Vinci • The only finite groups of isometries in the plane are the group of rosettes (cyclic groups Cn and dihedral groups Dn).
The Escher problem • There is a square stamp with asymmetric motif. • By 90 degree rotations we obtain 4 different aspects. • By combining 4 aspects in a square 2 x 2 block, a translational unit is obtained that is used for plane tilnig. Such a tiling is called a pattern. • Question: What is the number of different patterns ? • Answer: 23.
Example
Recall Burnside Lemma. • Let G be a group, acting on space S. • For g 2 G let fix(g) denote the number of points form S fixed by g. • Let N denote the number of orbits of G on S. • Then:
Application • Determine the group (G) and the sapce (S). • Pattern can be translated and rotated. • Basic observatrion: • Instead of pattern consired the block (signature). • Group operations: • H – horizontal translation • V – vertical translation • R – 90 degrees rotation.
(Abstract) group G hv hvr 2 hvr 3 h hr hr 2 hr 3 1 r r 2 r 3 v vr vr 2 vr 3 • h 2 = v 2 = r 4 = 1. • hv = vh • hr = rv
Space S • Space S consists of 4 4 = 256 signatures. • Count fix(g) for g 2 G. • For instance: • fix(1) = 256. • fix(r) = fix(r 3) = 4. • fix(h) = fix(v) = 16. • By Burnside Lemma we obtain N = 23.
Homework • Consider the Escher problem with the motiff on the left. • H 1. Determine the abstract group and its Cayley graph. • H 2. What is the number of different patterns? • H 3. What is the number of different patterns if we reflections are allowed? • H 4. What is the number of different patterns in the original Escher problem if reflections are allowed?
1 -dimensional Escher problem • Rectangular Asymmetric Motiff • Only Two Aspects. • 1 x n block (signature) • Determine the number of patterns: • Two more variations: • II two motiffs(mirror images) • III reflections are allowed.
Space S • Space S consists of 2 2. . 2 = 2 n signatures. • Count fix(g) for g 2 G. • For instance: • fix(1) = 256. • fix(r) = fix(r 3) = 4. • fix(h) = fix(v) = 16. • By Burnside Lemma we obtain N = 23.
Solution for the basic case • where g(n) = 0 for odd n and for even n:
Program in Mathematica • f[n_] : = (Apply[Plus, Map[Euler. Phi[#] 2^(n/#)&, Divisors[n]]] + If[Odd. Q[n], 0, (n/2) 2^(n/2)])/(2 n) • f[n_, m_] : = (Apply[Plus, Map[Euler. Phi[#] (2 m)^(n/#)&, Divisors[n]]] + If[Odd. Q[n], 0, (n/2) (2 m)^(n/2)])/(2 n) • g[n_] : = (Apply[Plus, Map[If[Odd. Q[#], 1, 2] Euler. Phi[#] 4^(n/#)&, Divisors[n]]] + If[Odd. Q[n], 0, (n) 4^(n/2)])/(4 n)
Results for a tape • • • n 1 2 3 4 5 6 7 8 9 10 I II 1 2 2 6 2 12 4 39 4 104 9 366 10 1172 22 4179 30 14572 62 52740 III 1 4 6 23 52 194 586 2131 7286 26524
Exercise • Determine the Cayley graph of each of the Frieze groups. • Determine the crystallographic groups that may arise from the classical Escher problem
17 CRYSTALLOGRAPHIC GROUPS 6 -števna os? zrcaljenje? p 6 mm 4 -števna os? p 6 zrcaljenje? p 4 Zrcala v 4 smereh? p 4 mm 3 -števna os? 3 -osi na zrcalih? p 4 gm zrcaljenje? p 31 m p 3 m 1 Drugo zrcalo? Rombska mreža? c 2 mm p 3 2 -števna os? p 2 mm p 2 mg Rombska mreža? glide? c 2 gg zrcaljenje? p 2 cm pm glide? pg p 1
p 1 • p 1 = <a, b|ab=ba>
p 2 • p 2 = <a, b, c| b 2=c 2=(ab)2=(ac)2=1>
pm • pm = <a, b, c| b 2=c 2=1, ab=ba, ac=ca>
pg • pg = <a, b|ab=ba-1>
cm • cm = <a, b, c| b 2=c 2=1, ab=ca>
p 2 mm • p 2 mm = <a, b, c, d| a 2=b 2= c 2=d 2= 1, (ab)2=(ad)2=(bc)2=(cd)2=1>
p 2 mg • p 2 mg = <a, b, c| b 2 = c 2 = 1, (ab)2=(ac)2=1>
p 2 gg • p 2 gg = <a, b| (ab)2=1>
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