Proof of Exactly 7 Frieze Pattern Symmetries Background Notes • We adopt a “product notation” to describe the composition of any two or more transformations, so, for example, T H denotes the composition of a translation followed by a reflection in a horizontal line. • We adopt the symbol I to denote the identity transformation, that is, the transformation where every point in the image corresponds to the same point in the pre-image. • We let T denote the minimal translation to the right where the image corresponds to the pre-image. We let T −1 denote the minimal translation to the left where the image corresponds to the pre-image. • By definition, every frieze pattern has at least the following symmetries: ...T −3 , T −2 , T −1 , I, T, T 2 , T 3 , ... Where T n stands for the composition of n copies of T , while T −n stands for the composition of n copies of T −1 PROOF Based on the availability of 4 rigid transformations, any frieze pattern has available only 5 types of transformational symmetry: T - Translational Symmetry H - Reflection Symmetry in a Horizontal Line V - Reflection Symmetry in a Vertical Line R - 180 Degree Rotational Symmetry G - Glide Reflection Symmetry Every frieze pattern, by definition, has translational symmetry (T). So, in theory, there are 24 = 16 possible combinations of the other available symmetries: H, V, R and G. These combinations are shown the the following table. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 T √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ H R V G √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ Type T TH TR TV TG THR THV THG TRV TRG TVG THRV THRG THVG TRVG THRVG Of the 16 combinations, 7 can be shown to exist by example. Type T Type THG Type TR Type TRVG Type TV Type THRVG Type TG To show that exactly 7 such combinations exist, we must rule out the remaining 9. This can be accomplished in 5 steps: 1. If both T and H are present, then, G must also be present because T H = HT = G. That is, the composition of a translation and a horizontal reflection (in either order) results in a glide-reflection. We can rule out types T H, T HR, and T HRV 2. If both H and V are present, then, R must also be present because V H = HV = R. That is, the composition of a vertical reflection followed by a horizontal reflection (in either order) results in a 180 degree rotation about the point that the two lines of reflection share. We can rule out types T HV and T HV G. 3. If both R and G are present, then, V must be present because: RGT −1 = (V H)(HT )T −1 = V (HH)(T T −1 ) = V II = V That is, composing a 180 degree rotation with a glide reflection and a left-ward translation results in a reflection in a vertical line. We can rule out types T RG and T HRG 4. If both V and G are present, then, R must be present because: V GT −1 = V (HT )T −1 = V H(T T −1 ) = V HI = V H = R That is, composing vertical reflection with a glide reflection and a left-ward translation results in a reflection in a 180 degree rotation. We can rule out type T V G. 5. If both R and V are present, then, G must be present because: T RV = T (HV )V = T H(V V ) = T HI = T H = G That is, composing a translation and a 180 degree rotation and a reflection in a vertical line results in a glide-reflection. We can rule out type T RV . The following table summarizes our result. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 T H R V √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ 2. SYMMETRY IN GEOMETRY √ √ √ √ G Type Constructible? T Yes TH No (1) TR Yes TV Yes √ TG Yes THR No (1) THV No (2) √ THG Yes TRV No (5) √ TRG No (3) √ TVG No (4) THRV No (1) √ THRG No (3) √ THVG No (2) √ TRVG 2.4. Crystals, Friezes and Wallpapers Yes √ THRVG Yes The mathematician John Conway — who often has his own spin on certain mathematical theorems and proofs — has coined his own set of names for the types of frieze patterns. HOP (none) JUMP (HG) SIDLE (V) SPINNING SIDLE (VRG) (R) JUMP (HVRG) mathematician John Conway. Another representation ofSPINNING our HOP result provided bySPINNING the famous STEP (G) The following diagrams should hopefully explain Conway’s rather strange nomenclature for the frieze patterns. 4

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