 Proof of Exactly 7 Frieze Pattern Symmetries Background Notes

```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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