Fríz symmetries

11

m1

1m

1g

12

mg

mm

11 - translation

m1 - translation, reflection for the perpendicular line of the translation direction,
 reflection, distance of the reflection lines are half of the translation length

1m - translation, refletion to the paralel line of the direction of the translation 
and translating reflection

1g - translating and translating reflection, dashed arrow 
denotes translating reflection, the length is half of the translation

12 – translation and 180°-degree rotation, the centers of the 180°-degree rotation
is the half of the translation length

mg - translation, 180°-degree rotation, reflection to a perpendicular line of the 
translation direction is a translating reflection

mm – translation, 180°-degree rotation, reflection to a perpendicular line of the translation direction, reflection to a paralel line 
of the translation and translating reflection, the center of 180°-degree rotation is in the cross of the reflection lines

Animation

h = 0 (slider from 0 to 2, step 0.1)
a = true
b = false
c = false
d = false
e = false
f = false
g = false
A = Point(yAxis)
B = (2, 1)
C: (h, 2 + h / 2)
A1, B1, C1, D1 = Polygon(A, B, C)
u = Vector((0, 0), (3, 0))
A' = Translate(A, u)
B' = Translate(B, u)
C' = Translate(C, u)
A2, B2, C2, D2 = Polygon(A', B', C')
A'' = Translate(A', u)
B'' = Translate(B', u)
C'' = Translate(C', u)
A3, B3, C3, D3 = Polygon(A'', B'', C'')
A''' = Translate(A'', u)
B''' = Translate(B'', u)
C''' = Translate(C'', u)
A4, B4, C4, D4 = Polygon(A''', B''', C''')
D = Translate(A''', u)
E = Translate(B''', u)
F = Translate(C''', u)
A5, B5, C5, D5 = Polygon(D, E, F)
D' = Translate(D, u)
E' = Translate(E, u)
F' = Translate(F, u)
A6, B6, C6, D6 = Polygon(D', E', F')
D'' = Translate(D', u)
E'' = Translate(E', u)
F'' = Translate(F', u)
A7, B7, C7, D7 = Polygon(D'', E'', F'')
D''' = Translate(D'', u)
E''' = Translate(E'', u)
F''' = Translate(F'', u)
A8, B8, C8, D8 = Polygon(D''', E''', F''')
B10, C10, D10, E10 = Polygon(A, B, C)
A10 = Segment((2.5, 0), (2.5, 3))
A11 = Translate(A10, u)
A12 = Translate(A11, u)
A13 = Translate(A12, u)
A14 = Translate(A13, u)
A15 = Translate(A14, u)
A16 = Translate(A15, u)
A'_1 = Mirror(A, A10)
B'_1 = Mirror(B, A10)
C'_1 = Mirror(C, A10)
B11, C11, D11, E11 = Polygon(A'_1, B'_1, C'_1)
G = Mirror(A'_1, A11)
H = Mirror(B'_1, A11)
I = Mirror(C'_1, A11)
B12, C12, D12, E12 = Polygon(G, H, I)
G' = Mirror(G, A12)
H' = Mirror(H, A12)
I' = Mirror(I, A12)
B13, C13, D13, E13 = Polygon(G', H', I')
G'' = Mirror(G', A13)
H'' = Mirror(H', A13)
I'' = Mirror(I', A13)
B14, C14, D14, E14 = Polygon(G'', H'', I'')
G''' = Mirror(G'', A14)
H''' = Mirror(H'', A14)
I''' = Mirror(I'', A14)
B15, C15, D15, E15 = Polygon(G''', H''', I''')
J = Mirror(G''', A15)
K = Mirror(H''', A15)
L = Mirror(I''', A15)
B16, C16, D16, E16 = Polygon(J, K, L)
N = Point(A12)
O = (5.5, 3.48)
P = (11.5, 3.48)
w = Vector(O, P)
Q = (5, 3.5)
R = (8, 3.5)
v = Vector(Q, R)
A18, B18, C18, D18 = Polygon(A, B, C)
A19, B19, C19, D19 = Polygon(A', B', C')
A20, B20, C20, D20 = Polygon(A'', B'', C'')
A21, B21, C21, D21 = Polygon(A''', B''', C''')
A22, B22, C22, D22 = Polygon(D, E, F)
A23, B23, C23, D23 = Polygon(D', E', F')
A24, B24, C24, D24 = Polygon(D'', E'', F'')
A25, B25, C25, D25 = Polygon(D''', E''', F''')
S = Intersect(xAxis, yAxis)
T = Point(xAxis)
j = Line(S, T)
A'_2 = Mirror(A, j)
B'_2 = Mirror(B, j)
C'_2 = Mirror(C, j)
E18, F18, G18, H18 = Polygon(A'_2, B'_2, C'_2)
A''_1 = Mirror(A', j)
B''_1 = Mirror(B', j)
C''_1 = Mirror(C', j)
E19, F19, G19, H19 = Polygon(A''_1, B''_1, C''_1)
A'''_1 = Mirror(A'', j)
B'''_1 = Mirror(B'', j)
C'''_1 = Mirror(C'', j)
E20, F20, G20, H20 = Polygon(A'''_1, B'''_1, C'''_1)
U = Mirror(A''', j)
V = Mirror(B''', j)
W = Mirror(C''', j)
E21, F21, G21, H21 = Polygon(U, V, W)
D'_1 = Mirror(D, j)
E'_1 = Mirror(E, j)
F'_1 = Mirror(F, j)
E22, F22, G22, H22 = Polygon(D'_1, E'_1, F'_1)
D''_1 = Mirror(D', j)
E''_1 = Mirror(E', j)
F''_1 = Mirror(F', j)
E23, F23, G23, H23 = Polygon(D''_1, E''_1, F''_1)
D'''_1 = Mirror(D'', j)
E'''_1 = Mirror(E'', j)
F'''_1 = Mirror(F'', j)
E24, F24, G24, H24 = Polygon(D'''_1, E'''_1, F'''_1)
Z = Mirror(D''', j)
A_1 = Mirror(E''', j)
B_1 = Mirror(F''', j)
C_1 = Mirror(D''', j)
D_1 = Mirror(E''', j)
E_1 = Mirror(F''', j)
E25, F25, G25, H25 = Polygon(C_1, D_1, E_1)
k = false
A27, B27, C27, D27 = Polygon(A, B, C)
A28, B28, C28, D28 = Polygon(Translate[A, u], Translate[B, u], Translate[C, u])
A29 = Translate(A28, u)
A30 = Translate(A29, u)
A31 = Translate(A30, u)
A32 = Translate(A31, u)
A33 = Translate(A32, u)
A34 = Translate(A33, u)
E28 = Mirror(A28, y = 2)
E32 = Mirror(A32, y = 2)
E30 = Mirror(A30, y = 2)
E34 = Mirror(A34, y = 2)
F_1 = (2, 2)
G_1 = (5, 2)
l = Vector(F_1, G_1)
B36, C36, D36, E36 = Polygon(A, B, C)
A36 = (2.5, 2)
A37 = Translate(A36, u)
A38 = Translate(A37, u)
A39: (2 * A38) - A37
A40: (2 * A39) - A38
A41: (2 * A40) - A39
A42: (2 * A41) - A40
A'_3 = Mirror(A, A36)
B'_3 = Mirror(B, A36)
C'_3 = Mirror(C, A36)
B37, C37, D37, E37 = Polygon(A'_3, B'_3, C'_3)
H_1 = Mirror(A'_3, A37)
I_1 = Mirror(B'_3, A37)
J_1 = Mirror(C'_3, A37)
B38, C38, D38, E38 = Polygon(H_1, I_1, J_1)
K_1 = Mirror(H_1, A38)
L_1 = Mirror(I_1, A38)
M_1 = Mirror(J_1, A38)
B39, C39, D39, E39 = Polygon(K_1, L_1, M_1)
N_1 = Mirror(K_1, A39)
O_1 = Mirror(L_1, A39)
P_1 = Mirror(M_1, A39)
B40, C40, D40, E40 = Polygon(N_1, O_1, P_1)
Q_1 = Mirror(N_1, A40)
R_1 = Mirror(O_1, A40)
S_1 = Mirror(P_1, A40)
B41, C41, D41, E41 = Polygon(Q_1, R_1, S_1)
T_1 = Mirror(Q_1, A41)
U_1 = Mirror(R_1, A41)
V_1 = Mirror(S_1, A41)
B42, C42, D42, E42 = Polygon(T_1, U_1, V_1)
W_1 = Mirror(T_1, A42)
Z_1 = Mirror(U_1, A42)
A_2 = Mirror(V_1, A42)
B43, C43, D43, E43 = Polygon(W_1, Z_1, A_2)
B45, C45, D45, E45 = Polygon(A, B, C)
A46: A37
A48: A39
A45 = Segment((2.5, 0), (2.5, 3))
A47 = Translate(A45, Vector[(2 * u)])
A49 = Translate(A47, Vector[(2 * u)])
A51 = Translate(A49, Vector[(2 * u)])
A50: A41
A'_4 = Mirror(A, A45)
B'_4 = Mirror(B, A45)
C'_4 = Mirror(C, A45)
B46, C46, D46, E46 = Polygon(A'_4, B'_4, C'_4)
B_2 = Mirror(A'_4, A46)
C_2 = Mirror(B'_4, A46)
D_2 = Mirror(C'_4, A46)
B47, C47, D47, E47 = Polygon(B_2, C_2, D_2)
E_2 = Mirror(B_2, A47)
F_2 = Mirror(C_2, A47)
G_2 = Mirror(D_2, A47)
B48, C48, D48, E48 = Polygon(E_2, F_2, G_2)
H_2 = Mirror(E_2, A48)
I_2 = Mirror(F_2, A48)
J_2 = Mirror(G_2, A48)
B49, C49, D49, E49 = Polygon(H_2, I_2, J_2)
K_2 = Mirror(H_2, A49)
L_2 = Mirror(I_2, A49)
M_2 = Mirror(J_2, A49)
B50, C50, D50, E50 = Polygon(K_2, L_2, M_2)
N50 = 0
B51, C51, D51, E51 = Polygon(Mirror[K_2, A50], Mirror[L_2, A50], Mirror[M_2, A50])
B52 = Mirror(B51, A51)
m = Vector((2.5, 2), (8.5, 2))
n = Vector((2.5, 3.5), (14.5, 3.5))
B54, C54, D54, E54 = Polygon(A, B, C)
A54 = Segment((2.5, -3), (2.5, 3))
A55 = Translate(A54, u)
A56 = Translate(A55, u)
A57 = Translate(A56, u)
A58 = Translate(A57, u)
A59 = Translate(A58, u)
A60 = Translate(A59, u)
A'_5 = Mirror(A, A54)
B'_5 = Mirror(B, A54)
C'_5 = Mirror(C, A54)
A'_6 = Mirror(A, A54)
B'_6 = Mirror(B, A54)
C'_6 = Mirror(C, A54)
B55, C55, D55, E55 = Polygon(A'_6, B'_6, C'_6)
B56, C56, D56, E56 = Polygon(Mirror[A'_6, A55], Mirror[B'_6, A55], Mirror[C'_6, A55])
B57 = Mirror(B56, A56)
B58 = Mirror(B57, A57)
B59 = Mirror(B58, A58)
B60 = Mirror(B59, A59)
B61 = Mirror(B60, A60)
A'_7 = Mirror(A, j)
B'_7 = Mirror(B, j)
C'_7 = Mirror(C, j)
M = Mirror(A'_6, j)
N_2 = Mirror(B'_6, j)
O_2 = Mirror(C'_6, j)
F56 = Mirror(B56, j)
F57 = Mirror(B57, j)
F58 = Mirror(B58, j)
F59 = Mirror(B59, j)
F60 = Mirror(B60, j)
F61 = Mirror(B61, j)
J54 = (2.5, 0)
J55 = Translate(J54, u)
J56 = Translate(J55, u)
J57: (2 * J56) - J55
J58: (2 * J57) - J56
J59: (2 * J58) - J57
J60: (2 * J59) - J58
P_2 = Mirror(A'_6, j)
Q_2 = Mirror(B'_6, j)
R_2 = Mirror(C'_6, j)
A'_8 = Mirror(A, j)
B'_8 = Mirror(B, j)
C'_8 = Mirror(C, j)
F54, G54, H54, I54 = Polygon(A'_8, B'_8, C'_8)
S_2 = Mirror(A'_6, j)
T_2 = Mirror(B'_6, j)
U_2 = Mirror(C'_6, j)
F55, G55, H55, I55 = Polygon(S_2, T_2, U_2)
I57 = 3
I58 = 3
I59 = 3
I60 = 3
I61 = 3
I62 = 3
I63 = 3
I64 = 3
I65 = 3
I66 = 3
I67 = 3
I68 = 3
I69 = 3
I70 = 3
I71 = 3
I72 = 3
I73 = 3
I74 = 3
I75 = 3
I76 = 3
I77 = 3
I78 = 3
I79 = 3
I80 = 3
I81 = 3
I82 = 3
I83 = 3
I84 = 3
I85 = 3
I86 = 3
I87 = 3
I88 = 3
I89 = 3
I90 = 3
I91 = 3
I92 = 3
I93 = 3
I94 = 3
I95 = 3
I96 = 3
I97 = 3
I98 = 3
I99 = 3
I100 = 3
I101 = 3
I102 = 3
I103 = 3
I104 = 3
I105 = 3
I106 = 3
I107 = 3
I108 = 3
I109 = 3
I110 = 3
I111 = 3
I112 = 3
I113 = 3
I114 = 3
I115 = 3
I116 = 3
I117 = 3
I118 = 3
I119 = 3
I120 = 3
I121 = 3
I122 = 3
I123 = 3
I124 = 3
I125 = 3
I126 = 3
I127 = 3
I128 = 3
I129 = 3
I130 = 3
I131 = 3
I132 = 3
I133 = 3
I134 = 3