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211 lines (190 loc) · 8.99 KB
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# This file stores the representations and automatic structures for a number of groups
# we're interested in testing
# Graphs should be saved as {0: {a: A, b: B, ...}, ... , n: {...}} where a needs to map into 0 by A and so on.
import numpy as np
def group(name = 'cyclic', *args):
'''Returns the automatic structure of the desired group
name: cyclic, triangle, surface
*args: orders for cyclic group orders'''
if name == 'cyclic':
generators = []
for order in args:
theta = np.pi / order
generators += [np.array([[np.cos(theta), -np.sin(theta)], [np.sin(theta), np.cos(theta)]])]
l = 1.5 # Interesting: if we increase this, it decreases the number of search steps
C = np.array([[l, 0], [0, 1/l]])
A, B = generators
A = np.linalg.inv(C) @ A @ C
B = C @ B @ np.linalg.inv(C)
graph = {0: {1: B},
1: {0: A, 2: B},
2: {0: A}}
return graph
elif name == 'triangle':
# Triangle Group <a, b, c | a^2 = b^2 = c^2 = 1, (ab)^3 = (cb)^3 = (ac)^4 = 1>
A = np.array([[ 0.923879532511287, -0.217284326304659],
[-0.673986071141597, -0.923879532511287]])
B = np.array([[0., 1.219308768593441],
[0.820136806818482, 0. ]])
C = np.array([[ 0.923879532511287, 0.21728432630466 ],
[ 0.673986071141597, -0.923879532511286]])
# graph = {0: {},
# 1: {2: A, 3: B, 4: C},
# 2: {5: B, 6: C},
# 3: {7: A, 8: C},
# 4: {9: A, 10: B},
# 5: {11: A, 8: C},
# 6: {12: A, 10: B},
# 7: {11: B, 6: C},
# 8: {9: A, 13: B},
# 9: {5: B, 14: C},
# 10: {7: A, 13: C},
# 11: {15: C},
# 12: {5: B, 16: C},
# 13: {17: A},
# 14: {16: A, 10: B},
# 15: {12: A, 13: B},
# 16: {18: B},
# 17: {11: B, 14: C},
# 18: {11: A, 13: C}}
graph = {0: {1: A, 2: B, 3: C},
1: {2: B, 3: C},
2: {4: A, 3: C},
3: {5: A, 6: B},
4: {3: C},
5: {2: B, 7: C},
6: {8: A},
7: {6: B},
8: {9: C},
9: {10: A, 6: B},
10: {11: B},
11: {4: A, 12: C},
12: {13: A},
13: {4: B, 7: C}}
return graph
elif name == 'surface':
# l = 2 * (2 ** (1/4)) * np.cos(np.pi/8) + (2 ** (1/2)) + 1
# X = np.array([[l, 0], [0, 1/l]])
# R = lambda theta: np.array([[np.cos(theta), -np.sin(theta)], [np.sin(theta), np.cos(theta)]])
# a = X
# b = R(-np.pi/8) @ X @ R(np.pi/8)
# c = R(-np.pi/4) @ X @ R(np.pi/4)
# d = R(-3*np.pi/8) @ X @ R(3*np.pi/8)
# A = np.linalg.inv(a)
# B = np.linalg.inv(b)
# C = np.linalg.inv(c)
# D = np.linalg.inv(d)
# graph = {
# 0: {},
# 1: {2: d, 3: D, 4: c, 5: C, 6: b, 7: B, 8: a, 9: A},
# 2: {2: d, 4: c, 5: C, 6: b, 7: B, 8: a, 9: A},
# 3: {3: D, 4: c, 5: C, 6: b, 7: B, 8: a, 9: A},
# 4: {2: d, 10: D, 4: c, 6: b, 11: B, 8: a, 9: A},
# 5: {12: d, 3: D, 5: C, 13: b, 7: B, 8: a, 9: A},
# 6: {2: d, 3: D, 4: c, 14: C, 6: b, 8: a, 15: A},
# 7: {2: d, 3: D, 16: c, 5: C, 7: B, 17: a, 9: A},
# 8: {18: d, 3: D, 4: c, 5: C, 6: b, 19: B, 8: a},
# 9: {2: d, 20: D, 4: c, 5: C, 21: b, 7: B, 9: A},
# 10: {3: D, 4: c, 5: C, 6: b, 7: B, 8: a, 22: A},
# 11: {2: d, 3: D, 16: c, 5: C, 7: B, 23: a, 9: A},
# 12: {2: d, 4: c, 5: C, 6: b, 7: B, 24: a, 9: A},
# 13: {2: d, 3: D, 4: c, 14: C, 6: b, 8: a, 25: A},
# 14: {26: d, 3: D, 5: C, 13: b, 7: B, 8: a, 9: A},
# 15: {2: d, 27: D, 4: c, 5: C, 21: b, 7: B, 9: A},
# 16: {2: d, 28: D, 4: c, 6: b, 11: B, 8: a, 9: A},
# 17: {29: d, 3: D, 4: c, 5: C, 6: b, 19: B, 8: a},
# 18: {2: d, 4: c, 30: C, 6: b, 7: B, 8: a, 9: A},
# 19: {2: d, 3: D, 31: c, 5: C, 7: B, 17: a, 9: A},
# 20: {3: D, 32: c, 5: C, 6: b, 7: B, 8: a, 9: A},
# 21: {2: d, 3: D, 4: c, 33: C, 6: b, 8: a, 15: A},
# 22: {2: d, 20: D, 4: c, 5: C, 34: b, 7: B, 9: A},
# 23: {3: D, 4: c, 5: C, 6: b, 19: B, 8: a},
# 24: {18: d, 3: D, 4: c, 5: C, 6: b, 35: B, 8: a},
# 25: {2: d, 4: c, 5: C, 21: b, 7: B, 9: A},
# 26: {2: d, 4: c, 5: C, 6: b, 7: B, 36: a, 9: A},
# 27: {3: D, 5: C, 6: b, 7: B, 8: a, 9: A},
# 28: {3: D, 4: c, 5: C, 6: b, 7: B, 8: a, 37: A},
# 29: {2: d, 4: c, 6: b, 7: B, 8: a, 9: A},
# 30: {12: d, 3: D, 5: C, 7: B, 8: a, 9: A},
# 31: {2: d, 4: c, 6: b, 11: B, 8: a, 9: A},
# 32: {2: d, 10: D, 4: c, 6: b, 8: a, 9: A},
# 33: {3: D, 5: C, 13: b, 7: B, 8: a, 9: A},
# 34: {2: d, 10: D, 4: c, 6: b, 8: a, 15: A},
# 35: {12: d, 3: D, 5: C, 7: B, 17: a, 9: A},
# 36: {18: d, 3: D, 4: c, 14: C, 6: b, 8: a},
# 37: {2: d, 20: D, 16: c, 5: C, 7: B, 9: A}
# }
s = 6
a = np.array([
[ s, 0],
[ 0, 1/s]
])
b = np.array([
[1/2*s + 1/2/s, 1/2*s - 1/2/s],
[1/2*s - 1/2/s, 1/2*s + 1/2/s]
])
c = np.array([
[ -1/2*(s**6 - s**4 + 11*s**2 - 3)/(s**5 - 6*s**3 + s), 1/2*(3*s**6 - s**4 - 3*s**2 + 1)/(s**5 - 6*s**3 + s)],
[ -1/2*(s**6 - 3*s**4 - s**2 + 3)/(s**5 - 6*s**3 + s), 1/2*(3*s**6 - 11*s**4 + s**2 - 1)/(s**5 - 6*s**3 + s)]
])
d = np.array([
[ (s**5 - 2*s**3 - 3*s)/(s**4 - 6*s**2 + 1), -2*(s**5 - 2*s**3 + s)/(s**4 - 6*s**2 + 1)],
[ 2*(s**4 - 2*s**2 + 1)/(s**5 - 6*s**3 + s), -(3*s**4 + 2*s**2 - 1)/(s**5 - 6*s**3 + s)]
])
A = np.array([
[1/s, 0],
[ 0, s]
])
B = np.array([
[ 1/2*s + 1/2/s, -1/2*s + 1/2/s],
[-1/2*s + 1/2/s, 1/2*s + 1/2/s]
])
C = np.array([
[1/2*(3*s**6 - 11*s**4 + s**2 - 1)/(s**5 - 6*s**3 + s), -1/2*(3*s**6 - s**4 - 3*s**2 + 1)/(s**5 - 6*s**3 + s)],
[ 1/2*(s**6 - 3*s**4 - s**2 + 3)/(s**5 - 6*s**3 + s), -1/2*(s**6 - s**4 + 11*s**2 - 3)/(s**5 - 6*s**3 + s)]
])
D = np.array([
[-(3*s**4 + 2*s**2 - 1)/(s**5 - 6*s**3 + s), 2*(s**5 - 2*s**3 + s)/(s**4 - 6*s**2 + 1)],
[-2*(s**4 - 2*s**2 + 1)/(s**5 - 6*s**3 + s), (s**5 - 2*s**3 - 3*s)/(s**4 - 6*s**2 + 1)]
])
print('IDENTITY')
print(a @ b @ A @ B @ c @ d @ C @ D)
graph = {
0: {1: d, 2: D, 3: c, 4: C, 5: b, 6: B, 7: a, 8: A},
1: {1: d, 3: c, 4: C, 5: b, 6: B, 7: a, 8: A},
2: {2: D, 3: c, 4: C, 5: b, 6: B, 7: a, 8: A},
3: {9: d, 10: D, 3: c, 5: b, 6: B, 7: a, 8: A},
4: {1: d, 11: D, 4: C, 12: b, 6: B, 7: a, 8: A},
5: {1: d, 2: D, 3: c, 4: C, 5: b, 13: a, 8: A},
6: {1: d, 2: D, 14: c, 4: C, 6: B, 7: a, 8: A},
7: {1: d, 2: D, 3: c, 4: C, 15: b, 16: B, 7: a},
8: {17: d, 2: D, 3: c, 4: C, 5: b, 18: B, 8: A},
9: {1: d, 3: c, 19: C, 5: b, 6: B, 7: a, 8: A},
10: {2: D, 3: c, 20: C, 5: b, 6: B, 7: a, 8: A},
11: {2: D, 3: c, 4: C, 5: b, 6: B, 21: a, 8: A},
12: {1: d, 2: D, 3: c, 4: C, 5: b, 22: a, 8: A},
13: {1: d, 2: D, 3: c, 4: C, 15: b, 23: B, 7: a},
14: {24: d, 10: D, 3: c, 5: b, 6: B, 7: a, 8: A},
15: {1: d, 2: D, 3: c, 4: C, 5: b, 13: a, 25: A},
16: {1: d, 2: D, 14: c, 4: C, 6: B, 7: a, 26: A},
17: {1: d, 27: c, 4: C, 5: b, 6: B, 7: a, 8: A},
18: {1: d, 2: D, 28: c, 4: C, 6: B, 7: a, 8: A},
19: {1: d, 29: D, 4: C, 12: b, 6: B, 7: a, 8: A},
20: {1: d, 11: D, 4: C, 6: B, 7: a, 8: A},
21: {1: d, 2: D, 14: c, 4: C, 16: B, 7: a},
22: {1: d, 2: D, 3: c, 4: C, 15: b, 7: a},
23: {1: d, 2: D, 14: c, 4: C, 30: B, 7: a},
24: {1: d, 3: c, 31: C, 5: b, 6: B, 7: a, 8: A},
25: {17: d, 2: D, 3: c, 4: C, 5: b, 8: A},
26: {2: D, 3: c, 4: C, 5: b, 18: B, 8: A},
27: {9: d, 3: c, 5: b, 6: B, 7: a, 8: A},
28: {10: D, 3: c, 5: b, 6: B, 7: a, 8: A},
29: {2: D, 3: c, 4: C, 5: b, 32: B, 8: A},
30: {1: d, 2: D, 14: c, 4: C, 6: B, 7: a, 33: A},
31: {1: d, 4: C, 12: b, 6: B, 7: a, 8: A},
32: {1: d, 2: D, 34: c, 4: C, 6: B, 7: a, 8: A},
33: {35: d, 2: D, 3: c, 4: C, 5: b, 18: B, 8: A},
34: {27: d, 10: D, 3: c, 5: b, 6: B, 7: a, 8: A},
35: {1: d, 4: C, 5: b, 6: B, 7: a, 8: A}
}
return graph