303 lines
7.3 KiB
Python
303 lines
7.3 KiB
Python
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from sympy.combinatorics.permutations import Permutation
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from sympy.core.symbol import symbols
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from sympy.matrices import Matrix
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from sympy.utilities.iterables import variations, rotate_left
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def symmetric(n):
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"""
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Generates the symmetric group of order n, Sn.
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Examples
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========
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>>> from sympy.combinatorics.generators import symmetric
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>>> list(symmetric(3))
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[(2), (1 2), (2)(0 1), (0 1 2), (0 2 1), (0 2)]
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"""
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for perm in variations(range(n), n):
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yield Permutation(perm)
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def cyclic(n):
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"""
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Generates the cyclic group of order n, Cn.
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Examples
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========
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>>> from sympy.combinatorics.generators import cyclic
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>>> list(cyclic(5))
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[(4), (0 1 2 3 4), (0 2 4 1 3),
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(0 3 1 4 2), (0 4 3 2 1)]
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See Also
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========
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dihedral
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"""
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gen = list(range(n))
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for i in range(n):
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yield Permutation(gen)
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gen = rotate_left(gen, 1)
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def alternating(n):
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"""
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Generates the alternating group of order n, An.
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Examples
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========
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>>> from sympy.combinatorics.generators import alternating
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>>> list(alternating(3))
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[(2), (0 1 2), (0 2 1)]
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"""
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for perm in variations(range(n), n):
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p = Permutation(perm)
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if p.is_even:
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yield p
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def dihedral(n):
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"""
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Generates the dihedral group of order 2n, Dn.
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The result is given as a subgroup of Sn, except for the special cases n=1
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(the group S2) and n=2 (the Klein 4-group) where that's not possible
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and embeddings in S2 and S4 respectively are given.
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Examples
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========
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>>> from sympy.combinatorics.generators import dihedral
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>>> list(dihedral(3))
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[(2), (0 2), (0 1 2), (1 2), (0 2 1), (2)(0 1)]
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See Also
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========
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cyclic
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"""
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if n == 1:
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yield Permutation([0, 1])
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yield Permutation([1, 0])
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elif n == 2:
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yield Permutation([0, 1, 2, 3])
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yield Permutation([1, 0, 3, 2])
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yield Permutation([2, 3, 0, 1])
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yield Permutation([3, 2, 1, 0])
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else:
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gen = list(range(n))
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for i in range(n):
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yield Permutation(gen)
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yield Permutation(gen[::-1])
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gen = rotate_left(gen, 1)
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def rubik_cube_generators():
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"""Return the permutations of the 3x3 Rubik's cube, see
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https://www.gap-system.org/Doc/Examples/rubik.html
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"""
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a = [
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[(1, 3, 8, 6), (2, 5, 7, 4), (9, 33, 25, 17), (10, 34, 26, 18),
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(11, 35, 27, 19)],
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[(9, 11, 16, 14), (10, 13, 15, 12), (1, 17, 41, 40), (4, 20, 44, 37),
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(6, 22, 46, 35)],
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[(17, 19, 24, 22), (18, 21, 23, 20), (6, 25, 43, 16), (7, 28, 42, 13),
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(8, 30, 41, 11)],
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[(25, 27, 32, 30), (26, 29, 31, 28), (3, 38, 43, 19), (5, 36, 45, 21),
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(8, 33, 48, 24)],
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[(33, 35, 40, 38), (34, 37, 39, 36), (3, 9, 46, 32), (2, 12, 47, 29),
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(1, 14, 48, 27)],
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[(41, 43, 48, 46), (42, 45, 47, 44), (14, 22, 30, 38),
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(15, 23, 31, 39), (16, 24, 32, 40)]
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]
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return [Permutation([[i - 1 for i in xi] for xi in x], size=48) for x in a]
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def rubik(n):
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"""Return permutations for an nxn Rubik's cube.
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Permutations returned are for rotation of each of the slice
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from the face up to the last face for each of the 3 sides (in this order):
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front, right and bottom. Hence, the first n - 1 permutations are for the
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slices from the front.
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"""
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if n < 2:
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raise ValueError('dimension of cube must be > 1')
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# 1-based reference to rows and columns in Matrix
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def getr(f, i):
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return faces[f].col(n - i)
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def getl(f, i):
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return faces[f].col(i - 1)
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def getu(f, i):
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return faces[f].row(i - 1)
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def getd(f, i):
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return faces[f].row(n - i)
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def setr(f, i, s):
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faces[f][:, n - i] = Matrix(n, 1, s)
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def setl(f, i, s):
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faces[f][:, i - 1] = Matrix(n, 1, s)
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def setu(f, i, s):
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faces[f][i - 1, :] = Matrix(1, n, s)
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def setd(f, i, s):
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faces[f][n - i, :] = Matrix(1, n, s)
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# motion of a single face
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def cw(F, r=1):
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for _ in range(r):
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face = faces[F]
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rv = []
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for c in range(n):
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for r in range(n - 1, -1, -1):
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rv.append(face[r, c])
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faces[F] = Matrix(n, n, rv)
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def ccw(F):
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cw(F, 3)
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# motion of plane i from the F side;
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# fcw(0) moves the F face, fcw(1) moves the plane
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# just behind the front face, etc...
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def fcw(i, r=1):
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for _ in range(r):
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if i == 0:
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cw(F)
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i += 1
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temp = getr(L, i)
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setr(L, i, list(getu(D, i)))
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setu(D, i, list(reversed(getl(R, i))))
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setl(R, i, list(getd(U, i)))
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setd(U, i, list(reversed(temp)))
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i -= 1
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def fccw(i):
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fcw(i, 3)
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# motion of the entire cube from the F side
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def FCW(r=1):
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for _ in range(r):
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cw(F)
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ccw(B)
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cw(U)
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t = faces[U]
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cw(L)
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faces[U] = faces[L]
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cw(D)
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faces[L] = faces[D]
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cw(R)
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faces[D] = faces[R]
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faces[R] = t
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def FCCW():
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FCW(3)
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# motion of the entire cube from the U side
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def UCW(r=1):
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for _ in range(r):
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cw(U)
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ccw(D)
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t = faces[F]
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faces[F] = faces[R]
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faces[R] = faces[B]
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faces[B] = faces[L]
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faces[L] = t
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def UCCW():
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UCW(3)
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# defining the permutations for the cube
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U, F, R, B, L, D = names = symbols('U, F, R, B, L, D')
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# the faces are represented by nxn matrices
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faces = {}
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count = 0
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for fi in range(6):
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f = []
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for a in range(n**2):
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f.append(count)
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count += 1
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faces[names[fi]] = Matrix(n, n, f)
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# this will either return the value of the current permutation
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# (show != 1) or else append the permutation to the group, g
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def perm(show=0):
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# add perm to the list of perms
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p = []
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for f in names:
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p.extend(faces[f])
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if show:
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return p
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g.append(Permutation(p))
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g = [] # container for the group's permutations
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I = list(range(6*n**2)) # the identity permutation used for checking
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# define permutations corresponding to cw rotations of the planes
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# up TO the last plane from that direction; by not including the
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# last plane, the orientation of the cube is maintained.
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# F slices
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for i in range(n - 1):
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fcw(i)
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perm()
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fccw(i) # restore
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assert perm(1) == I
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# R slices
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# bring R to front
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UCW()
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for i in range(n - 1):
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fcw(i)
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# put it back in place
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UCCW()
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# record
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perm()
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# restore
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# bring face to front
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UCW()
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fccw(i)
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# restore
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UCCW()
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assert perm(1) == I
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# D slices
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# bring up bottom
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FCW()
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UCCW()
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FCCW()
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for i in range(n - 1):
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# turn strip
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fcw(i)
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# put bottom back on the bottom
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FCW()
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UCW()
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FCCW()
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# record
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perm()
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# restore
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# bring up bottom
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FCW()
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UCCW()
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FCCW()
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# turn strip
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fccw(i)
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# put bottom back on the bottom
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FCW()
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UCW()
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FCCW()
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assert perm(1) == I
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return g
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