# ****************************************************************************
# This file is part of sage-flatsurf.
#
# Copyright (C) 2016-2019 Vincent Delecroix
# 2016-2019 W. Patrick Hooper
# 2023 Julian Rüth
#
# sage-flatsurf is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 2 of the License, or
# (at your option) any later version.
#
# sage-flatsurf is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with sage-flatsurf. If not, see <https://www.gnu.org/licenses/>.
# ****************************************************************************
from sage.groups.group import Group
from sage.categories.groups import Groups
from sage.structure.unique_representation import UniqueRepresentation
from sage.structure.element import MultiplicativeGroupElement
from sage.algebras.quatalg.quaternion_algebra import QuaternionAlgebra
from sage.rings.integer_ring import ZZ
from flatsurf.geometry.origami import AbstractOrigami
_Q = QuaternionAlgebra(-1, -1)
_i, _j, _k = _Q.gens()
[docs]class MegaWollmilchsauGroupElement(MultiplicativeGroupElement):
[docs] @staticmethod
def quat_to_tuple(r):
r"""Convert an element in the quaternion algebra to a quadruple"""
if isinstance(r, int):
return (r, 0, 0, 0)
else:
return (r[0], r[1], r[2], r[3])
[docs] @staticmethod
def wedge(r1, r2):
r"""Wedge two quaterions. Returns an integer."""
x = MegaWollmilchsauGroupElement.quat_to_tuple(r1)
y = MegaWollmilchsauGroupElement.quat_to_tuple(r2)
return -x[0] * y[3] + x[1] * y[2] - x[2] * y[1] + x[3] * y[0]
def __init__(self, parent, i, r, q):
if parent is None:
raise ValueError("The parent must be provided")
# I should assert that the element lives in the domain of the group.
if i not in ZZ:
raise ValueError
if r not in _Q:
raise ValueError
# Actually q should be in {1,-1,-i,i,j,-j,k,-k}. I'm not testing for that.
if q not in _Q:
raise ValueError
# There is one more condition. The group doesn't have full image...
self._i = i
self._r = r
self._q = q
self._parent = parent
MultiplicativeGroupElement.__init__(self, parent)
def _repr_(self):
return "[" + str(self._i) + ", " + str(self._r) + ", " + str(self._q) + "]"
def _cmp_(self, other):
return (
(self._i > other._i - self._i < other._i)
or (self._r > other._r - self._r < other._r)
or (self._q > other._q - self._q < other._q)
)
def _mul_(self, m):
return MegaWollmilchsauGroupElement(
self._parent,
self._i
+ m._i
+ MegaWollmilchsauGroupElement.wedge(self._r, self._q * m._r),
self._r + self._q * m._r,
self._q * m._q,
)
def __invert__(self):
q1 = ~self._q
r1 = -(q1 * self._r)
i1 = -(self._i + MegaWollmilchsauGroupElement.wedge(r1, q1 * self._r))
return MegaWollmilchsauGroupElement(self._parent, i1, r1, q1)
def _div_(self, m):
return self._mul_(m.__invert__())
def __hash__(self):
return (
67 * hash(self._i)
+ 23 * hash(MegaWollmilchsauGroupElement.quat_to_tuple(self._r))
- 17 * hash(MegaWollmilchsauGroupElement.quat_to_tuple(self._q))
)
[docs]class MegaWollmilchsauGroup(UniqueRepresentation, Group):
Element = MegaWollmilchsauGroupElement
def _element_constructor_(self, *args, **kwds):
if len(args) != 1:
return self.element_class(self, *args, **kwds)
x = args[0]
return self.element_class(self, x, **kwds)
def __init__(self):
Group.__init__(self, category=Groups().Infinite())
def _repr_(self):
return "MegaWollmilchsauGroup"
[docs] def a(self):
return self.element_class(self, 0, 1, _i)
[docs] def b(self):
return self.element_class(self, 0, 1, _j)
[docs] def one(self):
return self.element_class(self, 0, 0, 1)
[docs] def gens(self):
return (self.a(), self.b())
[docs] def is_abelian(self):
return False
def _an_element_(self):
return self.a()
[docs] def some_elements(self):
return [self.a(), self.b()]
def _test_relations(self, **options):
tester = self._tester(**options)
a, b = self.gens()
e = self.one()
tester.assertEqual(a**4, e)
tester.assertEqual(b**4, e)
tester.assertEqual((a * b) ** 4, e)
tester.assertEqual((a / b) ** 4, e)
tester.assertEqual((a * a * b) ** 4, e)
tester.assertEqual((a * a / b) ** 4, e)
tester.assertNotEqual((a * b / a / b) ** 2, e)
[docs]class MegaWollmilchsau(AbstractOrigami):
def __init__(self):
self._G = self._domain = MegaWollmilchsauGroup()
self._a, self._b = self._G.gens()
self._ai = ~self._a
self._bi = ~self._b
[docs] def up(self, label):
return self._b * label
[docs] def down(self, label):
return self._bi * label
[docs] def right(self, label):
return self._a * label
[docs] def left(self, label):
return self._ai * label
def _repr_(self):
return "MegaWollmilchsau Origami"