answerspace.sagews

A.<u,v,w>=QuaternionAlgebra(2,3)
A.maximal_order()

Id=identity_matrix(2)
U=matrix([[sqrt(2),0],[0,-sqrt(2)]])
V=matrix([[0,1],[3,0]])
W=U*V

E0=Id
E1=U
E2=(1/2)*Id+(1/2)*U+(1/2)*V
E3=(1/2)*U+(1/2)*W
Ba=[E0,E1,E2,E3]

# Robust way to generate the list of matrices
maxm = 2 #set the range
L = []
for i in range(-maxm,maxm):
    for j in range(-maxm,maxm):
        for k in range(-maxm,maxm):
            for l in range(-maxm,maxm):
                temp = i*Ba[0] + j*Ba[1] + k*Ba[2] + l*Ba[3]
                if temp.determinant() == 1:
                    L.append(temp)

print(str(len(L)) + " matrices are selected in special linear group for polyhedron 1")

maxm = 3
L2 = []
for i in range(-maxm,maxm):
    for j in range(-maxm,maxm):
        for k in range(-maxm,maxm):
            for l in range(-maxm,maxm):
                temp = i*Ba[0] + j*Ba[1] + k*Ba[2] + l*Ba[3]
                if temp.determinant() == 1:
                    L2.append(temp)

print(str(len(L2)) + " matrices are selected in special linear group for polyhedron 2")

Order of Quaternion Algebra (2, 3) with base ring Rational Field with basis (1, u, 1/2 + 1/2*u + 1/2*v, 1/2*u + 1/2*w) 41 matrices are selected in special linear group 1 96 matrices are selected in special linear group 2

INEQS=[g*g.transpose() - identity_matrix(2)  for g in L]
ineqs=[]
for F in INEQS:
    #C = [0,F[0,0],2*F[0,1],F[1,1]] #P = [[a,b],[b,c]]
    C = [F[1,1],F[0,0] - F[1,1],2*F[0,1]] #P = [[a,b],[b,1-a]]
    ineqs.append(C)

INEQS2=[g*g.transpose() - identity_matrix(2)  for g in L2]
ineqs2=[]
for F in INEQS2:
    #C = [0,F[0,0],2*F[0,1],F[1,1]] #P = [[a,b],[b,c]]
    C = [F[1,1],F[0,0] - F[1,1],2*F[0,1]] #P = [[a,b],[b,1-a]]
    ineqs2.append(C)

poly=Polyhedron(ieqs=ineqs,base_ring = AA)
poly2=Polyhedron(ieqs=ineqs2,base_ring = AA)
print("\nPolyhedron 1: ")
print(poly)
print("\nPolyhedron 2: ")
print(poly2)
print("\nTwo polyhedrons are the same ?")
print(poly == poly2)

print("\nVertices are:")
for j in poly.Vrepresentation():
    #print(j)
    print("(" + str(j[0].radical_expression()) + " , " + str(j[1].radical_expression()) + ")")
poly.plot()

Polyhedron 1: A 2-dimensional polyhedron in AA^2 defined as the convex hull of 6 vertices Polyhedron 2: A 2-dimensional polyhedron in AA^2 defined as the convex hull of 6 vertices Two polyhedrons are the same ? True Vertices are: (-1/4*sqrt(1/2) + 3/4 , -1/4*sqrt(1/2) - 1/4) (-3/14*sqrt(2) + 9/14 , 0) (1/4*sqrt(1/2) + 3/4 , 1/4*sqrt(1/2) - 1/4) (3/14*sqrt(2) + 9/14 , 0) (-1/4*sqrt(1/2) + 3/4 , 1/4*sqrt(1/2) + 1/4) (1/4*sqrt(1/2) + 3/4 , -1/4*sqrt(1/2) + 1/4)

def convertVertice(Vrep,trace1 = True):
    mList = []
    for i in Vrep:
        if trace1:
            a = i[0].radical_expression()
            b = i[1].radical_expression()
            temp = matrix([[a,b],[b,1-a]])
        else:
            a = i[0].radical_expression()
            b = i[1].radical_expression()
            c = i[2].radical_expression()
            temp = matrix([[a,b],[b,c]])
        mList.append(temp)
    return(mList)


matrixList = convertVertice(poly.Vrepresentation(),True)
print("\nVertices in matrix form:")
for i in matrixList:
    print(i)
    print("\n")

Vertices in matrix form: [-1/4*sqrt(1/2) + 3/4 -1/4*sqrt(1/2) - 1/4] [-1/4*sqrt(1/2) - 1/4 1/4*sqrt(1/2) + 1/4] [-3/14*sqrt(2) + 9/14 0] [ 0 3/14*sqrt(2) + 5/14] [ 1/4*sqrt(1/2) + 3/4 1/4*sqrt(1/2) - 1/4] [ 1/4*sqrt(1/2) - 1/4 -1/4*sqrt(1/2) + 1/4] [ 3/14*sqrt(2) + 9/14 0] [ 0 -3/14*sqrt(2) + 5/14] [-1/4*sqrt(1/2) + 3/4 1/4*sqrt(1/2) + 1/4] [ 1/4*sqrt(1/2) + 1/4 1/4*sqrt(1/2) + 1/4] [ 1/4*sqrt(1/2) + 3/4 -1/4*sqrt(1/2) + 1/4] [-1/4*sqrt(1/2) + 1/4 -1/4*sqrt(1/2) + 1/4]

RRmatlist=[M.change_ring(RR) for M in matrixList]

for r in RRmatlist:
    print(r)
    print("\n")

[ 0.573223304703363 -0.426776695296637] [-0.426776695296637 0.426776695296637] [0.339811379491480 0.000000000000000] [0.000000000000000 0.660188620508520] [ 0.926776695296637 -0.0732233047033631] [-0.0732233047033631 0.0732233047033631] [ 0.945902906222806 0.000000000000000] [ 0.000000000000000 0.0540970937771939] [0.573223304703363 0.426776695296637] [0.426776695296637 0.426776695296637] [ 0.926776695296637 0.0732233047033631] [0.0732233047033631 0.0732233047033631]

E1;E2;E3

[ sqrt(2) 0] [ 0 -sqrt(2)] [ 1/2*sqrt(2) + 1/2 1/2] [ 3/2 -1/2*sqrt(2) + 1/2] [ 1/2*sqrt(2) 1/2*sqrt(2)] [-3/2*sqrt(2) -1/2*sqrt(2)]