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RoseTTAFold-All-Atom/rf2aa/symmetry.py
Rohith Krishna f87f5b8cdf initial commit
2024-03-04 22:38:17 -08:00

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import sys
import numpy as np
import torch
SYMA = 1.0
def generateC(angs, eps=1e-6):
L = angs.shape[0]
Rs = torch.eye(3, device=angs.device).repeat(L,1,1)
Rs[:,1,1] = torch.cos(angs)
Rs[:,1,2] = -torch.sin(angs)
Rs[:,2,1] = torch.sin(angs)
Rs[:,2,2] = torch.cos(angs)
return Rs
def generateD(angs, eps=1e-6):
L = angs.shape[0]
Rs = torch.eye(3, device=angs.device).repeat(2*L,1,1)
Rs[:L,1,1] = torch.cos(angs)
Rs[:L,1,2] = -torch.sin(angs)
Rs[:L,2,1] = torch.sin(angs)
Rs[:L,2,2] = torch.cos(angs)
Rx = torch.tensor([[-1.,0,0],[0,-1,0],[0,0,1]],device=angs.device)
Rs[L:] = torch.einsum('ij,bjk->bik',Rx,Rs[:L])
return Rs
def find_symm_subs(xyz,Rs,metasymm):
com = xyz[:,:,1].mean(dim=-2)
rcoms = torch.einsum('sij,bj->si', Rs, com)
subsymms, nneighs = metasymm
subs = []
for i in range(len(subsymms)):
drcoms = torch.linalg.norm(rcoms[0,:] - rcoms[subsymms[i],:], dim=-1)
_,subs_i = torch.topk(drcoms,nneighs[i],largest=False)
subs_i,_ = torch.sort( subsymms[i][subs_i] )
subs.append(subs_i)
subs=torch.cat(subs)
xyz_new = torch.einsum('sij,braj->bsrai', Rs[subs], xyz).reshape(
xyz.shape[0],-1,xyz.shape[2],3)
return xyz_new, subs
def update_symm_subs(xyz,subs,Rs,metasymm):
xyz_new = torch.einsum('sij,braj->bsrai', Rs[subs], xyz).reshape(
xyz.shape[0],-1,xyz.shape[2],3)
return xyz_new
def get_symm_map(subs,O):
symmmask = torch.zeros(O,dtype=torch.long)
symmmask[subs] = torch.arange(1,subs.shape[0]+1)
return symmmask
def rotation_from_matrix(R, eps=1e-4):
w, W = torch.linalg.eig(R.T)
i = torch.where(abs(torch.real(w) - 1.0) < eps)[0]
if (len(i)==0):
i = torch.tensor([0])
print ('rotation_from_matrix w',torch.real(w))
print ('rotation_from_matrix R.T',R.T)
axis = torch.real(W[:, i[-1]]).squeeze()
cosa = (torch.trace(R) - 1.0) / 2.0
if abs(axis[2]) > eps:
sina = (R[1, 0] + (cosa-1.0)*axis[0]*axis[1]) / axis[2]
elif abs(axis[1]) > eps:
sina = (R[0, 2] + (cosa-1.0)*axis[0]*axis[2]) / axis[1]
else:
sina = (R[2, 1] + (cosa-1.0)*axis[1]*axis[2]) / axis[0]
angle = torch.atan2(sina, cosa)
return angle, axis
def kabsch(pred, true):
def rmsd(V, W, eps=1e-6):
L = V.shape[0]
return torch.sqrt(torch.sum((V-W)*(V-W)) / L + eps)
def centroid(X):
return X.mean(dim=-2, keepdim=True)
cP = centroid(pred)
cT = centroid(true)
pred = pred - cP
true = true - cT
C = torch.matmul(pred.permute(1,0), true)
V, S, W = torch.svd(C)
d = torch.ones([3,3], device=pred.device)
d[:,-1] = torch.sign(torch.det(V)*torch.det(W))
U = torch.matmul(d*V, W.permute(1,0)) # (IB, 3, 3)
rpred = torch.matmul(pred, U) # (IB, L*3, 3)
rms = rmsd(rpred, true)
return rms, U, cP, cT
# do lines X0->X and Y0->Y intersect?
def intersect(X0,X,Y0,Y,eps=0.1):
mtx = torch.cat(
(torch.stack((X0,X0+X,Y0,Y0+Y)), torch.ones((4,1))) , axis=1
)
det = torch.linalg.det( mtx )
return (torch.abs(det) <= eps)
def get_angle(X,Y):
angle = torch.acos( torch.clamp( torch.sum(X*Y), -1., 1. ) )
if (angle > np.pi/2):
angle = np.pi - angle
return angle
# given the coordinates of a subunit +
def get_symmetry(xyz, mask, rms_cut=2.5, nfold_cut=0.1, angle_cut=0.05, trans_cut=2.0):
nops = xyz.shape[0]
L = xyz.shape[1]//2
# PASS 1: find all symm axes
symmaxes = []
for i in range(nops):
# if there are multiple biomt records, this may occur.
# rather than try to rescue, we will take the 1st (typically author-assigned)
offset0 = torch.linalg.norm(xyz[i,:L,1]-xyz[0,:L,1], dim=-1)
if (torch.mean(offset0)>1e-4):
continue
# get alignment
mask_i = mask[i,:L,1]*mask[i,L:,1]
xyz_i = xyz[i,:L,1][mask_i,:]
xyz_j = xyz[i,L:,1][mask_i,:]
rms_ij, Uij, cI, cJ = kabsch(xyz_i, xyz_j)
if (rms_ij > rms_cut):
#print (i,'rms',rms_ij)
continue
# get axis and point symmetry about axis
angle, axis = rotation_from_matrix(Uij)
nfold = 2*np.pi/torch.abs(angle)
# a) ensure integer # of subunits per rotation
if (torch.abs( nfold - torch.round(nfold) ) > nfold_cut ):
#print ('nfold fail',nfold)
continue
nfold = torch.round(nfold).long()
# b) ensure rotation only (no translation)
delCOM = torch.mean(xyz_i, dim=-2) - torch.mean(xyz_j, dim=-2)
trans_dot_symaxis = nfold * torch.abs(torch.dot(delCOM, axis))
if (trans_dot_symaxis > trans_cut ):
#print ('trans fail',trans_dot_symaxis)
continue
# 3) get a point on the symm axis from CoMs and angle
cIJ = torch.sign(angle) * (cJ-cI).squeeze(0)
dIJ = torch.linalg.norm(cIJ)
p_mid = (cI+cJ).squeeze(0) / 2
u = cIJ / dIJ # unit vector in plane of circle
v = torch.cross(axis, u) # unit vector from sym axis to p_mid
r = dIJ / (2*torch.sin(angle/2))
d = torch.sqrt( r*r - dIJ*dIJ/4 ) # distance from mid-chord to center
point = p_mid - (d)*v
# check if redundant
toadd = True
for j,(nf_j,ax_j,pt_j,err_j) in enumerate(symmaxes):
if (not intersect(pt_j,ax_j,point,axis)):
continue
angle_j = get_angle(ax_j,axis)
if (angle_j < angle_cut):
if (nf_j < nfold): # stored is a subsymmetry of complex, overwrite
symmaxes[j] = (nfold, axis, point, i)
toadd = False
if (toadd):
symmaxes.append( (nfold, axis, point, i) )
# PASS 2: combine
symmgroup = 'C1'
subsymm = []
if len(symmaxes)==1:
symmgroup = 'C%d'%(symmaxes[0][0])
subsymm = [symmaxes[0][3]]
elif len(symmaxes)>1:
symmaxes = sorted(symmaxes, key=lambda x: x[0], reverse=True)
angle = get_angle(symmaxes[0][1],symmaxes[1][1])
subsymm = [symmaxes[0][3],symmaxes[1][3]]
# 2-fold and n-fold intersect at 90 degress => Dn
if (symmaxes[1][0] == 2 and torch.abs(angle-np.pi/2) < angle_cut):
symmgroup = 'D%d'%(symmaxes[0][0])
else:
# polyhedral rules:
# 3-Fold + 2-fold intersecting at acos(-1/sqrt(3)) -> T
angle_tgt = np.arccos(-1/np.sqrt(3))
if (symmaxes[0][0] == 3 and symmaxes[1][0] == 2 and torch.abs(angle - angle_tgt) < angle_cut):
symmgroup = 'T'
# 3-Fold + 2-fold intersecting at asin(1/sqrt(3)) -> O
angle_tgt = np.arcsin(1/np.sqrt(3))
if (symmaxes[0][0] == 3 and symmaxes[1][0] == 2 and torch.abs(angle - angle_tgt) < angle_cut):
symmgroup = 'O'
# 4-Fold + 3-fold intersecting at acos(1/sqrt(3)) -> O
angle_tgt = np.arccos(1/np.sqrt(3))
if (symmaxes[0][0] == 4 and symmaxes[1][0] == 3 and torch.abs(angle - angle_tgt) < angle_cut):
symmgroup = 'O'
# 3-Fold + 2-fold intersecting at 0.5*acos(sqrt(5)/3) -> I
angle_tgt = 0.5*np.arccos(np.sqrt(5)/3)
if (symmaxes[0][0] == 3 and symmaxes[1][0] == 2 and torch.abs(angle - angle_tgt) < angle_cut):
symmgroup = 'I'
# 5-Fold + 2-fold intersecting at 0.5*acos(1/sqrt(5)) -> I
angle_tgt = 0.5*np.arccos(1/np.sqrt(5))
if (symmaxes[0][0] == 5 and symmaxes[1][0] == 2 and torch.abs(angle - angle_tgt) < angle_cut):
symmgroup = 'I'
# 5-Fold + 3-fold intersecting at 0.5*acos((4*sqrt(5)-5)/15) -> I
angle_tgt = 0.5*np.arccos((4*np.sqrt(5)-5)/15)
if (symmaxes[0][0] == 5 and symmaxes[1][0] == 3 and torch.abs(angle - angle_tgt) < angle_cut):
symmgroup = 'I'
else:
pass
#fd: we could use a single symmetry here instead.
# But these cases mostly are bad BIOUNIT annotations...
#print ('nomatch',angle, [(x,y) for x,_,_,y in symmaxes])
return symmgroup, subsymm
def symm_subunit_matrix(symmid):
if (symmid[0]=='C'):
nsub = int(symmid[1:])
symmatrix = (
torch.arange(nsub)[:,None]-torch.arange(nsub)[None,:]
)%nsub
angles = torch.linspace(0,2*np.pi,nsub+1)[:nsub]
Rs = generateC(angles)
metasymm = (
[torch.arange(nsub)],
[min(3,nsub)]
)
if (nsub==1):
D = 0.0
else:
est_radius = 2.0*SYMA
theta = 2.0*np.pi/nsub
D = est_radius/np.sin(theta/2)
offset = torch.tensor([ 0.0,0.0,float(D) ])
elif (symmid[0]=='D'):
nsub = int(symmid[1:])
cblk=(torch.arange(nsub)[:,None]-torch.arange(nsub)[None,:])%nsub
symmatrix=torch.zeros((2*nsub,2*nsub),dtype=torch.long)
symmatrix[:nsub,:nsub] = cblk
symmatrix[:nsub,nsub:] = cblk+nsub
symmatrix[nsub:,:nsub] = cblk+nsub
symmatrix[nsub:,nsub:] = cblk
angles = torch.linspace(0,2*np.pi,nsub+1)[:nsub]
Rs = generateD(angles)
metasymm = (
[torch.arange(nsub), nsub+torch.arange(nsub)],
[min(3,nsub),2]
)
#metasymm = (
# [torch.arange(2*nsub)],
# [min(2*nsub,5)]
#)
est_radius = 2.0*SYMA
theta1 = 2.0*np.pi/nsub
theta2 = np.pi
D1 = est_radius/np.sin(theta1/2)
D2 = est_radius/np.sin(theta2/2)
offset = torch.tensor([ float(D2),0.0,float(D1) ])
#offset = torch.tensor([ 0.0,0.0,0.0 ])
elif (symmid=='T'):
symmatrix=torch.tensor(
[[ 0, 1, 2, 3, 8, 11, 9, 10, 4, 6, 7, 5],
[ 1, 0, 3, 2, 9, 10, 8, 11, 5, 7, 6, 4],
[ 2, 3, 0, 1, 10, 9, 11, 8, 6, 4, 5, 7],
[ 3, 2, 1, 0, 11, 8, 10, 9, 7, 5, 4, 6],
[ 4, 6, 7, 5, 0, 1, 2, 3, 8, 11, 9, 10],
[ 5, 7, 6, 4, 1, 0, 3, 2, 9, 10, 8, 11],
[ 6, 4, 5, 7, 2, 3, 0, 1, 10, 9, 11, 8],
[ 7, 5, 4, 6, 3, 2, 1, 0, 11, 8, 10, 9],
[ 8, 11, 9, 10, 4, 6, 7, 5, 0, 1, 2, 3],
[ 9, 10, 8, 11, 5, 7, 6, 4, 1, 0, 3, 2],
[10, 9, 11, 8, 6, 4, 5, 7, 2, 3, 0, 1],
[11, 8, 10, 9, 7, 5, 4, 6, 3, 2, 1, 0]])
Rs = torch.zeros(12,3,3)
Rs[ 0]=torch.tensor([[1.000000,0.000000,0.000000],[0.000000,1.000000,0.000000],[0.000000,0.000000,1.000000]])
Rs[ 1]=torch.tensor([[-1.000000,0.000000,0.000000],[0.000000,-1.000000,0.000000],[0.000000,0.000000,1.000000]])
Rs[ 2]=torch.tensor([[-1.000000,0.000000,0.000000],[0.000000,1.000000,0.000000],[0.000000,0.000000,-1.000000]])
Rs[ 3]=torch.tensor([[1.000000,0.000000,0.000000],[0.000000,-1.000000,0.000000],[0.000000,0.000000,-1.000000]])
Rs[ 4]=torch.tensor([[0.000000,0.000000,1.000000],[1.000000,0.000000,0.000000],[0.000000,1.000000,0.000000]])
Rs[ 5]=torch.tensor([[0.000000,0.000000,1.000000],[-1.000000,0.000000,0.000000],[0.000000,-1.000000,0.000000]])
Rs[ 6]=torch.tensor([[0.000000,0.000000,-1.000000],[-1.000000,0.000000,0.000000],[0.000000,1.000000,0.000000]])
Rs[ 7]=torch.tensor([[0.000000,0.000000,-1.000000],[1.000000,0.000000,0.000000],[0.000000,-1.000000,0.000000]])
Rs[ 8]=torch.tensor([[0.000000,1.000000,0.000000],[0.000000,0.000000,1.000000],[1.000000,0.000000,0.000000]])
Rs[ 9]=torch.tensor([[0.000000,-1.000000,0.000000],[0.000000,0.000000,1.000000],[-1.000000,0.000000,0.000000]])
Rs[10]=torch.tensor([[0.000000,1.000000,0.000000],[0.000000,0.000000,-1.000000],[-1.000000,0.000000,0.000000]])
Rs[11]=torch.tensor([[0.000000,-1.000000,0.000000],[0.000000,0.000000,-1.000000],[1.000000,0.000000,0.000000]])
nneigh = 5
metasymm = (
[torch.arange(12)],
[6]
)
est_radius = 4.0*SYMA
offset = torch.tensor([ 1.0,0.0,0.0 ])
offset = est_radius * offset / torch.linalg.norm(offset)
metasymm = (
[torch.arange(12)],
[6]
)
elif (symmid=='O'):
symmatrix=torch.tensor(
[[ 0, 1, 2, 3, 8, 11, 9, 10, 4, 6, 7, 5, 12, 13, 15, 14, 19, 17,
18, 16, 22, 21, 20, 23],
[ 1, 0, 3, 2, 9, 10, 8, 11, 5, 7, 6, 4, 13, 12, 14, 15, 18, 16,
19, 17, 23, 20, 21, 22],
[ 2, 3, 0, 1, 10, 9, 11, 8, 6, 4, 5, 7, 14, 15, 13, 12, 17, 19,
16, 18, 20, 23, 22, 21],
[ 3, 2, 1, 0, 11, 8, 10, 9, 7, 5, 4, 6, 15, 14, 12, 13, 16, 18,
17, 19, 21, 22, 23, 20],
[ 4, 6, 7, 5, 0, 1, 2, 3, 8, 11, 9, 10, 16, 18, 17, 19, 21, 22,
23, 20, 15, 14, 12, 13],
[ 5, 7, 6, 4, 1, 0, 3, 2, 9, 10, 8, 11, 17, 19, 16, 18, 20, 23,
22, 21, 14, 15, 13, 12],
[ 6, 4, 5, 7, 2, 3, 0, 1, 10, 9, 11, 8, 18, 16, 19, 17, 23, 20,
21, 22, 13, 12, 14, 15],
[ 7, 5, 4, 6, 3, 2, 1, 0, 11, 8, 10, 9, 19, 17, 18, 16, 22, 21,
20, 23, 12, 13, 15, 14],
[ 8, 11, 9, 10, 4, 6, 7, 5, 0, 1, 2, 3, 20, 23, 22, 21, 14, 15,
13, 12, 17, 19, 16, 18],
[ 9, 10, 8, 11, 5, 7, 6, 4, 1, 0, 3, 2, 21, 22, 23, 20, 15, 14,
12, 13, 16, 18, 17, 19],
[10, 9, 11, 8, 6, 4, 5, 7, 2, 3, 0, 1, 22, 21, 20, 23, 12, 13,
15, 14, 19, 17, 18, 16],
[11, 8, 10, 9, 7, 5, 4, 6, 3, 2, 1, 0, 23, 20, 21, 22, 13, 12,
14, 15, 18, 16, 19, 17],
[12, 13, 15, 14, 19, 17, 18, 16, 22, 21, 20, 23, 0, 1, 2, 3, 8, 11,
9, 10, 4, 6, 7, 5],
[13, 12, 14, 15, 18, 16, 19, 17, 23, 20, 21, 22, 1, 0, 3, 2, 9, 10,
8, 11, 5, 7, 6, 4],
[14, 15, 13, 12, 17, 19, 16, 18, 20, 23, 22, 21, 2, 3, 0, 1, 10, 9,
11, 8, 6, 4, 5, 7],
[15, 14, 12, 13, 16, 18, 17, 19, 21, 22, 23, 20, 3, 2, 1, 0, 11, 8,
10, 9, 7, 5, 4, 6],
[16, 18, 17, 19, 21, 22, 23, 20, 15, 14, 12, 13, 4, 6, 7, 5, 0, 1,
2, 3, 8, 11, 9, 10],
[17, 19, 16, 18, 20, 23, 22, 21, 14, 15, 13, 12, 5, 7, 6, 4, 1, 0,
3, 2, 9, 10, 8, 11],
[18, 16, 19, 17, 23, 20, 21, 22, 13, 12, 14, 15, 6, 4, 5, 7, 2, 3,
0, 1, 10, 9, 11, 8],
[19, 17, 18, 16, 22, 21, 20, 23, 12, 13, 15, 14, 7, 5, 4, 6, 3, 2,
1, 0, 11, 8, 10, 9],
[20, 23, 22, 21, 14, 15, 13, 12, 17, 19, 16, 18, 8, 11, 9, 10, 4, 6,
7, 5, 0, 1, 2, 3],
[21, 22, 23, 20, 15, 14, 12, 13, 16, 18, 17, 19, 9, 10, 8, 11, 5, 7,
6, 4, 1, 0, 3, 2],
[22, 21, 20, 23, 12, 13, 15, 14, 19, 17, 18, 16, 10, 9, 11, 8, 6, 4,
5, 7, 2, 3, 0, 1],
[23, 20, 21, 22, 13, 12, 14, 15, 18, 16, 19, 17, 11, 8, 10, 9, 7, 5,
4, 6, 3, 2, 1, 0]])
Rs = torch.zeros(24,3,3)
Rs[0]=torch.tensor([[ 1.000000, 0.000000,0.000000],[ 0.000000, 1.000000,0.000000],[ 0.000000, 0.000000,1.000000]])
Rs[1]=torch.tensor([[-1.000000, 0.000000,0.000000],[ 0.000000,-1.000000,0.000000],[ 0.000000, 0.000000,1.000000]])
Rs[2]=torch.tensor([[-1.000000, 0.000000,0.000000],[ 0.000000, 1.000000,0.000000],[ 0.000000 ,0.000000,-1.000000]])
Rs[3]=torch.tensor([[ 1.000000, 0.000000,0.000000],[ 0.000000,-1.000000,0.000000],[ 0.000000 ,0.000000,-1.000000]])
Rs[4]=torch.tensor([[ 0.000000, 0.000000,1.000000],[ 1.000000, 0.000000,0.000000],[ 0.000000, 1.000000,0.000000]])
Rs[5]=torch.tensor([[ 0.000000, 0.000000,1.000000],[-1.000000, 0.000000,0.000000],[ 0.000000,-1.000000,0.000000]])
Rs[6]=torch.tensor([[ 0.000000 ,0.000000,-1.000000],[-1.000000, 0.000000,0.000000],[ 0.000000, 1.000000,0.000000]])
Rs[7]=torch.tensor([[ 0.000000 ,0.000000,-1.000000],[ 1.000000, 0.000000,0.000000],[ 0.000000,-1.000000,0.000000]])
Rs[8]=torch.tensor([[ 0.000000, 1.000000,0.000000],[ 0.000000, 0.000000,1.000000],[ 1.000000, 0.000000,0.000000]])
Rs[9]=torch.tensor([[ 0.000000,-1.000000,0.000000],[ 0.000000, 0.000000,1.000000],[-1.000000, 0.000000,0.000000]])
Rs[10]=torch.tensor([[ 0.000000, 1.000000,0.000000],[ 0.000000 ,0.000000,-1.000000],[-1.000000, 0.000000,0.000000]])
Rs[11]=torch.tensor([[ 0.000000,-1.000000,0.000000],[ 0.000000 ,0.000000,-1.000000],[ 1.000000, 0.000000,0.000000]])
Rs[12]=torch.tensor([[ 0.000000, 1.000000,0.000000],[ 1.000000, 0.000000,0.000000],[ 0.000000 ,0.000000,-1.000000]])
Rs[13]=torch.tensor([[ 0.000000,-1.000000,0.000000],[-1.000000, 0.000000,0.000000],[ 0.000000 ,0.000000,-1.000000]])
Rs[14]=torch.tensor([[ 0.000000, 1.000000,0.000000],[-1.000000, 0.000000,0.000000],[ 0.000000, 0.000000,1.000000]])
Rs[15]=torch.tensor([[ 0.000000,-1.000000,0.000000],[ 1.000000, 0.000000,0.000000],[ 0.000000, 0.000000,1.000000]])
Rs[16]=torch.tensor([[ 1.000000, 0.000000,0.000000],[ 0.000000, 0.000000,1.000000],[ 0.000000,-1.000000,0.000000]])
Rs[17]=torch.tensor([[-1.000000, 0.000000,0.000000],[ 0.000000, 0.000000,1.000000],[ 0.000000, 1.000000,0.000000]])
Rs[18]=torch.tensor([[-1.000000, 0.000000,0.000000],[ 0.000000 ,0.000000,-1.000000],[ 0.000000,-1.000000,0.000000]])
Rs[19]=torch.tensor([[ 1.000000, 0.000000,0.000000],[ 0.000000 ,0.000000,-1.000000],[ 0.000000, 1.000000,0.000000]])
Rs[20]=torch.tensor([[ 0.000000, 0.000000,1.000000],[ 0.000000, 1.000000,0.000000],[-1.000000, 0.000000,0.000000]])
Rs[21]=torch.tensor([[ 0.000000, 0.000000,1.000000],[ 0.000000,-1.000000,0.000000],[ 1.000000, 0.000000,0.000000]])
Rs[22]=torch.tensor([[ 0.000000 ,0.000000,-1.000000],[ 0.000000, 1.000000,0.000000],[ 1.000000, 0.000000,0.000000]])
Rs[23]=torch.tensor([[ 0.000000 ,0.000000,-1.000000],[ 0.000000,-1.000000,0.000000],[-1.000000, 0.000000,0.000000]])
est_radius = 6.0*SYMA
offset = torch.tensor([ 1.0,0.0,0.0 ])
offset = est_radius * offset / torch.linalg.norm(offset)
metasymm = (
[torch.arange(24)],
[6]
)
elif (symmid=='I'):
symmatrix=torch.tensor(
[[ 0, 4, 3, 2, 1, 5, 33, 49, 41, 22, 10, 27, 51, 59, 38, 15, 16, 17,
18, 19, 40, 21, 9, 32, 48, 55, 39, 11, 28, 52, 45, 42, 23, 6, 34, 50,
58, 37, 14, 26, 20, 8, 31, 47, 44, 30, 46, 43, 24, 7, 35, 12, 29, 53,
56, 25, 54, 57, 36, 13],
[ 1, 0, 4, 3, 2, 6, 34, 45, 42, 23, 11, 28, 52, 55, 39, 16, 17, 18,
19, 15, 41, 22, 5, 33, 49, 56, 35, 12, 29, 53, 46, 43, 24, 7, 30, 51,
59, 38, 10, 27, 21, 9, 32, 48, 40, 31, 47, 44, 20, 8, 36, 13, 25, 54,
57, 26, 50, 58, 37, 14],
[ 2, 1, 0, 4, 3, 7, 30, 46, 43, 24, 12, 29, 53, 56, 35, 17, 18, 19,
15, 16, 42, 23, 6, 34, 45, 57, 36, 13, 25, 54, 47, 44, 20, 8, 31, 52,
55, 39, 11, 28, 22, 5, 33, 49, 41, 32, 48, 40, 21, 9, 37, 14, 26, 50,
58, 27, 51, 59, 38, 10],
[ 3, 2, 1, 0, 4, 8, 31, 47, 44, 20, 13, 25, 54, 57, 36, 18, 19, 15,
16, 17, 43, 24, 7, 30, 46, 58, 37, 14, 26, 50, 48, 40, 21, 9, 32, 53,
56, 35, 12, 29, 23, 6, 34, 45, 42, 33, 49, 41, 22, 5, 38, 10, 27, 51,
59, 28, 52, 55, 39, 11],
[ 4, 3, 2, 1, 0, 9, 32, 48, 40, 21, 14, 26, 50, 58, 37, 19, 15, 16,
17, 18, 44, 20, 8, 31, 47, 59, 38, 10, 27, 51, 49, 41, 22, 5, 33, 54,
57, 36, 13, 25, 24, 7, 30, 46, 43, 34, 45, 42, 23, 6, 39, 11, 28, 52,
55, 29, 53, 56, 35, 12],
[ 5, 33, 49, 41, 22, 0, 4, 3, 2, 1, 15, 16, 17, 18, 19, 10, 27, 51,
59, 38, 45, 42, 23, 6, 34, 50, 58, 37, 14, 26, 40, 21, 9, 32, 48, 55,
39, 11, 28, 52, 25, 54, 57, 36, 13, 35, 12, 29, 53, 56, 30, 46, 43, 24,
7, 20, 8, 31, 47, 44],
[ 6, 34, 45, 42, 23, 1, 0, 4, 3, 2, 16, 17, 18, 19, 15, 11, 28, 52,
55, 39, 46, 43, 24, 7, 30, 51, 59, 38, 10, 27, 41, 22, 5, 33, 49, 56,
35, 12, 29, 53, 26, 50, 58, 37, 14, 36, 13, 25, 54, 57, 31, 47, 44, 20,
8, 21, 9, 32, 48, 40],
[ 7, 30, 46, 43, 24, 2, 1, 0, 4, 3, 17, 18, 19, 15, 16, 12, 29, 53,
56, 35, 47, 44, 20, 8, 31, 52, 55, 39, 11, 28, 42, 23, 6, 34, 45, 57,
36, 13, 25, 54, 27, 51, 59, 38, 10, 37, 14, 26, 50, 58, 32, 48, 40, 21,
9, 22, 5, 33, 49, 41],
[ 8, 31, 47, 44, 20, 3, 2, 1, 0, 4, 18, 19, 15, 16, 17, 13, 25, 54,
57, 36, 48, 40, 21, 9, 32, 53, 56, 35, 12, 29, 43, 24, 7, 30, 46, 58,
37, 14, 26, 50, 28, 52, 55, 39, 11, 38, 10, 27, 51, 59, 33, 49, 41, 22,
5, 23, 6, 34, 45, 42],
[ 9, 32, 48, 40, 21, 4, 3, 2, 1, 0, 19, 15, 16, 17, 18, 14, 26, 50,
58, 37, 49, 41, 22, 5, 33, 54, 57, 36, 13, 25, 44, 20, 8, 31, 47, 59,
38, 10, 27, 51, 29, 53, 56, 35, 12, 39, 11, 28, 52, 55, 34, 45, 42, 23,
6, 24, 7, 30, 46, 43],
[10, 27, 51, 59, 38, 15, 16, 17, 18, 19, 0, 4, 3, 2, 1, 5, 33, 49,
41, 22, 50, 58, 37, 14, 26, 45, 42, 23, 6, 34, 55, 39, 11, 28, 52, 40,
21, 9, 32, 48, 30, 46, 43, 24, 7, 20, 8, 31, 47, 44, 25, 54, 57, 36,
13, 35, 12, 29, 53, 56],
[11, 28, 52, 55, 39, 16, 17, 18, 19, 15, 1, 0, 4, 3, 2, 6, 34, 45,
42, 23, 51, 59, 38, 10, 27, 46, 43, 24, 7, 30, 56, 35, 12, 29, 53, 41,
22, 5, 33, 49, 31, 47, 44, 20, 8, 21, 9, 32, 48, 40, 26, 50, 58, 37,
14, 36, 13, 25, 54, 57],
[12, 29, 53, 56, 35, 17, 18, 19, 15, 16, 2, 1, 0, 4, 3, 7, 30, 46,
43, 24, 52, 55, 39, 11, 28, 47, 44, 20, 8, 31, 57, 36, 13, 25, 54, 42,
23, 6, 34, 45, 32, 48, 40, 21, 9, 22, 5, 33, 49, 41, 27, 51, 59, 38,
10, 37, 14, 26, 50, 58],
[13, 25, 54, 57, 36, 18, 19, 15, 16, 17, 3, 2, 1, 0, 4, 8, 31, 47,
44, 20, 53, 56, 35, 12, 29, 48, 40, 21, 9, 32, 58, 37, 14, 26, 50, 43,
24, 7, 30, 46, 33, 49, 41, 22, 5, 23, 6, 34, 45, 42, 28, 52, 55, 39,
11, 38, 10, 27, 51, 59],
[14, 26, 50, 58, 37, 19, 15, 16, 17, 18, 4, 3, 2, 1, 0, 9, 32, 48,
40, 21, 54, 57, 36, 13, 25, 49, 41, 22, 5, 33, 59, 38, 10, 27, 51, 44,
20, 8, 31, 47, 34, 45, 42, 23, 6, 24, 7, 30, 46, 43, 29, 53, 56, 35,
12, 39, 11, 28, 52, 55],
[15, 16, 17, 18, 19, 10, 27, 51, 59, 38, 5, 33, 49, 41, 22, 0, 4, 3,
2, 1, 55, 39, 11, 28, 52, 40, 21, 9, 32, 48, 50, 58, 37, 14, 26, 45,
42, 23, 6, 34, 35, 12, 29, 53, 56, 25, 54, 57, 36, 13, 20, 8, 31, 47,
44, 30, 46, 43, 24, 7],
[16, 17, 18, 19, 15, 11, 28, 52, 55, 39, 6, 34, 45, 42, 23, 1, 0, 4,
3, 2, 56, 35, 12, 29, 53, 41, 22, 5, 33, 49, 51, 59, 38, 10, 27, 46,
43, 24, 7, 30, 36, 13, 25, 54, 57, 26, 50, 58, 37, 14, 21, 9, 32, 48,
40, 31, 47, 44, 20, 8],
[17, 18, 19, 15, 16, 12, 29, 53, 56, 35, 7, 30, 46, 43, 24, 2, 1, 0,
4, 3, 57, 36, 13, 25, 54, 42, 23, 6, 34, 45, 52, 55, 39, 11, 28, 47,
44, 20, 8, 31, 37, 14, 26, 50, 58, 27, 51, 59, 38, 10, 22, 5, 33, 49,
41, 32, 48, 40, 21, 9],
[18, 19, 15, 16, 17, 13, 25, 54, 57, 36, 8, 31, 47, 44, 20, 3, 2, 1,
0, 4, 58, 37, 14, 26, 50, 43, 24, 7, 30, 46, 53, 56, 35, 12, 29, 48,
40, 21, 9, 32, 38, 10, 27, 51, 59, 28, 52, 55, 39, 11, 23, 6, 34, 45,
42, 33, 49, 41, 22, 5],
[19, 15, 16, 17, 18, 14, 26, 50, 58, 37, 9, 32, 48, 40, 21, 4, 3, 2,
1, 0, 59, 38, 10, 27, 51, 44, 20, 8, 31, 47, 54, 57, 36, 13, 25, 49,
41, 22, 5, 33, 39, 11, 28, 52, 55, 29, 53, 56, 35, 12, 24, 7, 30, 46,
43, 34, 45, 42, 23, 6],
[20, 8, 31, 47, 44, 30, 46, 43, 24, 7, 35, 12, 29, 53, 56, 25, 54, 57,
36, 13, 0, 4, 3, 2, 1, 5, 33, 49, 41, 22, 10, 27, 51, 59, 38, 15,
16, 17, 18, 19, 40, 21, 9, 32, 48, 55, 39, 11, 28, 52, 45, 42, 23, 6,
34, 50, 58, 37, 14, 26],
[21, 9, 32, 48, 40, 31, 47, 44, 20, 8, 36, 13, 25, 54, 57, 26, 50, 58,
37, 14, 1, 0, 4, 3, 2, 6, 34, 45, 42, 23, 11, 28, 52, 55, 39, 16,
17, 18, 19, 15, 41, 22, 5, 33, 49, 56, 35, 12, 29, 53, 46, 43, 24, 7,
30, 51, 59, 38, 10, 27],
[22, 5, 33, 49, 41, 32, 48, 40, 21, 9, 37, 14, 26, 50, 58, 27, 51, 59,
38, 10, 2, 1, 0, 4, 3, 7, 30, 46, 43, 24, 12, 29, 53, 56, 35, 17,
18, 19, 15, 16, 42, 23, 6, 34, 45, 57, 36, 13, 25, 54, 47, 44, 20, 8,
31, 52, 55, 39, 11, 28],
[23, 6, 34, 45, 42, 33, 49, 41, 22, 5, 38, 10, 27, 51, 59, 28, 52, 55,
39, 11, 3, 2, 1, 0, 4, 8, 31, 47, 44, 20, 13, 25, 54, 57, 36, 18,
19, 15, 16, 17, 43, 24, 7, 30, 46, 58, 37, 14, 26, 50, 48, 40, 21, 9,
32, 53, 56, 35, 12, 29],
[24, 7, 30, 46, 43, 34, 45, 42, 23, 6, 39, 11, 28, 52, 55, 29, 53, 56,
35, 12, 4, 3, 2, 1, 0, 9, 32, 48, 40, 21, 14, 26, 50, 58, 37, 19,
15, 16, 17, 18, 44, 20, 8, 31, 47, 59, 38, 10, 27, 51, 49, 41, 22, 5,
33, 54, 57, 36, 13, 25],
[25, 54, 57, 36, 13, 35, 12, 29, 53, 56, 30, 46, 43, 24, 7, 20, 8, 31,
47, 44, 5, 33, 49, 41, 22, 0, 4, 3, 2, 1, 15, 16, 17, 18, 19, 10,
27, 51, 59, 38, 45, 42, 23, 6, 34, 50, 58, 37, 14, 26, 40, 21, 9, 32,
48, 55, 39, 11, 28, 52],
[26, 50, 58, 37, 14, 36, 13, 25, 54, 57, 31, 47, 44, 20, 8, 21, 9, 32,
48, 40, 6, 34, 45, 42, 23, 1, 0, 4, 3, 2, 16, 17, 18, 19, 15, 11,
28, 52, 55, 39, 46, 43, 24, 7, 30, 51, 59, 38, 10, 27, 41, 22, 5, 33,
49, 56, 35, 12, 29, 53],
[27, 51, 59, 38, 10, 37, 14, 26, 50, 58, 32, 48, 40, 21, 9, 22, 5, 33,
49, 41, 7, 30, 46, 43, 24, 2, 1, 0, 4, 3, 17, 18, 19, 15, 16, 12,
29, 53, 56, 35, 47, 44, 20, 8, 31, 52, 55, 39, 11, 28, 42, 23, 6, 34,
45, 57, 36, 13, 25, 54],
[28, 52, 55, 39, 11, 38, 10, 27, 51, 59, 33, 49, 41, 22, 5, 23, 6, 34,
45, 42, 8, 31, 47, 44, 20, 3, 2, 1, 0, 4, 18, 19, 15, 16, 17, 13,
25, 54, 57, 36, 48, 40, 21, 9, 32, 53, 56, 35, 12, 29, 43, 24, 7, 30,
46, 58, 37, 14, 26, 50],
[29, 53, 56, 35, 12, 39, 11, 28, 52, 55, 34, 45, 42, 23, 6, 24, 7, 30,
46, 43, 9, 32, 48, 40, 21, 4, 3, 2, 1, 0, 19, 15, 16, 17, 18, 14,
26, 50, 58, 37, 49, 41, 22, 5, 33, 54, 57, 36, 13, 25, 44, 20, 8, 31,
47, 59, 38, 10, 27, 51],
[30, 46, 43, 24, 7, 20, 8, 31, 47, 44, 25, 54, 57, 36, 13, 35, 12, 29,
53, 56, 10, 27, 51, 59, 38, 15, 16, 17, 18, 19, 0, 4, 3, 2, 1, 5,
33, 49, 41, 22, 50, 58, 37, 14, 26, 45, 42, 23, 6, 34, 55, 39, 11, 28,
52, 40, 21, 9, 32, 48],
[31, 47, 44, 20, 8, 21, 9, 32, 48, 40, 26, 50, 58, 37, 14, 36, 13, 25,
54, 57, 11, 28, 52, 55, 39, 16, 17, 18, 19, 15, 1, 0, 4, 3, 2, 6,
34, 45, 42, 23, 51, 59, 38, 10, 27, 46, 43, 24, 7, 30, 56, 35, 12, 29,
53, 41, 22, 5, 33, 49],
[32, 48, 40, 21, 9, 22, 5, 33, 49, 41, 27, 51, 59, 38, 10, 37, 14, 26,
50, 58, 12, 29, 53, 56, 35, 17, 18, 19, 15, 16, 2, 1, 0, 4, 3, 7,
30, 46, 43, 24, 52, 55, 39, 11, 28, 47, 44, 20, 8, 31, 57, 36, 13, 25,
54, 42, 23, 6, 34, 45],
[33, 49, 41, 22, 5, 23, 6, 34, 45, 42, 28, 52, 55, 39, 11, 38, 10, 27,
51, 59, 13, 25, 54, 57, 36, 18, 19, 15, 16, 17, 3, 2, 1, 0, 4, 8,
31, 47, 44, 20, 53, 56, 35, 12, 29, 48, 40, 21, 9, 32, 58, 37, 14, 26,
50, 43, 24, 7, 30, 46],
[34, 45, 42, 23, 6, 24, 7, 30, 46, 43, 29, 53, 56, 35, 12, 39, 11, 28,
52, 55, 14, 26, 50, 58, 37, 19, 15, 16, 17, 18, 4, 3, 2, 1, 0, 9,
32, 48, 40, 21, 54, 57, 36, 13, 25, 49, 41, 22, 5, 33, 59, 38, 10, 27,
51, 44, 20, 8, 31, 47],
[35, 12, 29, 53, 56, 25, 54, 57, 36, 13, 20, 8, 31, 47, 44, 30, 46, 43,
24, 7, 15, 16, 17, 18, 19, 10, 27, 51, 59, 38, 5, 33, 49, 41, 22, 0,
4, 3, 2, 1, 55, 39, 11, 28, 52, 40, 21, 9, 32, 48, 50, 58, 37, 14,
26, 45, 42, 23, 6, 34],
[36, 13, 25, 54, 57, 26, 50, 58, 37, 14, 21, 9, 32, 48, 40, 31, 47, 44,
20, 8, 16, 17, 18, 19, 15, 11, 28, 52, 55, 39, 6, 34, 45, 42, 23, 1,
0, 4, 3, 2, 56, 35, 12, 29, 53, 41, 22, 5, 33, 49, 51, 59, 38, 10,
27, 46, 43, 24, 7, 30],
[37, 14, 26, 50, 58, 27, 51, 59, 38, 10, 22, 5, 33, 49, 41, 32, 48, 40,
21, 9, 17, 18, 19, 15, 16, 12, 29, 53, 56, 35, 7, 30, 46, 43, 24, 2,
1, 0, 4, 3, 57, 36, 13, 25, 54, 42, 23, 6, 34, 45, 52, 55, 39, 11,
28, 47, 44, 20, 8, 31],
[38, 10, 27, 51, 59, 28, 52, 55, 39, 11, 23, 6, 34, 45, 42, 33, 49, 41,
22, 5, 18, 19, 15, 16, 17, 13, 25, 54, 57, 36, 8, 31, 47, 44, 20, 3,
2, 1, 0, 4, 58, 37, 14, 26, 50, 43, 24, 7, 30, 46, 53, 56, 35, 12,
29, 48, 40, 21, 9, 32],
[39, 11, 28, 52, 55, 29, 53, 56, 35, 12, 24, 7, 30, 46, 43, 34, 45, 42,
23, 6, 19, 15, 16, 17, 18, 14, 26, 50, 58, 37, 9, 32, 48, 40, 21, 4,
3, 2, 1, 0, 59, 38, 10, 27, 51, 44, 20, 8, 31, 47, 54, 57, 36, 13,
25, 49, 41, 22, 5, 33],
[40, 21, 9, 32, 48, 55, 39, 11, 28, 52, 45, 42, 23, 6, 34, 50, 58, 37,
14, 26, 20, 8, 31, 47, 44, 30, 46, 43, 24, 7, 35, 12, 29, 53, 56, 25,
54, 57, 36, 13, 0, 4, 3, 2, 1, 5, 33, 49, 41, 22, 10, 27, 51, 59,
38, 15, 16, 17, 18, 19],
[41, 22, 5, 33, 49, 56, 35, 12, 29, 53, 46, 43, 24, 7, 30, 51, 59, 38,
10, 27, 21, 9, 32, 48, 40, 31, 47, 44, 20, 8, 36, 13, 25, 54, 57, 26,
50, 58, 37, 14, 1, 0, 4, 3, 2, 6, 34, 45, 42, 23, 11, 28, 52, 55,
39, 16, 17, 18, 19, 15],
[42, 23, 6, 34, 45, 57, 36, 13, 25, 54, 47, 44, 20, 8, 31, 52, 55, 39,
11, 28, 22, 5, 33, 49, 41, 32, 48, 40, 21, 9, 37, 14, 26, 50, 58, 27,
51, 59, 38, 10, 2, 1, 0, 4, 3, 7, 30, 46, 43, 24, 12, 29, 53, 56,
35, 17, 18, 19, 15, 16],
[43, 24, 7, 30, 46, 58, 37, 14, 26, 50, 48, 40, 21, 9, 32, 53, 56, 35,
12, 29, 23, 6, 34, 45, 42, 33, 49, 41, 22, 5, 38, 10, 27, 51, 59, 28,
52, 55, 39, 11, 3, 2, 1, 0, 4, 8, 31, 47, 44, 20, 13, 25, 54, 57,
36, 18, 19, 15, 16, 17],
[44, 20, 8, 31, 47, 59, 38, 10, 27, 51, 49, 41, 22, 5, 33, 54, 57, 36,
13, 25, 24, 7, 30, 46, 43, 34, 45, 42, 23, 6, 39, 11, 28, 52, 55, 29,
53, 56, 35, 12, 4, 3, 2, 1, 0, 9, 32, 48, 40, 21, 14, 26, 50, 58,
37, 19, 15, 16, 17, 18],
[45, 42, 23, 6, 34, 50, 58, 37, 14, 26, 40, 21, 9, 32, 48, 55, 39, 11,
28, 52, 25, 54, 57, 36, 13, 35, 12, 29, 53, 56, 30, 46, 43, 24, 7, 20,
8, 31, 47, 44, 5, 33, 49, 41, 22, 0, 4, 3, 2, 1, 15, 16, 17, 18,
19, 10, 27, 51, 59, 38],
[46, 43, 24, 7, 30, 51, 59, 38, 10, 27, 41, 22, 5, 33, 49, 56, 35, 12,
29, 53, 26, 50, 58, 37, 14, 36, 13, 25, 54, 57, 31, 47, 44, 20, 8, 21,
9, 32, 48, 40, 6, 34, 45, 42, 23, 1, 0, 4, 3, 2, 16, 17, 18, 19,
15, 11, 28, 52, 55, 39],
[47, 44, 20, 8, 31, 52, 55, 39, 11, 28, 42, 23, 6, 34, 45, 57, 36, 13,
25, 54, 27, 51, 59, 38, 10, 37, 14, 26, 50, 58, 32, 48, 40, 21, 9, 22,
5, 33, 49, 41, 7, 30, 46, 43, 24, 2, 1, 0, 4, 3, 17, 18, 19, 15,
16, 12, 29, 53, 56, 35],
[48, 40, 21, 9, 32, 53, 56, 35, 12, 29, 43, 24, 7, 30, 46, 58, 37, 14,
26, 50, 28, 52, 55, 39, 11, 38, 10, 27, 51, 59, 33, 49, 41, 22, 5, 23,
6, 34, 45, 42, 8, 31, 47, 44, 20, 3, 2, 1, 0, 4, 18, 19, 15, 16,
17, 13, 25, 54, 57, 36],
[49, 41, 22, 5, 33, 54, 57, 36, 13, 25, 44, 20, 8, 31, 47, 59, 38, 10,
27, 51, 29, 53, 56, 35, 12, 39, 11, 28, 52, 55, 34, 45, 42, 23, 6, 24,
7, 30, 46, 43, 9, 32, 48, 40, 21, 4, 3, 2, 1, 0, 19, 15, 16, 17,
18, 14, 26, 50, 58, 37],
[50, 58, 37, 14, 26, 45, 42, 23, 6, 34, 55, 39, 11, 28, 52, 40, 21, 9,
32, 48, 30, 46, 43, 24, 7, 20, 8, 31, 47, 44, 25, 54, 57, 36, 13, 35,
12, 29, 53, 56, 10, 27, 51, 59, 38, 15, 16, 17, 18, 19, 0, 4, 3, 2,
1, 5, 33, 49, 41, 22],
[51, 59, 38, 10, 27, 46, 43, 24, 7, 30, 56, 35, 12, 29, 53, 41, 22, 5,
33, 49, 31, 47, 44, 20, 8, 21, 9, 32, 48, 40, 26, 50, 58, 37, 14, 36,
13, 25, 54, 57, 11, 28, 52, 55, 39, 16, 17, 18, 19, 15, 1, 0, 4, 3,
2, 6, 34, 45, 42, 23],
[52, 55, 39, 11, 28, 47, 44, 20, 8, 31, 57, 36, 13, 25, 54, 42, 23, 6,
34, 45, 32, 48, 40, 21, 9, 22, 5, 33, 49, 41, 27, 51, 59, 38, 10, 37,
14, 26, 50, 58, 12, 29, 53, 56, 35, 17, 18, 19, 15, 16, 2, 1, 0, 4,
3, 7, 30, 46, 43, 24],
[53, 56, 35, 12, 29, 48, 40, 21, 9, 32, 58, 37, 14, 26, 50, 43, 24, 7,
30, 46, 33, 49, 41, 22, 5, 23, 6, 34, 45, 42, 28, 52, 55, 39, 11, 38,
10, 27, 51, 59, 13, 25, 54, 57, 36, 18, 19, 15, 16, 17, 3, 2, 1, 0,
4, 8, 31, 47, 44, 20],
[54, 57, 36, 13, 25, 49, 41, 22, 5, 33, 59, 38, 10, 27, 51, 44, 20, 8,
31, 47, 34, 45, 42, 23, 6, 24, 7, 30, 46, 43, 29, 53, 56, 35, 12, 39,
11, 28, 52, 55, 14, 26, 50, 58, 37, 19, 15, 16, 17, 18, 4, 3, 2, 1,
0, 9, 32, 48, 40, 21],
[55, 39, 11, 28, 52, 40, 21, 9, 32, 48, 50, 58, 37, 14, 26, 45, 42, 23,
6, 34, 35, 12, 29, 53, 56, 25, 54, 57, 36, 13, 20, 8, 31, 47, 44, 30,
46, 43, 24, 7, 15, 16, 17, 18, 19, 10, 27, 51, 59, 38, 5, 33, 49, 41,
22, 0, 4, 3, 2, 1],
[56, 35, 12, 29, 53, 41, 22, 5, 33, 49, 51, 59, 38, 10, 27, 46, 43, 24,
7, 30, 36, 13, 25, 54, 57, 26, 50, 58, 37, 14, 21, 9, 32, 48, 40, 31,
47, 44, 20, 8, 16, 17, 18, 19, 15, 11, 28, 52, 55, 39, 6, 34, 45, 42,
23, 1, 0, 4, 3, 2],
[57, 36, 13, 25, 54, 42, 23, 6, 34, 45, 52, 55, 39, 11, 28, 47, 44, 20,
8, 31, 37, 14, 26, 50, 58, 27, 51, 59, 38, 10, 22, 5, 33, 49, 41, 32,
48, 40, 21, 9, 17, 18, 19, 15, 16, 12, 29, 53, 56, 35, 7, 30, 46, 43,
24, 2, 1, 0, 4, 3],
[58, 37, 14, 26, 50, 43, 24, 7, 30, 46, 53, 56, 35, 12, 29, 48, 40, 21,
9, 32, 38, 10, 27, 51, 59, 28, 52, 55, 39, 11, 23, 6, 34, 45, 42, 33,
49, 41, 22, 5, 18, 19, 15, 16, 17, 13, 25, 54, 57, 36, 8, 31, 47, 44,
20, 3, 2, 1, 0, 4],
[59, 38, 10, 27, 51, 44, 20, 8, 31, 47, 54, 57, 36, 13, 25, 49, 41, 22,
5, 33, 39, 11, 28, 52, 55, 29, 53, 56, 35, 12, 24, 7, 30, 46, 43, 34,
45, 42, 23, 6, 19, 15, 16, 17, 18, 14, 26, 50, 58, 37, 9, 32, 48, 40,
21, 4, 3, 2, 1, 0]])
Rs = torch.zeros(60,3,3)
Rs[0]=torch.tensor([[ 1.000000, 0.000000,0.000000],[ 0.000000, 1.000000,0.000000],[ 0.000000, 0.000000,1.000000]])
Rs[1]=torch.tensor([[ 0.500000,-0.809017,0.309017],[ 0.809017 ,0.309017,-0.500000],[ 0.309017, 0.500000,0.809017]])
Rs[2]=torch.tensor([[-0.309017,-0.500000,0.809017],[ 0.500000,-0.809017,-0.309017],[ 0.809017, 0.309017,0.500000]])
Rs[3]=torch.tensor([[-0.309017, 0.500000,0.809017],[-0.500000,-0.809017,0.309017],[ 0.809017,-0.309017,0.500000]])
Rs[4]=torch.tensor([[ 0.500000, 0.809017,0.309017],[-0.809017, 0.309017,0.500000],[ 0.309017,-0.500000,0.809017]])
Rs[5]=torch.tensor([[-0.809017, 0.309017,0.500000],[ 0.309017,-0.500000,0.809017],[ 0.500000, 0.809017,0.309017]])
Rs[6]=torch.tensor([[ 0.000000, 1.000000,0.000000],[ 0.000000, 0.000000,1.000000],[ 1.000000, 0.000000,0.000000]])
Rs[7]=torch.tensor([[ 0.809017 ,0.309017,-0.500000],[ 0.309017, 0.500000,0.809017],[ 0.500000,-0.809017,0.309017]])
Rs[8]=torch.tensor([[ 0.500000,-0.809017,-0.309017],[ 0.809017, 0.309017,0.500000],[-0.309017,-0.500000,0.809017]])
Rs[9]=torch.tensor([[-0.500000,-0.809017,0.309017],[ 0.809017,-0.309017,0.500000],[-0.309017, 0.500000,0.809017]])
Rs[10]=torch.tensor([[-0.500000,-0.809017,0.309017],[-0.809017 ,0.309017,-0.500000],[ 0.309017,-0.500000,-0.809017]])
Rs[11]=torch.tensor([[-0.809017, 0.309017,0.500000],[-0.309017 ,0.500000,-0.809017],[-0.500000,-0.809017,-0.309017]])
Rs[12]=torch.tensor([[ 0.000000, 1.000000,0.000000],[ 0.000000 ,0.000000,-1.000000],[-1.000000, 0.000000,0.000000]])
Rs[13]=torch.tensor([[ 0.809017 ,0.309017,-0.500000],[-0.309017,-0.500000,-0.809017],[-0.500000 ,0.809017,-0.309017]])
Rs[14]=torch.tensor([[ 0.500000,-0.809017,-0.309017],[-0.809017,-0.309017,-0.500000],[ 0.309017 ,0.500000,-0.809017]])
Rs[15]=torch.tensor([[ 0.309017 ,0.500000,-0.809017],[ 0.500000,-0.809017,-0.309017],[-0.809017,-0.309017,-0.500000]])
Rs[16]=torch.tensor([[ 0.309017,-0.500000,-0.809017],[-0.500000,-0.809017,0.309017],[-0.809017 ,0.309017,-0.500000]])
Rs[17]=torch.tensor([[-0.500000,-0.809017,-0.309017],[-0.809017, 0.309017,0.500000],[-0.309017 ,0.500000,-0.809017]])
Rs[18]=torch.tensor([[-1.000000, 0.000000,0.000000],[ 0.000000, 1.000000,0.000000],[ 0.000000 ,0.000000,-1.000000]])
Rs[19]=torch.tensor([[-0.500000 ,0.809017,-0.309017],[ 0.809017 ,0.309017,-0.500000],[-0.309017,-0.500000,-0.809017]])
Rs[20]=torch.tensor([[-0.500000,-0.809017,-0.309017],[ 0.809017,-0.309017,-0.500000],[ 0.309017,-0.500000,0.809017]])
Rs[21]=torch.tensor([[-1.000000, 0.000000,0.000000],[ 0.000000,-1.000000,0.000000],[ 0.000000, 0.000000,1.000000]])
Rs[22]=torch.tensor([[-0.500000 ,0.809017,-0.309017],[-0.809017,-0.309017,0.500000],[ 0.309017, 0.500000,0.809017]])
Rs[23]=torch.tensor([[ 0.309017 ,0.500000,-0.809017],[-0.500000, 0.809017,0.309017],[ 0.809017, 0.309017,0.500000]])
Rs[24]=torch.tensor([[ 0.309017,-0.500000,-0.809017],[ 0.500000 ,0.809017,-0.309017],[ 0.809017,-0.309017,0.500000]])
Rs[25]=torch.tensor([[ 0.000000 ,0.000000,-1.000000],[-1.000000, 0.000000,0.000000],[ 0.000000, 1.000000,0.000000]])
Rs[26]=torch.tensor([[-0.309017,-0.500000,-0.809017],[-0.500000 ,0.809017,-0.309017],[ 0.809017 ,0.309017,-0.500000]])
Rs[27]=torch.tensor([[-0.809017,-0.309017,-0.500000],[ 0.309017 ,0.500000,-0.809017],[ 0.500000,-0.809017,-0.309017]])
Rs[28]=torch.tensor([[-0.809017 ,0.309017,-0.500000],[ 0.309017,-0.500000,-0.809017],[-0.500000,-0.809017,0.309017]])
Rs[29]=torch.tensor([[-0.309017 ,0.500000,-0.809017],[-0.500000,-0.809017,-0.309017],[-0.809017, 0.309017,0.500000]])
Rs[30]=torch.tensor([[ 0.809017, 0.309017,0.500000],[-0.309017,-0.500000,0.809017],[ 0.500000,-0.809017,-0.309017]])
Rs[31]=torch.tensor([[ 0.809017,-0.309017,0.500000],[-0.309017, 0.500000,0.809017],[-0.500000,-0.809017,0.309017]])
Rs[32]=torch.tensor([[ 0.309017,-0.500000,0.809017],[ 0.500000, 0.809017,0.309017],[-0.809017, 0.309017,0.500000]])
Rs[33]=torch.tensor([[ 0.000000, 0.000000,1.000000],[ 1.000000, 0.000000,0.000000],[ 0.000000, 1.000000,0.000000]])
Rs[34]=torch.tensor([[ 0.309017, 0.500000,0.809017],[ 0.500000,-0.809017,0.309017],[ 0.809017 ,0.309017,-0.500000]])
Rs[35]=torch.tensor([[-0.309017, 0.500000,0.809017],[ 0.500000 ,0.809017,-0.309017],[-0.809017 ,0.309017,-0.500000]])
Rs[36]=torch.tensor([[ 0.500000, 0.809017,0.309017],[ 0.809017,-0.309017,-0.500000],[-0.309017 ,0.500000,-0.809017]])
Rs[37]=torch.tensor([[ 1.000000, 0.000000,0.000000],[ 0.000000,-1.000000,0.000000],[ 0.000000 ,0.000000,-1.000000]])
Rs[38]=torch.tensor([[ 0.500000,-0.809017,0.309017],[-0.809017,-0.309017,0.500000],[-0.309017,-0.500000,-0.809017]])
Rs[39]=torch.tensor([[-0.309017,-0.500000,0.809017],[-0.500000, 0.809017,0.309017],[-0.809017,-0.309017,-0.500000]])
Rs[40]=torch.tensor([[-0.500000, 0.809017,0.309017],[-0.809017,-0.309017,-0.500000],[-0.309017,-0.500000,0.809017]])
Rs[41]=torch.tensor([[ 0.500000 ,0.809017,-0.309017],[-0.809017 ,0.309017,-0.500000],[-0.309017, 0.500000,0.809017]])
Rs[42]=torch.tensor([[ 0.809017,-0.309017,-0.500000],[-0.309017 ,0.500000,-0.809017],[ 0.500000, 0.809017,0.309017]])
Rs[43]=torch.tensor([[ 0.000000,-1.000000,0.000000],[ 0.000000 ,0.000000,-1.000000],[ 1.000000, 0.000000,0.000000]])
Rs[44]=torch.tensor([[-0.809017,-0.309017,0.500000],[-0.309017,-0.500000,-0.809017],[ 0.500000,-0.809017,0.309017]])
Rs[45]=torch.tensor([[ 0.809017,-0.309017,0.500000],[ 0.309017,-0.500000,-0.809017],[ 0.500000 ,0.809017,-0.309017]])
Rs[46]=torch.tensor([[ 0.309017,-0.500000,0.809017],[-0.500000,-0.809017,-0.309017],[ 0.809017,-0.309017,-0.500000]])
Rs[47]=torch.tensor([[ 0.000000, 0.000000,1.000000],[-1.000000, 0.000000,0.000000],[ 0.000000,-1.000000,0.000000]])
Rs[48]=torch.tensor([[ 0.309017, 0.500000,0.809017],[-0.500000 ,0.809017,-0.309017],[-0.809017,-0.309017,0.500000]])
Rs[49]=torch.tensor([[ 0.809017, 0.309017,0.500000],[ 0.309017 ,0.500000,-0.809017],[-0.500000, 0.809017,0.309017]])
Rs[50]=torch.tensor([[-0.309017 ,0.500000,-0.809017],[ 0.500000, 0.809017,0.309017],[ 0.809017,-0.309017,-0.500000]])
Rs[51]=torch.tensor([[ 0.000000 ,0.000000,-1.000000],[ 1.000000, 0.000000,0.000000],[ 0.000000,-1.000000,0.000000]])
Rs[52]=torch.tensor([[-0.309017,-0.500000,-0.809017],[ 0.500000,-0.809017,0.309017],[-0.809017,-0.309017,0.500000]])
Rs[53]=torch.tensor([[-0.809017,-0.309017,-0.500000],[-0.309017,-0.500000,0.809017],[-0.500000, 0.809017,0.309017]])
Rs[54]=torch.tensor([[-0.809017 ,0.309017,-0.500000],[-0.309017, 0.500000,0.809017],[ 0.500000 ,0.809017,-0.309017]])
Rs[55]=torch.tensor([[ 0.000000,-1.000000,0.000000],[ 0.000000, 0.000000,1.000000],[-1.000000, 0.000000,0.000000]])
Rs[56]=torch.tensor([[-0.809017,-0.309017,0.500000],[ 0.309017, 0.500000,0.809017],[-0.500000 ,0.809017,-0.309017]])
Rs[57]=torch.tensor([[-0.500000, 0.809017,0.309017],[ 0.809017, 0.309017,0.500000],[ 0.309017 ,0.500000,-0.809017]])
Rs[58]=torch.tensor([[ 0.500000 ,0.809017,-0.309017],[ 0.809017,-0.309017,0.500000],[ 0.309017,-0.500000,-0.809017]])
Rs[59]=torch.tensor([[ 0.809017,-0.309017,-0.500000],[ 0.309017,-0.500000,0.809017],[-0.500000,-0.809017,-0.309017]])
est_radius = 10.0*SYMA
offset = torch.tensor([ 1.0,0.0,0.0 ])
offset = est_radius * offset / torch.linalg.norm(offset)
metasymm = (
[torch.arange(60)],
[6]
)
else:
print ("Unknown symmetry",symmid)
assert False
return symmatrix,Rs,metasymm,offset
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