Mathematica routines for sphere packings

### Mathematica Sphere Packing Code

This file gives descriptions of functions appearing in a Mathematica package designed to assist in the exploration of packings in three dimensions. Detailed descriptions of most of these functions can be found in Sphere Packings I. The source code is available. Sean McLaughlin has also implemented many of these functions in Maple.
zeta, pt, doct
Various constants
chi
A polynomial in six variables giving the orientation of a simplex along the fourth, fifth and sixth edges
eta
eta[x,y,z] is the circumradius of a triangle with edges x,y,z.
volR[a,b,c]
the volume of the Rogers simplex R[a,b,c]
solR[a,b,c]
the solid angle of the Rogers simplex R[a,b,c]
dihR[a,b,c]
the dihedral angle of the Rogers simplex R[a,b,c]
vorR[a,b,c]
the voronoi score of the Rogers simplex R[a,b,c]
denR[a,b,c]
the packing density of the Rogers simplex R[a,b,c]
tauR[a,b,c]
the amount squandered by the Rogers simplex R[a,b,c]
TRUNC
1.255
Dihedral[y1,...,y6]
The dihedral angle of a simplex with edges y1,...,y6.
Dihedral2[y1,...,y6]
The dihedral angle along the second edge of a simplex with edges y1,...,y6.
Dihedral3[y1,...,y6]
The dihedral angle along the third edge of a simplex with edges y1,...,y6.
Chi[y1,...,y6]
The function Chi along the top face of a simplex with edges y1,...,y6
tauAnalytic[y1,...,y6]
What the simplex S(y1,...,y6) squanders under the analytic Voronoi function.
tau[y1,...,y6]
What the simplex S(y1,...,y6) squanders under compression.
VorAnalytic[y1,..,y6]
The analytic voronoi function at the simplex S(y1,...,y6)
Delta[y1,...,y6]
The function delta expressed in terms of edge lengths.
U[x1,x2,x6]
The function U expressed in terms of squares of edge lengths.
Rho[y1,...,y6]
The function rho expressed in terms of edge lengths
The circumradius of a simplex S(y1,...,y6)
aSolid[y1,...,y6]
The function a(y1,...,y6) appearing in the definition of solid angles
Gamma[y1,...,y6]
The compression of the simplex S(y1,...,y6)
Solid[y1,...,y6]
The solid angle of a simplex S(y1,...,y6)
Density[x]
The packing density corresponding to a score of x
Norm[x]
The length squared of a vector x
Norm[x,y]
The distance squared from x to y
Distance[x,y]
The Euclidean distance between vectors x and y
FarFrom[v1,{u,v}]
This function returns vector u or v depending on which has the greatest distance from v1.
ExtremePoint[{v1,v2,v3}]
The circumcenter of the simplex with vertices 0,v1,v2,v3.
Vertex[{v1,dist1},{v2,dist2},{v3,dist3}]
This function returns a pairs of vectors {u1,u2}, where u1 and u2 are the two vectors at distance dist1 from vector 1, dist2 from vector2, and dist3 from vector3.
SimplexCoordinates[y1,...,y6]
This function returns {v1,v2,v3}, where 0,v1,v2,v3 are the coordinate vectors of the four vertices of a simplex with edges y1,...,y6.
Enclosed[y1,...,y6,z1,z2,z3]
The function \\Cal E from Sphere Packings papers
NullVector
The vector {0,0,0}
Keep[x]
Place the definition of x in the file keep.m
Fix[x]
Edit the definition of x
\$SpherePrecision
The working precision for calculations in the package
Ns[x]
Evaluate x to the precision \$SpherePrecision

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