Mathematica Sphere Packing Code

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

Rad[y1,...,y6]

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