A395321 Decimal expansion of the conjecturally maximum possible volume of a polyhedron with 16 vertices inscribed in the unit sphere.

2, 8, 8, 6, 4, 5, 5, 3, 9, 2, 2, 7, 4, 5, 1, 8, 5, 8, 2, 7, 4, 7, 3, 8, 1, 6, 7, 8, 2, 8, 5, 4, 8, 9, 0, 0, 8, 7, 9, 8, 8, 7, 9, 5, 0, 1, 2, 0, 8, 2, 3, 8, 6, 9, 2, 6, 6, 6, 0, 4, 8, 6, 5, 2, 6, 8, 7, 6, 9, 0, 6, 5, 2, 9, 3, 2, 6, 2, 5, 9, 5, 6, 8, 3, 9, 4, 9, 5, 3

Offset

1,1

Comments

Due to the high symmetry of the arrangement of the vertices (tetrahedral symmetry group S_4 with order 24), there is hope that an analytical representation of this constant exists, but this is not yet known.

From Natalia L. Skirrow, Jun 04 2026: (Start)

The polyhedron can be written with vertex coordinates:

( t, t, t),

( t,-t,-t) (3 perms),

( r,-s, s) (6 perms),

(-r, s, s) (3 perms),

(-r,-s,-s) (3 perms),

(closed under the symmetry operations "permute axes" and "flip two signs"), with t=1/sqrt(3), and r^2+2*s^2 = 1. Decimal expansions:

t = 0.57735026918962576450914878050196,

r = 0.91157938253114920597802631908894,

s = 0.29070864223834896670682754080185.

The polyhedron is constructible out of radial tetrahedra, in particular (the equivalence classes up to S_4 of):

I = {(0,0,0),(t,t,t),(r,-s,s),(r,s,-s)} (12 representatives),

J = {(0,0,0),(t,t,t),(s,-v,r),(r,-s,s)} (12 reps),

E = {(0,0,0),(-s,-s,-r),(-s,-r,-s),(-r,-s,-s)} (4 reps).

These have volumes:

vol(I) = det([[t,t,t],[r,-s,s],[r, s,-s]])/6 = 2*t*r*s/3,

vol(J) = det([[t,t,t],[s,-s,r],[r,-s, s]])/6 = t*(r-s)*(r+3*s)/6,

vol(E) = det([[r,s,s],[s, r,s],[s, s, r]])/6 = (r-s)^2*(r+2*s)/6,

so the total polyhedron's volume is V(r) = (8*r*s + 2*(r-s)*(r+3*s))*t + 2*(r-s)^2*(r+2*s)/3, where s is defined implicitly.

V'(r) = (10*r)*t + 5*r^2 + 8*s/t - 2*r*s - 2/(s*t) - 1, which Mathematica's Solve reduces to the minpoly:

69 - 73*sqrt(3)*r - 130*r^2 + 82*sqrt(3)*r^3 + 25*r^4 + 27*sqrt(3)*r^5 = 0; the other way around,

9 - 3*sqrt(3)*s - 94*s^2 - 6*sqrt(3)*s^3 + 113*s^4 - 27*sqrt(3)*s^5 = 0.

The polyhedron's volume has the minimal polynomial:

3784084398958182400 - 1457973565222551552*V^2 + 179253470562877440*V^4 - 7618104759214080*V^6 + 74727572486976*V^8 - 847288609443*V^10 = 0,

or where W = V^2*3^2/2^5,

3695394920857600 - 632801026572288*W + 34578215772160*W^2 - 653129694720*W^3 + 2847415504*W^4 - 14348907*W^5 = 0.

The field extensions over Q formed by the minpolies of r/sqrt(3), s/sqrt(3) and W all have Galois group S_5, so none have a closed form in terms of radicals.

(End)

Links

Table of n, a(n) for n=1..89.

Hugo Pfoertner, Visualization of Polyhedron, (1999).

Hugo Pfoertner, Number of edges incident with the 16 vertices, video (2021).

Polytope Wiki, Hexakis truncated tetrahedron.

Example

2.886455392274518582747381678285489...

Prog

(PARI) my(s3=sqrt(3), t=1/s3, s=vecmax(polrootsreal(9 - 3*s3*s - 94*s^2 - 6*s3*s^3 + 113*s^4 - 27*s3*s^5)), r=vecmax(polrootsreal(69 - 73*s3*r - 130*r^2 + 82*s3*r^3 +  25*r^4 + 27*s3*r^5))); (8*r*s + 2*(r-s)*(r+3*s))*t + 2*(r-s)^2*(r+2*s)/3 \\ Hugo Pfoertner, Jun 05 2026

Crossrefs

Cf. A081314, A081366, A339259-A339263, A395320.

Sequence in context: A155739 A197139 A198367 * A021780 A020769 A105388

Adjacent sequences: A395318 A395319 A395320 * A395322 A395323 A395324

Keyword

nonn,cons

Author

Hugo Pfoertner, Apr 20 2026

Status

approved