A381756 Decimal expansion of the smallest angular distance between two vertices of the equilateral square antiprism measured along the circumscribing sphere.

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

Offset

1,2

Comments

The equilateral square antiprism of side number n=4, lateral edge length a, and the two bases separated vertically by h has h = a*sqrt( 1-sec^2(Pi/(2n)) ) = a/2^(1/4). The 4 vertices of the top base have Cartesian coordinates (+-a/sqrt(2),0,h/2), (0,+-a/sqrt(2),h/2); the 4 vertices at the bottom base have (+-a/2,+-a/2,-h/2). The common distance of these 8 vertices from the origin is r = a*sqrt(8+2^(3/2))/4, the radius of the circumscribing sphere. The largest dot product between pairs of the 8 vertices is sqrt(2)*a^2/8 , which is equivalent to the smallest distance measured along the surface of the sphere of radius r. Dividing this dot product through r^2 gives 2^(3/2)/(8+2^(3/2)), the cosine of the angle between closest vertices. This here is the angle measured in radians.

Links

Table of n, a(n) for n=1..91.

Eric Weisstein's World of Mathematics, Antiprism.

Formula

Equals arccos(1/(2^(3/2)+1)) = arcsec(A086178).

Example

1.3065271617174372755341...

Maple

evalf( arccos(1/(2^(3/2)+1)) ) ;

Mathematica

RealDigits[ArcCos[1/(2^(3/2)+1)], 10, 91][[1]] (* Stefano Spezia, Jul 29 2025 *)

Crossrefs

Cf. A086178.

Sequence in context: A386492 A360173 A109693 * A188858 A199610 A285871

Adjacent sequences: A381753 A381754 A381755 * A381757 A381758 A381759

Keyword

nonn,cons

Author

R. J. Mathar, Mar 06 2025

Status

approved