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

Author

R. J. Mathar, Mar 06 2025

Keywords

• nonn
• cons

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.

Examples

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Links

• Eric Weisstein's World of Mathematics, Antiprism.

Crossrefs

Cf. A086178.

Programs

Maple

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

Mathematica

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

Formula

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