A352454 Decimal expansion of the volume of intersection of 8 unit-radius spheres that have the vertices of a unit-side cube as centers.

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

Offset

-1,2

Comments

The surface area of the solid formed by the intersection is A352455.

The solution to the two-dimensional version of this problem is A352453.

Links

Table of n, a(n) for n=-1..103.

Kee-wai Lau, Problem 1189, Crux Mathematicorum, Vol. 12, No. 9 (1986), p. 242; Solution to Problem 1189, by Rex Westbrook, ibid., Vol. 14, No. 2 (1988), pp. 51-53.

Mathematics Stackexchange, Intersection of 8 spheres: find the volume, 2015.

Missouri State University, Problem #8, Finding the Area (resp. Volume) of Overlapping Circles (resp. Spheres), Advanced Problem Archive; Solution to Problem #8, by Ross Millikan.

Formula

Equals 97*Pi/12 - 27*arctan(sqrt(2)) + sqrt(2) - 1.

Equals 9*arctan(sqrt(2)/5) - 11*Pi/12 + sqrt(2) - 1.

Equals 8 * Integral_{y=1/2..sqrt(2)/2} Integral_{x=1/2..sqrt(1-y^2-1/4)} (sqrt(1-x^2-y^2) - 1/2) dx dy.

Example

0.01520548952883988261724763779355369459126108838890...

Mathematica

RealDigits[97*Pi/12 - 27*ArcTan[Sqrt[2]] + Sqrt[2] - 1, 10, 100][[1]]

Crossrefs

Cf. A102888, A195696, A336198, A352453, A352455.

Sequence in context: A296493 A196821 A333419 * A351445 A147710 A153456

Adjacent sequences: A352451 A352452 A352453 * A352455 A352456 A352457

Keyword

nonn,cons

Author

Amiram Eldar, Mar 16 2022

Status

approved