A395392 Decimal expansion of the surface area of the solid of revolution formed by rotating a unit half-width lemniscate of Bernoulli about the horizontal axis.

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

Offset

1,1

Comments

The lemniscate of Bernoulli can be described by (x^2 + y^2)^2 = a^2*(x^2 - y^2) in Cartesian coordinates, or r^2 = a^2*cos(2*phi) in polar coordinates, where a is the half-width. Here we take a = 1.

References

J. Dennis Lawrence, A Catalog of Special Plane Curves, Dover Publications Inc., New York, 1972, p. 122.

Babu Ram, Engineering Mathematics, Pearson, 2009. See example 8.26.

Links

Table of n, a(n) for n=1..105.

G. N. Berman, A Problem Book in Mathematical Analysis, 2nd ed., Mir Publishers, Moscow, 1980, p. 177, Problem 2607 (solution on page 397).

Mandeep Beniwal, Surface of solid of revolution of Lemniscate, Smart study helper, YouTube video, 2021.

Amiram Eldar, Illustration of the solid of revolution.

Eric Weisstein's World of Mathematics, Lemniscate.

Eric Weisstein's World of Mathematics, Solid of Revolution.

Wikipedia, Lemniscate of Bernoulli.

Wikipedia, Solid of revolution.

Index entries for transcendental numbers.

Formula

Equals 2*Pi*(2 - sqrt(2)).

Equals 4*Pi*(1 - 1/sqrt(2)).

Example

3.680604738042440459818811552996624139559434218749042...

Mathematica

RealDigits[2*Pi*(2-Sqrt[2]), 10, 120][[1]]

Prog

(PARI) 2*Pi*(2-sqrt(2))

Crossrefs

Cf. A064853, A154739, A154747, A154743, A343392, A395393, A395394, A395395.

Sequence in context: A067697 A137128 A256372 * A330890 A337404 A133442

Adjacent sequences: A395389 A395390 A395391 * A395393 A395394 A395395

Keyword

nonn,cons,changed

Author

Amiram Eldar, Apr 21 2026

Status

approved