A273633 -id:A273633 - cp's OEIS frontend

A381671 Decimal expansion of the isoperimetric quotient of a regular tetrahedron.

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

Offset: 0

Paolo Xausa, Mar 03 2025

Polya (1954) defines the isoperimetric quotient of a solid as 36*Pi*V^2/(S^3), where V and S are the volume and surface area of the solid, respectively.

The isoperimetric quotient of a sphere is 1.

			0.30229989403903630843234637627369262204734437468212...

George Polya, Mathematics and Plausible Reasoning, Vol. 1: Induction and Analogy in Mathematics, Princeton University Press, Princeton, New Jersey, 1954. See pp. 188-189, exercise 43.

Paolo Xausa, Table of n, a(n) for n = 0..10000

Cf. A273633 (sphericity).
Cf. isoperimetric quotient of other Platonic solids: A019673 (cube), A073010 (octahedron), A374772 (dodecahedron), A381672 (icosahedron).
Cf. A002194, A019673.

First[RealDigits[Pi/(6*Sqrt[3]), 10, 100]]

Equals Pi/(6*sqrt(3)) = A019673/A002194.

A273637 Decimal expansion of ((Pi(3+sqrt(5))^2)/(60sqrt(3)))^(1/3), the sphericity of the icosahedron.

Original entry on oeis.org

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

Offset: 0

Felix Fröhlich, May 27 2016

			0.93932565156763615941610143241466136614288864170479161093295918...

Wikipedia, Sphericity.
Index entries for transcendental numbers

Cf. A273633, A273634, A273635, A273636.

RealDigits[Surd[(Pi (3+Sqrt[5])^2)/(60Sqrt[3]),3],10,120][[1]] (* Harvey P. Dale, Nov 16 2022 *)

default(realprecision, 50080); my(x=((Pi*(3+sqrt(5))^2)/(60*sqrt(3)))^(1/3)); for(k=1, 100, my(d=floor(x)); x=(x-d)*10; print1(d, ", "))

A273634 Decimal expansion of (Pi/6)^(1/3), the sphericity of the cube.

Original entry on oeis.org

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

Offset: 0

Felix Fröhlich, May 27 2016

			0.80599597700823482035848342331964246947230703616193077784614603...

Wikipedia, Sphericity.
Index entries for transcendental numbers.

Cf. A019673, A273633, A273635, A273636, A273637.

RealDigits[Surd[Pi/6, 3], 10, 120][[1]] (* Amiram Eldar, Jun 29 2023 *)

default(realprecision, 50080); my(x=(Pi/6)^(1/3)); for(k=1, 100, my(d=floor(x)); x=(x-d)*10; print1(d, ", "))

Equals cube root of A019673. - Michel Marcus, May 27 2016

A273635 Decimal expansion of (Pi/(3*sqrt(3)))^(1/3), the sphericity of the octahedron.

Original entry on oeis.org

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

Offset: 0

Felix Fröhlich, May 27 2016

			0.84558252053658756632718815977306625250200668234085959800699611...

Wikipedia, Sphericity.
Index entries for transcendental numbers

Cf. A093602, A273633, A273634, A273636, A273637.

RealDigits[Surd[Pi/(3Sqrt[3]),3],10,120][[1]] (* Harvey P. Dale, Jan 07 2023 *)

default(realprecision, 50080); my(x=(Pi/(3*sqrt(3)))^(1/3)); for(k=1, 100, my(d=floor(x)); x=(x-d)*10; print1(d, ", "))

Equals cube root of A093602. - Michel Marcus, May 27 2016

Definition corrected by Georg Fischer, Jul 12 2021

A273636 Decimal expansion of ((Pi(15+7sqrt(5))^2)/(12(25+10sqrt(5))^(3/2)))^(1/3), the sphericity of the dodecahedron.

Original entry on oeis.org

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

Offset: 0

Felix Fröhlich, May 27 2016

			0.91045318140924227916953803446625862857458658029433801265061554...

Wikipedia, Sphericity.
Index entries for transcendental numbers.

Cf. A273633, A273634, A273635, A273637.

RealDigits[((Pi*(15 + 7*Sqrt[5])^2)/(12*(25 + 10*Sqrt[5])^(3/2)))^(1/3), 10, 120][[1]] (* Amiram Eldar, Jun 29 2023 *)

default(realprecision, 50080); my(x=((Pi*(15+7*sqrt(5))^2)/(12*(25+10*sqrt(5))^(3/2)))^(1/3)); for(k=1, 100, my(d=floor(x)); x=(x-d)*10; print1(d, ", "))

cp's OEIS Frontend

A381671 Decimal expansion of the isoperimetric quotient of a regular tetrahedron.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

A273637 Decimal expansion of ((Pi*(3+sqrt(5))^2)/(60*sqrt(3)))^(1/3), the sphericity of the icosahedron.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A273634 Decimal expansion of (Pi/6)^(1/3), the sphericity of the cube.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A273635 Decimal expansion of (Pi/(3*sqrt(3)))^(1/3), the sphericity of the octahedron.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A273636 Decimal expansion of ((Pi*(15+7*sqrt(5))^2)/(12*(25+10*sqrt(5))^(3/2)))^(1/3), the sphericity of the dodecahedron.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A273637 Decimal expansion of ((Pi(3+sqrt(5))^2)/(60sqrt(3)))^(1/3), the sphericity of the icosahedron.

A273636 Decimal expansion of ((Pi(15+7sqrt(5))^2)/(12(25+10sqrt(5))^(3/2)))^(1/3), the sphericity of the dodecahedron.