A349578 -id:A349578 - cp's OEIS frontend

A349577 Decimal expansion of the volume of the solid formed by the intersection of 4 right circular unit-diameter cylinders whose axes pass through the diagonals of a cube.

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

Offset: 0

Amiram Eldar, Nov 22 2021

Equivalently, the axes of the cylinders can be placed along the lines joining the vertices of a regular tetrahedron with the centers of the faces on the opposite sides.
This constant was first calculated by Moore (1974).
The corresponding volumes in the analogous cases of 2 and 3 mutually orthogonal cylinders are 2/3 (A010722) and 2 - sqrt(2) (A101465), respectively.

			0.56840607294451799910914006057025714776009440514583...

Paul Bourke, Intersecting cylinders, 2003-2016.
Moreton Moore, Symmetrical Intersections of Right Circular Cylinders, The Mathematical Gazette, Vol. 58, No. 405 (1974), pp. 181-185.
Eric Weisstein's World of Mathematics, Steinmetz Solid.
Wikipedia, Steinmetz solid.

Cf. A010722, A101465, A349578, A349579, A349580.

RealDigits[(3/2) * Sqrt[2] * (2 - Sqrt[3]), 10, 100][[1]]

Equals (3/2) * sqrt(2) * (2 - sqrt(3)).

A349579 Decimal expansion of the 4-dimensional Steinmetz solid formed by the intersection of 4 unit-diameter 4-dimensional cylinders whose axes are mutually orthogonal and intersect at a single point.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Nov 22 2021

The constant given by Hildebrand et al. (2012) and Kong et al. (2013) is for unit-radius cylinders, and is thus larger by a factor of 2^4. The constant here, for a unit-diameter cylinders, is analogous to the 3-dimensional case given by Moore (1974).

			0.32966191362422503979540474867758757134334519333162...

A. J. Hildebrand, Lingyi Kong, Abby Turner and Ananya Uppal, Applications of n-dimensional Integrals: Random Points, Broken Sticks and Intersecting Cylinders, Illinois Geometry Lab Project Report, University of Illinois at Urbana-Champaign, December 11, 2012.
Lingyi Kong, Luvsandondov Lkhamsuren, Abigail Turner, Aananya Uppal and A. J. Hildebrand, Intersecting Cylinders: From Archimedes and Zu Chongzhi to Steinmetz and Beyond, Illinois Geometry Lab Project Report, University of Illinois at Urbana-Champaign, April 25, 2013.
Moreton Moore, Symmetrical Intersections of Right Circular Cylinders, The Mathematical Gazette, Vol. 58, No. 405 (1974), pp. 181-185.

Cf. A101465, A195696, A349577, A349578, A349580.

RealDigits[3 * (Pi/4 - ArcTan[Sqrt[2]]/Sqrt[2]), 10, 100][[1]]

Equals 3 * (Pi/4 - arctan(sqrt(2))/sqrt(2)).

A349580 Decimal expansion of the 5-dimensional Steinmetz solid formed by the intersection of 5 unit-diameter 5-dimensional cylinders whose axes are mutually orthogonal and intersect at a single point.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Nov 22 2021

The constant given by Hildebrand et al. (2012) and Kong et al. (2013) is for unit-radius cylinders, and is thus larger by a factor of 2^5. The constant here, for a unit-diameter cylinders, is analogous to the 3-dimensional case given by Moore (1974).

			0.17198723701328857806510936213684483040318641193634...

A. J. Hildebrand, Lingyi Kong, Abby Turner and Ananya Uppal, Applications of n-dimensional Integrals: Random Points, Broken Sticks and Intersecting Cylinders, Illinois Geometry Lab Project Report, University of Illinois at Urbana-Champaign, December 11, 2012.
Lingyi Kong, Luvsandondov Lkhamsuren, Abigail Turner, Aananya Uppal and A. J. Hildebrand, Intersecting Cylinders: From Archimedes and Zu Chongzhi to Steinmetz and Beyond, Illinois Geometry Lab Project Report, University of Illinois at Urbana-Champaign, April 25, 2013.
Moreton Moore, Symmetrical Intersections of Right Circular Cylinders, The Mathematical Gazette, Vol. 58, No. 405 (1974), pp. 181-185.

Cf. A101465, A188615, A349577, A349578, A349579.

RealDigits[8 * (Pi/12 - ArcCot[2*Sqrt[2]]/Sqrt[2]), 10, 100][[1]]

Equals 8 * (Pi/12 - arccot(2*sqrt(2))/sqrt(2)).

cp's OEIS Frontend

A349577 Decimal expansion of the volume of the solid formed by the intersection of 4 right circular unit-diameter cylinders whose axes pass through the diagonals of a cube.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A349579 Decimal expansion of the 4-dimensional Steinmetz solid formed by the intersection of 4 unit-diameter 4-dimensional cylinders whose axes are mutually orthogonal and intersect at a single point.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A349580 Decimal expansion of the 5-dimensional Steinmetz solid formed by the intersection of 5 unit-diameter 5-dimensional cylinders whose axes are mutually orthogonal and intersect at a single point.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula