A093591 -id:A093591 - cp's OEIS frontend

A093524 Decimal expansion of 3977/216000 - Pi^2/2160.

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

Offset: 0

Eric W. Weisstein, Mar 30 2004

Average volume of a tetrahedron picked at random in a unit cube.
From Amiram Eldar, Aug 25 2020: (Start)
The exact value of this constant was first calculated by Zinani (2003).
Equals (1/5) times the probability that 5 points independently and uniformly chosen in a cube are the vertices of a re-entrant (concave) polyhedron, i.e., one of the points falls within the tetrahedron formed by the other 4 points. Do and Solomon (1986) evaluated this probability using simulations, and their result is equivalent to an estimate of 0.0139 of this constant, with a 95% confidence interval of [0.01358, 0.01420] (Zinani, 2003). (End)

			0.0138427757...

Kim-Anh Do and Herbert Solomon, A simulation study of Sylvester's problem in three dimensions, Journal of applied probability, Vol. 23, No. 2 (1986), pp. 509-513, alternative link.
Johan Philip, The Expected Volume of a Random Tetrahedron in a Cube, 2007.
Eric Weisstein's World of Mathematics, Cube Tetrahedron Picking.
Alessandro Zinani, The expected volume of a tetrahedron whose vertices are chosen at random in the interior of a cube, Monatshefte für Mathematik, Vol. 139, No. 4 (2003), pp. 341-348.
Index entries for transcendental numbers

Cf. A093591.

RealDigits[3977/216000-Pi^2/2160,10,120][[1]] (* Harvey P. Dale, Oct 31 2013 *)

Added initial 0 to match offset. - N. J. A. Sloane, Feb 08 2015

A093525 Decimal expansion of 13/720 - Pi^2/15015.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Mar 30 2004

Average volume of a tetrahedron picked at random in a tetrahedron with unit volume.
Buchta & Reitzner announced this result in 1992, and Mannion (independently) proved it in 1994. Buchta & Reitzner proved a more general result in 2001. - Charles R Greathouse IV, Sep 04 2015
Klee (1969) conjectured that the average volume is 1/60 and stated that according to Monte Carlo experiments 1/57 is the integer-reciprocal closest to this value. - Amiram Eldar, Apr 09 2022

			0.0173982392...

Christian Buchta and Matthias Reitzner, What is the expected volume of a tetrahedron whose vertices are chosen at random from a given tetrahedron, Österreichische Akademie der Wissenschaften. Mathematisch-Naturwissenschaftliche Klasse, Vol. 129 (1992), pp. 63-68.
Christian Buchta and Matthias Reitzner, The convex hull of random points in a tetrahedron: Solution of Blaschke's problem and more general results, J. reine angew. Math., Vol. 536 (2001), pp. 1-29.
Victor Klee, What is the Expected Volume of a Simplex Whose Vertices are Chosen at Random from a Given Convex Body?, The American Mathematical Monthly, Vol. 76, No. 3 (1969), pp. 286-288.
David Mannion, The volume of a tetrahedron whose vertices are chosen at random in the interior of a parent tetrahedron, Advances in Applied Probability, Vol. 26, No. 3 (Sep 1994), pp. 577-596.
Eric Weisstein's World of Mathematics, Tetrahedron Tetrahedron Picking.
Index entries for transcendental numbers

Cf. A020829, A093524, A093591.

Join[{0}, RealDigits[13/720 - Pi^2/15015, 10, 100][[1]]] (* Amiram Eldar, Apr 09 2022 *)

13/720 - Pi^2/15015 \\ Charles R Greathouse IV, Sep 04 2015

Added initial 0 to make offset correct. - N. J. A. Sloane, Feb 08 2015

A178988 Decimal expansion of volume of golden tetrahedron.

Original entry on oeis.org

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

Offset: 2

Jonathan Vos Post, Jan 03 2011

Volume of tetrahedron with edges 1, phi, phi^2, phi^3, phi^4, phi^5 where phi is the golden ratio (1+sqrt(5))/2.

A152149 records more recent developments about side-golden and angle-golden triangles, both of which, like the golden rectangle, have generalizations that match continued fractions. There is a unique triangle which is both side-golden and angle-golden. Is there a comparable tetrahedron? - Clark Kimberling, Mar 31 2011

			75.7552212810...

Clark Kimberling, "A New Kind of Golden Triangle." In Applications of Fibonacci Numbers: Proceedings of the Fourth International Conference on Fibonacci Numbers and Their Applications,' Wake Forest University (Ed. G. E. Bergum, A. N. Philippou, and A. F. Horadam). Dordrecht, Netherlands: Kluwer, pp. 171-176, 1991.
Theoni Pappas, "The Pentagon, the Pentagram & the Golden Triangle." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 188-189, 1989.

Marjorie Bicknell and Verner E. Hoggatt Jr., Golden Triangles, Rectangles, and Cuboids, Fib. Quart. 7, 73-91, 1969.
Frank M. Jackson and Eric W. Weisstein, Tetrahedron.
Clark Kimberling, A New Kind of Golden Triangle, In: Bergum G.E., Philippou A.N., Horadam A.F. (eds), Applications of Fibonacci Numbers. Springer, Dordrecht, pp. 171-176, 1991.
Robert Schoen, The Fibonacci Sequence in Successive Partitions of a Golden Triangle, Fib. Quart. 20, 159-163, 1982.
Eric W. Weisstein, Golden Triangle.

Cf. A001622, A071399, A171973, A070169, A126766, A010524, A093524, A093525, A093591, A152149.

RealDigits[Sqrt[275465/96 + 369575*Sqrt[5]/288], 10, 120][[1]] (* Amiram Eldar, Jun 12 2023 *)

sqrt(275465/96 + (369575*sqrt(5))/288) \\ Charles R Greathouse IV, May 27 2016

Equals sqrt(275465/96 + (369575*sqrt(5))/288).

The minimal polynomial is 20736*x^4 - 119000880*x^2 + 73225. - Joerg Arndt, Jul 25 2021

a(101) corrected by Georg Fischer, Jul 25 2021

A363966 Decimal expansion of the probability that a sphere that is passing through 4 points uniformly and independently chosen at random in a 3D ball is completely lying inside the ball.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Jun 30 2023

The corresponding 2D probability, that a circle that is passing through 3 points uniformly and independently chosen at random in a 2D disk is completely lying inside the disk, is 2/5.

For the general solution in any number of dimensions see the solution of the user "joriki" in the Mathematics Stackexchange link.

			0.12304961331228291265040404363481954662209287572663...

Thomas Browning, Probability of random sphere lying inside the unit ball, Mathematics Stackexchange, 2020.

Cf. A093591.

Mathematica
```
RealDigits[24*Pi^2/1925, 10, 120][[1]]
```
PARI
```
24*Pi^2/1925
```

Equals 24*Pi^2/1925.

cp's OEIS Frontend

A093524 Decimal expansion of 3977/216000 - Pi^2/2160.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A093525 Decimal expansion of 13/720 - Pi^2/15015.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A178988 Decimal expansion of volume of golden tetrahedron.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A363966 Decimal expansion of the probability that a sphere that is passing through 4 points uniformly and independently chosen at random in a 3D ball is completely lying inside the ball.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula