A093524 -id:A093524 - cp's OEIS frontend

A093591 Decimal expansion of (12*Pi)/715.

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

Offset: 0

Eric W. Weisstein, Apr 02 2004

Mean volume of a tetrahedron formed by four random points in a unit ball.

Equals (4*Pi/15) times the probability (9/143) that 5 points independently and uniformly chosen in a ball 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. It was calculated by the Czech physicist and mathematician Bohuslav Hostinský (1884 - 1951) in 1925. - Amiram Eldar, Aug 25 2020

			0.0527260305...

Bohuslav Hostinský, Sur les probabilités géométriques, Brno: Publications de la Faculté des sciences de l'Université Masaryk, 1925.

Fernando Affentranger, The expected volume of a random polytope in a ball, Journal of Microscopy, Vol. 151, No. 3 (1988), pp. 277-287.
Herbert Solomon, Geometric Probability, Philadelphia, PA: SIAM, 1978, p. 124.
Eric Weisstein's World of Mathematics, Ball Tetrahedron Picking.
Index entries for transcendental numbers

Cf. A093524.

RealDigits[12*Pi/715, 10, 100][[1]] (* Amiram Eldar, Aug 25 2020 *)

12*Pi/715 \\ Charles R Greathouse IV, Sep 30 2022

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

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A093591 Decimal expansion of (12*Pi)/715.

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

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A178988 Decimal expansion of volume of golden tetrahedron.

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