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Decimal expansion of volume of golden tetrahedron.
1

%I #37 Jun 12 2023 02:57:35

%S 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,

%T 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,

%U 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

%N Decimal expansion of volume of golden tetrahedron.

%C 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.

%C 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

%D 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.

%D Theoni Pappas, "The Pentagon, the Pentagram & the Golden Triangle." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 188-189, 1989.

%H Marjorie Bicknell and Verner E. Hoggatt Jr., <a href="http://www.fq.math.ca/Scanned/7-1/bicknell-a.pdf">Golden Triangles, Rectangles, and Cuboids</a>, Fib. Quart. 7, 73-91, 1969.

%H Frank M. Jackson and Eric W. Weisstein, <a href="http://mathworld.wolfram.com/Tetrahedron.html">Tetrahedron</a>.

%H Clark Kimberling, <a href="https://doi.org/10.1007/978-94-011-3586-3_20">A New Kind of Golden Triangle</a>, In: Bergum G.E., Philippou A.N., Horadam A.F. (eds), Applications of Fibonacci Numbers. Springer, Dordrecht, pp. 171-176, 1991.

%H Robert Schoen, <a href="http://www.fq.math.ca/Scanned/20-2/schoen.pdf">The Fibonacci Sequence in Successive Partitions of a Golden Triangle</a>, Fib. Quart. 20, 159-163, 1982.

%H Eric W. Weisstein, <a href="http://mathworld.wolfram.com/GoldenTriangle.html">Golden Triangle</a>.

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

%F The minimal polynomial is 20736*x^4 - 119000880*x^2 + 73225. - _Joerg Arndt_, Jul 25 2021

%e 75.7552212810...

%t RealDigits[Sqrt[275465/96 + 369575*Sqrt[5]/288], 10, 120][[1]] (* _Amiram Eldar_, Jun 12 2023 *)

%o (PARI) sqrt(275465/96 + (369575*sqrt(5))/288) \\ _Charles R Greathouse IV_, May 27 2016

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

%K nonn,cons,easy

%O 2,1

%A _Jonathan Vos Post_, Jan 03 2011

%E a(101) corrected by _Georg Fischer_, Jul 25 2021