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A178988 Decimal expansion of volume of golden tetrahedron. 1
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, 7 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

2,1

COMMENTS

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.

Ed Pegg Jr pointed out when I asked that the volume of this type of golden tetrahedron, which I'd computed first my Cayley-Menger determinant, has the closed form formula given below.

The golden triangle, sometimes also called the sublime triangle, is an isosceles triangle such that the ratio of the hypotenuse a to base b is equal to the golden ratio.  But Kimberling (1991) defines a second type of golden triangle in which the ratio of angles is phi:1 where phi is the golden ratio. Thus there are several different kinds of golden tetrahedra. These also come up in icosahedral stellations, of course.

(1) an isosceles golden tetrahedron, or sublime tetrahedra, with an equilateral triangle base such that the ratio of the altitude a to base b is equal to the golden ratio, a/b=phi.

(2) a nonisosceles golden tetrahedron with a golden triangle base (ratio of the hypotenuse a to base b is equal to the golden ratio, a/b=phi) and a ratio of the altitude c to hypotenuse a is equal to the golden ratio, c/a=phi.

(3) a tetrahedron with all faces being golden triangles of this kind.

And then all the variations based on Kimberling's (1991) second type of golden triangle.

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? [From Clark Kimberling, Mar 31 2011]

REFERENCES

Kimberling, C. "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.

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

LINKS

Table of n, a(n) for n=2..101.

M. Bicknell and V. E. Hoggatt Jr.,  Golden Triangles, Rectangles, and Cuboids, Fib. Quart. 7, 73-91, 1969.

Frank M. Jackson and Eric W. Weisstein, Tetrahedron

R. Schoen, The Fibonacci Sequence in Successive Partitions of a Golden Triangle, Fib. Quart. 20, 159-163, 1982.

Eric W. Weisstein, Golden Triangle

FORMULA

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

EXAMPLE

75.7552212810...

PROG

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

CROSSREFS

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

Sequence in context: A171677 A021573 A080411 * A163505 A021136 A214444

Adjacent sequences:  A178985 A178986 A178987 * A178989 A178990 A178991

KEYWORD

nonn,cons,easy

AUTHOR

Jonathan Vos Post, Jan 03 2011

STATUS

approved

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Last modified April 28 12:37 EDT 2017. Contains 285575 sequences.