Golden Rhombus
GoldenRhombus
A golden rhombus is a rhombus whose diagonals are in the ratio p/q=phi, where phi is the golden ratio.
RhombicHexecontahedron
RhombicTriacontahedron
The faces of the acute golden rhombohedron, Bilinski dodecahedron, obtuse golden rhombohedron, rhombic hexecontahedron, and rhombic triacontahedron are golden rhombi.
The half-angle theta is given by
theta = cot^(-1)phi
(1)
= 1/2tan^(-1)2
(2)
approx 0.553574
(3)
approx 31.7175 degrees
(4)
(OEIS A195693).
RhombicTriacontahedronRhomb
Labeling the smaller interior angle as alpha and the larger as beta, then
| alpha+beta=pi |
(5)
|
and
alpha = 2theta
(6)
= [画像:cos^(-1)(1/(sqrt(5)))]
(7)
= sec^(-1)(sqrt(5))
(8)
= [画像:sin^(-1)(2/(sqrt(5)))]
(9)
= tan^(-1)2
(10)
= 1.10714...
(11)
= 63.4349 degrees...
(12)
beta = [画像:cos^(-1)(-1/(sqrt(5)))]
(13)
= sec^(-1)(-sqrt(5))
(14)
= arg(2i-1)
(15)
= 2.0344...
(16)
= 116.6550 degrees...
(17)
The diagonal lengths of a golden rhombus with edge length a are given by
p = [画像:(2a)/(sqrt(1+phi^(-2)))]
(18)
= [画像:acsc(pi/5)]
(19)
= [画像:asqrt(2+2/(sqrt(5)))]
(20)
= 1.70130...a
(21)
q = [画像:(2a)/(sqrt(1+phi^2))]
(22)
= [画像:acsc((2pi)/5)]
(23)
= [画像:asqrt(2-2/(sqrt(5)))]
(24)
= 1.05146...a
(25)
(OEIS A121570 and A179290), the inradius by
| [画像: r=a/(sqrt(5)), ] |
(26)
|
and the area by
| [画像: A=(2a^2)/(sqrt(5)). ] |
(27)
|
See also
Acute Golden Rhombohedron, Bilinski Dodecahedron, Golden Angle, Golden Isozonohedron, Golden Ratio, Golden Rectangle, Golden Rhombohedron, Obtuse Golden Rhombohedron, Rhombic Hexecontahedron, Rhombic Triacontahedron, RhombusExplore with Wolfram|Alpha
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References
Kabai, S. Mathematical Graphics I: Lessons in Computer Graphics Using Mathematica. Püspökladány, Hungary: Uniconstant, pp. 177, 179, and 187, 2002.Sloane, N. J. A. Sequences A105199, A121570, A137218, A179290, and A195693 in "The On-Line Encyclopedia of Integer Sequences."Referenced on Wolfram|Alpha
Golden RhombusCite this as:
Weisstein, Eric W. "Golden Rhombus." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GoldenRhombus.html