Golden triangle (mathematics)

There are two type of triangles known as golden triangles. Type 1 is the "classical" type, and type 2 is analogous to the golden rectangle.

Type 1 golden triangle: a/b = φ, the golden ratio.

A type 1 golden triangle is an isosceles triangle in which the two longer sides have equal lengths and in which the ratio of this length to that of the third, smaller side is the golden ratio

$\varphi ={1+{\sqrt {5}} \over 2}.$

File:Goldentriangles-pentagram.png

A pentagram. Each corner is a type 1 golden triangle.

This is the shape of the triangles found in the points of pentagrams, in which every vertex angle is equal to

$\theta =\cos ^{-1}\left({\varphi \over 2}\right)={\pi \over 5}=36^{\circ }.$

A type 2 golden triangle is a triangle, one of whose angles is φ times another of the angles. A type 2 golden triangle is analogous to a golden rectangle, as both are closely related to the infinite continued fraction [1, 1, 1, ...]: the "all ones" count certain squares nested within a golden rectangle, and they also count certain isosceles triangles nested within a type 2 golden triangle. Type 2 golden triangles were introduced¹ in 1980 as follows:

Suppose T = ABC is a triangle having sides AB < AC < BC and a point B' on segment BC satisfying AB' = AB. Call T admissible if the shortest side of triangle T' = AB'C does not touch the shortest side of T; i.e., the shortest side of T' is B'C. Then the sequence

T₁ = T, T₂ = T'₁, T₃ = T'₂, . . .

consists exclusively of admissible triangles if and only if B/C = the golden ratio.

The connection that the continued fraction [1, 1, 1, ...] has to golden rectangles and type 2 golden triangles generalizes² to a connection between arbitrary continued fractions [a₁,a₂, a₃, ...] and arbitrary rectangles, and arbitrary triangles, in the following manner. For a rectangle R having length L and width W, the number a_n counts removable squares in the nth-level nest, and the continued fraction converges to L/W; for a triangle T, the number a_n counts removable isosceles triangles in the nth-level nest, and the continued fraction converges to a quotient B/C of two of the angles of T. Thus, the distinguishing feature of type 2 golden triangles - among all triangles - is that one and only one isosceles triangle of the prescribed sort exists at each level of nesting.

References

1. Clark Kimberling, American Mathematical Monthly, Problem S29: "A Fibonacci Sequence of Nested Triangles", proposed 87 (1980) 302, solved 89 (1982) 496-7.

2. Clark Kimberling, "A New Kind of Golden Triangle," in G. E. Bergum et al. (eds.) Applications of Fibonacci Numbers, vol. 4, Kluwer, 1991, 171-176.

External links

• Golden triangle article at Wolfram MathWorld