A supergolden rectangle contains three scaled copies of itself, ψ = ψ-1 + 2ψ-3 + ψ-5
Rationality
irrational algebraic
Symbol
ψ
Representations
Decimal
1.4655712318767680266567312...
Algebraic form
real root of x3 = x2 + 1
Continued fraction (linear)
[1;2,6,1,3,5,4,22,1,1,4,1,2,84,...] not periodic infinite
In mathematics, the supergolden ratio is a geometrical proportion close to 85/58. Its true value is the real solution of the equation x3 = x2 + 1.
The name supergolden ratio results from analogy with the golden ratio, the positive solution of the equation x2 = x + 1.
A triangle with side lengths ψ, 1, and ψ-1 has an angle of exactly 120 degrees opposite the side of length ψ. (The other angles are ≈23.778° and ≈36.222°.)[1]
Two quantities a > b > 0 are in the supergolden ratio-squared if
.
The ratio is commonly denoted
Based on this definition, one has
It follows that the supergolden ratio is found as the unique real solution of the cubic equation The decimal expansion of the root begins as (sequence A092526 in the OEIS).
Rectangles in aspect ratios ψ, ψ2 and ψ3 (from left to right) tile the square.
Many properties of are related to golden ratio . For example, the supergolden ratio can be expressed in terms of itself as the infinite geometric series[4]
Notably, the continued fraction of begins as permutation of the first six natural numbers; the next term is equal to their sum + 1.
The supergolden ratio is the fourth smallest Pisot number.[6] Because the absolute value of the algebraic conjugates is smaller than 1, powers of generate almost integers. For example: . After eleven rotation steps the phases of the inward spiraling conjugate pair – initially close to – nearly align with the imaginary axis.
A Rauzy fractal associated with the supergolden ratio-cubed. The central tile and its three subtiles have areas in the ratios ψ4 : ψ2 : ψ : 1.
A Rauzy fractal associated with the supergolden ratio-squared, with areas as above.
Narayana's cows is a recurrence sequence originating from a problem proposed by the 14th century Indian mathematician Narayana Pandita.[9] He asked for the number of cows and calves in a herd after 20 years, beginning with one cow in the first year, where each cow gives birth to one calf each year from the age of three onwards.
The Narayana sequence has a close connection to the Fibonacci and Padovan sequences and plays an important role in data coding, cryptography and combinatorics. The number of compositions of n into parts 1 and 3 is counted by the nth Narayana number.
The first few terms are 1, 1, 1, 2, 3, 4, 6, 9, 13, 19, 28, 41, 60, 88,... (sequence A000930 in the OEIS). The limit ratio between consecutive terms is the supergolden ratio.
The first 11 indices n for which is prime are n = 3, 4, 8, 9, 11, 16, 21, 25, 81, 6241, 25747 (sequence A170954 in the OEIS). The last number has 4274 decimal digits.
The sequence can be extended to negative indices using
The characteristic equation of the recurrence is . If the three solutions are real root and conjugate pair and , the Narayana numbers can be computed with the Binet formula[10]
, with real and conjugates and the roots of .
Since and , the number is the nearest integer to , with n ≥ 0 and 0.2846930799753185027474714...
Coefficients result in the Binet formula for the related sequence .
The first few terms are 3, 1, 1, 4, 5, 6, 10, 15, 21, 31, 46, 67, 98, 144,... (sequence A001609 in the OEIS).
This anonymous sequence has the Fermat property: if p is prime, . The converse does not hold, but the small number of odd pseudoprimes makes the sequence special.[11] The 8 odd composite numbers below 108 to pass the test are n = 1155, 552599, 2722611, 4822081, 10479787, 10620331, 16910355, 66342673.
A supergolden Rauzy fractal of type a ↦ ab, with areas as above. The fractal boundary has box-countingdimension 1.50
The Narayana numbers are obtained as integral powers n > 3 of a matrix with real eigenvalue[9]
and initiator . The series of words produced by iterating the substitution have the property that the number of c's, b's and a's are equal to successive Narayana numbers. The lengths of these words are
Associated to this string rewriting process is a compact set composed of self-similar tiles called the Rauzy fractal, that visualizes the combinatorial information contained in a multiple-generation three-letter sequence.[12]
Nested supergolden rectangles with perpendicular diagonals and side lengths in ratio ψ.
A supergolden rectangle is a rectangle whose side lengths are in a ratio. Compared to the golden rectangle, containing linear ratios , the supergolden rectangle has one more degree of self-similarity.
Given a rectangle of height 1, length and diagonal length (according to ). On the left-hand side, cut off a square of side length 1 and mark the intersection with the falling diagonal. The remaining rectangle now has aspect ratio (according to ). The upper right triangle has altitude and the perpendicular foot coincides with the intersection point. Divide the original rectangle into four parts by a second, horizontal cut passing through the intersection point.
Numbering counter-clockwise starting from the upper right, the resulting first, second and fourth parts are all supergolden rectangles; while the third has aspect ratio . The original rectangle and successively the second, first and fourth parts have diagonal lengths in the ratios or, equivalently, areas . The areas of the diagonally opposite first and third parts are equal.[13][4]
In the first part supergolden rectangle perpendicular to the original one, the process can be repeated at a scale of .