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.
Definition
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).
Dividing the defining trinomial by one obtains , and the
conjugate elements of are
Properties
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.[5] 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.
(which is less than 1/10 the
eccentricity of the orbit of Venus).
Narayana sequence
Narayana's cows is a
recurrence sequence originating from a problem proposed by the 14th century Indian mathematician
Narayana Pandita.[8] 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[9]
, 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.[10] The 8 odd composite numbers below 108 to pass the test are n = 1155, 552599, 2722611, 4822081, 10479787, 10620331, 16910355, 66342673.
The Narayana numbers are obtained as integral powers n > 3 of a
matrix with real
eigenvalue[8]
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 ). 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.[11][4]
In the first part supergolden rectangle perpendicular to the original one, the process can be repeated at a scale of .
See also
Solutions of equations similar to :
Golden ratio – the only positive solution of the equation
Plastic ratio – the only real solution of the equation