A supergolden rectangle contains three scaled copies of itself, ψ = ψ−1 + 2ψ−3 + ψ−5
Rationality
irrational algebraic
Symbol
ψ
Representations
Decimal
1.46557123187676802665...
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 equal to 1.46557123187676802665...(sequence A092526 in the OEIS); it is the real solution of the equation x3 = x2 + 1.
Dividing the defining trinomial by one obtains , and the conjugate elements of are with and
Properties
Rectangles in aspect ratios ψ, ψ and ψ (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]
and
in comparison to the golden ratio identity
and vice versa.
Additionally, , while
For every integer one has From this an infinite number of further relations can be found.
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.
(which is less than 1/10 the eccentricity of the orbit of Venus).
Narayana sequence
A Rauzy fractal associated with the supergolden ratio-cubed. The central tile and its three subtiles have areas in the ratios ψ: ψ: 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.[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 Narayana sequence is defined by the third-order recurrence relation with initial values
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.
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[8]
Alternatively, can be interpreted as incidence matrix for a D0LLindenmayer system on the alphabet with corresponding substitution rule 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.[11]
Supergolden rectangle
Nested supergolden rectangles with perpendicular diagonals and side lengths in powers of ψ.
A supergolden rectangle is a rectangle whose side lengths are in a ratio. Compared to the golden rectangle, the supergolden rectangle has one more degree of self-similarity.
Given a rectangle of height 1, length and diagonal length (according to ). The triangles on the diagonal have altitudes each perpendicular foot divides the diagonal in ratio .
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.[12][4]
The rectangle below the diagonal has aspect ratio , the other three are all supergolden rectangles, with a fourth one between the feet of the altitudes. The parent rectangle and the four scaled copies have linear sizes in the ratios It follows from the theorem of the gnomon that the areas of the two rectangles opposite the diagonal are equal.
In the supergolden rectangle above the diagonal, the process is repeated at a scale of .
Supergolden spiral
Supergolden spirals with different initial radii on a ψ− rectangle.
A supergolden spiral is a logarithmic spiral that gets wider by a factor of for every quarter turn. It is described by the polar equation with initial radius and parameter If drawn on a supergolden rectangle, the spiral has its pole at the foot of altitude of a triangle on the diagonal and passes through vertices of rectangles with aspect ratio which are perpendicularly aligned and successively scaled by a factor
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
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