Supergolden ratio: Difference between revisions
Cardano formula with prime coefficients & explicit discriminant |
Dan Leonard (talk | contribs) Removed non-decimal approximations: see Wikipedia talk:WikiProject Mathematics#Mathematical constant infobox |
||
Line 6: | Line 6: | ||
| symbol=<math>\psi</math> |
| symbol=<math>\psi</math> |
||
| decimal={{gaps|1.46557|12318|76768|02665|67312|...}} |
| decimal={{gaps|1.46557|12318|76768|02665|67312|...}} |
||
| binary={{gaps|1.0111|0111|0010|1111|1010|...}} |
|||
| hexadecimal={{gaps|1.772|FAD1|EDE8|0B46|...}} |
|||
| continued_fraction_linear=[1;2,6,1,3,5,4,22,1,1,4,1,2,84,...] |
|||
| continued_fraction_periodic=not periodic |
|||
| continued_fraction_finite=infinite |
|||
| algebraic=real root of <math> x^{3}=x^{2}+1 </math> |
| algebraic=real root of <math> x^{3}=x^{2}+1 </math> |
||
}} |
}} |
Revision as of 00:04, 23 December 2023
![]() A supergolden rectangle contains three scaled copies of itself, | |
Rationality | irrational algebraic |
---|---|
Symbol | |
Representations | |
Decimal | 1.4655712318767680266567312... |
Algebraic form | real root of |
In mathematics, the supergolden ratio is a geometrical proportion close to 85/58. Its true value is the real solution of the equation .
The name supergolden ratio results from analogy with the golden ratio, the positive solution of the equation .
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).
Solving the equation with Cardano's formula,[1]
or, using the hyperbolic cosine,[2]
is the stable fixed point of the iteration .
The iteration results in the continued radical
Dividing the defining trinomial by one obtains , and the conjugate elements of are
Properties
![](http://upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Supergolden_triangle.png/220px-Supergolden_triangle.png)
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 [5]
- ,
in comparison to the golden ratio identity
- .
Additionally, , while
For all powers
Continued fraction pattern of a few low powers
- (13/19)
- (22/15)
- (15/7)
- (22/7)
- (60/13)
- (115/17)
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.[4] 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 ±13π/22 – nearly align with the imaginary axis.
The minimal polynomial of the supergolden ratio has discriminant . The Hilbert class field of imaginary quadratic field can be formed by adjoining . With argument a generator for the ring of integers of , one has the special value of Dedekind eta quotient
- .
Expressed in terms of the Weber-Ramanujan class invariant Gn
- .[6]
Properties of the related Klein j-invariant result in near identity . The difference is < 1/143092.
Narayana sequence
Narayana's cows is a recurrence sequence originating from a problem proposed by the 14th century Indian mathematician Narayana Pandita.[7] 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
- for n > 2,
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 generating function of the Narayana sequence is given by
- for
The Narayana numbers are related to sums of binomial coefficients by
- .
The characteristic equation of the recurrence is . If the three solutions are real root α and conjugate pair β and γ, the Narayana numbers can be computed with the Binet formula [8]
- , with real and conjugates and the roots of .
Since and , the number is the nearest integer to , with n ≥ 0 and 0.2846930799753185027474714...
The Narayana numbers are obtained as integral powers n > 3 of a matrix with real eigenvalue [7]
Supergolden rectangle
![](http://upload.wikimedia.org/wikipedia/commons/thumb/9/9c/Supergolden_Ratio.png/220px-Supergolden_Ratio.png)
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.[9][5]
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
References
- ^ Penn, Michael (2022). "What is the super-golden ratio?". YouTube. Retrieved 17 November 2023.
- ^ Sloane, N. J. A. (ed.). "Sequence A092526". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ 3/2√x = x2/3
- ^ a b Sloane, N. J. A. (ed.). "Sequence A092526". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ a b Koshy, Thomas (2017). Fibonacci and Lucas Numbers with Applications (2 ed.). John Wiley & Sons. ISBN 9781118742174. Retrieved 14 August 2018.
- ^ Ramanujan G-function (in German)
- ^ a b Sloane, N. J. A. (ed.). "Sequence A000930". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Lin, Xin (2021). "On the Recurrence Properties of Narayana's Cows Sequence". Symmetry. 13 (149). doi:10.3390/sym13010149.
- ^ Crilly, Tony (1994). "A Supergolden Rectangle". The Mathematical Gazette. 78 (483): 320–325. doi:10.2307/3620208.