Supersilver ratio
![]() A supersilver rectangle contains two scaled copies of itself, ς = ((ς − 1)2 + 2(ς − 1) + 1) / ς | |
Rationality | irrational algebraic |
---|---|
Symbol | ς |
Representations | |
Decimal | 2.2055694304005903117020286... |
Algebraic form | real root of x3 = 2x2 + 1 |
Continued fraction (linear) | [2;4,1,6,2,1,1,1,1,1,1,2,2,1,2,1,...] not periodic infinite |
In mathematics, the supersilver ratio is a geometrical proportion close to 75/34. Its true value is the real solution of the equation x3 = 2x2 + 1.
The name supersilver ratio results from analogy with the silver ratio, the positive solution of the equation x2 = 2x + 1, and the supergolden ratio.
Definition
Two quantities a > b > 0 are in the supersilver ratio-squared if
- .
The ratio is here denoted
Based on this definition, one has
It follows that the supersilver ratio is found as the unique real solution of the cubic equation The decimal expansion of the root begins as (sequence A356035 in the OEIS).
The minimal polynomial for the reciprocal root is the depressed cubic thus the simplest solution with Cardano's formula,
or, using the hyperbolic sine,
is the superstable fixed point of the iteration
Rewrite the minimal polynomial as , then the iteration results in the continued radical
- [1]
Dividing the defining trinomial by one obtains , and the conjugate elements of are
with and
Properties
![](http://upload.wikimedia.org/wikipedia/commons/thumb/5/53/SuperSilverSquare_6.png/280px-SuperSilverSquare_6.png)
The growth rate of the average value of the n-th term of a random Fibonacci sequence is .[2]
The supersilver ratio can be expressed in terms of itself as the infinite geometric series
- and
in comparison to the silver ratio identities
- and
For every integer one has
Continued fraction pattern of a few low powers
- (5/24)
- (5/11)
- (53/24)
- (73/15)
- (118/11)
The supersilver ratio is a Pisot number.[3] Because the absolute value of the algebraic conjugates is smaller than 1, powers of generate almost integers. For example: After ten rotation steps the phases of the inward spiraling conjugate pair – initially close to – nearly align with the imaginary axis.
The minimal polynomial of the supersilver ratio has discriminant and factors into the imaginary quadratic field has class number Thus, the Hilbert class field of can be formed by adjoining [4] With argument a generator for the ring of integers of , the real root j(τ) of the Hilbert class polynomial is given by [5][6]
The Weber-Ramanujan class invariant is approximated with error < 3.5 ∙ 10−20 by
while its true value is the single real root of the polynomial
The elliptic integral singular value[7] for has closed form expression
(which is less than 1/294 the eccentricity of the orbit of Venus).
Third-order Pell sequences
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These numbers are related to the supersilver ratio as the Pell numbers and Pell-Lucas numbers are to the silver ratio.
The fundamental sequence is defined by the third-order recurrence relation
- for n > 2,
with initial values
The first few terms are 1, 2, 4, 9, 20, 44, 97, 214, 472, 1041, 2296, 5064,... (sequence A008998 in the OEIS). The limit ratio between consecutive terms is the supersilver ratio.
The first 8 indices n for which is prime are n = 1, 6, 21, 114, 117, 849, 2418, 6144. The last number has 2111 decimal digits.
The sequence can be extended to negative indices using
- .
The generating function of the sequence is given by
- for [8]
The third-order Pell numbers are related to sums of binomial coefficients by
- .[9]
The characteristic equation of the recurrence is If the three solutions are real root and conjugate pair and , the supersilver numbers can be computed with the Binet formula
- with real and conjugates and the roots of
Since and the number is the nearest integer to with n ≥ 0 and 0.1732702315504081807484794...
Coefficients result in the Binet formula for the related sequence
The first few terms are 3, 2, 4, 11, 24, 52, 115, 254, 560, 1235, 2724, 6008,... (sequence A332647 in the OEIS).
This third-order Pell-Lucas 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. The 14 odd composite numbers below 108 to pass the test are n = 32, 52, 53, 315, 99297, 222443, 418625, 9122185, 32572, 11889745, 20909625, 24299681, 64036831, 76917325.[10]
![](http://upload.wikimedia.org/wikipedia/commons/thumb/4/41/Supersilver_Rauzy_aba.png/260px-Supersilver_Rauzy_aba.png)
The third-order Pell numbers are obtained as integral powers n > 3 of a matrix with real eigenvalue
The trace of gives the above
Alternatively, can be interpreted as incidence matrix for a D0L Lindenmayer 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 third-order Pell numbers. The lengths of these words are given by [11]
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]
Supersilver rectangle
![](http://upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Supersilver_ratio.svg/330px-Supersilver_ratio.svg.png)
A supersilver rectangle is a rectangle whose side lengths are in a ratio. Compared to the silver rectangle, containing a single scaled copy of itself, the supersilver rectangle has one more degree of self-similarity.
Given a rectangle of height 1 and length . On the right-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.[13]
Along the diagonal are two supersilver rectangles. The original rectangle and the scaled copies have diagonal lengths in the ratios or, equivalently, areas The areas of the rectangles opposite the diagonal are both equal to with aspect ratios (below) and (above).
The process can be repeated in the smallest supersilver rectangle at a scale of
The lower right triangle has altitude relative to the upper-left corner, the perpendicular foot is obtained from the intersection point by multiplication with factor This gives an alternative tiling in aspect ratios with diagonal lengths in ratios and tiles of equal area opposite the diagonal.
See also
- Solutions of equations similar to :
- Silver ratio – the only positive solution of the equation
- Golden ratio – the only positive solution of the equation
- Supergolden ratio – the only real solution of the equation
References
- ^ Sloane, N. J. A. (ed.). "Sequence A272874". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ (sequence A137421 in the OEIS)
- ^ Panju, Maysum (2011). "A systematic construction of almost integers" (PDF). The Waterloo Mathematics Review. 1 (2): 35–43.
- ^ "Hilbert class field of a quadratic field whose class number is 3". Mathematics stack exchange. 2012. Retrieved May 1, 2024.
- ^ Berndt, Bruce C.; Chan, Heng Huat (1999). "Ramanujan and the modular j-invariant". Canadian Mathematical Bulletin. 42 (4): 427–440. doi:10.4153/CMB-1999-050-1.
- ^ Johansson, Fredrik (2021). "Modular j-invariant". Fungrim. Retrieved April 30, 2024.
Table of Hilbert class polynomials
- ^ Weisstein, Eric W. "Elliptic integral singular value". MathWorld.
- ^ (sequence A008998 in the OEIS)
- ^ Mahon, Br. J. M.; Horadam, A. F. (1990). "Third-order diagonal functions of Pell polynomials". The Fibonacci Quarterly. 28 (1): 3–10.
- ^ Only one of these is a 'restricted pseudoprime' as defined in: Adams, William; Shanks, Daniel (1982). "Strong primality tests that are not sufficient". Mathematics of Computation. 39 (159). American Mathematical Society: 255–300. doi:10.1090/S0025-5718-1982-0658231-9. JSTOR 2007637.
- ^ for n ≥ 2 (sequence A193641 in the OEIS)
- ^ Siegel, Anne; Thuswaldner, Jörg M. (2009). "Topological properties of Rauzy fractals". Mémoires de la Société Mathématique de France. 2. 118: 1–140. doi:10.24033/msmf.430.
- ^ Analogue to the construction in: Crilly, Tony (1994). "A supergolden rectangle". The Mathematical Gazette. 78 (483): 320–325. doi:10.2307/3620208.
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- t
- e
- Algebraic integer
- Chebyshev nodes
- Constructible number
- Conway's constant
- Cyclotomic field
- Doubling the cube
- Eisenstein integer
- Gaussian integer
- Golden ratio (φ)
- Perron number
- Pisot–Vijayaraghavan number
- Plastic ratio (ρ)
- Quadratic irrational number
- Rational number
- Root of unity
- Salem number
- Silver ratio (δS)
- Square root of 2
- Square root of 3
- Square root of 5
- Square root of 6
- Square root of 7
- Supergolden ratio (ψ)
- Supersilver ratio (ς)
- Twelfth root of 2
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