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 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 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 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]
The third-order Pell numbers are obtained as integral powers n > 3 of a matrix with real eigenvalue
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
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
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
^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
^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.