Silver ratio

Rationality	irrational algebraic
Symbol	σ
Representations
Decimal	2.41421356237309504880...
Algebraic form	${\displaystyle 1+\sqrt{2}}$
Continued fraction	${\displaystyle 2+\frac{1}{2+\frac{1}{2+\frac{1}{\ddots }}}}$ purely periodic infinite

In mathematics, thesilver ratio is a geometricalproportion with exact value1 + √2, the positivesolution of the equationx² = 2x + 1.

The namesilver ratio is by analogy with thegolden ratio, the positive solution of the equationx² =x + 1.

Although its name is recent, the silver ratio (or silver mean) has been studied since ancient times because of its connections to thesquare root of 2, almost-isoscelesPythagorean triples,square triangular numbers,Pell numbers, theoctagon, and sixpolyhedra withoctahedral symmetry.

If the ratio of two quantitiesa > b > 0 is proportionate to the sum of two and their reciprocal ratio, they are in the silver ratio:$${\displaystyle \frac{a}{b}=\frac{2a+b}{a}}$$The ratio${\displaystyle \frac{a}{b}}$ is here denoted⁠${\displaystyle \sigma .}$⁠^[a]

Substituting${\displaystyle a=\sigma b}$ in the second fraction,$${\displaystyle \sigma =\frac{b(2\sigma +1)}{\sigma b}.}$$ It follows that the silver ratio is the positive solution ofquadratic equation${\displaystyle {\sigma }^2-2\sigma -1=0.}$ Thequadratic formula gives the two solutions${\displaystyle 1\pm \sqrt{2},}$ the decimal expansion of the positiveroot begins with2.414213562373095... (sequenceA014176 in theOEIS).

Using thetangent function ^[4]$${\displaystyle \sigma =\tan \left(\frac{3\pi }{8}\right)=\cot \left(\frac{\pi }{8}\right),}$$or thehyperbolic sine$${\displaystyle \sigma =\exp (\mathrm{arsinh}(1)).}$$

⁠${\displaystyle \sigma }$⁠ and itsalgebraic conjugate can be written as sums of eighthroots of unity:which is guaranteed by theKronecker–Weber theorem.

⁠${\displaystyle \sigma }$⁠ is the superstablefixed point of theNewton iteration${\displaystyle x\leftarrow \frac{1}{2}(x^2+1)/(x-1),\text{ with }{x}_0\in [2,3]}$

The iteration${\displaystyle x\leftarrow \sqrt{1+2x\phantom{/}}}$ results in thecontinued radical$${\displaystyle \sigma =\sqrt{1+2\sqrt{1+2\sqrt{1+\cdots }}}}$$

The defining equation can be written

The silver ratio can be expressed in terms of itself as fractions

Similarly as the infinitegeometric series

For every integer⁠${\displaystyle n}$⁠ one hasfrom this an infinite number of further relations can be found.

Continued fraction pattern of a few low powers

$${\displaystyle {\sigma }^{-n}\equiv (-1{)}^{n-1}{\sigma }^{n}mod1.}$$

The silver ratio is aPisot number, the next quadratic Pisot number after the golden ratio.^[5] By definition of these numbers, theabsolute value${\displaystyle \sqrt{2}-1}$ of thealgebraic conjugate is smaller than1, thus powers of⁠${\displaystyle \sigma }$⁠ generatealmost integers and the sequence${\displaystyle {\sigma }^{n}mod1}$ is dense at the borders of theunit interval.^[6]

⁠${\displaystyle \sigma }$⁠ is thefundamental unit of realquadratic field${\displaystyle K=\mathbb{Q}\left(\sqrt{2}\right)}$ with discriminant${\displaystyle {\mathrm{\Delta }}_k=8.}$ Theintegers${\displaystyle \mathbb{Z}[\sigma ]\text{ of }K}$ are the numbers${\displaystyle \xi =a+b\sigma (a,b\in \mathbb{Z}),}$ with conjugate${\displaystyle \overline{\xi }=(a+2b)-b\sigma ,}$ norm${\displaystyle \xi \overline{\xi }=(a+b{)}^2-2b^2}$ and trace${\displaystyle \xi +\overline{\xi }=2(a+b).}$^[7]The first few positive numbers occurring as norm are 1, 2, 4, 7, 8, 9, 14, 16, 17, 18, 23, 25.^[8] Arithmetic in thering${\displaystyle O_k=\mathbb{Z}[\sigma ]}$ resembles that of the rational integers,i.e. the elements of⁠${\displaystyle \mathbb{Z}.}$⁠ Prime factorization isunique up to order and unit factors${\displaystyle \pm {\sigma }^{\pm n}(n=0,1,2,\dots ),}$ and there is aEuclidean function on the absolute value of the norm.^[9] The primes of⁠${\displaystyle O_k}$⁠ are of three types:

• ⁠${\displaystyle \sigma -1}$⁠ with norm⁠${\displaystyle 2,}$⁠ the single rational prime thatdividesΔ_k ,
• the factors⁠${\displaystyle a+b\sigma }$⁠ of rational primes${\displaystyle p=8n\pm 1}$ with norm⁠${\displaystyle p,}$⁠^[10]
• the rational primes${\displaystyle p=8n\pm 3}$ withnorm⁠${\displaystyle p^2,}$⁠^[11]

and any one of these numbers multiplied by a unit.^[12]

The silver ratio can be used asbase of anumeral system, here called thesigmary scale.^[b] Everyreal numberx in[0,1] can be represented as aconvergent series$${\displaystyle x=\sum _{n=1}^{\mathrm{\infty }}\frac{a_n}{{\sigma }^n},}$$ withweights⁠${\displaystyle a_n\in [0,1,2].}$⁠

Sigmary expansions are not unique. Due to the identitiesdigit blocks${\displaystyle {21}_{\sigma }\text{ and }{22}_{\sigma }}$carry to the next power of⁠${\displaystyle \sigma ,}$⁠ resulting in${\displaystyle {100}_{\sigma }\text{ and }{101}_{\sigma }.}$ The number one has finite and infinite representations${\displaystyle {1.0}_{\sigma },{0.21}_{\sigma }}$ and${\displaystyle 0.{\overline{20}}_{\sigma },0.1{\overline{2}}_{\sigma },}$ where the first of each pair is incanonical form. Thealgebraic number⁠${\displaystyle 2(3\sigma -7)}$⁠ can be written⁠${\displaystyle {0.101}_{\sigma },}$⁠ or non-canonically as⁠${\displaystyle {0.022}_{\sigma }.}$⁠ Thedecimal number${\displaystyle 10={111.12}_{\sigma },}$${\displaystyle 7\sigma +3={1100}_{\sigma }}$ and${\displaystyle \frac{1}{\sigma -1}=0.{\overline{1}}_{\sigma }.}$

Properties of canonical sigmary expansions, with coefficients${\displaystyle a,b,c\in \mathbb{Z}:}$

• Everyalgebraic integer${\displaystyle \xi =a+b\sigma \text{ in }K}$ has a finite expansion.^[13]
• Everyrational number${\displaystyle \rho =\frac{a+b\sigma }{c}\text{ in }K}$ has a purely periodic expansion.^[14]
• All numbers that do not lie in⁠${\displaystyle K}$⁠ have chaotic expansions.

Remarkably, the same holdsmutatis mutandis for all quadratic Pisot numbers that satisfy the general equation⁠${\displaystyle x^2=nx+1,}$⁠ with integern > 0.^[15] It follows by repeated substitution of⁠${\displaystyle x=n+\frac{1}{x}}$⁠ that all positive solutions${\displaystyle \frac{1}{2}\left(n+\sqrt{n^2+4\phantom{/}}\right)}$ have a purely periodic continued fraction expansion$${\displaystyle {\sigma }_n=n+\frac{1}{n+\frac{1}{n+\frac{1}{\ddots }}}}$$Vera de Spinadel described the properties of these irrationals and introduced the monikermetallic means.^[16]

The silver ratio is related to thecentral Delannoy numbers⁠${\displaystyle D_n}$⁠ = 1, 3, 13, 63, 321, 1683, 8989,... that count the number of "king walks" between one pair of opposite corners of a squaren × n lattice. The sequence hasgenerating function ^[17]from which are obtained theintegral representation ^[18]$${\displaystyle D_n=\frac{1}{\pi }{\int }_{{\sigma }^{-2}}^{{\sigma }^2}\frac{\mathrm{d}t}{\sqrt{(t-{\sigma }^{-2})({\sigma }^2-t)}{t}^{n+1}}}$$andasymptotic formula ^[19]$${\displaystyle D_n\sim \frac{{\sigma }^{2n+1}}{2\sqrt{\pi (\sigma -1)n}}\left(1-\frac{11-3\sigma }{32n}+\frac{221-36{\sigma }^2}{(32n{)}^2}+\mathcal{O}(n^{-3})\right).}$$

For an application of the sigmary scale, consider the problem of writing a possible third-order coefficientc in terms of the silver ratio. The decimal value ofc is approximately0.006865233, which can be found with themethod of dominant balance using therecurrence relation for the central Delannoy numbers,${\displaystyle n{D}_n=(6n-3)D_{n-1}-(n-1)D_{n-2},}$^[20] with${\displaystyle D_{-1}=D_0=1,n_{max}={10}^5.}$ "The coefficients all lie in⁠${\displaystyle \mathbb{Q}\left(\sqrt{2}\right)}$⁠ and have denominators equal to some power of the prime⁠${\displaystyle \sqrt{2}\mathbb{Z}[\sigma ].}$⁠"^[21]Choosing denominatord = 32768, the approximate numeratordc has sigmary expansion⁠${\displaystyle {\mathrm{1001201.010201012000000110...}}_{\sigma }}$⁠ and is truncated to aquadratic integer by dropping all digits of order⁠⁠ Write the remaining powers⁠${\displaystyle {\sigma }^k}$⁠ in linear form withPell numbers as coefficients (see the following section), take the weighted sum and simplify, giving term${\displaystyle -\frac{4123-309{\sigma }^3}{(32n{)}^3}.}$ A certified value forc is however as yet unknown.

These numbers are related to the silver ratio as theFibonacci numbers andLucas numbers are to thegolden ratio.

The fundamental sequence is defined by therecurrence relation$${\displaystyle P_n=2P_{n-1}+P_{n-2}\text{ for }n>1,}$$with initial values$${\displaystyle P_0=0,P_1=1.}$$

The first few terms are 0, 1, 2, 5, 12, 29, 70, 169,...OEIS: A000129.
The limit ratio of consecutive terms is the silver mean.

Fractions of Pell numbers providerational approximations of⁠${\displaystyle \sigma }$⁠ with error

The sequence is extended to negative indices using$${\displaystyle P_{-n}=(-1{)}^{n-1}{P}_n.}$$

Powers of⁠${\displaystyle \sigma }$⁠ can be written with Pell numbers as linear coefficients$${\displaystyle {\sigma }^n=\sigma P_n+P_{n-1},}$$ which is proved bymathematical induction onn. The relation also holds forn < 0.

Thegenerating function of the sequence is given by ^[22]

Thecharacteristic equation of the recurrence is${\displaystyle x^2-2x-1=0}$ withdiscriminant⁠${\displaystyle D=8.}$⁠ If the two solutions are silver ratio⁠${\displaystyle \sigma }$⁠ and conjugate⁠${\displaystyle \overline{\sigma },}$⁠ so that${\displaystyle \sigma +\overline{\sigma }=2\text{ and }\sigma \cdot \overline{\sigma }=-1,}$ the Pell numbers are computed with theBinet formula$${\displaystyle P_n=a({\sigma }^n-{\overline{\sigma }}^n),}$$with⁠${\displaystyle a}$⁠ the positive root of${\displaystyle 8x^2-1=0.}$

Since the number⁠${\displaystyle P_n}$⁠ is the nearest integer to${\displaystyle a{\sigma }^n,}$ with${\displaystyle a=1/\sqrt{8}}$ andn ≥ 0.

The Binet formula${\displaystyle {\sigma }^n+{\overline{\sigma }}^n}$ defines the companion sequence${\displaystyle Q_n=P_{n+1}+P_{n-1}.}$

The first few terms are 2, 2, 6, 14, 34, 82, 198,...OEIS: A002203.

ThisPell-Lucas sequence has theFermat property: if p is prime,${\displaystyle Q_p\equiv Q_{1}modp.}$ The converse does not hold, the least oddpseudoprimes${\displaystyle n\mid (Q_n-2)}$ are 13², 385, 31², 1105, 1121, 3827, 4901.^[23][c]

Pell numbers are obtained as integral powersn > 2 of amatrix with positiveeigenvalue⁠${\displaystyle \sigma }$⁠

Thetrace of⁠${\displaystyle M^n}$⁠ gives the above⁠${\displaystyle Q_n.}$⁠

A rectangle with edges in ratio√2 ∶ 1 can be created from a square piece of paper with anorigami folding sequence. Considered a proportion of great harmony inJapanese aesthetics —Yamato-hi (大和比) —the ratio is retained if the√2 rectangle is folded in half, parallel to the short edges.Rabatment produces a rectangle with edges in the silver ratio (according to⁠1/σ⁠ = √2 − 1).^[d]

• Fold a square sheet of paper in half, creating a falling diagonal crease (bisect 90° angle), then unfold.
• Fold the right hand edge onto the diagonal crease (bisect 45° angle).
• Fold the top edge in half, to the back side (reduce width by⁠1/σ + 1⁠), and open out the triangle. The result is a√2 rectangle.
• Fold the bottom edge onto the left hand edge (reduce height by⁠1/σ − 1⁠). The horizontal part on top is a silver rectangle.

If the folding paper is opened out, the creases coincide with diagonal sections of a regularoctagon. The first two creases divide the square into asilver gnomon with angles in the ratios5 ∶ 2 ∶ 1, between two right triangles with angles in ratios4 ∶ 2 ∶ 2 (left) and4 ∶ 3 ∶ 1 (right). The unit angle is equal to⁠22+1/2⁠ degrees.

If the octagon has edge length⁠${\displaystyle 1,}$⁠ its area is⁠${\displaystyle 2\sigma }$⁠ and the diagonals have lengths${\displaystyle \sqrt{\sigma +1\phantom{/}},\sigma }$ and${\displaystyle \sqrt{2(\sigma +1)\phantom{/}}.}$ The coordinates of the vertices are given by the8permutations of${\displaystyle \left(\pm \frac{1}{2},\pm \frac{\sigma }{2}\right).}$^[26] The paper square has edge length⁠${\displaystyle \sigma -1}$⁠ and area⁠${\displaystyle 2.}$⁠ The triangles have areas${\displaystyle 1,\frac{\sigma -1}{\sigma }}$ and${\displaystyle \frac{1}{\sigma };}$ the rectangles have areas${\displaystyle \sigma -1\text{ and }\frac{1}{\sigma }.}$

Divide a rectangle with sides in ratio1 ∶ 2 into four congruentright triangles with legs of equal length and arrange these in the shape of a silver rectangle, enclosing asimilar rectangle that is scaled by factor⁠${\displaystyle \frac{1}{\sigma }}$⁠ and rotated about the centre by⁠${\displaystyle \frac{\pi }{4}.}$⁠ Repeating the construction at successively smaller scales results in four infinite sequences of adjoining right triangles, tracing awhirl of converging silver rectangles.^[27]

The logarithmic spiral through the vertices of adjacent triangles haspolar slope${\displaystyle k=\frac{4}{\pi }\ln (\sigma ).}$Theparallelogram between the pair of grey triangles on the sides has perpendicular diagonals in ratio⁠${\displaystyle \sigma }$⁠, hence is asilverrhombus.

If the triangles have legs of length⁠${\displaystyle 1}$⁠ then each discrete spiral has length${\displaystyle \frac{\sigma }{\sigma -1}=\sum _{n=0}^{\mathrm{\infty }}{\sigma }^{-n}.}$ The areas of the triangles in each spiral region sum to${\displaystyle \frac{\sigma }{4}=\frac{1}{2}\sum _{n=0}^{\mathrm{\infty }}{\sigma }^{-2n};}$ the perimeters are equal to⁠${\displaystyle \sigma +2}$⁠ (light grey) and⁠${\displaystyle 2\sigma -1}$⁠ (silver regions).

Arranging the tiles with the fourhypotenuses facing inward results in the diamond-in-a-square shape. Roman architectVitruvius recommended the impliedad quadratura ratio as one of three for proportioning a town houseatrium. The scaling factor is⁠${\displaystyle \frac{1}{\sigma -1},}$⁠ and iteration on edge length2 gives an angular spiral of length⁠${\displaystyle \sigma +1.}$⁠

The silver mean has connections to the followingArchimedean solids withoctahedral symmetry; all values are based on edge length= 2.

• Rhombicuboctahedron

The coordinates of the vertices are given by 24 distinct permutations of${\displaystyle (\pm \sigma ,\pm 1,\pm 1),}$ thus three mutually-perpendicular silver rectangles touch six of its square faces.
Themidradius is${\displaystyle \sqrt{2(\sigma +1)\phantom{/}},}$ the centre radius for the square faces is⁠${\displaystyle \sigma .}$⁠^[28]

• Truncated cube

Coordinates: 24 permutations of${\displaystyle (\pm \sigma ,\pm \sigma ,\pm 1).}$
Midradius:⁠${\displaystyle \sigma +1,}$⁠ centre radius for the octagon faces:⁠${\displaystyle \sigma .}$⁠^[29]

• Truncated cuboctahedron

Coordinates: 48 permutations of${\displaystyle (\pm (2\sigma -1),\pm \sigma ,\pm 1).}$
Midradius:${\displaystyle \sqrt{6(\sigma +1)\phantom{/}},}$ centre radius for the square faces:⁠${\displaystyle \sigma +2,}$⁠ for the octagon faces:⁠${\displaystyle 2\sigma -1.}$⁠^[30]

See also the dualCatalan solids

• Tetragonal trisoctahedron
• Trisoctahedron
• Hexakis octahedron

Theacuteisosceles triangle formed by connecting two adjacent vertices of aregular octagon to its centre point, is here called thesilver triangle. It is uniquely identified by its angles in ratios⁠${\displaystyle 2:3:3.}$⁠ Theapex angle measures⁠${\displaystyle 360/8=45,}$⁠ eachbase angle⁠${\displaystyle 67\frac{1}{2}}$⁠ degrees. It follows that theheight to base ratio is${\displaystyle \frac{1}{2}\tan (67\frac{1}{2})=\frac{\sigma }{2}.}$

Bytrisecting one of its base angles, the silver triangle is partitioned into a similar triangle andanobtusesilvergnomon. The trisector is collinear with amedium diagonal of the octagon. Sharing the apex of the parent triangle, the gnomon has angles of${\displaystyle 67\frac{1}{2}/3=22\frac{1}{2},45\text{ and }112\frac{1}{2}}$ degrees in the ratios⁠${\displaystyle 1:2:5.}$⁠ From thelaw of sines, its edges are in ratios${\displaystyle 1:\sqrt{\sigma +1}:\sigma .}$

The similar silver triangle is likewise obtained by scaling the parent triangle in base to leg ratio⁠${\displaystyle 2\cos (67\frac{1}{2})}$⁠, accompanied with an⁠${\displaystyle 112\frac{1}{2}}$⁠ degree rotation. Repeating the process at decreasing scales results in an infinite sequence of silver triangles, which converges at thecentre of rotation. It is assumed without proof that the centre of rotation is the intersection point of sequentialmedian lines that join corresponding legs and base vertices.^[31]The assumption is verified by construction, as demonstrated in the vector image.

The centre of rotation hasbarycentric coordinates$${\displaystyle \left(\frac{\sigma +1}{\sigma +5}:\frac{2}{\sigma +5}:\frac{2}{\sigma +5}\right)\sim \left(\frac{\sigma +1}{2}:1:1\right),}$$the three whorls of stacked gnomons have areas in ratios$${\displaystyle {\left(\frac{\sigma +1}{2}\right)}^2:\frac{\sigma +1}{2}:1.}$$

Thelogarithmic spiral through the vertices of all nested triangles haspolar slope$${\displaystyle k=\frac{4}{5\pi }\ln \left(\frac{\sigma }{\sigma -1}\right),}$$or an expansion rate of⁠${\displaystyle \frac{\sigma +1}{2}}$⁠ for every⁠${\displaystyle 225}$⁠ degrees of rotation.

Silvertriangle centers:affine coordinates on the axis of symmetry

circumcenter	${\displaystyle \left(\frac{2}{\sigma +1}:\frac{1}{\sigma }\right)\sim (\sigma -1:1)}$
centroid	${\displaystyle \left(\frac{2}{3}:\frac{1}{3}\right)\sim (2:1)}$
nine-point center	${\displaystyle \left(\frac{1}{\sigma -1}:\frac{1}{\sigma +1}\right)\sim (\sigma :1)}$
incenter,α =⁠3π/8⁠	${\displaystyle \left([1+\cos (\alpha ){]}^{-1}:[1+\sec (\alpha ){]}^{-1}\right)\sim (\sec (\alpha ):1)}$
symmedian point	${\displaystyle \left(\frac{\sigma +1}{\sigma +2}:\frac{1}{\sigma +2}\right)\sim (\sigma +1:1)}$
orthocenter	${\displaystyle \left(\frac{2}{\sigma }:\frac{1}{{\sigma }^2}\right)\sim (2\sigma :1)}$

The long, medium and short diagonals of the regular octagon concur respectively at the apex, the circumcenter and the orthocenter of a silver triangle.

Assume a silver rectangle has been constructed as indicated above, with height1, length⁠${\displaystyle \sigma }$⁠ anddiagonal length${\displaystyle \sqrt{{\sigma }^2+1}}$. The triangles on the diagonal havealtitudes${\displaystyle 1/\sqrt{1+{\sigma }^{-2}};}$ each perpendicular foot divides the diagonal in ratio⁠${\displaystyle {\sigma }^2.}$⁠

If an horizontal line is drawn through the intersection point of the diagonal and the internaledge of arabatment square, the parent silver rectangle and the two scaled copies along the diagonal have areas in the ratios${\displaystyle {\sigma }^2:2:1,}$ the rectangles opposite the diagonal both have areas equal to${\displaystyle \frac{2}{\sigma +1}.}$^[32]

Relative tovertexA, the coordinates of feet of altitudesU andV are$${\displaystyle \left(\frac{\sigma }{{\sigma }^2+1},\frac{1}{{\sigma }^2+1}\right)\text{ and }\left(\frac{\sigma }{1+{\sigma }^{-2}},\frac{1}{1+{\sigma }^{-2}}\right).}$$

If the diagram is further subdivided by perpendicular lines throughU andV, the lengths of the diagonal and its subsections can be expressed astrigonometric functions of argument${\displaystyle \alpha =67\frac{1}{2}}$ degrees, the base angle of the silver triangle:

with⁠${\displaystyle \sigma =\tan (\alpha ).}$⁠

Both the lengths of the diagonal sections and the trigonometric values are elements ofbiquadraticnumber field${\displaystyle K=\mathbb{Q}\left(\sqrt{2+\sqrt{2}}\right).}$

Thesilver rhombus with edge⁠${\displaystyle 1}$⁠ has diagonal lengths equal to⁠${\displaystyle \overline{UV}}$⁠ and⁠${\displaystyle 2\overline{AU}.}$⁠ The regularoctagon with edge⁠${\displaystyle 2}$⁠ has long diagonals of length⁠${\displaystyle 2\overline{AB}}$⁠ that divide it into eight silver triangles. Since the regular octagon is defined by its side length and the angles of the silver triangle, it follows that all measures can be expressed in powers ofσ and the diagonal segments of the silver rectangle, as illustrated above,pars pro toto on a single triangle.

The leg to base ratio⁠${\displaystyle \overline{AB}/2\approx 1.306563}$⁠ has been dubbed theCordovan proportion by Spanish architect Rafael de la Hoz Arderius. According to his observations, it is a notable measure in thearchitecture andintricate decorations of themediævalMosque of Córdoba,Andalusia.^[33]

A silver spiral is alogarithmic spiral that gets wider by a factor of⁠${\displaystyle \sigma }$⁠ for every quarter turn. It is described by thepolar equation${\displaystyle r(\theta )=a\exp (k\theta ),}$ with initial radius⁠${\displaystyle a}$⁠ and parameter${\displaystyle k=\frac{2}{\pi }\ln (\sigma ).}$ If drawn on a silver rectangle, the spiral has its pole at the foot of altitude of a triangle on the diagonal and passes through vertices of paired squares which are perpendicularly aligned and successively scaled by a factor⁠${\displaystyle {\sigma }^{-1}.}$⁠

The silver ratio appears prominently in theAmmann–Beenker tiling, anon-periodic tiling of the plane with octagonal symmetry, build from a square andsilver rhombus with equal side lengths. Discovered byRobert Ammann in 1977, its algebraic properties were described by Frans Beenker five years later.^[34]If the squares are cut into two triangles, the inflation factor forAmmann A5-tiles is⁠${\displaystyle {\sigma }^2,}$⁠ the dominanteigenvalue of substitutionmatrix

• Solutions of equations similar to${\displaystyle x^2=2x+1}$:
  • Golden ratio – the positive solution of the equation${\displaystyle x^2=x+1}$
  • Metallic means – positive solutions of the general equation${\displaystyle x^2=nx+1}$
  • Supersilver ratio – the real solution of the equation${\displaystyle x^3=2x^2+1}$

• YouTube lecture on the silver ratio, Pell sequence and metallic means
• Silver rectangle and Pell sequence at Tartapelago by Giorgio Pietrocola

• Quadratic irrational numbers
• Mathematical constants
• History of geometry
• Metallic means

• CS1 German-language sources (de)
• Articles with short description
• Short description is different from Wikidata