Draft:golden diamond

The golden diamond. A fractal containing triangular tiles whose proportions are equal to various powers of the golden ratio $\phi$ with respect to the outer triangle. This illustrates the well-known identity $\phi =\Sigma _{n=1}^{\infty }\phi ^{-n}$.

The golden diamond (also, Fibonacci Representation Graph or $S_{2,3}$) is a fractal attractive fixed set and tree graph featuring proportions of the golden ratio.. It is notable for its unique embedding of the Fibonacci diatomic sequence [A000119 - OEIS], Fibonacci words [A003849 - OEIS], and phinary integers, representing a quasicrystal in the hyperbolic upper half-plane, and as an instance of projective geometry.

As a fractal, the golden diamond contains infinitely many copies of itself, representing one of the basic examples of self-similar sets—that is, it is a mathematically generated pattern that is reproducible at any magnification or reduction.

The golden diamond was discovered and named by American mathematician Scott V. Tezlaf, noting its resemblance to "the profile of a cut diamond with an infinite number of facets." The fractal geometry was discovered independently by mathematician Sam Northshield, who referred to the graph as the Fibonacci representation graph or $S_{2,3}$ in reference to the fractal's branching structure and its relation to the Fibonacci diatomic sequence, Fibonacci words and Fibonacci numbers.

Classification

Hyperbolic graph

The fractal is considered a "hyperbolic graph" due to each tile having exactly the same size when embedded in the hyperbolic upper half-plane. The tree graph features a unique representation of the Fibonacci diatomic sequence (discussed below).

The hyperbolic tree graph underlying the golden diamond.

Quasicrystal

The golden diamond represents a quasicrystal in the hyperbolic upper half-plane.

Tessellation

The golden diamond is a tessellation of quadrilaterals or triangles, depending on the choice of tile. In the latter case, it is an asymptotic monomorphic dissection of the equilateral triangle, irrep-$\infty$.

Construction

One can generate the fractal through an iterative process of duplication. Beginning with a single, upward-pointing equilateral triangle, one stacks two similar copies upon the "shoulders" of the first. Each of the similar copies is smaller than the original triangle by a factor of the golden ratio and intersects the original triangle at a distance from the base vertex equal to a side-length of one of the new triangles. The process is repeated for every newly-generated triangle and continues indefinitely. After an infinite number of iterations, a final equilateral triangle can be drawn around the image, intersecting the vertices of the perimeter triangles.

A step-by-step construction of the golden diamond. Two similar copies of an equilateral triangle—each smaller by a factor of the golden ratio—are stacked upon the “shoulders” of the first. A shoulder is located at a distance from a base vertex equal to the length of the new generation of triangles. The process is repeated for each new generation of triangles. After an infinite number of iterations, a final equilateral triangle can be drawn around the image, intersecting the vertices of the perimeter triangles.

Properties

Mathematical Identities

Several mathematical identities involving the golden ratio and Fibonacci numbers can be recovered from the geometry of the golden diamond by measuring its features.

Outer edges

The well-known identity

$\phi =\sum _{n=1}^{\infty }\phi ^{-n}$,

can be found by equating the sum of the tile edges along a side of the fractal to the fractal's total edge length.

Area

Each row of the fractal contains pairs of equilateral triangles, and the total number of triangles in the $n$th row is $2(F_{n+2}-1)$, where $F_{n}$ is the $n$th Fibonacci number. If an outer edge of the golden diamond has length $\phi$, then each triangle in the $n$th row has a side length $s_{n}=\phi ^{-n}$. Therefore, the area of a triangle in the $n$th row is

$a_{n}={\frac {\sqrt {3}}{4}}\phi ^{-2n}$

and the sum of triangle areas in row n is

$A_{n}=2(F_{n+2}-1)a_{n}={\frac {\sqrt {3}}{2}}(F_{n+2}-1)\phi ^{-2n}$,

for an outer triangle of area

$A={\frac {\sqrt {3}}{4}}\phi ^{2}$.

It follows that

This figure illustrates the method for calculating row lengths of the golden diamond.

$A=\sum _{n=1}^{\infty }A_{n}$

and therefore,

${\frac {\sqrt {3}}{4}}\phi ^{2}=\sum _{n=1}^{\infty }{\frac {{\sqrt {3}}(F_{n+2}-1)}{2\phi ^{2n}}}$,

such that

${\frac {1}{2}}=\sum _{n=1}^{\infty }{\frac {F_{n+2}-1}{\phi ^{2n+2}}}.$

Row lengths

Let $L_{n}$ be the length of the $n$th row of the fractal, equal to the sum of side lengths $s_{n}=\phi ^{-n}$ shared by the pairs of tiles in the given row, plus the sum of the "gap" lengths occurring between facet pairs of length $s_{n+1}$. Therefore, for length of row $n$ is

${\begin{aligned}L_{n}&=(F_{n+2}-1)s_{n}+(F_{n+1}-1)s_{n+1}\\&={\frac {F_{n+2}-1}{\phi ^{n}}}+{\frac {F_{n+1}-1}{\phi ^{n+1}}}.\end{aligned}}$

As this length is equivalent to the distance from the bottom vertex of the golden diamond to the start of the $n$th row, one finds that

${\frac {F_{n+2}-1}{\phi ^{n}}}+{\frac {F_{n+1}-1}{\phi ^{n+1}}}=\sum _{i=1}^{n}{\frac {1}{\phi ^{i}}}$

Letting $n$ approach infinity, one finds

$\lim _{n\to \infty }{\frac {F_{n+2}-1}{\phi ^{n}}}+{\frac {F_{n+1}-1}{\phi ^{n+1}}}=\phi$,

such that

$\lim _{n\to \infty }{\frac {F_{n+1}}{\phi ^{n}}}+{\frac {F_{n}}{\phi ^{n+1}}}=1.$

Embedding of Fibonacci words

The golden diamond geometrically embeds the complete set of Fibonacci words via the patterns of triangular tiles within its rows. Unlike the Fibonacci word fractal, which is constructed of Fibonacci words by design, the golden diamond naturally contains each Fibonacci word as an emergent feature of its geometry.

A Fibonacci word is formed by repeated concatenation in the same way that the Fibonacci numbers are formed by repeated addition. Each successive Fibonacci word is a string of letters or numbers generated from the previous, two Fibonacci words. If the first Fibonacci words are $w_{1}=A$ and $w_{2}=AB$, one generates the third word $w_{3}$ via the concatenation of $w_{1}$ with $w_{2}$, such that $w_{3}=w_{2}w_{1}=ABA$. Alternatively, one can generate a successive Fibonacci word via the substitution rule $A\rightarrow AB$ and $B\rightarrow A$. This substitution rule can be illustrated in the golden diamond and represents an example of a proof without words.

A subset of the golden diamond contains the set of all Fibonacci words. If the initial words are w₁ = A and w₂ = AB, then the Fibonacci word substitution rule A → AB and B → A can be observed as is illustrated by the dashed lines. The above is an example of a proof without words.

Fibonacci word spelling

The choice of characters used to form the "spelling" of a Fibonacci word has traditionally been an arbitrary one, with the most common choice being the values 0 and 1. However, the emergence of Fibonacci words in the golden diamond suggests a preferred spelling; that is, the ratio of the interval lengths associated with each character in the Fibonacci words of the golden diamond are such that $A/B=\phi$, as is apparent from the geometry of the fractal. Consequently, the ratio of lengths of any two consecutive words $w_{n}$ and $w_{n+1}$ is such that $w_{n}/w_{n+1}=\phi$.

Perspective projection

The golden diamond can be expressed as the perspective projection of infinitely many standing triangles within a quadrant of the Cartesian plane as observed from focal point near the origin. The resulting image produces a version of the golden diamond with a 90° angle at its bottom vertex, i.e., such that each tile in the fractal is an isosceles right triangle.

The golden diamond can be expressed as the perspective projection of infinitely many isosceles right triangles of altitude $a={\frac {1}{\sqrt {2}}}$, standing within the plane, as viewed near the origin, where $h=d={\frac {\phi }{\sqrt {2}}}$. The resulting image is scaled such that $\theta =90^{\circ }$.

By connecting aligned triangles through their base vertices, grid lines are generated in the plane. Through calculating the cross ratio, the distance between grid lines can been shown to alternate between a value of 1 and ${\frac {1}{\phi }}$ via the infinite Fibonacci word pattern.

Grid lines connecting vertices of the triangular tiles illustrate the plane in which the projected triangles stand.

By projecting the tile lengths along the edge of the fractal, one can determine the lengths of interval markings in the domain of the projected image.

A "bird's eye view" of the phinary (base-$\phi$) integer domain within which the projected triangles stand. Points indicate the base vertices of each triangle and the rays indicate the "shadow" of triangles as cast from the focal-point observer. The grid lines alternate in intervals of 1 and 1/$\phi$ via the infinite Fibonacci word pattern.

Embedding of phinary (base-$\phi$) integers

A remarkable property of the above perspective projection is its embedding of Bergman's base-$\phi$ positional number system. The grid lines generated between the aforementioned standing triangles, alternate in intervals of 1 and 1/$\phi$ via the infinite Fibonacci word pattern. Consequently, along any given axis, the distance $d_{n}$ of the $n$th grid line from the origin produces the following list of values:

$d_{1}$= 1,

$d_{2}$= 1 + 1/$\phi$,

$d_{3}$= 1 + 1/$\phi$ + 1,

$d_{4}$= 1 + 1/$\phi$ + 1 + 1,

$d_{5}$= 1 + 1/$\phi$ + 1 + 1 + 1/$\phi$,

$d_{6}$= 1 + 1/$\phi$ + 1 + 1 + 1/$\phi$ + 1,

$d_{7}$= 1 + 1/$\phi$ + 1 + 1 + 1/$\phi$ + 1 + 1/$\phi$,

$\vdots$

which simplifies to the following:

$d_{1}$= 1,

$d_{2}$= $\phi$,

$d_{3}$= $\phi$ + 1

$d_{4}$= $\phi$ + 2,

$d_{5}$= 2$\phi$ + 1,

$d_{6}$= 2$\phi$ + 2,

$d_{7}$= 3$\phi$ + 1,

$\vdots$

The grid lines in the pre-image projective plane. Distances from the origin are displayed along each axis, resulting in the list of phinary integers.

Written yet another way:

$d_{1}$= $\phi ^{0}$,

$d_{2}$= $\phi ^{1}$,

$d_{3}$= $\phi ^{2}$

$d_{4}$= $\phi ^{2}$ + 1,

$d_{5}$= $\phi ^{3}$,

$d_{6}$= $\phi ^{3}$ + $\phi ^{0}$,

$d_{7}$= $\phi ^{3}$ + $\phi ^{1}$,

$\vdots$

These values are precisely the integers of the phinary (base-$\phi$) positional number system, as is clear when written in the standard form convention:

$d_{1}$= $1_{\phi }$,

$d_{2}$= $10_{\phi }$,

$d_{3}$= $100_{\phi }$

$d_{4}$= $101_{\phi }$,

$d_{5}$= $1000_{\phi }$,

$d_{6}$= $1001_{\phi }$,

$d_{7}$= $1010_{\phi }$,

$\vdots$

Therefore, the pre-image domain that generates the golden diamond perspective projection is partioned by the phinary integers.

Applications

Number Theory

Embedding of Fibonacci diatomic sequence

The tree graph of the fractal embeds the well-studied Fibonacci diatomic sequence (1,1,1,2,1,2,2,1,3,2,2,...) [A000119 - OEIS], which counts the partitions of a positive integer $n$ as a sum of distinct Fibonacci numbers.

This sequence can be recovered from the tree graph via a random walk downward from the uppermost vertex. That is, one counts the number of possible paths to any vertex, yielding the Fibonacci diatomic array.

1 1

1 2 1

1 2 2 1

1 3 2 2 3 1

1 3 3 2 4 2 3 3 1

1 4 3 3 5 2 4 4 2 5 3 3 4 1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

If $N(i,j)=1$ if $j\in \{\lfloor \phi i\rfloor ,\lceil \phi i\rceil \}$ and 0 otherwise, then $N$ is the adjacency matrix of the graph and $b_{n}:=lim_{n\to \infty }N^{n}(1,n),$where $b_{n}$ is the Fibonacci diatomic sequence. A similar process can be applied to the graph $S_{3,2}$, to generate Stern's diatomic array and thus Stern's diatomic sequence.

Analogues to the Stern-Brocot and Calkin-Wilf trees

The tree graph of the golden diamond occurs as a subset of trees related to the Calkin-Wilf tree and Stern-Brocot trees. While the fractions appearing in the nodes of the Calkin-Wilf and Stern-Brocot trees result from values in Stern's diatomic sequence, the fractions in the trees associated with the golden diamond result analogously from values occurring in the Fibonacci diatomic sequence.

The phinary recurrence tree, of which the golden diamond occurs as a subtree, indicated by gray lines. The Calkin-Wilf tree is a subtree, indicated by the darkened branches.

The phinary even-recurrence tree, of which the golden diamond occurs as a subtree, indicated by gray lines. The Stern-Brocot tree is a subtree, indicated by the darkened branches.

Variations

By introducing extra lines into the original fractal, one may produce the projected image of an arrangement of infinitely many octahedra in the plane as observed without gaps or overlaps.

A variation of the golden diamond, depicting the projected image of infinitely many octahedra as observed without gaps or overlaps.

References

This article "Golden diamond" is from Wikipedia. The list of its authors can be seen in its historical and/or the page Edithistory:Golden diamond. Articles copied from Draft Namespace on Wikipedia could be seen on the Draft Namespace of Wikipedia and not main one.

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