# Metallic mean fractal systems and their tilings

Sam Coates  Surface Science Research Centre and Department of Physics,
University of Liverpool, Liverpool L69 3BX, UK  Corresponding author:
<EMAIL_ADDRESS>
###### Abstract

Fractals and quasiperiodic structures share self-similarity as a structural
property. Motivated by the link between Fibonacci fractals and quasicrystals
which are scaled by the golden mean ratio $\frac{1+\sqrt{5}}{2}$, we introduce
and characterize a family of metallic-mean ratio fractals. We calculate the
spatial properties of this generalized family, including their boundaries,
which are also fractal. We then demonstrate how these fractals can be related
to aperiodic tilings, and show how we can decorate them to produce new,
fractal tilings.

##  I Introduction

Fractals can be used to understand the behaviours of a range of complex
systems, both natural and manufactured [1, 2, 3]. Their application as models
for biological, chemical, and physical systems is varied and wide, and there
is also particular interest in the properties of fabricated fractal materials.
Within the context of physical media and/or condensed matter physics [4], the
use or exploration of fractals includes but is not limited to the study of
diffusive properties [5, 6, 7, 8], topological states (electronic, photonic,
acoustic etc.) [9, 10, 11, 12, 13, 14], and tunable metamaterials and antennas
[15, 16, 17, 18, 19, 20].

The self-similarity of fractals is an attribute shared by many
quasicrystalline materials or aperiodic tilings, and, the properties of such
systems can be fractal [21, 22, 23, 24, 25, 26, 27, 28, 29]. In particular,
the 1D Fibonacci sequence is a ubiquitous aperiodic structure which is
routinely used as a model for quasicrystals/aperiodic media, and its links
with fractality are strong [30, 31]. It can be generated by two letters $A$
and $B$, using the substitutions $A\rightarrow AB$ and $B\rightarrow A$, and,
when the sequence of A and B letters are drawn with geometric Lindenmayer
system (L-system) rules, it produces a fractal structure [32, 33, 34].

Fibonacci systems and quasicrystalline structures are also directly linked by
the self-similar golden-mean $\frac{1+\sqrt{5}}{2}$ – the ratio of $A$:$B$
letters gives $\frac{1+\sqrt{5}}{2}$, and structural elements of
quasicrystalline intermetallic alloys are commonly scaled by
$\frac{1+\sqrt{5}}{2}$ (e.g. atom-atom distances, surface step-edge heights
etc.). However, aperiodic tilings are not restricted to golden-mean scaling;
tilings can be generated which are scaled by higher-order metallic-mean ratios
[35, 25, 36, 37]. Here, the entire family of metallic-mean sequences are
produced by expanding the Fibonacci sequence substitution rules as
$A\rightarrow A^{m}B,B\rightarrow A$, where $m=1$ returns the golden-mean
$m=2$ is silver, $m=3$ is bronze etc.

Motivated by Fibonacci word geometries [33, 34] and these metallic-mean
structures, we introduce a family of fractals with a wide variety of
customisable parameters which we generalize over all metallic-mean ratios.
First, we present the L-system used to create a golden-mean fractal, calculate
its properties, and demonstrate its geometric restrictions. Then, we move on
to generalising this fractal over all metallic-means, repeating our analysis
in the general case. Lastly, we discuss how these fractals can be decorated to
give aperiodic tilings.

##  II Fractal system

We will start by introducing the golden-mean fractal system using a basic set
of parameters, before analysing the effect of varying these same parameters.
Then, we will discuss the generalisation of the system (and its properties)
over all of the metallic means. Here, we will refer to a metallic mean
$\varphi$ in terms of $m$:

$\varphi_{m}=\frac{m+\sqrt{m^{2}+4}}{2}$ (1)

where $m=1$ is the golden mean, $m=2$ is silver, and so on. Similarly, we will
discuss the fractal systems in terms of F${}_{m}^{n}$, where $n$ is the
generation of the fractal, and $m$ is the metallic mean value as above. So,
the first generation of the golden-mean is F${}_{1}^{1}$, the second is
F${}_{1}^{2}$, and so on.

###  II.1 Golden-mean

F1 is described as an L-system, with the following variables, constants, and
rules:

Variables: $\displaystyle\quad A,B,C,A^{\prime},B^{\prime},C^{\prime}$ (2)
Constants: $\displaystyle\quad+,-$ Rules: $\displaystyle\quad\begin{aligned}
A\rightarrow&C-B+C^{\prime}+B^{\prime}-C\\\ B\rightarrow&C-B+C^{\prime}\\\
C\rightarrow&A\\\
A^{\prime}\rightarrow&C^{\prime}+B^{\prime}-C-B+C^{\prime}\\\
B^{\prime}\rightarrow&C^{\prime}+B^{\prime}-C\\\
C^{\prime}\rightarrow&A^{\prime}\end{aligned}$

$\displaystyle\text{F${}_{1}^{1}$}:$ $\displaystyle\,A$ (3)
$\displaystyle\text{F${}_{1}^{2}$}:$
$\displaystyle\,C\\!-\\!B\\!+\\!C^{\prime}\\!+\\!B^{\prime}\\!-\\!C$
$\displaystyle\text{F${}_{1}^{3}$}:$
$\displaystyle\,A\\!-\\!C\\!-\\!B\\!+\\!C^{\prime}\\!+\\!A^{\prime}\\!+\\!C^{\prime}\\!+\\!B^{\prime}\\!-\\!C\\!-\\!A$
$\displaystyle\text{F${}_{1}^{4}$}:$
$\displaystyle\,C\\!-\\!B\\!+\\!C^{\prime}\\!+\\!B^{\prime}\\!-\\!C\\!-\\!A\\!-\\!C\\!-\\!B\\!+\\!C^{\prime}\\!+\\!A^{\prime}\\!+\\!C^{\prime}\\!+\\!B^{\prime}\\!-\\!C\\!-\\!B\\!+\\!C^{\prime}\\!+\\!A^{\prime}\\!+\\!C^{\prime}\\!+\\!B^{\prime}\\!-\\!C\\!-\\!A\\!-\\!C\\!-\\!B\\!+\\!C^{\prime}\\!+\\!B^{\prime}\\!-\\!C$
$\displaystyle\text{F${}_{1}^{5}$}:$
$\displaystyle\,A\\!-\\!C\\!-\\!B\\!+\\!C^{\prime}\\!+\\!A^{\prime}\\!+\\!C^{\prime}\\!+\\!B^{\prime}\\!-\\!C\\!-\\!A\\!-\\!C\\!-\\!B\\!+\\!C^{\prime}\\!+\\!B^{\prime}\\!-\\!C\\!-\\!A\\!-\\!C\\!-\\!B\\!+\\!C^{\prime}\\!+\\!A^{\prime}\\!+\\!C^{\prime}\\!+\\!B^{\prime}\\!-\\!C\\!-\\!B\\!+\\!C^{\prime}\\!+\\!A^{\prime}\\!$
$\displaystyle+\\!C^{\prime}\\!+\\!B^{\prime}\\!-\\!C\\!-\\!B\\!+\\!C^{\prime}\\!+\\!A^{\prime}\\!+\\!C^{\prime}\\!+\\!B^{\prime}\\!-\\!C\\!-\\!A\\!-\\!C\\!-\\!B\\!+\\!C^{\prime}\\!+\\!B^{\prime}\\!-\\!C\\!-\\!A\\!-\\!C\\!-\\!B\\!+\\!C^{\prime}\\!+\\!A^{\prime}\\!+\\!C^{\prime}\\!+\\!B^{\prime}\\!-\\!C\\!-\\!B\\!$
$\displaystyle+\\!C^{\prime}\\!+\\!A^{\prime}\\!+\\!C^{\prime}\\!+\\!B^{\prime}\\!-\\!C\\!-\\!A\\!-\\!C\\!-\\!B\\!+\\!C^{\prime}\\!+\\!A^{\prime}\\!+\\!C^{\prime}\\!+\\!B^{\prime}\\!-\\!C\\!-\\!B\\!+\\!C^{\prime}\\!+\\!A^{\prime}\\!+\\!C^{\prime}\\!+\\!B^{\prime}\\!-\\!C\\!-\\!A\\!-\\!C\\!-\\!B\\!+\\!C^{\prime}\\!+\\!B^{\prime}\\!-\\!C$
Figure 1: Generations $n=4,5,6,7,8$ and $15$ of the F1 system, drawn as an
L-system according to the rules of Eq. 2, where $j=3$, and $A=B=C=1$. The
scale of each generation has been normalised to 1 in $x$. Arrows highlight the
approximate polygonal edges of F${}_{1}^{15}$, where $\varphi_{1}$ corresponds
to Eq. 1 with $m=1$.

where variables $A$, $B$, $C$ ($A^{\prime},B^{\prime},C^{\prime}$) are
segments which move forward along a heading by an integer value, and constants
$+,-$ rotate the heading by an angle $\theta$ clockwise and counter-clockwise
respectively. We note that $\theta$ is controlled by an extra parameter $j$,
such that $\theta=\frac{\pi}{j}$ where $j>2$. The rules for the $A$, $B$, and
$C$ segments have mirror-symmetric counterparts in $A^{\prime}$, $B^{\prime}$,
and $C^{\prime}$: the $+$ and $-$ signs are flipped, while the rule
‘structure’ stays the same. For the purposes of clarity, when we discuss
specific properties of segments, we will refer to both $A$ and $A^{\prime}$
simply as $A$ and so on. We initially set the lengths of $A=B=C=1$ and $j=3$.
We discuss the condition for $j$ and the impact of changes to $j$ and the
lengths of $A,B,C$ further on.

The first few alphabet substitution generations of F1 are shown in Eq. (3),
which we initialise with an $A$ segment. We can start with any letter segment:
starting with $B$ truncates the length of each generation with respect to Eq.
3, while C simply ‘delays’ the generation by 1 (as $C\rightarrow A$).
Similarly, if we start using the mirror-symmetric components, the results are
identical except a simple replacement for each symmetric pair (e.g.
$A:A^{\prime}$, $+:-$). For simplicity, we only discuss F1 as initialised from
$A$ in the main text. It should be noted, however, that the properties we
discuss in the following are applicable regardless of the starting segment.

Finally, as a point of interest, if we take F${}_{1}^{4}$, remove the $+$/$-$
constants and replace the $A$, $B$, and $C$ segments with integers 2, 1, and
0, we find a section of the ternary Fibonacci sequence:
0101020102010102010201010. This sequence takes the Fibonacci word (comprised
of 1’s and 0’s) and replaces 00 pairs with 020 [38, 39, 40]. This appears true
for all even $n$ \- for odd $n$, the first segment needs to be removed.

Figure 2: F${}_{1}^{7}$ decomposed into three sub-units: two F${}_{6}^{1}$
(black) and one F${}_{1}^{5}$ (red). The projected contribution to the length
of F${}_{1}^{7}$ in $x$ is shown as a line, colour-coded to the sub-unit.
Figure 3: (a)F${}_{1}^{10}$, where the bottom boundary is overlaid and
highlighted in red. (b) Generations of the L-system which describes the
boundary of the F1 system. $Q,P,$ and $R$ refer to segment lengths described
in the text.

#### Geometry and the Hausdorff dimension of F1

Figure 1 shows the geometry of successive generations of F1 starting from
F${}_{1}^{4}$, with F${}_{1}^{15}$ also shown to indicate the characteristic
features of the curve at higher generations. Each curve has had its maximum
scale in $x$ normalised to 1. The polygonal boundary of F${}_{1}^{15}$ is a
trapezoid with edge lengths 1, $1/\varphi_{1}$, and $1/\varphi_{1}^{2}$, as
indicated on Figure 1. We note that initialising the sequence with a $B$
segment creates a fractal with a parallelogram polygonal boundary with edge
lengths 1, $1/\varphi_{1}$.

The Hausdorff dimension is a key property of a fractal, and is a measure of
the scaling behaviour of a set: how the size of a set grows as its scale
increases. In other words, it is the roughness of the curve, in this case. It
is calculated by [41]:

$H^{d}=\frac{\ln(N)}{\ln(S)}$ (4)

where $N$ represents the increase in the number of segments created in
successive generations, and $S$ is the scaling factor between generations.
Similar to the analysis used in the substitution rules of quasiperiodic
tilings [42], inspecting the substitution matrix of the L-system rules can
give us the number of segments generated after each substitution, or, $N$. Eq.
(5) shows the substitution matrix of F1:

$M_{N}=\hbox{}\vbox{\kern 0.86108pt\hbox{$\kern 0.0pt\kern
2.5pt\kern-5.0pt\left(\kern
0.0pt\kern-2.5pt\kern-6.66669pt\vbox{\kern-0.86108pt\vbox{\vbox{
\halign{\kern\arraycolsep\hfil\@arstrut$\kbcolstyle#$\hfil\kern\arraycolsep&
\kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep&&
\kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep\cr
5.0pt\hfil\@arstrut$\scriptstyle$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
A$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle B$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle C$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
A^{\prime}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle B^{\prime}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle C^{\prime}\\\A$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0\\\B$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0\\\C$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 2$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0\\\A^{\prime}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1\\\B^{\prime}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0\\\C^{\prime}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 2$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern
5.0pt\crcr}}}}\right)$}}$ (5)

As the absolute number of segments is independent of their ’symmetry’, we can
simply sum the contributions of the symmetric components within each
substitution rule ($A+A^{\prime}$ etc.), to give the equivalent:

$M_{N}=\hbox{}\vbox{\kern 0.86108pt\hbox{$\kern 0.0pt\kern
2.5pt\kern-5.0pt\left(\kern
0.0pt\kern-2.5pt\kern-6.66669pt\vbox{\kern-0.86108pt\vbox{\vbox{
\halign{\kern\arraycolsep\hfil\@arstrut$\kbcolstyle#$\hfil\kern\arraycolsep&
\kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep&&
\kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep\cr
5.0pt\hfil\@arstrut$\scriptstyle$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
A$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle B$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle C\\\A$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 1\\\B$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
2$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0\\\C$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
3$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 2$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0\\\$\hfil\kern 5.0pt\crcr}}}}\right)$}},$ (6)

The largest eigenvalue, $N$, is $1+\sqrt{2}$, or $\varphi_{2}$, which is
consistent with other Fibonacci fractals [33]. For completeness, the
corresponding eigenvector of the eigenvalue $\varphi_{2}$ is proportional to
(1, $\sqrt{2}$, $\varphi_{2}$), which tells us that the $A$ segments appear
least frequently, $B$ segments $\sqrt{2}$ more often, and $C$ segments
$\varphi_{2}$ more often than $A$ segments.

We calculate the scaling factor $S$ by deconstructing F1 after some generation
$n$. Figure 2 shows F${}_{1}^{7}$, which has been subdivided into three
labelled segments, highlighted by their parallelogram boundaries: two are
F${}_{1}^{6}$ (black), one is F${}_{1}^{5}$ (red). These segments are rotated
(-)$\frac{2\pi}{3}$ and $\pi$ with respect to F${}_{1}^{7}$. The length of
F${}_{1}^{7}$, $L^{7}$, can therefore be described by the sum of the black and
red segments projected onto $x$, as indicated below F${}_{1}^{7}$ in Figure 2.
These projected lengths are determined by the rotation of the segments, such
that we can generally describe:

$\displaystyle L^{n}=$ $\displaystyle
L^{n-1}\absolutevalue{\cos(-\frac{2\pi}{3})}+L^{n-1}\absolutevalue{\cos(\frac{2\pi}{3})}$
(7) $\displaystyle+$ $\displaystyle L^{n-2}\absolutevalue{\cos(\pi)},$
$\displaystyle=$ $\displaystyle L^{n-1}+L^{n-2}$

Then, as we know that

$\displaystyle S=\frac{L^{n}}{L^{n-1}}=\frac{L^{n-1}}{L^{n-2}}$ (8)

such that,

$\displaystyle L^{n}=SL^{n-1},\qquad L^{n-2}=\frac{L^{n-1}}{S}$ (9)

we can therefore combine Eqs. Eq. 7 and 9 to give:

$\displaystyle SL^{n-1}=$ $\displaystyle L^{n-1}+\frac{L^{n-1}}{S}$ (10)
$\displaystyle S=$ $\displaystyle 1+\frac{1}{S}$ $\displaystyle S^{2}-S=$
$\displaystyle 1$

whose solution gives $S=\frac{1+\sqrt{5}}{2}$, meaning:

$H^{d}=\frac{\ln(\varphi_{2})}{\ln(\varphi_{1})}=1.8316...$ (11)

which, following Monnerot-Dumaine [33], we can write as:

$H^{d}=\frac{\ln(2+\frac{2}{2+\frac{2}{2+\frac{2}{2+...}}})}{\ln(1+\frac{1}{1+\frac{1}{1+\frac{1}{1+...}}})}$
(12)

#### Boundary

The boundary of F1 is also a fractal. Figure 3(a) shows F${}_{1}^{10}$, where
the bottom boundary is overlaid and highlighted in red. The left, right, and
top boundaries are the bottoms of F${}_{1}^{9}$, F${}_{1}^{9}$, and
F${}_{1}^{8}$ respectively, as expected from Eq. 7. We also note that the
perimeters of the internal triangular-like holes follow these boundary forms.
The boundaries can be constructed as an L-system by considering three segments
$Q,P,$ and $R$ and the same angular parameters $+$ and $-$. Eq. 13 shows the
substitution rules:

$\displaystyle Q\rightarrow$ $\displaystyle\enspace P-R++R-P$ (13)
$\displaystyle P\rightarrow$ $\displaystyle\enspace Q$ $\displaystyle
R\rightarrow$ $\displaystyle\enspace P$

and Figure 3(b) shows how the boundary develops for selected generations,
where we have initialised with a $Q$ segment, and the lengths of the segments
are $Q=P=1$, $R=1/\varphi_{1}$. As with the fractal itself, we can calculate
$H^{d}$ by considering the number of segments in the boundary $NB$ at each
generation by finding the largest eigenvalue of the substitution matrix:

$M_{NB}=\hbox{}\vbox{\kern 0.86108pt\hbox{$\kern 0.0pt\kern
2.5pt\kern-5.0pt\left(\kern
0.0pt\kern-2.5pt\kern-6.66669pt\vbox{\kern-0.86108pt\vbox{\vbox{
\halign{\kern\arraycolsep\hfil\@arstrut$\kbcolstyle#$\hfil\kern\arraycolsep&
\kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep&&
\kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep\cr
5.0pt\hfil\@arstrut$\scriptstyle$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
Q$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle P$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle R\\\Q$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0\\\P$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
2$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 1\\\R$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
2$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0\\\$\hfil\kern 5.0pt\crcr}}}}\right)$}},$ (14)

Figure 4: Schematic to show self-intersection after four successive $-$
segments when $j=2$. Sequence reads from left to right: $B^{\prime}-C-A-C-B$,
where the red circle indicates the end of segment $C$ intersecting with the
start of $B^{\prime}$, and the red arrow shows the $B$ segment being overlaid
on $B^{\prime}$.

which is:

$NB=\frac{1}{3}\left(27-3\sqrt{57}\right)^{\frac{1}{3}}+\frac{\left(9+\sqrt{57}\right)^{\frac{1}{3}}}{3^{\frac{2}{3}}}=1.7692...$

Then, as $SB$ is simply the scaling factor of F1, we find

$H^{dB}=\frac{\ln(NB)}{\ln(\varphi_{1})}=1.1857...$ (15)

#### Effect of $j$ on the Hausdorff dimension of $F^{1}$

Figure 5: (a-c) F1 for $j=4,5,6$ respectively. Each curve has been normalized
so that their lengths in $x$ are equal. As $j$ increases, the curve flattens,
so that at the infinite limit we produce a straight line.

So far we have fixed $j=3$. Here, we show why $j>2$, and investigate the
impact on $H^{d}$ when $j$ is increased.

Trivially, when $j=1$, F1 is a straight line. To show that $j\neq 2$, it is
convenient to analyse the distribution of the angular changes in a
sufficiently long sequence. Eq. (16) shows the $+$ and $-$ signs only from
F${}_{1}^{5}$:

$\displaystyle-$ $\displaystyle-++++----++----++++--$ (16) $\displaystyle+$
$\displaystyle+++----++++--++++----$ $\displaystyle+$
$\displaystyle+----++++--$

By inspecting Eq. (16) we can see that we have, at most, four successive + or
$-$ segments. This is true over arbitrary generations, and can be proven by
analysing the behaviour of adjacent segments under substitution, which is
demonstrated in Appendix A. The resultant effect of this behaviour can be
shown by taking a sequence of F1 which includes a quadruplet of angle changes:
$B^{\prime}-C-A-C-B$. We note that the $-$ quadruplet always appears in this
letter sequence, and the $+$ in the mirror inverse
($B+C^{\prime}+A^{\prime}+C^{\prime}+B^{\prime}$). The $-$ sequence is shown
in Figure 4, where the arrows indicate the direction of the sequence as read
from left-to-right. If we set the initial orientation of the $B^{\prime}$
segment to be 0∘, when $j$=2 ($\theta=\frac{\pi}{2}$), the sequence is in
self-contact with its initial position after the second $C$ segment, as
indicated by the red circle. Then, the final $B$ segment self-intersects with
the sequence, as indicated by the overlaid red section and arrow. In other
words, the cumulative operation of this quadruplet on the heading is equal to
$0+\frac{4\pi}{2}=2\pi\equiv 0$, or zero displacement, such that the fractal
is guaranteed to loop and self-intersect. This looping behaviour cannot occur
for $j>2$, as $\frac{4\pi}{j}<2\pi$ for this condition.

$j$ | $m=2\cos(\frac{\pi}{j})$ | $S(m)$ | $H^{d}$
---|---|---|---
3 | 1 | 1.618 | 1.832
4 | 1.414 | 1.932 | 1.338
5 | 1.618 | 2.095 | 1.192
6 | 1.732 | 2.189 | 1.125
7 | 1.802 | 2.247 | 1.089
8 | 1.848 | 2.285 | 1.066
9 | 1.879 | 2.312 | 1.052
10 | 1.902 | 2.331 | 1.041
1000 | $\sim 2$ | 2.414 | $\sim 1$
Table 1: The $H^{d}$ of F1 tends to 1 as $j$ increases. While the number of
segments $N$ is constant, the value of $S$ is calculated by Eq. 1, where
$m=2\cos(\frac{\pi}{j})$.

Figures 5(a-c) show the effect of increasing $j$ for $j=4,5,6$ respectively,
where the scale in $x$ is normalized to 1 for each of the curves. It is
immediately obvious that we obtain curves of different forms to when $j=3$.
This can be related back to Eq. 16, where we see that angular changes also
come in pairs i.e. $++$ or $--$. When $j=3$, both the pair and quadruplet
angle changes sum to be $>\pi$ but $<2\pi$, leading to a dense, winding form:
a consequence of construction using obtuse and reflex angles. When $j=4$, the
pair changes sum to equal $\frac{\pi}{2}$, and the quadruplet $\pi$, such that
we obtain a ‘flatter’ form essentially built from acute and right angles,
which removes the ‘winding’ element. This effect only intensifies as $j$
increases.

Figure 6: An example of self-contact in the F1 fractal curve when $j=3$ and
the length of $B$ $\geq$ $A+C$. Here, $B=2$, and $A=C=1$. Arrows indicate the
‘direction’ of the fractal curve as read from left to right in the L-system.
Letters and $+/-$ indicate the segments of the L-system, and a red circle
indicates the point of self-contact.  Figure 7: (a-c) F${}_{2}^{6}$,
F${}_{3}^{5}$, and F${}_{4}^{4}$ respectively, where $j=3$ and $A=B=C=1$.
Enlarged areas correspond to those areas indicated by black arrows.

We now consider the change of $H^{d}$ of F1 as $j\rightarrow\infty$; the
number of segments $N$ is a constant value for each generation of $n$
regardless of $j$, meaning we only have to consider the scaling factor $S$. As
the angle between segments decreases, the absolute magnitude of the length of
the curve along the $x$-axis increases. This is evident when we consider that
the maximum length of a single segment along the positive $x$-direction is
determined by $\cos(\frac{\pi}{j})$, which is equal to $[0.5,1]$ for
$j=[3,\infty]$. So, in the infinite limit of $j$, $S$ is determined by $N$
parallel segments laid end-to-end. In other words, for $j=\infty$, $S\equiv
N\equiv\varphi_{2}$. As such, $S$ is simply equal to Eq. (1) for $m$ values
between 1 and 2, where to find $m$ for arbitrary $j$ we can use
$m=2\cos(\frac{\pi}{j})$. Table 1 shows the values for $j,m$, and $H^{d}$ for
a few values of $j$.

#### Different length values for A, B, C

Finally, we discuss the valid values for $A,B$, and $C$, which so far have
been fixed as $A=B=C=1$. The limiting factor for arbitrarily setting these
lengths is self-intersection, as with our argument on the limits for $j$, this
occurs when the curve ‘loops’ during angle change segments. In other words, we
need to find the conditions for the lengths of $A,B$, and $C$ which guarantee
a non-zero displacement from a certain position after successive identical
angle operations. We again note that we focus on systems where the lengths of
the segments and their mirror-symmetric components are equal; the conditions
we find likely hold when this is not the case and only require some simple
extension to our proof.

We can use our analysis in the limits of $j$ to find valid lengths of $A,B$,
and $C$: it is trivial that this looping can only occur when we encounter our
quadruplet of angle changes. To self-intersect, and starting from an arbitrary
heading, we require that the total sum of angle changes is $\geq\pi$, which
can only be achieved by four angle changes when $j>2$. Similarly, if $j>4$,
the curve cannot self-intersect ($\frac{4\pi}{j}<\pi$ for this condition).
Therefore, for any $j>4$, $A$, $B$, and $C$ can be any value so long as
$A+B+C\geq 1$. We show a few examples for arbitrary parameters in Appendix B.

We can take a geometric approach for when $j=3,4$, where we again consider the
$B^{\prime}-C-A-C-B$ sequence. When $j=4$, we only require that one of $A$ or
$C$ are non-zero, to prevent $B$ segments directly overlapping after a
cumulative heading change of $\pi$. For $j=3$, we show this sequence in Figure
6 with a few preceding segments: $C-B+C^{\prime}$. Here, the lengths are
$A=C=1$, and $B=2$, which leads to a self-intersection point marked in red. As
stated, we require non-zero displacement from the starting point of the
$B^{\prime}$ segment; in other words the total displacement $\Delta>0$, which
we can write as:

$\displaystyle\Delta=$ $\displaystyle
B^{\prime}\cdot\begin{pmatrix}\cos(-\frac{\pi}{3})\\\
\sin(-\frac{\pi}{3})\end{pmatrix}+C\cdot\begin{pmatrix}\cos(0)\\\
\sin(0)\end{pmatrix}+A\cdot\begin{pmatrix}\cos(\frac{\pi}{3})\\\
\sin(\frac{\pi}{3})\end{pmatrix}$ (17)
$\displaystyle+C\begin{pmatrix}\cos(\frac{2\pi}{3})\\\
\sin(\frac{2\pi}{3})\end{pmatrix}+B\begin{pmatrix}\cos(\pi)\\\
\sin(\pi)\end{pmatrix}>0$

Considering the $x$ and $y$ components separately, we find:

$\Delta_{x}=\frac{A}{2}-\frac{B}{2}+\frac{C}{2}>0$ (18)

$\Delta_{y}=\frac{\sqrt{3}A}{2}-\frac{\sqrt{3}B}{2}+\frac{\sqrt{3}C}{2}>0$
(19)

both of which can be rearranged to give:

$A+C>B$ (20)

which is our condition for segment lengths when $j=3$. We show some examples
for arbitrary lengths which both meet and fail this condition in Appendix B.

###  II.2 Generalising over $\varphi_{m}$

The fractal can be generalised over all metallic means by extending the
L-system rules we defined in Eq. 2. Using $\mathbin{+\mkern-10.0mu+}$ to
indicate the concatenation of strings, the generalised substitution rules of
the L-system can be written as:

$\displaystyle A\rightarrow$ $\displaystyle
C-B+C^{\prime}+B^{\prime}-C\mathbin{+\mkern-10.0mu+}(m-1)\times(-B+C^{\prime}+B^{\prime}-C)$
(21) $\displaystyle B\rightarrow$ $\displaystyle
C-B+C^{\prime}\mathbin{+\mkern-10.0mu+}(m-1)\times(+B^{\prime}-C-B+C^{\prime})$
$\displaystyle C\rightarrow$ $\displaystyle A$ $\displaystyle
A^{\prime}\rightarrow$ $\displaystyle
C^{\prime}+B^{\prime}-C-B+C^{\prime}\mathbin{+\mkern-10.0mu+}(m-1)\times(+B^{\prime}-C-B+C^{\prime})$
$\displaystyle B^{\prime}\rightarrow$ $\displaystyle
C^{\prime}+B^{\prime}-C\mathbin{+\mkern-10.0mu+}(m-1)\times(-B+C^{\prime}+B^{\prime}-C)$
$\displaystyle C^{\prime}\rightarrow$ $\displaystyle A^{\prime}$

where $m$ is the metallic mean ratio as before, and the $\times$ operation
multiplies the additional string section to be concatenated. For example, for
$m=3$ and segment $A$, we would add
$-B+C^{\prime}+B^{\prime}-C-B+C^{\prime}+B^{\prime}-C$ to the original rule.
The conditions we have discussed for $j$ and segment lengths still hold, as
these are not affected by the additional sequences added.

#### Geometry and the Hausdorff dimension of Fn

Figure 7(a-c) shows the resultant geometry for F${}_{2}^{6}$, F${}_{3}^{5}$,
and F${}_{4}^{4}$ where $j=3$ and $A=B=C=1$. We choose these generations to
more directly compare each fractal’s geometry: when $m$ increases, the length
of the sequences increases, so that the density of the fractal also increases
for identical generations. Each of the curves has been normalised so that
their length in $x$ = 1, which has the effect of ‘shrinking’ each curve in
$y$. Broadly speaking, from a macroscopic view, the curves can be described as
$m$ conjoined triangular morphologies. To show the finer structure of the
curves, we have enlarged areas specified by the black arrows.

To calculate $H^{d}$ over all $m$, we note that, trivially, $S_{m}$ will
always be $\varphi_{m}$; what remains is to calculate $N_{m}$ for each $m$. As
before, we transform Eq. 21 into a substitution matrix:

$M_{N_{m}}=\hbox{}\vbox{\kern 0.86108pt\hbox{$\kern 0.0pt\kern
2.5pt\kern-5.0pt\left(\kern
0.0pt\kern-2.5pt\kern-6.66669pt\vbox{\kern-0.86108pt\vbox{\vbox{
\halign{\kern\arraycolsep\hfil\@arstrut$\kbcolstyle#$\hfil\kern\arraycolsep&
\kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep&&
\kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep\cr
5.0pt\hfil\@arstrut$\scriptstyle$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
A$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle B$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle C\\\A$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 1\\\B$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
2m$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 2m-1$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0\\\C$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
2m+1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 2m$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0\\\$\hfil\kern 5.0pt\crcr}}}}\right)$}},$ (22)

whose largest eigenvalue for arbitrary $m$ values is:

${N_{m}}=\sqrt{m^{2}+1}+m\equiv\frac{2m+\sqrt{(2m)^{2}+4}}{2},$ (23)

meaning we can write $N_{m}$ = $\varphi_{2m}$, such that:

$H^{d}_{m}=\frac{\ln(\varphi_{2m})}{\ln(\varphi_{m})}$ (24)

therefore as $m\rightarrow\infty$, $H^{d}_{m}\rightarrow 1$, or,
F${}_{\infty}^{n}$ is a straight line.

Figure 8: (a) The Hausdorff dimension $H^{d}$ for the fractals Fm and their
boundaries Bm for $m=[1,100]$. (b) In descending order, the boundaries of
F${}_{2}^{5}$, F${}_{3}^{5}$, F${}_{4}^{5}$, and F${}_{5}^{5}$, where $j=3$
and $A=B=C=1$.

#### Boundaries

Figure 9: $j=3$ initiators and their fractals. (a) $A-A-$ hexagon with
$m=A=B=C=1$. (b) $B-B-$ hexagon with $m=A=B=C=1$. (c) $A-A-$ hexagon with
$m=2$, $A=3$, $B=2$, $C=1$.

We can similarly extend the L-system rules which describe the Fm boundaries as
so:

$\displaystyle Q\rightarrow$ $\displaystyle\enspace
P-R++R-P\mathbin{+\mkern-10.0mu+}(m-1)\times(-R++R-P)$ (25) $\displaystyle
P\rightarrow$ $\displaystyle\enspace Q$ $\displaystyle R\rightarrow$
$\displaystyle\enspace P\mathbin{+\mkern-10.0mu+}(m-1)\times(-R++R-P)$

such that we can describe the number of segments in the boundary $NB_{m}$ by
the following matrix:

$M_{NB_{m}}=\hbox{}\vbox{\kern 0.86108pt\hbox{$\kern 0.0pt\kern
2.5pt\kern-5.0pt\left(\kern
0.0pt\kern-2.5pt\kern-6.66669pt\vbox{\kern-0.86108pt\vbox{\vbox{
\halign{\kern\arraycolsep\hfil\@arstrut$\kbcolstyle#$\hfil\kern\arraycolsep&
\kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep&&
\kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep\cr
5.0pt\hfil\@arstrut$\scriptstyle$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
Q$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle P$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle R\\\Q$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0\\\P$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
m+1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle m\\\R$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
2m$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 2m-2)\\\$\hfil\kern 5.0pt\crcr}}}}\right)$}},$
(26)

whose real eigenvalue is:

$NB_{m}=\frac{4m^{2}-5m+7}{3c}+\frac{2(m-1)+c}{3},$ (27)

where c is111 $\displaystyle c^{3}=$ $\displaystyle
8m^{3}-15m^{2}+3\sqrt{3}\sqrt{-m^{4}+15m^{3}-31m^{2}+45m-9}$
$\displaystyle+24m+10$ .

As before SBm is simply $\varphi_{m}$, meaning we can also find $H^{dB}$ of
the boundary for any $m$. We previously showed that as $m\rightarrow\infty$,
$H^{d}_{m}\rightarrow\infty$, so it also follows that
$H^{dB}_{m}\rightarrow\infty$ as the fractal approaches a straight line. We
plot $H^{d}_{m}$ and $H^{dB}_{m}$ for $m=[1,100]$ in Figure 8(a), which shows
that both values tend towards both each other, and 1. Out of interest, we note
that the roughness of the boundaries initially increases, and peaks at F3.
Figure 8(b) shows, in descending order, the boundaries of F${}_{2}^{5}$,
F${}_{3}^{5}$, F${}_{4}^{5}$, and F${}_{5}^{5}$. Upon inspection of these
boundaries, it becomes clear that the branch-like structures of F${}_{2}^{5}$
and F${}_{3}^{5}$ gives way to a saw-tooth like curve (where of course the
‘teeth’ also form a saw-tooth) as $m$ increases and the complexity decreases.

##  III Tiling system

It is possible to create a wide array of fractal geometries by initialising
our L-system with any arbitrary sequence of segments. However, we are
motivated by the decorations of fractal curves which give aperiodic tilings
[43, 44, 45], and, in particular, the work on the dimer model of the Penrose
and Ammann-Beenker tilings which found fractal membranes that can be decorated
with the constituent tiles of rhombs (and squares) [46, 47, 48]. Therefore, we
focus on developing closed curve ‘skeletons’ which can be decorated with
rhombuses to form the framework of aperiodic tilings. Here, we show how to
develop these skeletons using simple initiators, demonstrate the geometrical
constraints for $j,A,B,$ and $C$ when decorating, and finally present and
discuss some of the tilings produced.

###  III.1 Skeletons and their decorations

Figure 10: (a,b) Schematic for decorating segments of lengths 2 and 3,
respectively. (a) shows how a $-$ angle change causes the next segment to
intersect with a rhombus corner, while a $+$ segment allows a valid tile
placement. (b) The inverse of (a).

First, we focus on creating the simplest closed curve initiators which exhibit
some rotational symmetry at their centre point. In this case we can create
closed curves in terms of $j$, where all we have to do is define two ‘edges’
of a curve and multiply this by $j$. For example, for $j=3$, we can write:

$j\times(A-A-)=A-A-A-A-A-A-$ (28)

to create a hexagon, where we drop the superfluous $-$ segment at the end of
the sequence. From here, we can apply the rules of Eq. 21 to generate some
fractal geometries. Figure 9(a, b) show $A-A-$ and $B-B-$ hexagons where
$m=A=B=C=1$. Figure 9(c) shows an $A-A-$ hexagon where $m=2$ and
$A=3,B=2,C=1$. Appendix C shows other examples with different $j$ values and
arbitrary parameters.

We are motivated to find initiators and parameters which allow us to create
tiling systems based on the decoration of skeletons. As such, we consider
geometric arguments to find how to decorate the segments of these skeletons
with respect to $j$ and the angular changes $+/-$. While it may be possible
that we can decorate our skeletons with any set of polygons, we initially
choose rhombus tiles whose opposing internal angles are set to either
$\frac{\pi}{j}$ or $\frac{2\pi}{j}$. We note that to fill space the fractal
tilings need at least two tiles, which we discuss in the next section. In this
case, we envisage that the rhombus decorations act as guides for where to
place additional tiles.

Figure 10(a, b) each show two arbitrary segments where we have set the lengths
as 2 and 3 respectively, and taken $j=3$. Here, we decorate the start and end
points of each unit length within the segment, where adjacent rhombuses are
mirror symmetric and we have chosen their initial orientations arbitrarily. We
also show potential next steps in the L-system as red and green lines
respectively. Figure 10(a) shows that for length 2 and an angle change of $-$,
the next step coincides with a rhombus vertex, whereas a $+$ change allows a
potential tile placement. The converse is true for Figure 10 (b). As such, we
can deduce that segments which are succeeded by a $+$ change must have an even
length, and those by a $-$ change must be odd. We note that by inspection of
Eq. 2, $-$ segments always follow an $A$ or $C$, and a $+$ always follows a
$B$ (the inverse is true for the mirror symmetric components as expected).
Therefore, valid decorations can be found when $A,C$ are odd, and $B$ is even,
for any $m$ or $j$. Of course, the inverse condition is true if we decorate
the segments with mirror flipped rhombuses about the x-axis.

###  III.2 Fractal tilings

Despite the conditions on length parameters, $j$, and the requirement that we
start with an isotropic initiator, there is still an infinite set of unique
fractals and therefore tiling decorations which can be generated; there are no
upper bounds or limits for any of our parameters. Therefore, to limit our
scope, we restrict our investigations to tilings consisting of only two types
of tiles: rhombuses with opposing internal angles set to either
$\frac{\pi}{j}$ or $\frac{2\pi}{j}$ as mentioned, and polygons with all
internal angles set to $\frac{2\pi}{j}$. Here, we focus on a few examples of
these tilings, and discuss the natural limitations as we increase $j$.

####  j = 3 tilings

Figure 11: (a) An $A-C-$ initiator after 6 generations (left to right). Here,
$m=1,j=3,A=3,B=2,$and $C=1$. (b) Left: rhombuses decorate the sixth generation
of (a). Some empty spaces are indicated by black lines. Right: the empty
spaces are filled by hexagonal and rhombus tiles. The black lines point to
each of the corresponding spaces which are now filled, and white outlines
serve to highlight the motifs. (c) Each of the motifs are enlarged to show
their inner structure. The larger motifs appear to form early generations of
the fractal form used to generate the skeleton.

To illustrate the decoration of a skeleton, Figure 11(a) shows the first six
generations of a curve initiated by an $A-C-$ hexagon, where $m$ = 1,
$A=3,B=2$, and $C=1$. We have chosen these lengths as they are the first
integers in the Fibonacci sequence which obey the even/odd length rules.
Appendix D shows other rhombus skeletons generated using arbitrary parameters.
Figure 11(b), left, is the $n=6$ generation decorated by rhombs, which shows
empty spaces or holes. Figure 11(b), right, displays how these holes can be
filled by regular hexagons (internal angles of $\frac{2\pi}{j}$) and other
rhombs; five of the motifs have been linked by black lines, and are also
indicated by white outlines.

Figure 12: (a) Generation 3 skeletons decorated with rhombuses for $j=3,4,5$
in descending order. Each skeleton has been initiated using a $A-A-$ polygon
for their respective $j$ values. Circles indicate sections which correspond to
five successive angle changes, and green (red) lines indicate valid (invalid)
geometries for a two-tile system for $j=4,5$. (b) Decorated generation 3
skeletons for $A-C-$ polygon initiators which allow a two-tile system. (c) The
creation of a $j=4$ tiling based on successive generations of the fractal
system. Empty spaces have been filled with squares and rhombuses, ensuring
that we recreate the earlier fractals, which are indicated by white outlines.
(d) The generation 3, $j=5$ skeleton can be filled by fat rhombuses, but not
in such a way that recreates a connected set of thin rhombus fractals.

Technically, any arrangement of space-filling hexagons and rhombs could be
used to pack these spaces, and the examples in Appendix D have been left blank
for those interested in doing so. However, here we found that the closed curve
form of the rhombus tiles matches exactly with the distribution of rhombus
tiles in the $SEH_{00}$ tiling we recently introduced [49], as demonstrated in
Appendix E. Therefore, we used this work as a guide to fill in the missing
pieces. Figure 11(c) shows each of the motifs we have used, in ascending order
of size, where we have rotated the final motif by $\frac{2\pi}{3}$ for easier
comparison. We note that, perhaps unsurprisingly, the latter two motifs
contain curves of rhombus tiles which replicate the first two generations of
Figure 11(a), indicated by the overlaid black lines. The ‘generation’ of these
connected curves within the tiling may indicate a kind of self-similar
substitution system, which is still an open question with regards to the
$SEH_{00}$ tiling. It may be interesting to explore whether a set of rules
exist which could produce the edges of tiles as their own fractal system, as
with the Penrose tiling [29].

Considering the link between the fractal and tiling system, an alternative
method to investigate arbitrary $j=3$, $m$ metallic ratio fractal systems
would be to generate a tiling first, then extract the linked rhombuses.
Appendix E shows several examples, where we use de Bruijn’s dual-grid method
[50, 51] to produce $m$-mean single edge-length tilings and extract some
$A,B,C$ length parameters. Recent work has considered that metallic mean
hexagonal tilings with multiple edge-length tiles can approach modulated
honeycomb crystals as $m\rightarrow\infty$ [36]. In this work, domains of
large hexagon tile honeycombs are bounded by parallelograms and small
hexagonal tiles; in our case, these same domains are bounded by fractals of
rhombuses.

####  j $>3$ tilings

Here we consider fractal tilings when $j>3$, where again we restrict ourselves
to a two-tile scheme. First, we consider further geometrical conditions when
constructing an initiator. Figure 12(a) shows decorated skeletons after 3
generations which have been initiated with an $A-A-$ sequence for
$m=1,j=3,4,5$, and $A=C=1,B=0$. For $j=3$, it is clear that for both systems,
holes can be filled by rhombuses or hexagons. However, for $j=4$, while some
gaps can be filled by squares (indicated by green lines), other gaps are
incongruent (red lines). A similar situation is found for $j=5$. This arises
from the fact that after (twice) substituting Eq. 28, we find more than four
successive angle changes:

$A-C-B+C+A+C+\underline{B-C-A-A-C-}B...$ (29)

which produces the areas circled in Figure 12(a). In other words, $A-A-$
sequences do not occur naturally in Eq. 21. While $5\times\frac{2\pi}{3}$
leaves a $\frac{2\pi}{3}$ ‘gap’ associated with $j=3$ geometry,
$5\times\frac{2\pi}{4}$ and $5\times\frac{2\pi}{5}$ do not, and the same can
be found for larger values of $j$. Changing the initiator to an $A-C-$
sequence as in Figure 12(b), however, prevents this: $A-C-$ sequences are
commonly found in Eq. 21.

Figure 12(c) shows a possible tiling for the $j=4$, $A-C-$ initiator, where we
take into consideration the previous observations of the $j=3$/$SEH_{00}$
tiling: at each generation we fill empty spaces with squares and rhombuses,
taking care to connect rhombuses so that they recreate previous fractal curves
– examples are highlighted in white in the final generation. In the same way
that the $j=3$ case is related to the $SEH_{00}$ tiling, it would be
interesting to explore whether the $j=4$ example has any relation to existing
tilings.

Finally, Figure 12(d) shows that we cannot continue this specific type of
tiling construction when $j=5$. From a geometric standpoint, our system
doesn’t allow a two-tile tiling where our second tile is a polygon with all
internal angles set to $\frac{2\pi}{5}$, and the same holds for any $j>4$.
However, the gaps can still be filled such that we can tile the plane, as
demonstrated, although we cannot replicate the closed-curve forms of connected
thin rhombuses. Whether the higher order $j$ skeletons can be decorated in a
more sophisticated manner – i.e. with more than one tile [47, 48, 46] – is an
open question to explore.

##  IV Conclusions

We have introduced a generalized metallic-mean fractal L-system which can be
tuned by a series of parameters: the metallic-mean $m$, the symmetry $j$, and
some length-scales ($A,B,C$). First, we thoroughly explored the limits of the
geometric parameters and calculated the Hausdorff dimension, $H^{d}$, of the
golden-mean fractal, as an example. We also defined and briefly explored its
boundary in terms of another L-system. Then, we generalised this system over
all metallic-means, by way of expanding the original L-system with additional
terms. By doing so, we were able to calculate $H^{d}$ for both fractal and
boundary for $k\rightarrow\infty$, showing that these systems approach a
straight line.

Finally, we showed that, for some conditions, we can use these fractals as a
basis for decoration, with the end goal to create fractal tilings. In our
restricted system, we showed the link between a fractal system and the
$SEH_{00}$ tilings, demonstrated a quick example for the $j=4$ case, and
showed the limitations of our decoration method as $j$ increases.

For future work, it may be interesting to compare the experimental and
theoretical states of these systems to the abundance of work on existing
fractals – particularly those that share symmetries or are similar in
morphology. For instance, the $j=3$ fractal is reminiscent of the Sierpiński
triangle [52] or the Koch curve [53]. Similarly, it might be fruitful to
explore whether the tuning of our various parameters can yield properties of
interest for a given symmetry or metallic mean.

##  V Acknowledgements

This work was supported by EPSRC grant EP/X011984/1.

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##  Appendix A Maximum number of successive +/– segments in F1

First, we note that Eq. 2 does not allow identical letters between + or $-$
segments, which means that an $A$ can only be preceded or succeeded by a $B$
or $C$, and so on. If we take $A\rightarrow
C-\underline{B+C^{\prime}+B^{\prime}}-C$, we see that the middle $C^{\prime}$
segment is neighboured by two $+$, and $B/B^{\prime}$ segments, as underlined.
This sequence of angular changes is extended to four after we substitute for
each of the segments:
$A-C-B\underline{+C^{\prime}+A^{\prime}+C^{\prime}+}B^{\prime}-C-A$. The
right-hand side of the $B$ substitution, $C-B\underline{+C^{\prime}}$
concatenates with the $+C^{\prime}+$ substitution $+A^{\prime}+$, and the
left-hand side of the $B^{\prime}$ substitution
$\underline{C^{\prime}+}B^{\prime}-C$, giving four successive $+$ segments,
and no more. Similar arguments can be made for other sections under
substitution.

##  Appendix B Fractals with arbitrary j, A, B, and C for m = 1

Figures A1(a-c) show a series of F1 curves for $j=3,4,5$ for arbitrary
lengths. Lengths are labelled on each curve. The first two curves in Figure
A1(a), and the second in (b) break the length conditions we discuss in the
main text.

Figure A1: (a-c) $j=3,4,5$ fractals with the length of their segments
labelled. In some cases, these lengths break the conditions we set in the main
text, resulting in self-intersection.

##  Appendix C Arbitrary initiators

Figure A2 shows initiators for arbitrary parameters. The sequences used to
generate the initiators are labelled inside the first generation, and the
parameters used for the curves can be found in the figure caption.

Figure A2: Arbitrary initiators, where the initial sequence used to create the
first generation polygons are labelled, and the parameters are: (a)
$m=2,j=4,A=1,B=1,C=1$. (b) $m=1,j=4,A=0,B=1,C=1$. (c) $m=1,j=5,A=1,B=6,C=1$.
(d) $m=3,j=6,A=1,B=1,C=1$.

##  Appendix D Skeleton decoration with arbitrary m, A, B, and C for j = 3

Figure A3 shows a few examples of different initiators and parameters which
can be decorated with rhombus tiles for $j=3$. Parameters can be found in the
figure caption.

Figure A3: (a) $A-A-$ initiator, $m=1,A=3,B=2,C=1$. (b) $A-C-$ initiator,
$m=3,A=3,B=2,C=1$. (c) $A-C-A-A-C-A$ initiator, $m=1,A=3,B=2,C=3$.

##  Appendix E Fractal and tiling relationships for j = 3 and m

Figure A4: (a-f) Single edge-length hexagonal tilings generated for $m=1,...6$
respectively. Overlaid in black are the fractal curves which connect rhombus
tiles.

Figures A4(a-f) show single edge-length tilings generated using de Bruijn’s
dual-grid method [50, 51] for $m=1,..6$. We have rotated the tilings by
$30^{\circ}$ for graphical convenience. We overlay the fractal forms in black
which are found when connecting rhombus tiles. Table 2 shows the length
parameters extracted from these forms after a sufficient number of generations
(typically 3).

$m$ | $A$ | $B$ | $C$
---|---|---|---
1 | 3 | 2 | 1
2 | 5 | 4 | 1
3 | 7 | 6 | 1
4 | 9 | 8 | 1
5 | 11 | 10 | 1
6 | 13 | 12 | 1
Table 2: Length scales of the fractal forms found in metallic-mean tilings.


# An Autoethnographic Exploration of XAI in Algorithmic Composition

Ashley Noel-Hirst<EMAIL_ADDRESS>0009-0001-7987-9905 and  Nick
Bryan-Kinns<EMAIL_ADDRESS>0000-0002-1382-2914 Queen Mary
University of LondonMile End Rd, Bethnal GreenLondonUKE1 4NS

(2023)

###### Abstract.

Machine Learning models are capable of generating complex music across a range
of genres from folk to classical music. However, current generative music AI
models are typically difficult to understand and control in meaningful ways.
Whilst research has started to explore how explainable AI (XAI) generative
models might be created for music, no generative XAI models have been studied
in music making practice. This paper introduces an autoethnographic study of
the use of the MeasureVAE generative music XAI model with interpretable latent
dimensions trained on Irish folk music. Findings suggest that the exploratory
nature of the music-making workflow foregrounds musical features of the
training dataset rather than features of the generative model itself. The
appropriation of an XAI model within an iterative workflow highlights the
potential of XAI models to form part of a richer and more complex workflow
than they were initially designed for.

ethnographic artificial intelligence, music generation, explainable artificial
intelligence

††copyright: acmcopyright††journalyear: 2023††doi:
XXXXXXX.XXXXXXX††conference: Make sure to enter the correct conference title
from your rights confirmation emai; June 19, 2023; Online††booktitle: The 1st
International Workshop on Explainable AI for the Arts (XAIxArts), ACM
Creativity and Cognition (C&C). 2023, Online††price: 15.00††isbn:
978-1-4503-XXXX-X/18/06††ccs: Human-centered computing Interactive systems and
tools††ccs: Human-centered computing Graphical user interfaces††ccs: Applied
computing Sound and music computing

ACM Reference Format:
Ashley Noel-Hirst and Nick Bryan-Kinns. 2023. An Autoethnographic Exploration
of XAI in Algorithmic Composition. In The 1st International Workshop on
Explainable AI for the Arts (XAIxArts), ACM Creativity and Cognition (C&C)
2023. Online, 3 pages. https://xaixarts.github.io

##  1\. Introduction

In recent years, computational approaches to music generation and creativity
support have prompted and adopted a range of Artificial Intelligence (AI)
techniques, especially Machine Learning. Composers who utilize novel musical
representations, timbres, and pitch and time divisions go so far as to create
new software for their works to be realized often using AI tools support this
process (Lewis, 1999). These approaches broadly fall into two paradigms: the
appropriation of materials from existing deep learning models (Knotts and
Collins, 2021), and the development of novel models e.g. (Fiebrink and
Caramiaux, 2018). However, regardless of the paradigm or application of AI for
music, the underlying processes are typically hard to understand or control in
meaningful ways (Bryan-Kinns et al., 2021). Whilst explainable AI (XAI)
(Vilone and Longo, 2021) and Human-Centred AI (HCAI) (Shneiderman, 2020) aim
to make AI models more understandable to humans, there have been no studies of
how XAI approaches might be used in artistic practice.

##  2\. Exploring an Explainable Variational Autoencoder

To explore the use of XAI in creative practice we report on an
autoethnographic exploration of a generative music XAI system – MeasureVAE
(Pati et al., 2019) implemented as a Max4Live plugin (Banar et al., 2023) \-
as an illustrative case study of how such XAI models might be appropriated in
music-making practice. The autoethographic exploration is reported in first
person by the first author whose music making practice involves composing
interactive and generative musical systems (Clarice Hilton et al., 2023) ,
which are then rendered as fixed electronic works.

In my practice, I utilize a range of rule-based composition systems with other
AI methods. These are typically focused on the evolution of rhythms over time,
and integrate a number of Max/MSP tools – such as Euclidean sequencers, which
generate complex rhythms from a few parameters. These rhythms are found in a
number of folk traditions, but interestingly they are not represented in the
Irish folk data that MeasureVAE was trained on. To explore this dissonance
between musical practice and AI training data I reflexively iterated through
several approaches to music making. Here I report on one approach informed by
my music making practice in which Euclidean rhythms are used to drive and
explore the generation of melodies with MeasureVAE – as illustrated by the
workflow in figrue 1.

Figure 1. Musical workflow for generating musical measures from Euclidean
Rhythms.

A map, of algorithmic procedures. Links include: Euclidean rhythm generator to
piano roll; User chord choice to piano roll; Monophonic processor to and from
the piano roll; MeasureVAE to and from the piano roll.

Euclidean rhythms are lists of pulses [x] and rests [.]. These lists,
expressed as E(i,j,k), describe i pulses dividing the j total beats as evenly
as possible, displaced to the right by k positions. For example, the phrase
[x..x.x..x.x..] is the Euclidean Rhythm expressed by E(5,13,0). Toussaint
(Toussaint, 2005) outlined how this algorithm can be used to generate, analyse
and permute a range of rhythmic patterns from across the world. It can also be
found in use in minimalist composition (Lawton Hall, 2020) as well as being
used by live coders and modular synth users (Roberts and Sisman, 2021).

In my own practice, I typically use a range of parameterised sequences in
parallel to drive the creation of new musical phrases. For this study I
created a number of Euclidean sequences: E(3,7,2) [. . X . X . X]; E(4,16,0)
[X . . .]; and E(2,5,2) [. . X . X]. I then applied each rhythm to a note from
a C minor chord, resulting in the poly-rhythm-based melody illustrated in
figure 2a. Since MeasureVAE needs monophonic input, monophonic melodies were
extracted using a lowest note policy – restraint brings tension to the
infrequent high notes. When applied to the pattern in figure 2a, this produces
the legato melody in figure 2b. This monophonic melody is then fed in to
MeasureVAE which generates an output illustrated in figure 2c. Permuting the
parameters of each Euclidean rhythm produces a number of similar melodies with
comparable divergence from the model. I then explored modulations of my
Euclidean system, composing ‘into’ the fixed MeasureVAE to produce new
melodies as illustrated in figure 2d. Through this process I found musical
artifacts in the MeasureVAE – encoded elements which reflected the underlying
training data (or lack thereof) more than the input itself.

Figure 2. a) Three Euclidean rhythms are played in parallel using notes from a
C minor chord; b) This is made monophonic, generating a melody; c) Its
encoding without any modification; d) Its encoding after the Euclidean
parameters were altered.

###  2.1. Feedback and Explainability

Through exploration, I found that employing MeasureVAE in tandem with other
computational composition systems extended MeasureVAE’s explainability beyond
its intended scope. First, we gain new avenues for explainability. MeasureVAE
provides a navigable heatmap of the latent space, highlighting areas of the
source distribution which are more or less dense in the original dataset. In
the denser areas, regularisation makes for useful ‘contrastive feature’
explanation and navigation. By feeding a series of permuting Euclidean
sequences into MeasureVAE, I was able to explore the under-defined contours of
the latent space through example. My parametrically-defined melodies would be
approximated within the distribution of the original dataset, and the
artifacts which arose during this process gave some idea of the extent to
which my melody was congruent with the distribution of the training set.
Conversely, this could also be thought of as an evaluation of the
appropriateness of the original dataset for my task, or a qualitative
understanding of the high dimensional topology of the latent space, going
beyond the 4 regularised dimensions originally explored (Bryan-Kinns et al.,
2021).

Second, there is some appropriation of the explainability of one system to
another. For example, Euclidean systems are difficult to navigate and
evaluative feedback from MeasureVAE in the music making process helped to
inform the tweaking of my Euclidean system. Specifically, the MeasureVAE
encoder gave me feedback about the extent to which my melody caused activation
in its regularised dimensions. With this, I was able to more fluidly modify my
Euclidean parameters so as to create melodies which were encoded as being less
dense or complex e.g. 2d.

##  3\. Conclusions

Explainable AI music systems promise increased control and transparency to the
computer music composer. As illustrated in this paper, the location of the
explainability in a musical workflow has implications for the types of
feedback it can give - in our case, exploring the features of the training
set, and appropriating explainability from an XAI to an algorithmic
composition process. Future work includes exploring how we can leverage such
re-contextualised feedback for algorithmic surprise, qualitative evaluation of
datasets, and cross-genre adaptation of pre-trained models.

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