# Jordan Algebras over icosahedral cut-and-project quasicrystals

Daniele Corradetti  Departamento de Matematica
Universidade do Algarve
Campus de Gambelas
Faro, PT<EMAIL_ADDRESS>,  David Chester  Quantum Gravity Research
Los Angeles, California
CA 90290, USA
<EMAIL_ADDRESS>,  Raymond Aschheim  Quantum Gravity
Research
Los Angeles, California
CA 90290, USA
<EMAIL_ADDRESS>and  Klee Irwin  Quantum Gravity Research
Los Angeles, California
CA 90290, USA
<EMAIL_ADDRESS>
###### Abstract.

In this paper present a general setting for aperiodic Jordan algebras arising
from icosahedral quasicrystals that are obtainable as model sets of a cut-and-
project scheme with a convex acceptance window. In these hypothesis, we show
the existence of an aperiodic Jordan algebra structure whose generators are in
one-to-one correspondence with elements of the quasicrystal. Moreover, if the
acceptance window enjoys a non-crystallographic symmetry arising from
$H_{2},H_{3}$ or $H_{4}$ then the resulting Jordan algebra enjoys the same
$H_{2},H_{3}$ or $H_{4}$ symmetry. Finally, we present as special cases some
examples of Jordan algebras over a Fibonacci-chain quasicrystal, a Penrose
tiling, and the Elser-Sloane quasicrystal.

MSC2020: 52C23, 17B65, 17C50

##  1\. introduction and motivation

Crystallographic Coxeter groups are an indispensable tool in classification of
simple Lie algebras and thus play a prominent role in mathematical physics.
The Chevalley-Serre theorem [Se, Kac] notoriously prevents the existence of
finite-dimensional Lie algebras arising from non-crystallographic groups such
as $H_{2},H_{3}$ and $H_{4}$. Therefore, even though numerous attempts have
been made to encounter algebraic structures (e.g. [PT02, PPT98, Tw99a, DBT12])
and integrable systems (e.g. [Shc, FrKo, Tw99a, Tw00]) based non-
crystallographic groups, those are far less used and known in their physical
and algebraic applications. Nevertheless, the interest on non-crystallographic
groups has incredibily grown since Setchman in 1982 found a metallic alloy
with icosahedral point group symmetry [ShBGC], that was later defined by
Levine and Steinhard [LeSt] as _quasicrystal_ , in the sense of a crystal with
quasi-periodic translational symmetry. Indeed, a very general mathematical
set-up for the modelization of such type of quasicrystals was introduced by
Moody and Patera in [MP93], making use of an interesting embedding of the non
crystallographic group $H_{4}$ into $E_{8}$ noticed by Elser and Sloane in
[ES] and then developed noticed by Shcherbak in [Shc]. In their work, Moody
and Patera showed that a large class of icosahedral quasicrystals are
obtainable as a cut and project scheme on the lattice of icosians or,
equivalently, over an $E_{8}$-lattice. Here, we will have their cut-and-
project scheme as a starting point, focusing on a special class of
quasicrystals arising from an acceptance window that must be a convex set. In
this framework, we show in a constructive way the existence of a Jordan
algebra structure whose generator are in one-to-one correspondence with
elements of the quasicrystal. These Jordan algebras thus fits into the class
of _aperiodic algebras_ that were first introduced by Patera et al. in [PPT98]
and later on studied by Twarock[Tw00] and Mazochurck [Ma02], even though their
analysis was limited to aperiodic Lie algebras and never extended to Jordan
algebras. Moreover, due to the $H_{4}$-symmetry of icosians, if the acceptance
window enjoys an $H_{1}\cong A_{1},H_{2},H_{3}$ or $H_{4}$ symmetry, then the
resulting Jordan algebra enjoys the same $H_{1}\cong A_{1},H_{2},H_{3}$ or
$H_{4}$ symmetry. In fact, this class of Jordan algebras are an algebraic
structure naturally endowed with these non-crystallographic symmetries.

The present work is organized as follows. In section 2 we review basic
definitions in order to fix notations and present crystallographic groups
$H_{2},H_{3}$ and $H_{4}.$ In section 3 we present a reformulation of the cut-
and-project scheme in [MP93, ES] for both an icosian lattice and an
$E_{8}$-lattice. Section 4 is entirely devoted to a binary operation, called
_quasiaddition_ , first introduced in [BM94] and which will be crucial in the
definition of the aperiodic Jordan algebra. Section 5 deals with aperiodic
algebras, reviewing some known aperiodic algebras from [PPT98, Tw00, Ma02],
then we present the definition of the aperiodic Jordan algebra
$\mathfrak{J}\left(\Xi\right)$ defined for every quasicrystal $\Xi$ from
convex cut-and-project schemes, and we investigate basic properties of such
class of algebras. Finally, to show that the selected class of quasicrystals
is not only broad, but also highly significant, in section 6 we present some
well-known examples of icosahedral quasicrystals, and thus of aperiodic Jordan
algebras, that fall in this category: the _Fibonacci-chain quasicrystal_ , a
_Penrose tiling_ , a $\mathbb{Z}_{6}$-_quasicrystal_ and the _Elser-Sloane
quasicrystal_.

In the whole paper symbol $\tau$ indicates the _golden mean_ , i.e.
$\tau=\frac{1}{2}\left(1+\sqrt{5}\right)\approx 1.618$ and while, to avoid any
ambiguity, the other root of the equation $x^{2}+x+1$ is referred as $1-\tau$
or $-\tau^{-1}$, i.e.
$1-\tau=\frac{1}{2}\left(1-\sqrt{5}\right)\approx-0.618$.

.

##  2\. icosians and non-crystallographic root systems

The cut-and-project scheme that we will use in later section is a
reformulation of that proposed by Moody and Patera in [MP93] and that makes
use of icosians developing Elser-Sloane construction in [ES]. The goal of this
section is to review all concepts needed to fully express such mathematical
set-up. _Icosians_ are a special set of quaternions that are vertices of a
convex regular 4-polytope $\left\\{3,3,5\right\\}$, i.e. a 600-cell, which
forms a group under quaternionic multiplication and, also, a non-
crystallographic root system of type $H_{4}$ with the Euclidean norm. In fact,
it can be easily shown [De21] that the group arising from icosians is the
binary icosahedral group $2I$, the order-120 dicyclic group, which is double
cover of the rotational symmetry group of the icosahedron. The
$H_{4}$-symmetry of the icosian group allows the Jordan algebras that we will
define in sec. 5 to be $H_{4}$-symmetric, and it is of paramount importance in
order to model icosahedral quasicrystals.

###  2.1. Euclidean spaces and root systems

Let $\mathbb{E}^{n}$ be an Euclidean space, i.e. a vector space over the real
numbers $\mathbb{R}$ endowed with a positive defined symmetric bilinear form
$\left\langle\cdot\mid\cdot\right\rangle$ called _inner product_. Then the
_lenght_ of a vector $v\in\mathbb{E}^{n}$ is given by
$\left\|v\right\|=\sqrt{\left\langle v\mid v\right\rangle},$ and the _angle_
$\theta\left(v,w\right)$ of two vectors $v,w\in\mathbb{E}^{n}$ is defined as

(2.1) $\cos\theta\left(v,w\right)=\frac{\left\langle v\mid
w\right\rangle}{\left\|v\right\|\left\|w\right\|}.$

A _reflection_ in the Euclidean space $\mathbb{E}^{n}$ is a linear
transformation $r_{\alpha}$ that fixes all vectors orthogonal to the vector
$\alpha$, i.e.

(2.2) $r_{\alpha}\left(v\right)=v-\frac{2\left\langle
v\mid\alpha\right\rangle}{\left\langle\alpha\mid\alpha\right\rangle}\alpha,$

for every $\alpha,v\in\mathbb{E}^{n}$ . Obviously, the composition of two
reflections is still a reflection. Moreover, the inverse of any reflection is
the reflection itself, i.e. $r_{\alpha}^{-1}=r_{\alpha}$, and, finally,
identity itself is a reflection once we set $\alpha=0$. As a consequence, the
reflections of an Euclidean space form a group. Even more, a straightforward
computation shows that $\left\langle r_{\alpha}\left(v\right)\mid
r_{\alpha}\left(w\right)\right\rangle=\left\langle v\mid w\right\rangle$ for
every $v,w\in\mathbb{E}^{n}$, so that every reflection $r_{\alpha}$ is an
isometry. A group generated by reflections (2.2) is called a _reflection
group._

Let $\mathbb{E}^{n}$ be an Euclidean space. Then a subset $\Phi$ of vectors in
$\mathbb{E}^{n}$ is called a _root system_ , if it contains the opposite of
every elements and it is invariant to every reflection generated by its
elements, i.e.

(2.3) $\pm\alpha\in\Phi\,\,\text{ and }\,r_{\alpha}\left(\Phi\right)=\Phi,$

for every $\alpha\in\Phi$. If the subset $\Phi$ is a root system, then all
vectors $\alpha$ belonging to $\Phi$ are called _roots_ and the dimension $n$
of the embedding space $n$ is called the _rank_ of the system. Finally, for
every root system $\Phi$, we call the _root lattice_ of $\Phi$ in
$\mathbb{E}^{n}$ as the $\mathbb{Z}$-span of $\Phi$, i.e.

(2.4)
$\Lambda\left(\Phi\right)=\left\\{\beta\in\mathbb{E}^{n}:\beta=n\alpha,\alpha\in\Phi,n\in\mathbb{Z}\right\\}.$

It is important to stress out that, by definition (2.3), root systems and
reflection groups are linked one another, so that any root system give rise to
the reflection group that leave its roots invariant and the converse is also
true. Finally, all root systems are notoriously classified in the following
families

  * •

four one-parameter families of type $A_{n},B_{n},C_{n},D_{n}$ of rank $n\geq
1$;

  * •

a one-parameter family of type $I_{2}\left(m\right)$ of rank $2$ with
$m\in\mathbb{N}$. Special cases
$I_{2}\left(3\right),I_{2}\left(4\right),I_{2}\left(5\right)$ and
$I_{2}\left(6\right)$ are historically labeled as $A_{2},B_{2},H_{2}$ and
$G_{2}$ respectively;

  * •

and six exceptional systems of type $E_{6},E_{7},E_{8},F_{4},H_{3}$ and
$H_{4}$.

A special class of root systems, called crystallographic systems, found a
major role in Lie theory. Indeed, let us consider the quantities generated by
$\left\langle\beta\mid\alpha\right\rangle$ and
$\left\langle\alpha\mid\alpha\right\rangle$ where $\alpha,\beta$ are roots of
the same system $\Phi$. If the ratio between the two roots is always an
integer, i.e.

(2.5)
$\frac{\left\langle\beta\mid\alpha\right\rangle}{\left\langle\alpha\mid\alpha\right\rangle}\in\mathbb{Z},$

for every $\alpha,\beta\in\Phi$, then the root system is called
_crystallographic_ , while is called _non crystallographic_ otherwise. As
famously proved by Serre [Se], any crystallographic system allows the
definition of a finite dimensional Lie algebra with Cartan decomposition
related to the root system. On the other hand, non-crystallographic root
system do not satisfy Serre relations and thus do not give rise to finite
dimensional Lie algebra [Kac].

###  2.2. non-crystallographic root systems

A concrete example of a type-$H_{4}$ root system is
$\triangle_{4}\subset\mathbb{R}^{4}$ given by the following 120 vertices of a
600-cell $C_{600}$, i.e.

(2.6) $\triangle_{4}=\left\\{\begin{array}[]{cc}\left(\pm
1,0,0,0\right),\frac{1}{2}\left(\pm 1,\pm 1,\pm 1,\pm 1\right)&\text{\,\,and
all permutation}\\\ \frac{1}{2}\left(0,\pm
1,\pm\frac{1}{\tau},\pm\tau\right)&\text{\,\,and all even
permutation}\end{array}\right\\}.$

A subsystem of $\triangle_{4}\subset\mathbb{R}^{4}$ is given by
$\triangle_{3}\subset\mathbb{R}^{3}$, which is a root system of type $H_{3}$,
defined by the following 30 vertices of an icosahedron, i.e.

(2.7) $\triangle_{3}=\left\\{\begin{array}[]{cc}\left(\pm
1,0,0\right)&\text{\,\,and all permutation}\\\ \frac{1}{2}\left(\pm
1,\pm\frac{1}{\tau},\pm\tau\right)&\text{\,\,and all even
permutation}\end{array}\right\\}.$

Finally, $\triangle_{3}\subset\mathbb{R}^{3}$ contains a type-$H_{2}$ root
system $\triangle_{2}\subset\mathbb{R}^{2}$ given by the 10 vertices of a
decagon, i.e.

(2.8) $\triangle_{2}=\left\\{\begin{array}[]{cc}\left(\pm 1,0\right)\\\
\frac{1}{2}\left(\pm 1,\pm\tau\right)&\text{\,\,and all
permutation}\end{array}\right\\},$

so that we have the following chain of root systems
$\triangle_{2}\subset\triangle_{3}\subset\triangle_{4}$ that are of type
$H_{2},H_{3}$ and $H_{4}$ respectively.

###  2.3. non-crystallographic coxeter groups

In order to study reflection groups and root systems in an efficient way, it
is a common practice to define Coxeter groups which are abstract presentations
of reflection groups. Let a _Coxeter matrix_ $M$ be a symmetric matrix of
finite dimension $n$ with integer off diagonal entries $\geq 2$ and diagonal
entries equal to $1$, i.e. $m_{ij}\in\mathbb{N}$,
$m_{ij}=m_{ji},\,\,$$m_{ii}=1$ and $m_{ij}\geq 2$ if $i\neq j$. Then a
_Coxeter group_ is an abstract group with presentation

(2.9) $\left\langle
R_{1},...,R_{n}\mid\left(R_{i}R_{j}\right)^{m_{ij}}=1\right\rangle.$

As Coxeter himself proved in [Co34], every reflection group is a Coxeter group
and, even more, every finite Coxeter group admits a faithful representation as
a finite reflection group of some Euclidean space. In case of the non-
crystallographic root systems of type $H_{2},H_{3}$ and $H_{4}$, we have a
presentation of the respective Coxeter groups given by $\left\langle
R_{1},...,R_{n}\mid\left(R_{i}R_{j}\right)^{m_{ij}}=1\right\rangle$, with $n$
equal to $2,3$ and $4$ respectively, and with coefficients $m_{ij}$ of the
Coxeter matrices given by the following

(2.10) $\left(H_{2}\right)_{ij}=\left(\begin{array}[]{cc}1&5\\\
5&1\end{array}\right),\left(H_{3}\right)_{ij}=\left(\begin{array}[]{ccc}1&3&2\\\
3&1&5\\\
2&5&1\end{array}\right),\left(H_{4}\right)_{ij}=\left(\begin{array}[]{cccc}1&3&2&2\\\
3&1&3&2\\\ 2&3&1&5\\\ 2&2&5&1\end{array}\right).$

To give some concrete realisations of such groups, the Coxeter group $H_{2}$
of order 10 can be realised as the symmetry group of a decagon, i.e. the
reflection group that leaves invariant the roots of the system
$\triangle_{2}$; the group $H_{3}$ of order 120 can be realised as the full
symmetry group of an icosahedron, i.e. the reflection group that leaves
invariant the roots of the system $\triangle_{3}$; finally, the group $H_{4}$
of order 14400 can be realised as the symmetry group of a 600-cell, i.e. the
reflection group that leaves invariant the roots of the system
$\triangle_{4}$.

###  2.4. the icosian ring

Root of $\triangle_{4}$ | Icosian in $\mathbb{H}$ | Permutation in $A_{5}$ | Isometry of $\mathbb{R}^{3}$
---|---|---|---
$\pm\left(0,1,0,0\right)$ | $\pm\text{i}$ | $\left(2,3\right)\left(4,5\right)$ | $\frac{1}{2}\left(\begin{array}[]{ccc}-1&\frac{1}{\tau}&-\tau\\\ \frac{1}{\tau}&-\tau&-1\\\ -\tau&-1&\frac{1}{\tau}\end{array}\right)$
$\pm\left(0,0,1,0\right)$ | $\pm\text{j}$ | $\left(2,4\right)\left(5,3\right)$ | $\frac{1}{2}\left(\begin{array}[]{ccc}-\tau&1&-\frac{1}{\tau}\\\ 1&-\frac{1}{\tau}&\tau\\\ -\frac{1}{\tau}&\tau&-1\end{array}\right)$
$\pm\left(0,0,0,1\right)$ | $\pm\text{k}$ | $\left(2,5\right)\left(3,4\right)$ | $\frac{1}{2}\left(\begin{array}[]{ccc}-\frac{1}{\tau}&-\tau&1\\\ -\tau&-1&-\frac{1}{\tau}\\\ 1&-\frac{1}{\tau}&-\tau\end{array}\right)$
$\pm\frac{1}{2}\left(-1,1,1,1\right)$ | $\pm\frac{1}{2}\left(1+\text{i}+\text{j}+\text{k}\right)$ | $\left(3,4,5\right)$ | $\frac{1}{2}\left(\begin{array}[]{ccc}-1&\frac{1}{\tau}&\tau\\\ -\frac{1}{\tau}&\tau&-1\\\ -\tau&-1&-\frac{1}{\tau}\end{array}\right)$
$\pm\frac{1}{2}\left(0,1,-\frac{1}{\tau},\tau\right)$ | $\pm\frac{1}{2}\left(\text{i}-\frac{1}{\tau}\text{j}+\tau\text{k}\right)$ | $\left(1,3\right)\left(4,5\right)$ | $\frac{1}{2}\left(\begin{array}[]{ccc}-1&-\frac{1}{\tau}&\tau\\\ -\frac{1}{\tau}&-\tau&-1\\\ \tau&-1&\frac{1}{\tau}\end{array}\right)$
Table 1. Correspondence between root of the system $\triangle_{4}$ of type
$H_{4}$, element of the icosian group $\mathscr{I}$, elements of the
alternating group $A_{5}$ and the icosahedral symmetries in $\mathbb{R}^{3}$
of which the icosian group $\mathscr{I}$ is a double cover.

In section 2.2 we showed concrete examples of root systems of type of type
$H_{2},H_{3}$ and $H_{4}$. Here, we show a notable quaternionic realisation of
the root system $\triangle_{4}$ of type $H_{4}$, and thus of its subsystems
$\triangle_{3}$ and $\triangle_{2}$ of type $H_{3}$ and $H_{2}$ respectively.
With this in mind we identify $\mathbb{R}^{4}$ with quaternions $\mathbb{H}$
through the canonical identification $\imath$ from $\mathbb{R}^{4}$ to
$\mathbb{H}$ given by

(2.11) $\imath\left(a,b,c,d\right)\longrightarrow
a+b\text{i}+c\text{j}+d\text{k},$

and define the subset of _icosians_ $\mathscr{I}\subset\mathbb{H}$ as the
quaternions that are image of $\imath\left(\triangle_{4}\right)$, where
elements of $\triangle_{4}\subset\mathbb{R}^{4}$ are explicitly defined in
(2.6). Obviously $\mathscr{I}$ is root system of type $H_{4}$. Notice that the
quaternionic units $1,\text{i},\text{j},\text{k}$ belong to
$\mathscr{I}\subset\mathbb{H}$ since they are image of even permutations of
$\left(1,0,0,0\right)$ that belong to $\triangle_{4}$.

Moreover, it is easy to see that the 120 elements of $\mathscr{I}$ form a
group under quaternionic multiplication. This group is known as the _icosian
group_ [CoSl] and is isomorphic to the binary icosahedral group. The reason
behind the name it is due to Hamilton [Ha] and lies in its relationship with
the icosahedral rotational group $A_{5}$ of order 60 of which $\mathscr{I}$ is
the double cover. Indeed, the homomorphism from $\mathscr{I\subset\mathbb{H}}$
to $A_{5}$ given by

(2.17)
$\displaystyle\begin{array}[]{ccc}\left(0,1,0,0\right)&\longrightarrow&\left(2,3\right)\left(4,5\right),\\\
\left(0,0,1,0\right)&\longrightarrow&\left(2,4\right)\left(5,3\right),\\\
\left(0,0,0,1\right)&\longrightarrow&\left(2,5\right)\left(3,4\right),\\\
\frac{1}{2}\left(-1,1,1,1\right)&\longrightarrow&\left(3,4,5\right),\\\
\frac{1}{2}\left(0,1,-\frac{1}{\tau},\tau\right)&\longrightarrow&\left(1,3\right)\left(4,5\right),\end{array}$

it is well defined and has kernel $\left\\{1,-1\right\\}$. Table 1 shows the
correspondence between roots of $\triangle_{4}$, elements of the icosians
group $\mathscr{I}$, elements of the alternating group $A_{5}$ and, elements
of the icosahedral group as isometries in $\mathbb{R}^{3}$.

From the icosian group, we define the _icosian ring_ $\mathbb{I}$ as the
$\mathbb{Z}-$span, i.e. all finite sums of the icosian group $\mathscr{I}$.
Obviously, the icosian ring $\mathbb{I}$ is a subset of the algebra of
quaternions $\mathbb{H}$, and from such algebra inherits a conjugation and a
norm, that we will call _quaternionic norm_ $n$, given by

(2.18) $n\left(x\right)=\left\langle x,x\right\rangle=x\overline{x},$

for every $x\in\mathbb{I}$. It is worth noting that elements of the icosians
group $\mathscr{I}$ are icosians with unitary norm, i.e.

(2.19) $\mathscr{I}=\left\\{x\in\mathbb{I}:n\left(x\right)=1\right\\},$

so that icosians $\mathbb{I}$ of unitary quaternionic norm form a root system
of type $H_{4}$. In a next section we will see that this is not the only norm
we set over the ring of icosians and that ring of icosians $\mathbb{I}$
endowed with the _Euclidean norm_ $n_{\mathbb{E}}$ that we will later define
is isomorphic to an $E_{8}$-lattice [CoSl].

Now, consider the map $*$ on the Dirichlet ring $\mathbb{Z}\left[\tau\right]$
that sends $\tau$ in $1-\tau$, i.e. defined by

(2.20) $\left(a+\tau b\right)^{*}=\left(a+b\right)-\tau b,$

with $a,b\in\mathbb{Z}$. Clearly, the map $*$ can be extended to the whole
icosian ring $\mathbb{I}$ and it will be of paramount importance in defining
icosahedral quasicrystals in the next section.

Finally, an useful way of working with icosians is through their
$\mathbb{Z}^{8}$ decomposition, which is the following

(2.21) $x=\left(a+\tau b\right)+\left(c+\tau d\right)\text{i}+\left(e+\tau
f\right)\text{j}+\left(g+\tau h\right)\text{k},$

where $a,b,c,d,e,f,g\in\mathbb{Z}$. Thus, from now on we will use ambivalently
what we will call _integral coordinates_ of a generic icosian
$x\in\mathbb{I}$, i.e. $x=\left(a,b,c,d,e,f,g,h\right)$, or the quaternionic
notation depending on the convenience. Moreover, it is straightforward to note
that

(2.22) $x^{*}=\left(a+b,-b,c+d,-d,e+f,-f,g+h,-h\right),$

which gives us an explicit formula for the star map in integral coordinates.

##  3\. icosahedral quasicrystals as model sets

It is well-known that a general mathematical setup for cut-and-project
quasicrystals is given by _model sets_ that arise from cut-and-project schemes
[Mo00]. More specifically, a broad class of icosahedral quasicrystals, which
is the one of our interest, is linked to cut-and-project schemes over an
icosian lattice (e.g. see [MP93, Mo00]). In this section we present the notion
of _convex cut-and-project scheme_ , introducing a scheme inspired by [Mo00,
sec. 4.1] and that make use of an icosian lattice. Moreover, we present an
equivalent scheme that make use of an $E_{8}$-lattice. Since for the
definition of the aperiodic Jordan algebra the acceptance window of the scheme
must be convex, we will restrict ourself to this condition. It is worth noting
that the setting of the cut-and-project scheme through the use of the “icosian
lattice” or its equivalent $E_{8}$-lattice is endowed with an $H_{4}$ symmetry
that will be useful for Theorem 5.9 on the non-crystallographic symmetries of
the Jordan algebra.

###  3.1. model sets and cut-and-project schemes

A _cut-and-project scheme_ is a lattice $\Lambda$ in an _embedding space_
$\mathbb{R}^{d_{1}}\times G$, where $\mathbb{R}^{d_{1}}$ is a real euclidean
space called the _physical space_ and, $G$ is a locally compact abelian group
that, for our purposes, will simply be $\mathbb{R}^{d_{2}}$ and that is called
the _inner space_. Along with the previous embedding, a cut-and-project scheme
is endowed with two projections $\pi_{1}$ and $\pi_{2}$ that have image into
$\mathbb{R}^{d_{1}}$ and $\mathbb{R}^{d_{2}}$ respectively. In brief, a cut-
and-project scheme is given by the following collection of spaces and mappings

(3.1) $\begin{array}[]{ccccc}\Xi&&\Lambda&&\Omega\\\ \cap&&\cap&&\cap\\\
\mathbb{R}^{d_{1}}&\overset{\pi_{1}}{\longleftarrow}&\mathbb{R}^{d_{1}}\times\mathbb{R}^{d_{2}}&\overset{\pi_{2}}{\longrightarrow}&\mathbb{R}^{d_{2}}\end{array}$

such that the restriction $\pi_{1}\mid_{\Lambda}$ is injective in
$\mathbb{R}^{d}$ and the image $\pi_{2}\left(\Lambda\right)$ is dense in
$\mathbb{R}^{d_{2}}$. Given a cut-and-project scheme and a set $\Omega$ called
_acceptance window_ , such that the closure of the interior
$\overline{\text{int}\left(\Omega\right)}$ is compact, then a _model set_
$\Xi$ is defined as the points in physical space that are image of points that
once are projected in the inner space belong to the window $\Omega$, i.e.

(3.2)
$\Xi=\left\\{\pi_{1}\left(x\right)\in\mathbb{R}^{d_{1}}:x\in\Lambda,\pi_{2}\left(x\right)\in\Omega\right\\}.$

In practice, it is often useful to define the model set $\Xi$ through a _star
map_ $*$ that goes directly from $\mathbb{R}^{d_{1}}$ to $\mathbb{R}^{d_{2}}$,
i.e.

(3.3) $x^{*}=\pi_{2}\left(\pi_{1}\mid_{\Lambda}^{-1}\left(x\right)\right),$

so that the model set is then given by points in the physical space that are
sent by the star map into the acceptance window, i.e.
$\Xi=\left\\{x\in\pi_{1}\left(\Lambda\right):x^{*}\in\Omega\right\\}$. In
accordance with a common practice, we will call the model set $\Xi$ a _cut-
and-project quasicrystal_ ; moreover, we will add the adjective _convex_ if
the window $\Omega$ is a convex subset of $\mathbb{R}^{d_{2}}$.

###  3.2. icosahedral quasicrystals from icosians

Consider the cut-and-project scheme obtained embedding a lattice
$\mathbb{\widetilde{I}}$ made by two copies of the ring of icosians
$\mathbb{I}$, i.e. $\mathbb{\widetilde{I}}\cong\mathbb{I}\times\mathbb{I}$, in
the embedding space
$\mathbb{H}\times\mathbb{H}\cong\mathbb{R}^{4}\times\mathbb{R}^{4}$ through
the map $x\mapsto\left(x,x^{*}\right)$ where $*$ is the star map defined in
(2.20). We then have the following cut-and-project scheme

(3.4) $\begin{array}[]{ccccc}\Xi&&\mathbb{\widetilde{I}}&&\Omega\\\
\cap&&\cap&&\cap\\\
\mathbb{\mathbb{H}}&\overset{\pi_{\parallel}}{\longleftarrow}&\mathbb{H}\times\mathbb{H}&\overset{\pi_{\perp}}{\longrightarrow}&\mathbb{\mathbb{H}}\end{array}$

where $\pi_{\parallel}$ and $\pi_{\perp}$ are the trivial projections and
$\Omega$ is the acceptance window which, in this case, is a subset whose
closure is compact. The model set $\Xi$ is then given by

(3.5) $\Xi=\left\\{x\in\mathbb{I}:x^{*}\in P\right\\}.$

This cut-and-project scheme essentially encompasses the generic situation for
icosahedral symmetry in model sets. In fact, almost all known icosahedral
quasicrystals, obtainable from cut-and-project methods, arise from this
mathematical set up [Mo00, sec. 4.1]. Nevertheless, in order to build an
aperiodic Jordan algebra over the model set, we will restrict ourself to the
special cases where $\Omega$ is a convex set.

###  3.3. icosahedral quasicrystals from an $E_{8}$ lattice

The ring of icosians enjoys a deep relation with the $E_{8}$ lattice, which
can be made explicit with the aid of a different norm over icosians (see
[CoSl, MP93]). Indeed, in a previous section we have seen that icosian ring
$\mathbb{I}$ inherits, from its natural embedding in $\mathbb{H}$, a
_quaternionic norm_ $n$. We now notice that the norm of a generic element
$x\in\mathbb{I}$, i.e. $n\left(x\right)$, is a real number of the form
$p+q\sqrt{5}$ with $p,q$ being rational numbers. Thus, we can define a new
_Euclidean norm_ $n_{\mathbb{E}}$, given by

(3.6) $n_{\mathbb{E}}\left(x\right)=p+q.$

To be more explicit, in integral coordinates the quaternionic norm
$n\left(x\right)$ of an icosian
$x=\left(a_{0},b_{0},a_{1},b_{1},a_{2},b_{2},a_{3},b_{3}\right)$ is

(3.7) $\displaystyle
n\left(x\right)=\stackrel{{\scriptstyle[}}{{k}}=1]{3}{\sum}\left(a_{k}+\sqrt{5}b_{k}\right)^{2}\in\mathbb{Z}\left[\tau\right],$

while the Euclidean norm $n_{\mathbb{E}}\left(x\right)$ is

(3.8)
$n_{\mathbb{E}}\left(x\right)=\stackrel{{\scriptstyle[}}{{k}}=1]{3}{\sum}a_{k}^{2}+5b_{k}^{2}+2a_{k}b_{k}\in\mathbb{Z}.$

It can be shown that the icosian ring, with the Euclidean norm, is isomorphic
to an $E_{8}$ lattice $\Lambda\left(E_{8}\right)$ (see [CoSl, Ch. 8]). As a
consequence of this isomorphism, icosahedral quasicrystals, that are
obtainable as model set of a cut-and-project scheme over icosians, can also be
modeled through a cut-and-project scheme over an $E_{8}$ lattice.

Indeed, following [Pa95] let $a_{1},a_{2},a_{3},a_{4}\in\mathbb{R}^{4}$ be

(3.9)
$\begin{array}[]{cccc}a_{1}=&\frac{1}{2}\left(\frac{1}{\tau},-\tau,0,-1\right),&a_{2}=&\frac{1}{2}\left(0,\frac{1}{\tau},-\tau,1\right),\\\
a_{3}=&\frac{1}{2}\left(0,1,\frac{1}{\tau},-\tau\right),&a_{4}=&\frac{1}{2}\left(0,-1,\frac{1}{\tau},\tau\right).\end{array}$

A cut-and-project scheme equivalent to (3.4) is obtained embedding the
$E_{8}$-lattice $\Lambda\left(E_{8}\right)$ in
$\mathbb{R}^{8}\cong\mathbb{R}^{4}\times\mathbb{R}^{4}$, with the following
maps, i.e.

(3.10) $\begin{array}[]{ccccc}\Xi&&\Lambda\left(E_{8}\right)&&\Omega\\\
\cap&&\cap&&\cap\\\
\mathbb{R}^{4}&\overset{\pi_{\parallel}}{\longleftarrow}&\mathbb{R}^{4}\times\mathbb{R}^{4}&\overset{\pi_{\perp}}{\longrightarrow}&\mathbb{R}^{4}\end{array}$

where $\Omega$ is the acceptance window and the projections $\pi_{\parallel}$
and $\pi_{\perp}$ are defined as follows. Given an element
$X\in\Lambda\left(E_{8}\right)$, i.e.

(3.11) $X=c_{1}\alpha_{1}+...+c_{8}\alpha_{8},$

with $c_{1},...,c_{8}\in\mathbb{Z}$ and $\alpha_{1},...,\alpha_{8}$ simple
roots of $E_{8}$, we define

(3.14)
$\displaystyle\begin{array}[]{cc}\pi_{\parallel}\left(X\right)&=\left(c_{1}+\tau
c_{7}\right)a_{1}+\left(c_{2}+\tau c_{6}\right)a_{2}+\left(c_{3}+\tau
c_{5}\right)a_{3}+\left(c_{8}+\tau c_{4}\right)a_{4},\\\
\pi_{\perp}\left(X\right)&=\left(c_{1}-\frac{1}{\tau}c_{7}\right)a_{1}^{*}+\left(c_{2}-\frac{1}{\tau}c_{6}\right)a_{2}^{*}+\left(c_{3}-\frac{1}{\tau}c_{5}\right)a_{3}^{*}+\left(c_{8}-\frac{1}{\tau}c_{4}\right)a_{4}^{*},\end{array}$

where $a_{1},a_{2},a_{3},a_{4}$ are the elements defined in (3.9) and
$a_{1}^{*},a_{2}^{*},a_{3}^{*},a_{4}^{*}$ are their respective images through
the the star map in (2.20). Notice that through the canonical identification
between $\mathbb{R}^{4}$ and $\mathbb{H}$ in (2.11), we have that

(3.15)
$\begin{array}[]{cccc}\iota\left(a_{1}\right)=&\frac{1}{2}\left(\frac{1}{\tau}-\tau\text{i}-\text{k}\right),&\iota\left(a_{2}\right)=&\frac{1}{2}\left(\frac{1}{\tau}\text{i}-\tau\text{j}+\text{k}\right),\\\
\iota\left(a_{3}\right)=&\frac{1}{2}\left(\text{i}+\frac{1}{\tau}\text{j}-\tau\text{k}\right),&\iota\left(a_{4}\right)=&\frac{1}{2}\left(-\text{i}+\frac{1}{\tau}\text{j}+\tau\text{k}\right),\end{array}$

are all icosians, therefore the image of the $E_{8}$-lattice
$\Lambda\left(E_{8}\right)$ is the icosian ring $\mathbb{I}$, thus making the
model set $\Xi$ an aperiodic subset of the icosian ring.

##  4\. Quasiaddition

All convex cut-and-project quasicrystals enjoy a simmetry that it can be
easily expressed by a binary operation called _quasiaddition_ [BM94, Pa95,
CCAI] as contraction of “quasicrystal addition” and that can be defined for
the icosian ring $\mathbb{I}$, i.e.

(4.1) $x\vdash y=\tau^{2}x-\tau y,$

for every $x,y\in\mathbb{\mathbb{I}}$. Such binary operation is not
commutative, nor associative, and does not have a unit. Nevertheless is power-
associative and flexible [CCAI2]. The major properties of quasiaddition are
summarized in the following

######  Proposition 4.1.

Let $x,y\in\mathbb{\mathbb{\mathbb{I}}}$. Then

(4.2) $\displaystyle x$
$\displaystyle\vdash\left(x\vdash\left(x\vdash\left(\ldots\vdash
x\right)\right)\right)=x,$ (4.3) $\displaystyle x$
$\displaystyle\vdash\left(y\vdash x\right)=\left(x\vdash y\right)\vdash x,$
(4.4) $\displaystyle x$ $\displaystyle\vdash\left(y\vdash
y\right)=\left(x\vdash x\right)\vdash y=x\vdash y,$ (4.5) $\displaystyle x$
$\displaystyle\vdash\left(x\vdash y\right)=y\vdash x.$

###### Proof.

The first property is just the plain result of $\tau$ being solution of
$\tau^{2}-\tau-1=0$. Indeed,

$x\vdash x=\left(1+\tau\right)x-\tau x=x,$

and thus the (4.2) follows. As corollary, quasiaddition is _power-
associative_. _Flexibility_ is expressed by (4.3) which is a straightforward
calculation. Since is flexible and not associative, then obviously
$x\vdash\left(x\vdash y\right)\neq\left(x\vdash x\right)\vdash y$. But, since
$x\vdash\left(x\vdash y\right)=\left(1+\tau\right)y-\tau x,$ then, in a rather
interesting way, we have

(4.6) $\displaystyle x\vdash\left(x\vdash y\right)$ $\displaystyle=y\vdash x,$
(4.7) $\displaystyle\left(x\vdash x\right)\vdash y$ $\displaystyle=x\vdash y,$

from which (4.4) and (4.5) follow. ∎

An useful technical Lemma, whose straightforward proof we omit but can found
in [CCAI2], is the following

######  Lemma 4.2.

Let $x,y\in\mathbb{\mathbb{\mathbb{I}}}$. Then

(4.8) $\displaystyle\left(\left(\left(\left(y\vdash x\right)\vdash
x\right)\vdash x\right)\vdash\ldots\right)\vdash x$
$\displaystyle=\left(1+\tau\right)^{k}\left(y-x\right)+x,$

where $k$ is the number of times the element $x$ is quasiadded to $y$.

Another interesting property of quasiaddition, which is again the result of a
straightforward calculation, is called the _translational invariance_ [BM94],
i.e.

(4.9) $\left(x+u\right)\vdash\left(y+u\right)=\left(x\vdash y\right)+u,$

for every $x,y,u\in\mathbb{\mathbb{\mathbb{I}}}$. This property is useful when
working with model sets resulting from a cut-and-project scheme. Indeed, if
quasiaddition acts over quasicrystals as special kind of inflation of the
quasicrystal. Thus, the translational invariance in (4.9) ensures that the
inflation of a translated quasicrystal matches the translation of the already
inflated quasicrystal.

Finally, the most important feature of quasiaddition is expressed by the
following

######  Theorem 4.3.

If $x,y$ belong to a convex cut-and-project icosahedral quasicrystal $\Xi$
then also $x\vdash y$ belongs to it, i.e. $x,y\in\Xi$ implies

(4.10) $x\vdash y\in\Xi.$

###### Proof.

Let $x,y\in\Xi$ where $\Xi$ is the model set of a cut-and-project scheme as in
(3.4) with the acceptance window $\Omega$ that is a convex set, i.e.
$\left(1-t\right)x-ty\in\Omega$ for every $x,y\in\Omega$ and
$t\in\left[0,1\right]$. Then

$\displaystyle\left(x\vdash y\right)^{*}$
$\displaystyle=\left(1-\tau\right)^{2}x-\left(1-\tau\right)y$
$\displaystyle=\left(1-t\right)x+ty,$

where $t=1-\tau\approx 0.618...$ Since $\Omega$ is a convex, then
$\left(x\vdash y\right)^{*}\in\Omega$, thus $x\vdash y\in\Xi.$ ∎

Now, an useful Lemma is the one that shows how quasiaddition behaves on the
integral coordinates of two icosians as defined in (2.21).

######  Lemma 4.4.

Let $x,y\in\mathbb{\mathbb{\mathbb{I}}}$ with $x=\left(a,b,c,d,e,f,g,h\right)$
and
$y=\left(a^{\prime},b^{\prime},c^{\prime},d^{\prime},e^{\prime},f^{\prime},g^{\prime},h^{\prime}\right)$.
Then

(4.11) $x\vdash
y=\left(\begin{array}[]{c}\left(a-a^{\prime}\right)+\left(b-b^{\prime}\right)+a^{\prime}\\\
\left(a-a^{\prime}\right)+\left(b-b^{\prime}\right)+b\\\
\left(c-c^{\prime}\right)+\left(d-d^{\prime}\right)+c^{\prime}\\\
\left(c-c^{\prime}\right)+\left(d-d^{\prime}\right)+d\\\
\left(e-e^{\prime}\right)+\left(f-f^{\prime}\right)+e^{\prime}\\\
\left(e-e^{\prime}\right)+\left(f-f^{\prime}\right)+f\\\
\left(g-g^{\prime}\right)+\left(h-h^{\prime}\right)+g^{\prime}\\\
\left(g-g^{\prime}\right)+\left(h-h^{\prime}\right)+h\end{array}\right)^{t}.$

###### Proof.

The proof is a straightforward once we notice that for an Dirichlet number
$z=\left(a,b\right)=a+\tau b\in\mathbb{Z}\left[\tau\right]$, then $\tau z=\tau
a+\tau^{2}b$ and $\tau^{2}z=\left(1+\tau\right)\left(a+\tau b\right)$ so that
multiplication by $\tau$ and $\tau^{2}$ act on integral coordinates as

(4.12)
$\begin{array}[]{cc}\tau:\left(a,b\right)\longrightarrow&\left(b,a+b\right),\\\
\tau^{2}:\left(a,b\right)\longrightarrow&\left(a+b,a+2b\right).\end{array}$

In case of $x,y\in\mathbb{\mathbb{\mathbb{I}}}$, as in the hypothesis of the
Lemma, we have

(4.13)
$\begin{array}[]{cc}\tau^{2}:x\longrightarrow&\left(a+b,a+2b,c+d,c+2d,e+f,e+2f,g+h,g+2h\right),\\\
\tau:y\longrightarrow&\left(b^{\prime},a^{\prime}+b^{\prime},c^{\prime},c^{\prime}+d^{\prime},e^{\prime},e^{\prime}+f^{\prime},g^{\prime},g^{\prime}+h^{\prime}\right),\end{array}$

from which follows the Lemma. ∎

An immediate Corollary of the previous Lemma, which will be useful in the next
section is the following

######  Corollary 4.5.

Let $x,y\in\mathbb{\mathbb{\mathbb{I}}}$.Then $x\vdash y=x$ if and only if
$y=x$.

##  5\. Aperiodic Jordan Algebras from icosahedral quasicrystals

In this section we briefly review some important aperiodic algebras studied by
Patera and Twarock in [PPT98, Tw00, MT03]. Then we show how to define an
aperiodic Jordan algebra for convex cut-and-project icosahedral quasicrystals.
Even though specific quasicrystals give rise to different features of the
algebra, all Jordan algebras arising from this process enjoy some common
properties that we present here. Finally, to give concreteness to the
definition we analyze a three aperiodic Jordan algebra: one defined over the
palindromic Fibonacci-chain quasicrystal; one for the Penrose tiling in (6.6);
and, finally, one for the Elser-Sloane quasicrystal.

###  5.1. aperiodic algebras

Aperiodic algebras were introduced by Patera, Pelantova and Twarock in [PPT98]
with a specific class of aperiodic algebras named Quasicrystal Lie algebras
whose generators are in one to one correspondence with a one-dimensional
quasicrystal which was a modification of a Fibonacci-chain quasicrystal. Such
quasicrystal Lie algebras were inspired by the usual Witt algebra, but they
did not possess a Virasoro extension, which was an interesting feature that
the authors were aiming for. Later on Twarock defined an aperiodic Witt
algebra which indeed had a Virasoro extension [Tw00] and thus allowing some
interesting physical applications in the study of the breaking of Virasoro
symmetry and in the realisation of a class of integrable systems [Tw99a,
TW00b]. A similar algebra, related to the Penrose tiling we reviewed in (6.6),
was then studied by Twarock and Mazorchuk in [MT03]. Since in (3.4) we
introduced a cut-and-project scheme that allows the definition of both
algebras as a special cases we present them here in this general framework.

Let $\mathbb{I}$ be the ring of icosians and
$\left\\{L_{x}\right\\}_{x\in\mathbb{I}}$ be a basis for the infinite
dimensional algebra $\mathfrak{W}$ with bilinear product defined by

(5.1) $\left[L_{x},L_{y}\right]=\left(x-y\right)L_{x+y},$

for every $x,y\in\mathbb{I}$. This is a Witt-type algebra which is obviously
well defined, since $x,y\in\mathbb{I}$ imply $x+y\in\mathbb{I}$, and
antisymmetric. Moreover, since

(5.2)
$\left[L_{x},\left[L_{y},L_{z}\right]\right]=\left(z-y\right)\left(y+z-x\right)L_{x+y+z},$

then the Jacobi identity is fulfilled, i.e.

(5.3)
$\left[L_{x},\left[L_{y},L_{z}\right]\right]+\left[L_{y},\left[L_{z},L_{x}\right]\right]+\left[L_{z},\left[L_{x},L_{y}\right]\right]=0,$

and thus $\mathfrak{W}$ is an infinite dimensional Lie algebra.

Now, consider the cut-and-project scheme in (3.4). Since the quasicrystal
$\Xi$ is a subset of the ring of icosians $\mathbb{I}$, Patera et al.
considered an algebra generated by the subset
$\left\\{L_{x}\right\\}_{x\in\Xi}$ and looked for modification of (5.1) that
would still be a Lie algebra. Nevertheless, the first problem to overcome was
that, in general, quasicrystals are not closed under addition, so that
$x,y\in\Xi$ would not imply $x+y\in\Xi$. To avoid this problem they required
$L_{x}$ and $L_{y}$ to commute if $x+y$ did not belong to the quasicrystal,
i.e. $\left[L_{x},L_{y}\right]=0$ if $x+y\notin\Xi$. Considering the cut-and-
project scheme in (3.4), this is the same as requiring that the image of $x+y$
through the star map belongs to the acceptance window, i.e. redefining the
bilinear product in (5.1) as

(5.4)
$\left[L_{x},L_{y}\right]=\left(x-y\right)\chi_{\Omega}\left(x^{*}+y^{*}\right)L_{x+y},$

where $\chi_{\Omega}\left(x\right)$ is the characteristic function of the
acceptance window which is $1$ if $x\in\Omega$ and $0$ if $x\notin\Omega$.
Even though the product (5.4) is well defined and antisymmetric, the Jacobi
identity is not always satisfied. Indeed, considering

(5.5) $\displaystyle\left[L_{x},\left[L_{y},L_{z}\right]\right]$
$\displaystyle=\left(z-y\right)\left(y+z-x\right)\chi_{\Omega}\left(x^{*}+y^{*}+z^{*}\right)\chi_{\Omega}\left(y^{*}+z^{*}\right)L_{x+y+z},$
(5.6) $\displaystyle\left[L_{y},\left[L_{z},L_{x}\right]\right]$
$\displaystyle=\left(x-z\right)\left(z+x-y\right)\chi_{\Omega}\left(x^{*}+y^{*}+z^{*}\right)\chi_{\Omega}\left(z^{*}+x^{*}\right)L_{x+y+z},$
(5.7) $\displaystyle\left[L_{z},\left[L_{x},L_{y}\right]\right]$
$\displaystyle=\left(y-x\right)\left(x+y+z\right)\chi_{\Omega}\left(x^{*}+y^{*}+z^{*}\right)\chi_{\Omega}\left(x^{*}+y^{*}\right)L_{x+y+z},$

we see that the Jacobi identity is satisfied if and only if
$\chi_{\Omega}\left(x^{*}+y^{*}+z^{*}\right)=1$ and
$\chi_{\Omega}\left(x^{*}+y^{*}\right)=1$ imply
$\chi_{\Omega}\left(x^{*}+z^{*}\right)=1$ for all
$x^{*},y^{*},z^{*}\in\Omega$. This led to the notion of _acceptable window
$\Omega$ _that are those for which the Jacobi identity is satisfied.

Patera et al. in [Tw00, PPT98] showed that the window
$\Omega=\left[0,1\right]$ is an acceptable window, thus constructing an
aperiodic Witt algebra over the Fibonacci-chain quasicrystal. Mazorchuk and
Twarock in [MT03] showed that the window $\Omega$ defined as the convex
pentagon with vertices $\left\\{1,\xi,\xi^{2},\xi^{3},\xi^{4}\right\\}$ used
for the Penrose tiling in (6.6) is an acceptable window, but a general
statement on acceptability of such windows was never formulated.

An simple proposition that, neverthless is useful for our purposes, and that
can be proved in our general setting is the following

######  Proposition 5.1.

If the acceptance window $\Omega$ is convex in a region of all non-negative
coordinates, i.e.
$\left\\{a+b\text{i}+c\text{j}+d\text{k}\in\mathbb{H}:a,b,c,d\geq 0\right\\}$,
then the window is acceptable.

###### Proof.

We have to prove that for any given $x^{*},y^{*},z^{*}\in\Omega$, with
$\chi_{\Omega}\left(x^{*}+y^{*}+z^{*}\right)=1$ and
$\chi_{\Omega}\left(x^{*}+y^{*}\right)=1$, then we also have
$\chi_{\Omega}\left(x^{*}+z^{*}\right)=1$. The acceptance window $\Omega$ is
convex, and by hypothesis both $x^{*}+y^{*}+z^{*}$ and $z^{*}$ belong to
$\Omega$. Let $x^{*},y^{*}$ and $z^{*}$ have coordinates
$x^{*}=x_{0}^{*}+x_{1}^{*}\text{i}+x_{2}^{*}\text{j}+x_{3}^{*}\text{k}$,
$y^{*}=y_{0}^{*}+y_{1}^{*}\text{i}+y_{2}^{*}\text{j}+y_{3}^{*}\text{k}$ and
$z^{*}=z_{0}^{*}+z_{1}^{*}\text{i}+z_{2}^{*}\text{j}+z_{3}^{*}\text{k}$
respectively. Then by convexity of $\Omega$, we have that
$\left(1-t_{k}\right)\left(x_{k}^{*}+y_{k}^{*}+z_{k}^{*}\right)+tz_{k}^{*}\in\Omega$
for every $t_{k}\in\left[0,1\right]$ with $k=0,...,3$. Since all $x_{k}$ and
$y_{k}$ are positive, then

(5.8) $0<t_{k}=\frac{y_{k}^{*}}{\left(x_{k}^{*}+y_{k}^{*}\right)}<1,$

for $k=0,...,3.$ But, since setting
$t_{k}=y_{k}^{*}/\left(x_{k}^{*}+y_{k}^{*}\right)$, we have

(5.9)
$\left(1-\frac{y_{k}^{*}}{x_{k}^{*}+y_{k}^{*}}\right)\left(x_{k}^{*}+y_{k}^{*}+z_{k}^{*}\right)+\frac{y_{k}^{*}}{x_{k}^{*}+y_{k}^{*}}z_{k}^{*}=x_{k}^{*}+z_{k}^{*},$

then convexity implies $x^{*}+z^{*}\in\Omega$ and, thus,
$\chi_{\Omega}\left(x^{*}+z^{*}\right)=1$. ∎

A Corollary of Proposition 5.1 is the following

######  Corollary 5.2.

Given any icosahedral quasicrystal arising as model set of the convex cut-and-
project scheme in (3.4), it exists a translation of the acceptance window
$\Omega$ that turns it into an acceptable window $\Omega^{\prime}$ .

######  Remark 5.3.

Unfortunately, the previous Corollary does not state is possible to define an
aperiodic Witt algebra for every model set of the the cut-and-project scheme
in (3.4). Indeed, translations of the acceptance window can sometimes change
some crucial property of the quasicrystal, as it happens with the _palindromic
Fibonacci-chain quasicrystal_ which is a special Fibonacci chain quasicrystal
that enjoys a 180º symmetry at the origin, symmetry that is not preserve when
the acceptance window is translated in the inner space.

###  5.2. jordan algebras from quasicrystals

In the previous section we saw that is not always possible to define an
aperiodic Witt algebras over a model set $\Xi$, since the acceptability of the
window $\Omega$ is needed. On the other hand, the definition of an aperiodic
Jordan algebra is possible for every model set arising from the cut-and-
project scheme (3.4) with a convex window $\Omega$. Indeed, let $\Xi$ be the
convex cut-and-project quasicrystal arising from a convex window $\Omega$ and
let $\mathfrak{J}\left(\Xi\right)$ be the real vector space spanned by
$\left\\{L_{x}\right\\}_{x\in\Xi}$ equipped with the bilinear product given by

(5.10) $L_{x}\circ L_{y}=\frac{1}{2}\left(L_{x\vdash y}+L_{y\vdash x}\right),$

for every $x,y\in\Xi$. Since, by Theorem 4.3 we have that $x\vdash y\in\Xi$,
then the product is well defined. Moreover, we have the following

######  Theorem 5.4.

The algebra $\mathfrak{J}\left(\Xi\right)$ is a Jordan algebra.

###### Proof.

The product in (5.10) is bilinear by definition and is clearly commutative,
i.e.

(5.11) $L_{x}\circ L_{y}=L_{y}\circ L_{x}.$

Therefore, it is sufficient to show that the Jordan identity is valid. Indeed,
since (4.2) and the commutativity of the product, we have that

$\displaystyle\left(L_{x}\circ L_{y}\right)\circ\left(L_{x}\circ L_{x}\right)$
$\displaystyle=\frac{1}{2}\left(L_{x}\circ L_{y}\right)\circ\left(L_{x\vdash
x}+L_{x\vdash x}\right)$ $\displaystyle=\left(L_{x}\circ L_{y}\right)\circ
L_{x}$ $\displaystyle=L_{x}\circ\left(L_{y}\circ L_{x}\right)$
$\displaystyle=L_{x}\circ\left(L_{y}\circ\left(L_{x}\circ
L_{x}\right)\right),$

thus fulfilling the Jordan identity. ∎

It is well-known that Jordan algebras have an abundance of idempotent
elements. This algebra is no exception since (4.2) implies that all elements
of the basis used for its definition are idempotents, i.e.

(5.12) $\left(L_{x}\circ L_{x}\right)=L_{x},$

for every $x\in\Xi$.

Clearly there are specific features of the Jordan algebra
$\mathfrak{J}\left(\Xi\right)$ that depends on the choice of the quasycrystal
$\Xi$ from which is originated. Nevertheless there are few basic properties
that are common to all Jordan algebras arising from (5.10). Indeed, an
interesting feature is expressed by the the following

######  Lemma 5.5.

Let $x,x^{\prime},y,y^{\prime}\in\Xi$. If

(5.13) $L_{x}\circ
L_{y}=\frac{1}{2}\left(L_{x^{\prime}}+L_{y^{\prime}}\right),$

then $x+y=x^{\prime}+y^{\prime}$.

###### Proof.

Is a straightforward implication from Lemma 4.4, but it can be seen directly
since, by definition, $x^{\prime}=x\vdash y$ and $y^{\prime}=y\vdash x$. Thus
$x^{\prime}+y^{\prime}=\left(1+\tau\right)x-\tau y+\left(1+\tau\right)y-\tau
x=x+y$. ∎

From now on, we will suppose the Jordan algebra $\mathfrak{J}\left(\Xi\right)$
to be infinite dimensional, since the only case when this does not happen is
in the trivial case of the acceptance window $\Omega$ being a point, for which
the resulting algebra can hardly be considered “aperiodic” being a one-
dimensional algebra.

Beside being infinite dimensional, the algebras $\mathfrak{J}\left(\Xi\right)$
are not unital. This is a consequence of Corollary 4.5 for which $x\vdash y=x$
if and only if $y=x$, as shown by the following

######  Proposition 5.6.

The Jordan algebras $\mathfrak{J}\left(\Xi\right)$ are not unital.

###### Proof.

The proof is given by contraddiction. Let $I\in\mathfrak{J}\left(\Xi\right)$
be the identity, i.e. $I\circ X=X$ for every
$X\in\mathfrak{J}\left(\Xi\right)$ , and let $I=\sum\xi_{x}L_{x},$ with
$\xi_{x}\in\mathbb{R}$ and $x\in\Xi$, be a decomposition of $I$. Then, let us
consider the product $I\circ L_{0}=L_{0}$. For the bilinearity we have

(5.14) $I\circ L_{0}=\underset{x\in\Xi}{\sum}\xi_{x}L_{x}\circ L_{0}=L_{0},$

but, for Corollary 4.5 we have that $L_{x}\circ L_{0}=L_{0}$ if and only if
$x=0$, therefore the $L_{0}$ component $\xi_{0}$ is equal to $1$ and $I$ is of
the form $I=L_{0}+\underset{x\neq 0}{\sum}\xi_{x}L_{x}$. Repeating the same
argument for all $x\in\Xi$ we then have that all $\xi_{x}=1$ and $I$ must be
of the form

(5.15) $I=\underset{x\in\Xi}{\sum}L_{x}.$

But, let us consider again the product $I\circ L_{0}$ that we now know to be
of the form

(5.16) $I\circ L_{0}=L_{0}\circ L_{0}+\underset{x\neq 0}{\sum}L_{x}\circ
L_{0}=L_{0},$

this means that the remaining $\sum L_{x}\circ L_{0}$ must be equal to $0$.
Nevertheless, this is impossible since the terms $L_{x}\circ L_{0}$ give rise
only to terms of the form $\frac{1}{2}L_{x\vdash 0}$ and
$\frac{1}{2}L_{0\vdash x}$ thus with positive coefficients for every
$x\in\Xi$. ∎

######  Remark 5.7.

Even though the algebra $\mathfrak{J}\left(\Xi\right)$ is not unital, it can
be extended to an unital algebra with a one-dimensional extension. Indeed, let
$\mathfrak{J}\left(\Xi\right)_{\text{ext}}$ be the vector space
$\mathfrak{J}\left(\Xi\right)\oplus\mathbb{K}$ endowed with the bilinear
product defined by

(5.17) $\left(L_{x}+\alpha\right)*\left(L_{y}+\beta\right)=L_{x}\circ
L_{y}+\alpha v+\beta u+\alpha\beta,$

where $L_{x},L_{y}\in\mathfrak{J}\left(\Xi\right)$ and
$\alpha,\beta\in\mathbb{K}$. It is easy to show that this is a Jordan algebra
with $1\in\mathbb{K}$ as unit element and, that
$\mathfrak{J}\left(\Xi\right)_{\text{ext}}$ contains a copy of the original
algebra $\mathfrak{J}\left(\Xi\right)$. Nevertheless, we will not pursue the
study of such algebra and we hold to the the original aperiodic Jordan algebra
$\mathfrak{J}\left(\Xi\right)$ .

An interesting feature of these algebras is the lack of finite dimensional
sub-algebras that are not the one-dimensional ones generated by an element
$L_{x}$. Indeed, we have the following

######  Theorem 5.8.

The algebras $\mathfrak{J}\left(\Xi\right)$ do not have finite dimensional
subalgebras with dimension greater than 1.

###### Proof.

Let $\mathfrak{A}$ be a subalgebra in $\mathfrak{J}\left(\Xi\right)$. If
$\text{dim}\mathfrak{A}=1$ then $\mathfrak{A}$ is just the algebra generated
by $L_{x}$ with $x\in\Xi$. If $\text{dim}\mathfrak{A}>1$ then it exist
$x,y\in\Xi$ such that $L_{x}$ and $L_{y}$ belong to $\mathfrak{A}$. If $L_{x}$
and $L_{y}$ are in $\mathfrak{A}$, then also all elements of the form

(5.18) $L_{x\vdash y},L_{\left(x\vdash y\right)\vdash
y},\ldots,L_{\left(\left(x\vdash y\right)\vdash y\right)\cdots\vdash
y},\ldots$

belong to $\mathfrak{A}$. By Lemma 4.2 the series if infinite since

(5.19)
$\left(1+\tau\right)^{k_{1}}\left(x-y\right)+y=\left(1+\tau\right)^{k_{2}}\left(x-y\right)+y$

if and only if $k_{1}=k_{2}$, therefore implying that $\mathfrak{A}$ is
infinite dimensional and thus proving the theorem. ∎

As for the simplicity, or the non-simplicity, of the algebra
$\mathfrak{J}\left(\Xi\right)$, at the moment no model set $\Xi$ led to the
construction of of a simple aperiodic Jordan algebra
$\mathfrak{J}\left(\Xi\right)$ (see [CCAI, CCAI2]). In fact, this is no
surprise, since simple infinite-dimensional Jordan algebras are all classified
by Zel’manov in [Ze]. To have a simple aperiodic Jordan algebra of the type in
(5.10), would mean to have a quasicrystal $\Xi$ arising from a convex window
$\Omega$ that would give rise to an algebra $\mathfrak{J}\left(\Xi\right)$ of
the Clifford type or of the Hermitian type, since those are the only allowed
types from Zel’manov classification theorem. In both cases that would be
astonishing and if possible that would be the result of a very special _ad
hoc_ construction which would not be relevant for our practical purposes.

As a final note, even though the definition of an aperiodic Jordan Algebra
$\mathfrak{J}\left(\Xi\right)$ is always possible for a convex cut-and-project
icosahedral quasicrystal, the $H_{n}$-symmetry of the quasicrystal, and thus
of the Jordan algebra, is the result of the joint properties of the lattice
and the acceptance window $\Omega$. In fact we have the following

######  Theorem 5.9.

Let $\mathfrak{J}\left(\Xi\right)$ be an aperiodic Jordan algebra, where the
model set $\Xi$ the set-up defined in (3.4) with a convex window $\Omega$.
Then $\mathfrak{J}\left(\Xi\right)$ has an $H_{1}\cong A_{1},H_{2},H_{3}$ or
$H_{4}$ if and only if the acceptance window $\Omega$ enjoys an $H_{1}\cong
A_{1},H_{2},H_{3}$ or $H_{4}$ respectively.

###### Proof.

Consider a quasicrystal resulting from the set-up defined in (3.4) such that

(5.20) $\Xi=\left\\{x\in\mathbb{I}:x^{*}\in\Omega\right\\}.$

Now, let $\rho$ be a reflection of the Coxeter group of $H_{n}$ with
$n\in\left\\{1,2,3,4\right\\}$. Since the icosians $\mathbb{I}$ enjoy the
$H_{4}$ symmetry, then $\rho\left(\mathbb{I}\right)=\mathbb{I}$. Thus, if we
also have that the acceptance window is invariant through this reflection,
i.e. $\rho\left(\Omega\right)=\Omega$, then also the quasicrystal is
invariant, i.e. $\rho\left(\Xi\right)=\Xi$. Then, from the definition of a
reflection in (2.2) follows that

(5.21) $\rho\left(x\vdash
y\right)=\rho\left(x\right)\vdash\rho\left(y\right),$

for every $x,y\in\Xi$. The Jordan algebra $\mathfrak{J}\left(\Xi\right)$ it is
thus invariant under $\rho$ since canonically isomorphic to the algebra
spanned by
$\left\\{L_{\rho\left(x\right)}\right\\}_{\rho\left(x\right)\in\Xi}$ equipped
with the bilinear product given by

(5.22) $L_{\rho\left(x\right)}\circ
L_{\rho\left(y\right)}=\frac{1}{2}\left(L_{\rho\left(x\right)\vdash\rho\left(y\right)}+L_{\rho\left(y\right)\vdash\rho\left(x\right)}\right),$

for every $\rho\left(x\right),\rho\left(y\right)\in\Xi$. ∎

##  6\. A few examples

In the previous section we showed some common properties of aperiodic Jordan
Algebra $\mathfrak{J}\left(\Xi\right)$. In this section we want to give a hint
of what these algebras are. Thus, making use of cut-and-project schemes
defined in section 3 we define here four aperiodic Jordan algebras related to
four well-known quasicrystal: the palindromic _Fibonacci-chain_ quasicrystal;
a _Penrose tiling_ ; a $\mathbb{Z}^{6}$-quasicrystal; the _Elser-Sloane_
quasicrystal. It is worth noting that the Fibonacci-chain quasicrystal enjoy a
reflection symmetry, i.e. an $H_{1}\cong A_{1}$ symmetry; the Penrose tiling a
fivefold symmetry which is a subgroup of $H_{2}$; the specific
$\mathbb{Z}^{6}$-quasicrystal enjoys an $H_{3}$-symmetry; while the Elser-
Slone quasicrystal, and thus the Jordan algebra defined over it, enjoys an
$H_{4}$-symmetry.

###  6.1. Fibonacci-chain quasicrystal

Let us start from the cut-and-project scheme in (3.4) and set the acceptance
window $\Omega$ as $\left(0,1\right]$. Then, since the star map $*$ leaves
$1,\text{i},\text{j},\text{k}$ invariant, then the resulting model set

(6.1) $\Xi=\left\\{x\in\mathbb{I}:x^{*}\in\Omega\right\\},$

is equivalent to

(6.2)
$\Xi\cong\mathcal{F}=\left\\{x\in\mathbb{Z}\left[\tau\right]:x^{*}\in\left(0,1\right]\right\\},$

which is the definition of the Fibonacci-chain quasicrystal as model set (e.g.
see [PPT98, CCAI]). While the acceptance window $\Omega=\left(0,1\right]$
corresponds to the most known Fibonacci-chain quasicrystal, we are interested
here on its symmetric version, i.e. the palindromic Fibonacci-chain
quasicrystal $\mathcal{F}_{\text{pal}}$ which can be obtained setting
$\Omega=\left[-\frac{1}{2},\frac{1}{2}\right]$. A sample of element of the
palindromic Fibonacci-chain quasicrystal can be found in Tab. 2.

$n$ |  | $\cdots$ | -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | $\cdots$
---|---|---|---|---|---|---|---|---|---|---|---|---
$\mathcal{F}_{\text{pal}}\left(n\right)$ |  | $\cdots$ | $-2-4\tau$ | $-2-3\tau$ | $-1-2\tau$ | $-1-\tau$ | 0 | $1+\tau$ | $1+2\tau$ | $2+3\tau$ | $2+4\tau$ | $\cdots$
Table 2. Elements of the Fibonacci-chain Quasicrystal
$\mathcal{F}_{\text{pal}}\left(n\right)$.

Since both the icosians $\mathbb{I}$ and the acceptance window
$\Omega=\left[-\frac{1}{2},\frac{1}{2}\right]$ enjoy the 180º symmetry given
by $\rho\left(x\right)=-x$, then also the quasicrystal enjoys such symmetry,
i.e. $\rho\left(\Xi\right)=\Xi$, and more specifically, the palindromic
Fibonacci-chain quasicrystal is such that

(6.3)
$\mathcal{F}_{\text{pal}}\left(-n\right)=-\mathcal{F}_{\text{pal}}\left(n\right).$

The Jordan algebra $\mathfrak{J}\left(\mathcal{F}_{\text{pal}}\right)$, is
defined as the real vector space spanned by
$\left\\{L_{x}\right\\}_{x\in\mathcal{F}_{\text{pal}}}$ equipped with the
bilinear product given by

(6.4) $L_{x}\circ L_{y}=\frac{1}{2}\left(L_{x\vdash y}+L_{y\vdash x}\right).$

An equivalent definition of the Fibonacci-chain Jordan algebra
$\mathfrak{J}\left(\mathcal{F}_{\text{pal}}\right)$ can be obtained on a
vector space indexed by integral numbers, i.e. spanned by
$\left\\{L_{n}\right\\}_{n\in\mathbb{Z}}$, with bilinear product defined as

(6.5) $L_{n}\circ
L_{m}=\frac{1}{2}\left(L_{n^{\prime}-m^{\prime}+2n-m}+L_{m^{\prime}-n^{\prime}+2m-n}\right),$

where $n^{\prime},m^{\prime}\in\mathbb{Z}$ and
$n^{\prime}=\mathcal{F}_{\text{pal}}\left(n\right)-\tau n$ and
$m^{\prime}=\mathcal{F}_{\text{pal}}\left(m\right)-\tau m$.

$L_{n}\circ L_{m}$ |  | $L_{-2}$ | $L_{-1}$ | $L_{0}$ | $L_{1}$ | $L_{2}$
---|---|---|---|---|---|---
$L_{-4}$ |  | $\frac{1}{2}\left(L_{1}+L_{-7}\right)$ | $\frac{1}{2}\left(L_{-8}+L_{3}\right)$ | $\frac{1}{2}\left(L_{6}+L_{-10}\right)$ | $\frac{1}{2}\left(L_{9}+L_{-12}\right)$ | $\frac{1}{2}\left(L_{11}+L_{-13}\right)$
$L_{-3}$ |  | $\frac{1}{2}\left(L_{0}+L_{-5}\right)$ | $\frac{1}{2}\left(L_{-6}+L_{2}\right)$ | $\frac{1}{2}\left(L_{5}+L_{-8}\right)$ | $\frac{1}{2}\left(L_{8}+L_{-10}\right)$ | $\frac{1}{2}\left(L_{10}+L_{-11}\right)$
$L_{-2}$ |  | $L_{-2}$ | $\frac{1}{2}\left(L_{-3}+L_{0}\right)$ | $\frac{1}{2}\left(L_{3}+L_{-5}\right)$ | $\frac{1}{2}\left(L_{6}+L_{-7}\right)$ | $\frac{1}{2}\left(L_{8}+L_{-10}\right)$
$L_{-1}$ |  | $\frac{1}{2}\left(L_{-3}+L_{0}\right)$ | $L_{-1}$ | $\frac{1}{2}\left(L_{2}+L_{-3}\right)$ | $\frac{1}{2}\left(L_{5}+L_{-5}\right)$ | $\frac{1}{2}\left(L_{7}+L_{-6}\right)$
$L_{0}$ |  | $\frac{1}{2}\left(L_{-5}+L_{3}\right)$ | $\frac{1}{2}\left(L_{2}+L_{-3}\right)$ | $L_{0}$ | $\frac{1}{2}\left(L_{3}+L_{-2}\right)$ | $\frac{1}{2}\left(L_{5}+L_{-3}\right)$
$L_{1}$ |  | $\frac{1}{2}\left(L_{-7}+L_{6}\right)$ | $\frac{1}{2}\left(L_{5}+L_{-5}\right)$ | $\frac{1}{2}\left(L_{-2}+L_{3}\right)$ | $L_{1}$ | $\frac{1}{2}\left(L_{3}+L_{0}\right)$
$L_{2}$ |  | $\frac{1}{2}\left(L_{-8}+L_{8}\right)$ | $\frac{1}{2}\left(L_{-6}+L_{7}\right)$ | $\frac{1}{2}\left(L_{-3}+L_{5}\right)$ | $\frac{1}{2}\left(L_{0}+L_{3}\right)$ | $L_{2}$
$L_{3}$ |  | $\frac{1}{2}\left(L_{-10}+L_{11}\right)$ | $\frac{1}{2}\left(L_{-8}+L_{10}\right)$ | $\frac{1}{2}\left(L_{-5}+L_{8}\right)$ | $\frac{1}{2}\left(L_{-2}+L_{6}\right)$ | $\frac{1}{2}\left(L_{0}+L_{5}\right)$
$L_{4}$ |  | $\frac{1}{2}\left(L_{-11}+L_{13}\right)$ | $\frac{1}{2}\left(L_{-9}+L_{12}\right)$ | $\frac{1}{2}\left(L_{-6}+L_{10}\right)$ | $\frac{1}{2}\left(L_{-3}+L_{8}\right)$ | $\frac{1}{2}\left(L_{-1}+L_{7}\right)$
Table 3. Multiplication table for a sample of generators of the aperiodic
Jordan algebra $\mathfrak{J}\left(\mathcal{F}_{\text{pal}}\right)$.

We will call algebra the integral formulation of the Jordan algebra
$\mathfrak{J}\left(\mathcal{F}_{\text{pal}}\right)$. An analysis of Tab. 3,
along with few computations allows to see that
$\mathfrak{J}\left(\mathcal{F}_{\text{pal}}\right)$ is not simple since
$L_{0}\mathfrak{J}\left(\mathcal{F}_{\text{pal}}\right)=\left\\{L_{0}\circ
x:x\in\mathfrak{J}\left(\mathcal{F}_{\text{pal}}\right)\right\\}$ is a proper
ideal of the algebra. Similar results holds for all the examples presented
and, as already discussed before, we doubt the existence of a simple Jordan
algebra from this set-up.

###  6.2. A Penrose tiling

Penrose tilings are aperiodic tilings of the plane, introduced by Roger
Penrose in [Pen], making use of four different prototiles: two different
rhombi and two different tiles called kites and darts. In this context,
different cut-and-project schemes were proposed by Baake [BaHu, Ba98] and
recently refined [ShuMa], to give rise to different Penrose tilings. For our
purposes it is interesting the specific tiling presented in [MT03] that is a
special case of the general setting proposed by Baake. Indeed, the vertices of
such tilings are given by the aperiodic set

(6.6)
$\Xi=\left\\{z\in\mathbb{Z}\left[\tau\right]\triangle\subset\mathbb{C}:z^{*}\in\Omega\right\\},$

where the acceptance window $\Omega$ is the convex pentagon with vertices
$\left\\{1,\xi,\xi^{2},\xi^{3},\xi^{4}\right\\}$, with
$\xi=\text{exp}\left(2\pi\text{i}/5\right)$, and $\triangle$ is the root
system of type $H_{2}$ given by

(6.7) $\triangle=\left\\{\pm
1,\pm\xi^{2},\pm\left(1+\tau\xi^{2}\right),\pm\left(\tau+\xi^{2}\right),\pm\left(\tau+\tau\xi^{2}\right)\right\\}.$

The aperiodic set $\Xi$ is a model set of the following cut-and-project scheme

(6.8) $\begin{array}[]{ccccc}\Xi&&\Lambda&&\Omega\\\ \cap&&\cap&&\cap\\\
\mathbb{C}&\overset{\pi_{1}}{\longleftarrow}&\mathbb{C}\times\mathbb{C}&\overset{\pi_{2}}{\longrightarrow}&\mathbb{C}\end{array}$

where the lattice $\Lambda$ is given by
$\left(\mathbb{Z}+\text{i}\mathbb{Z}\right)\times\left(\mathbb{Z}+\text{i}\mathbb{Z}\right)$
and

(6.9) $\displaystyle\pi_{1}\left(x,y,z,w\right)$
$\displaystyle=x+y\tau+\left(z+w\tau\right)\xi^{2},$ (6.10)
$\displaystyle\pi_{2}\left(x,y,z,w\right)$
$\displaystyle=x+y\left(1-\tau\right)+\left(z+w\left(1-\tau\right)\right)\xi^{4},$

so that the star-map $*$ sends

(6.11) $\begin{cases}\begin{array}[]{cc}\xi^{*}\longrightarrow&\xi^{2},\\\
\tau^{*}\longrightarrow&1-\tau.\end{array}\end{cases}$

Such cut-and-project scheme allows the labeling of elements of the
quasicrystal $\Xi$ through the use of quadruples $\left(a,b,c,d\right)$ of
$\mathbb{Z}^{4}$, i.e.

(6.12) $\left(a,b,c,d\right)\longrightarrow
z=a+b\tau+\left(c+d\tau\right)\xi^{2},$

where $a,b,c,d\in\mathbb{Z}$ which are also called the _integer coordinates_
of $z$. A sample of the quasicrystal in its integer coordinates is given in
Tab. 4.

$\Xi$ | $z_{2}\left(n_{2}\right)$ | $-3$ | $-2$ | $-1$ | $0$ | $1$ | $2$
---|---|---|---|---|---|---|---
$z_{1}\left(n_{1}\right)$ |  | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$
$-5$ | $\ldots$ | $\left(-2,-2,-2-4\right)$ | $\left(-2,-2,-1,-2\right)$ | $\left(-2,-2,0,1\right)$ | $\left(-2,-2,0,0\right)$ | $\left(-2,-2,1,1\right)$ | $\left(-2,-2,2,3\right)$
$-4$ | $\ldots$ | $\left(-1,-3,-4,-6\right)$ | $\left(-1,-3,-3,-5\right)$ | $\left(-1,-3,-2,-3\right)$ | $\left(-1,-3,0,0\right)$ | $\left(-1,-3,1,2\right)$ | $\left(-1,-3,2,3\right)$
$-3$ | $\ldots$ | $\left(-1,-2,-1,-2\right)$ | $\left(-1,-2,-1,-1\right)$ | $\left(-1,-2,0,-1\right)$ | $\left(-1,-2,0,0\right)$ | $\left(-1,-2,0,1\right)$ | $\left(-1,-2,1,1\right)$
$-2$ | $\ldots$ | $\left(-1,-1,-1,-2\right)$ | $\left(-1,-1,-1,-1\right)$ | $\left(-1,-1,0,-1\right)$ | $\left(-1,-1,0,0\right)$ | $\left(-1,-1,0,1\right)$ | $\left(-1,-1,1,1\right)$
$-1$ | $\ldots$ | $\left(0,-1,-2,-2\right)$ | $\left(0,-1,-1,-2\right)$ | $\left(0,-1,-1,-1\right)$ | $\left(0,-1,0,0\right)$ | $\left(0,-1,0,1\right)$ | $\left(0,-1,1,1\right)$
$0$ | $\ldots$ | $\left(0,0,-1,-2\right)$ | $\left(0,0,-1,-1\right)$ | $\left(0,0,0,-1\right)$ | $0$ | $\left(0,0,0,1\right)$ | $\left(0,0,1,0\right)$
$1$ | $\ldots$ | $\left(0,1,-1,-2\right)$ | $\left(0,1,-1,-1\right)$ | $\left(0,1,0,-1\right)$ | $\left(0,1,0,0\right)$ | $\left(0,1,1,1\right)$ | $\left(0,1,1,2\right)$
2 | $\ldots$ | $\left(1,1,-1,-2\right)$ | $\left(1,1,-1,-1\right)$ | $\left(1,1,0,-1\right)$ | $\left(1,1,0,0\right)$ | $\left(1,1,0,1\right)$ | $\left(1,1,1,1\right)$
3 | $\ldots$ | $\left(1,2,-1,-2\right)$ | $\left(1,2,-1,-1\right)$ | $\left(1,2,0,-1\right)$ | $\left(1,2,0,0\right)$ | $\left(1,2,0,1\right)$ | $\left(1,2,1,1\right)$
4 | $\ldots$ | $\left(1,3,-4,-7\right)$ | $\left(1,3,-2,-4\right)$ | $\left(1,3,-1,-2\right)$ | $\left(1,3,0,-1\right)$ | $\left(1,3,1,1\right)$ | $\left(1,3,3,4\right)$
$5$ | $\ldots$ | $\left(2,2,-2,-3\right)$ | $\left(2,2,-1,-2\right)$ | $\left(2,2,-1,-1\right)$ | $\left(2,2,0,0\right)$ | $\left(2,2,1,2\right)$ | $\left(2,2,2,3\right)$
Table 4. A sample of elements $z$ in the quasicrystal. Every element $z$ is
given by $z\left(n,m\right)=z_{1}\left(n\right)+z_{2}\left(m\right)\xi^{2}$,
with $z_{1}\left(n\right),z_{2}\left(m\right)\in\mathbb{Z}\left[\tau\right]$ .
Moreover, we intended $\left(a,b,c,d\right)\in\mathbb{Z}^{4}$ as a short
notation that corresponds to $z=a+b\tau+\left(c+d\tau\right)\xi^{2}$.

The aperiodic Jordan algebra $\mathfrak{J}\left(\Xi\right)$ is defined to be
the real vector space spanned by $\left\\{L_{z}\right\\}_{z\in\Xi}$ equipped
with the bilinear product given by

(6.13) $L_{z}\circ L_{w}=\frac{1}{2}\left(L_{z\vdash w}+L_{w\vdash z}\right),$

for every $z,w\in\Xi$. An integral formulation of the same algebra, i.e. an
isomorphic algebra with generators
$\left\\{L_{\left(a,b,c,d\right)}\right\\}_{\left(a,b,c,d\right)\in\mathbb{Z}^{4}}$,
can be obtained from (4.11) as

(6.14) $\displaystyle L_{\left(a,b,c,d\right)}\circ
L_{\left(a^{\prime},b^{\prime},c^{\prime},d^{\prime}\right)}=$
$\displaystyle\frac{1}{2}\left(L_{\left(a^{\prime}+\left(a-a^{\prime}\right)+\left(b-b^{\prime}\right),b+\left(a-a^{\prime}\right)+\left(b-b^{\prime}\right),c^{\prime}+\left(c-c^{\prime}\right)+\left(d-d^{\prime}\right),d+\left(c-c^{\prime}\right)+\left(d-d^{\prime}\right)\right)}+\right.$
(6.15)
$\displaystyle\left.+L_{\left(a+\left(a-a^{\prime}\right)+\left(b-b^{\prime}\right),b^{\prime}+\left(a-a^{\prime}\right)+\left(b-b^{\prime}\right),c+\left(c-c^{\prime}\right)+\left(d-d^{\prime}\right),d^{\prime}+\left(c-c^{\prime}\right)+\left(d-d^{\prime}\right)\right)}\right).$

###  6.3. A $\mathbb{Z}^{6}$_-quasicrystal_

Three-dimensional icosahedral quasicrystals can be obtained by projection from
three different lattices in a six-dimensional embedding space
$\mathbb{R}^{6}$. These lattices are the primitive cubic lattice $P$, i.e.
$\mathbb{Z}^{6}$ lattice, the face-centred cubic lattice $2F$, i.e. the root
lattice $D_{6}$, and the body-centred cubic lattice $I$ (reciprocal to $2F$),
i.e. the weight lattice $D_{6}$, respectively [PKK97]. Here we consider a cut
and project scheme analogous to that of the $\mathbb{Z}^{6}$_-_ quasicrystal
obtained in [KD86], where the embedding space is $\mathbb{R}^{6}$ and the
lattice is given by the integral lattice $\mathbb{Z}^{6}$, i.e.

(6.16) $\begin{array}[]{ccccc}\Xi&&\mathbb{Z}^{6}&&\Omega_{3}\\\
\cap&&\cap&&\cap\\\
\mathbb{R}^{3}&\overset{\pi_{1}}{\longleftarrow}&\mathbb{R}^{3}\times\mathbb{R}^{3}&\overset{\pi_{2}}{\longrightarrow}&\mathbb{R}^{3}\end{array}$

where the maps $\pi_{1}$ and $\pi_{2}$ are given by

(6.17) $\pi_{1}=\left(\begin{array}[]{cccccc}1&\tau&0&0&0&0\\\
0&0&1&\tau&0&0\\\
0&0&0&0&1&\tau\end{array}\right),\pi_{2}=\left(\begin{array}[]{cccccc}1&\left(1-\tau\right)&0&0&0&0\\\
0&0&1&\left(1-\tau\right)&0&0\\\
0&0&0&0&1&\left(1-\tau\right)\end{array}\right),$

and the acceptance window $\Omega_{3}$ is given by the rhombic
triacontahedron, i.e. the convex hull obtained from the following $32$
vertices

(6.18) $\begin{array}[]{ccccc}\left(0,\pm
1,\pm\tau\right),&&\left(\pm\tau,0,\pm 1\right),&&\left(\pm
1,\pm\tau,0\right),\\\ &&\left(\pm 1,\pm 1,\pm 1\right),\\\
\left(0,\pm\tau,\pm\tau^{-1}\right),&&\left(\pm\tau^{-1},0,\pm\tau\right),&&\left(\pm\tau,\pm\tau^{-1},0\right).\end{array}$

We thus have that the $\mathbb{Z}^{6}$-quasicrystal is given by the model set

(6.19)
$\Xi=\left\\{\pi_{1}\left(x\right)\in\mathbb{R}^{3}:x\in\mathbb{Z}^{6},\pi_{2}\left(x\right)\in\Omega_{3}\right\\}.$

An equivalent definition of $\Xi$ can be achieved in a straightforward way
through the aid of an icosian cut and project set-up as in (3.4), considering
the embedding of $\mathbb{R}^{3}$ in the pure immaginary quaternions
$\text{Pu}\left[\mathbb{H}\right]$ given by

(6.20) $\mathbb{R}^{3}\ni\left(x,y,z\right)\longrightarrow
x\text{i}+y\text{j}+z\text{k}\in\text{Pu}\left[\mathbb{H}\right].$

We thus obtain the icosian formulation of the model set of $\Xi$ as

(6.21)
$\Xi=\left\\{x\in\text{Pu}\left[\mathbb{I}\right]:x^{*}\in\Omega_{3}\right\\},$

where the star-map is the usual one defined in (2.20).

###  6.4. The Elser-Sloane quasicrystal

The Elser-Sloane quasicrystal $\mathscr{C}$ is a famous quasicrystal with an
$H_{4}$ symmetry that was introduced by Elser and Sloane in [ES] obtained as a
projection of the eight dimensional lattice $\Lambda\left(E_{8}\right)$. In
order to follow Elser and Sloane construction in [ES], we use for this
quasicrystal the cut-and-project scheme over the $E_{8}$ lattice as in (3.10),
which is equivalent to that in (3.4) over the icosians lattice
$\mathbb{\widetilde{I}}$. Therefore, let $c^{*}=1/\sqrt{\left(4+2\tau\right)}$
and $\kappa=\tau c^{*}$. Moreover, let the acceptance window $\Omega_{ES}$ be
the convex hull of the following 720 vertices, i.e.

(6.22) $\begin{array}[]{ccccc}\frac{1}{2}\kappa\left(\pm
2,0,0,0\right),&&\frac{1}{2}\kappa\left(\pm 1,\pm 1,\pm 1,\pm
1\right),&&\frac{1}{2}\kappa\left(0,\pm 1,\pm\tau,\pm\tau^{-1}\right),\\\
\frac{1}{3}\kappa\left(\pm\tau^{2},\pm\tau^{-2},\pm
1,0\right),&&\frac{1}{3}\kappa\left(\pm\tau^{2},\pm\tau^{-1},\pm\tau^{-1},\pm\tau^{-1}\right),&&\frac{1}{2}\kappa\left(0,\pm
1,\pm\tau,\pm\frac{1}{\tau}\right),\\\
\frac{1}{3}\kappa\left(\pm\left(2\tau-1\right),\pm\tau^{-1},\pm\tau,0\right),&&\frac{1}{3}\kappa\left(\pm\left(2\tau-1\right),\pm
1,\pm 1,\pm
1\right),&&\frac{1}{3}\kappa\left(\pm\tau,\pm\tau,\pm\tau,\pm\tau^{-2}\right),\\\
\frac{1}{3}\kappa\left(\pm 2,\pm 2,0,0\right),&&\frac{1}{3}\kappa\left(\pm
2,\pm 1,\pm\tau,\pm\tau^{-1}\right),\end{array}$

considered with all choices of signs and any even permutation of the
coordinates. Then, the Elser-Sloane quasicrystal $\mathscr{C}$ is the model
set from the cut-and-project scheme in (3.10) given by

(6.23)
$\Xi\cong\mathcal{\mathscr{C}}=\left\\{x\in\pi_{\parallel}\left(X\right):X\in\Lambda\left(E_{8}\right),\pi_{\perp}\left(X\right)\in\Omega_{ES}\right\\},$

for a complete treatment reference see [ES, sec. 4].

Even though from an historical point of view the Elser-Sloane quasicrystal was
defined as (6.23), an equally valid quaternionic realisation of the
quasicrystal that make use of the cut-and-project scheme in (3.4) is

(6.24)
$\mathcal{\mathscr{C}}\cong\Xi=\left\\{x\in\mathbb{I}:x^{*}\in\iota\left(\Omega_{ES}\right)\right\\},$

where the star map $*$ is that in (2.20) and $\iota\left(\Omega_{ES}\right)$
is the image of the acceptance window $\Omega_{ES}$ through the canonical
identification (2.11) between $\mathbb{R}^{4}$ and $\mathbb{H}$.

##  7\. conclusions and further developments

In this work we presented a Jordan structure naturally linked to a class of
icosahedral quasicrystal arising from a cut-and-project setup with convex
acceptance window. The resulting Jordan algebras are infinite dimensional and
fits into the class of aperiodic algebras introduced by Patera et al. in
[PPT98]. Aperiodic Jordan algebras were firstly defined by the authors in
[CCAI] for a Fibonacci chain quasicrystal and later on extended to a Penrose
tiling in [CCAI2], but never fully studied in a general setting as this one.
In this work we showed some common features of such algebras, such as the lack
of the unit element (even though they can obviously be extended to a unital
Jordan algebra) and the lack of finite dimensional subalgebras with dimension
greater than $1$. We also showed that such algebras enjoy a non-
crystallographic symmetry when the acceptance window does. As for the physical
applications and future works, aperiodic algebras found applications in
integrable systems [TW00b] and in theoretical frameworks [Tw99a]. On the other
hand, it is well known that Jordan algebras are relevant in exactly solvable
models involving propagation of solitons. Indeed, it is well known a one-to-
one correspondence between finite dimensional Jordan algebra and multifield
Korteweg-de Vries equations[Sv91]. Similar results hold for the modified KdV
equation [Sv93], for the Sine-Gordon equation and for generalisations of the
non-linear Schrödinger equation [Sv92]. We thus expect that propagation of
certain kind of excitations in quasicrystal structures might be related with
the Jordan algebras presented here.

##  8\. Acknowledgment

This work was funded by the Quantum Gravity Research institute. Authors would
like to thank Fang Fang, Marcelo Amaral, Dugan Hammock, and Richard Clawson
for insightful discussions and suggestions.

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