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Affine extensions of non-crystallographic Coxeter groups induced by projection

In this paper, we show that affine extensions of non-crystallographic Coxeter groups can be derived via Coxeter-Dynkin diagram foldings and projections of affine extended versions of the root systems E_8, D_6 and A_4. We show that the induced affine extensions of the non-crystallographic groups H_4, H_3 and H_2 correspond to a distinguished subset of the Kac-Moody-type extensions considered in Dechant et al. This class of extensions was motivated by physical applications in icosahedral systems in biology (viruses), physics (quasicrystals) and chemistry (fullerenes). By connecting these here to extensions of E_8, D_6 and A_4, we place them into the broader context of crystallographic lattices such as E_8, suggesting their potential for applications in high energy physics, integrable systems and modular form theory. By inverting the projection, we make the case for admitting different number fields in the Cartan matrix, which could open up enticing possibilities in hyperbolic geometry and rational conformal field theory.


I. INTRODUCTION
The classification of finite-dimensional simple Lie algebras by Cartan and Killing is one of the mile stones of modern mathematics. The study of these algebras is essentially reduced to that of root systems and their Weyl groups, and all their geometric content is contained in Cartan matrices and visualised in Dynkin diagrams. The problem ultimately amounts to classifying all possible Cartan matrices 2 .
Coxeter groups describe (generalised) reflections 3 , and thus encompass the above Weyl groups, which are the reflective symmetry groups of the relevant root systems. In fact, the finite Coxeter groups are precisely the finite Euclidean reflection groups 4 . However, since the root systems arising in Lie Theory are related to lattices, the Weyl groups are automatically crystallographic in nature. Non-crystallographic Coxeter groups, i.e. those that do not stabilise any lattice (in the dimension equal to their rank), therefore cannot arise in the Lie Theory context, and as a consequence, they have not been studied as intensely. They include the groups H 2 , H 3 and the largest non-crystallographic group H 4 ; the icosahedral group H 3 and its rotational subgroup I are of particular practical importance as H 3 is the largest discrete symmetry group of physical space. Thus, many 3-dimensional systems with 'maximal symmetry', like viruses in biology [5][6][7][8][9] , fullerenes in chemistry [10][11][12][13] and quasicrystals in physics [14][15][16][17] , can be modeled using Coxeter groups.
Affine Lie algebras have also been studied for a long time, and many of the salient features of the theory of simple Lie algebras carry over to the affine case. More recently, Kac-Moody Theory has provided another framework in which generalised Cartan matrices induce interesting algebraic structures that preserve many of the features encountered in the simple and affine cases 18 . However, such considerations again only give rise to extensions of crystallographic Coxeter groups.
These infinite Coxeter groups are usually constructed directly from the finite Coxeter groups by introducing affine reflection planes (planes not containing the origin). While these infinite counterparts to the crystallographic Coxeter groups have been intensely studied 19 , much less is known about their non-crystallographic counterparts 20 . Recently, we have derived novel affine extensions of the non-crystallographic Coxeter groups H 2 , H 3 and H 4 in two, three and four dimensions, based on an extension of their Cartan matrices following the Kac-Moody formalism in Lie Theory 1 .
In this paper, we develop a different approach and induce such affine extensions of the noncrystallographic groups H 2 , H 3 and H 4 from affine extensions of the crystallographic groups A 4 , D 6 and E 8 , via projection from the higher-dimensional setting. Specifically, there exists a projection from the root system of E 8 , the largest exceptional Lie algebra, to the root system of H 4 , the largest non-crystallographic Coxeter group 19 , and, due to the inclusions A 4 ⊂ D 6 ⊂ E 8 and H 2 ⊂ H 3 ⊂ H 4 , also corresponding projections for the other non-crystallographic Coxeter groups.
We apply these projections here to the extended root systems of the groups A 4 , D 6 and E 8 . As expected, extending by a single node recovers only those affine extensions known in the literature.
However, we also consider simply-laced extensions with two additional nodes in the Kac-Moody formalism, and consider their compatibility with the projection formalism. Specifically, we use the projection of the affine root as an affine root for the projected root system, and thereby find a distinguished subset of the solutions in the classification scheme presented in 1 .
The E 8 root system, and the related structures: the E 8 lattice, the Coxeter group, the Lie algebra and the Lie group, are 'exceptional' structures, and are of critical importance in mathematics and in theoretical physics 2 . For instance, they occur in the context of Lie algebras, simple group theory and modular form theory, as well as lattice packing theory [21][22][23] . In theoretical physics, E 8 is central to String Theory, as it is the gauge group for the E 8 ×E 8 heterotic string 24 . More recently, via the Hořava-Witten picture [25][26][27] and other developments [28][29][30][31][32] , E 8  The structure of this paper is as follows. Section II reviews some standard results to provide the necessary background for our novel construction. Section II A discusses the basics of Coxeter groups. Section II B introduces the relationship between E 8 and H 4 , and discusses how it manifests itself on the level of the root systems, the representation theory, and the Dynkin diagram foldings and projection formalism. Section II C introduces affine extensions of crystallographic Coxeter groups, and presents the standard affine extensions of the groups relevant in our context. In Section III A, we compute where the affine roots of the standard extensions of the crystallographic groups map under the projection formalism and examine the resulting induced affine extensions of the non-crystallographic groups. Section III B discusses Coxeter-Dynkin diagram automorphisms of the simple and affine groups, and shows that the induced affine extensions are invariant under these automorphisms. In Section III C, we consider affine extending the crystallographic groups by two nodes and show that these do not induce any further affine extensions. In Section IV A, we briefly review the novel Kac-Moody-type extensions of non-crystallographic Coxeter groups from a recent paper and compare the induced extensions with the classification scheme presented there (Section IV B). In Section V, we conclude that in a wide class of extensions (single extensions or simply-laced double extensions with trivial projection kernel), the ten induced cases considered here are the only ones that are compatible with the projection. We also discuss how lifting affine extensions of non-crystallographic groups to the crystallographic setting, as well as symmetrisability of the resulting matrices, motivate a study of Cartan matrices over extended number fields.

II. MATHEMATICAL PRELIMINARIES
In this section, we introduce the context of our construction, with the relevant concepts and the known links between them, as illustrated in Fig. 1. We introduce Coxeter groups and their root systems in Section II A, and discuss how certain crystallographic and non-crystallographic groups are related via projection (Section II B). Affine extensions of the crystallographic Coxeter groups are introduced in Section II C. Affine extensions of the non-crystallographic Coxeter groups in dimensions two, three and four have been discussed in our previous papers 1,20 (see Section IV A).
Here, we present a different construction of such affine extensions, by inducing them from the known affine extensions of the crystallographic Coxeter groups via projection from the higherdimensional setting. These induced extensions will be shown to be a subset of those derived in 1 .

A. Finite Coxeter groups and root systems
Definition II.1 (Coxeter group). A Coxeter group is a group generated by some involutive generators s i , s j ∈ S subject to relations of the form (s i s j ) m i j = 1 with m i j = m ji ≥ 2 for i = j. The matrix A with entries A i j = m i j is called the Coxeter matrix.
The finite Coxeter groups have a geometric representation where the involutions are realised as reflections at hyperplanes through the origin in a Euclidean vector space E . In particular, let (·|·) denote the inner product in E , and λ , α ∈ E .
at a hyperplane perpendicular to the root vector α.
The action of the Coxeter group is to permute these root vectors, and its structure is thus encoded in the collection Φ ∈ E of all such root vectors, the root system: Definition II.3 (Root system). A root system Φ is a finite set of non-zero vectors in E such that the following two conditions hold: 1. Φ only contains a root α and its negative, but no other scalar multiples:

Φ is invariant under all reflections corresponding to vectors in
For a crystallographic Coxeter group, a subset ∆ of Φ, called simple roots, is sufficient to express every element of Φ via a Z-linear combination with coefficients of the same sign. Φ is therefore completely characterised by this basis of simple roots, which in turn completely characterises the Coxeter group. In the case of the non-crystallographic Coxeter groups H 2 , H 3 and H 4 , the same holds for the extended integer ring Z[τ] = {a + τb|a, b ∈ Z}, where τ is the golden . Note that together with its Galois conjugate τ ′ ≡ σ = 1 2 (1 − √ 5), τ satisfies the quadratic equation x 2 = x + 1. In the following, we will call the exchange of τ and σ Galois conjugation, and denote it by The structure of the set of simple roots is encoded in the Cartan matrix, which contains the geometrically invariant information of the root system as follows: Definition II.4 (Cartan matrix and Coxeter-Dynkin diagram). The Cartan matrix of a set of simple roots α i ∈ ∆ is defined as the matrix A graphical representation of the geometric content is given by Coxeter-Dynkin diagrams, in which nodes correspond to simple roots, orthogonal roots are not connected, roots at π 3 have a simple link, and other angles π m have a link with a label m.
Note that Cartan matrix entries of τ and σ yield Coxeter diagram labels of 5 and 5 2 , respectively, since in the simply-laced setting A i j = −2 cos π m i j , τ = 2 cos π 5 and −σ = 2 cos 2π 5 . Such fractional values can also be understood as angles in hyperbolic space 36 . By the crystallographic restriction theorem, there are no lattices (i.e. periodic structures) with such non-crystallographic symmetry H 2 , H 3 and H 4 in two, three and four dimensions, respectively. For these non-crystallographic Coxeter groups one therefore needs to move from a lattice to a quasilattice setting.

B. From E 8 to H 4 : standard Dynkin diagram foldings and projections
The largest exceptional (crystallographic) Coxeter group E 8 and the largest non-crystallographic Coxeter group H 4 are closely related. This connection between E 8 and H 4 can be exhibited in various ways, including Coxeter-Dynkin diagram foldings in the Coxeter group picture 37 , relating the root systems 16,38,39 , and in terms of the representation theory 14,16,[37][38][39] . For illustrative purposes, we focus on the folding picture first.
There are similar diagrams for A 4 and D 6 that can be obtained from the E 8 diagram by deleting nodes. We display these in Fig. 3 in order to set out our notation, as the conventional way of numbering the roots in the Dynkin diagrams differs from the natural numbering in the folding It has been observed that the E 8 root vectors can be realised in terms of unit quaternions with coefficients in Z[τ] 16,38,39,41 . Specifically, the set of 120 icosians forms a discrete group under standard quaternionic multiplication, and is a realisation of the H 4 root system 16,42,43 . The 240 roots of E 8 have been shown to be in 1-1-correspondence with the 120 icosians and their 120 τmultiples, so that, schematically, E 8 ∼ H 4 + τH 4 holds. The projection considered above therefore exhibits this mapping of the simple roots of E 8 onto the simple roots of H 4 and their τ-multiples.
Corresponding results hold for the other groups D 6 and H 3 , as well as A 4 and H 2 by inclusion.
From a group theoretic point of view, E 8 has two conjugate H 4 -invariant subspaces. Following the terminology in 37 , we make the following definition.
Definition II.5 (Standard and non-standard representations). We denote by the standard representation of a Coxeter group the representation generated by mirrors forming angles π m i j , where m i j are the entries of the Coxeter matrix, i.e. 5, 3 and 2 for the cases relevant here. One can also achieve a non-standard representation of the Coxeter groups H i by instead taking mirrors at angles 2π/5, i.e. by schematically going from a pentagon to a pentagram.
Note that these therefore have Coxeter diagram labels of 5 and 5 2 , respectively. These are non-equivalent irreducible representations of the non-crystallographic Coxeter groups, i.e. there is no similarity transformation that takes one to the other. Their characters are exchanged under the Galois automorphism τ ↔ σ , and the simple roots in the non-standard representation are the Galois conjugates of the simple roots of the standard representation, for instance a 1 = . This different set of simple rootsā i yields a Cartan matrix that is the Galois conjugate of the Cartan matrix of the simple roots a i , and leads to the label of 5 2 in the Coxeter diagram. However, in this finite-dimensional case the groups generated by the two sets of generators are isomorphic, as both sets of roots give rise to the same root system, but positioned differently in space. The non-trivial Coxeter relations are therefore in both Hence the Coxeter matrix is identical in both cases; in particular, it is still symmetric. Therefore, one usually restricts analysis to the standardrepresentation, where the simple roots form obtuse angles, and whose Cartan matrix therefore satisfies the usual negativity requirements. The projection π ⊥ into the second H 4 -invariant subspace in terms of the bases α i andā i is the Galois conjugate of π in Eq. (3), which, however, is with respect to the bases α i and a i . The relationship between E 8 and H 4 is best understood as the standard representation of E 8 inducing both the standard and non-standard representations of the subgroup H 4 . That is, E 8 decomposes under an H 4 subgroup as 8 = 4 +4 (c.f. also 38,39 , in particular in the wider context of similar constructions unified in the Freudenthal-Tits magic square 44 ). Equivalent statements 6 = 3 +3 and 4 = 2 +2 hold true for the lower-dimensional cases, and have found applications in the quasicrystal literature 14,15 . In the quasicrystal setting, one usually only considers the projection π ; in our setting one can consider projection into either invariant subspace, using π as well as π ⊥ .

C. Affine extensions of crystallographic Coxeter groups
For a crystallographic Coxeter group, an affine Coxeter group can be constructed by defining affine hyperplanes H α 0 ,i as solutions to the equations The nontrivial isometry of E that fixes H α 0 ,i pointwise is unique and called an affine reflection Definition II.6 (Affine Coxeter group). An affine Coxeter group is the extension of a Coxeter group by an affine reflection s a f f α 0 whose geometric action is given by

acts as a translation of twice the length.
It is in fact possible to construct the affine Coxeter group directly from an extension of the Cartan matrix.

Definition II.7 (Kac-Moody-type affine extension). A Kac-Moody-type affine extension A a f f of a Cartan matrix is an extension of the Cartan matrix A of a Coxeter group by further rows and
columns such that the following conditions hold: • The diagonal entries are normalised as A a f f ii = 2 according to the definition in Eq. (2).

• The additional matrix entries of A a f f take values in the same integer ring as the entries of
A. This includes potentially integer rings of extended number fields as in the case of H 3 .
• The affine extended matrix fulfils the determinant constraint det A a f f = 0.
In our previous paper 1 we have laid out a rationale for Kac-Moody-type extensions of Cartan matrices, as well as consistency conditions that lead to a somewhat improved algorithm for numerically searching for such matrices. This was necessitated by our search for novel asymmetric affine extensions of H 2 , H 3 and H 4 . Here, our algorithm simply recovers the affine extensions of E 8 , D 6 and A 4 that are well known in the literature for affine extensions by a single node. However, based on Definition II.7, we will also consider extending by two nodes in the context of the projection.
We begin with the case of E 8 , which is the most interesting from a high energy physics point of view, and the largest exceptional Coxeter group. Various notations are used in the literature to denote its unique (standard) affine extension, but here we shall use E = 8 , where the equality sign is meant to signify that the extra root has the same length as the other roots, i.e. the affine extension is simply-laced (see Fig. 4 for our notation). The affine root α 0 that gives rise to this affine extension can be expressed in terms of the root vectors of E 8 as which will prove important in the projection context later.
Likewise, D 6 has a simply-laced affine extension, here denoted D = 6 and depicted in Fig. 5. Again, the affine root can be expressed in terms of the other roots as However, D 6 is unusual in that it also has two affine extensions with a different root length, one which we shall denote by D < 6 , because the new root is shorter than the others. In this case, the affine root is given by There is, moreover, one with a longer root, which we denote by D > 6 (both are shown in Fig. 6). Its 6. Dynkin diagrams and Cartan matrices for the standard affine extensions of D 6 with a short affine root (here denoted D < 6 ), and that with a long affine root, here denoted D > 6 . The arrow conventionally points to the shorter root. affine root is similarly expressible in terms of the other roots as A 4 also has a unique standard affine extension, which is simply-laced and hence will be denoted by A = 4 (see Fig. 7). The affine root is given by

INDUCED BY PROJECTION
In this section, we present a novel construction of affine extensions of non-crystallographic Coxeter groups, as illustrated in Fig. 1 Proof. We consider the five cases in turn.
1. We begin with the case of E 8 . We have shown above that the root vectors can be projected onto the H 4 root vectors a i by the projection π shown in Fig. 2. The projection in Eq. (3) of the affine root in Eq. (5) is therefore This is one of the Kac-Moody-type affine extensions of H 4 that we derived in our previous paper 1 , in the context of non-crystallographic Coxeter groups. It was listed there as the first non-trivial example of affine extensions of this type and corresponds to an affine extension of length τ along the highest root α H of H 4 . We will briefly review the results from 1 in Section IV A, which we will use to classify all induced affine extensions in Section IV B.
Projecting with π ⊥ into the other H 4 -invariant subspace spanned by the basis of simple roots a i yields the Galois conjugate of the affine root in Eq. (10) Using the inner products (ā 1 |ā 2 ) = − 1 2 , (ā 2 |ā 3 ) = − 1 2 and (ā 3 |ā 4 ) = − σ 2 , the inner products of the affine root with the H 4 roots are (ā 0 |ā 1 ) = − 1 2 and (ā 0 |ā 2 ) = (ā 0 |ā 3 ) = (ā 0 |ā 4 ) = 0, and the resulting Cartan matrix A (H = 4 ) is thus the Galois conjugate of that in Eq. (11) from the other invariant subspace. Since both sets of roots a i andā i generate the same abstract group H 4 , one has a pair of Galois conjugate induced affine roots α 0 andᾱ 0 parallel to the highest root α H with Galois conjugate lengths τ and σ , respectively. Note that (A (H = 4 )) T would also generate the same translation of length σ along α H , and was contained in the results of 1 . We will consider whether (A (H = 4 )) T could also arise from projection in Section V.
2. Using the same procedure as above -i.e. employing linearity to project the affine root of D = 6 and using it as an affine extension of H 3 -generates the analogue of the previous case in three dimensions This is also the first non-trivial example of asymmetric affine extensions of H 3 considered in our previous paper 1 , corresponding to an affine extension of length τ along the highest root α H of H 3 (i.e. along a 2-fold axis of icosahedral symmetry). One choice of simple roots for 3. When projecting D < 6 we find In

A similar result is obtained when
The respective projections again yield the Galois conjugate pair α 0 = T 5 andᾱ 0 = −σ T 5 .
We note that even though the affine extensions D < 6 and D > 6 are related by transposition, the correspondence between the two induced affine extensions H < 3 and H > 3 (andH < 3 and H > 3 ) is not so straightforward, i.e. the operations of transposition and projection do not commute, schematically [T, P] = 0. However, the transposed versions of these induced lower-dimensional Cartan matrices, for instance A H > 3 T , were among the affine extensions derived in 1 . One might therefore wonder which higher-dimensional Cartan matrices could give rise to these transposed versions after projection. We will revisit these issues later.
5. The affine root of A 4 is given by Eq. (9) and upon projection with π yields an affine extension of H 2 analogous to the other simply-laced cases considered above This likewise corresponds to an affine root of length τ along α H , the highest root of H 2 given by α H = τ(α 1 + α 2 ), which was also found in 1 , where we also visualised its action on a pentagon. Projection with π ⊥ yields an induced extensionH = 2 with the Galois-conjugate length −σ along α H . This completes the proof.
As is well known 19   is also an affine extension corresponding to λ = −σ , and is contained in the solutions found in 1 . We will discuss how this could be lifted to the higher-dimensional case in Section V. We note that all three translations in each case are distinguished, i.e. they give rise to less than maximal cardinality.
In order to further explore this interesting relation with quasilattices, we begin by introducing some terminology 20 . We recall that a generic affine non-crystallographic Coxeter group H + i is generated by the s j s from Section II A together with the translation T that we identified in Definition II.6.

Definition III.3 (Quasicrystal fragment). Let Φ denote the root polytope of the non-crystallographic
Coxeter group H i , and let W m (s j ; T ) denote the set of all words w(s j ; T ) in the alphabet formed from the letters s j and T in which T appears precisely m times. The set of points is called an H + i -induced quasicrystal fragment; n is the cut-off-parameter. The cardinality of such a quasicrystal fragment will be denoted by |Q i (n)|, and a generic translation yields the maximal cardinality |Q max i (n)|. We will say that a quasicrystal fragment with less than maximal cardinality has coinciding/degenerate points/vertices and we call the corresponding translation distinguished.
This degeneracy implies non-trivial relations w 1 (s j ; T )v = w 2 (s j ; T )v (for v ∈ Φ) amongst the (words in the) generators. The set of points P i (n) := {W m (s j ; T )R|m = n} will denote the shell of the quasicrystal fragment determined by the words that contain T precisely n times.
The affine roots relevant here are all parallel to the respective highest root α H but have various different lengths λ , which we write as α 0 = −λ α H . Therefore, in Table I we Table I. The cardinalities of the shells |P 2 (n)| are also given. We note that Galois conjugate translations yield the same cardinalities.
set of cardinality 36 in Table I, and panels (d) and (g) correspond to the entries with cardinality 40. We again note that Galois conjugate affine roots yield the same cardinalities, as in the higherdimensional cases before. Our novel construction thus leads to different types of quasicrystalline point arrays. We will later consider whether the extensions from 20 could similarly be induced from a higher-dimensional setting. We will see that they would correspond to Cartan matrices with positive and fractional (c.f. H < 3 and H > 3 in Eqs (14) and (15)) off-diagonal entries (Section V), making the case for a suitable generalisation of the standard approach by analysing generalised Cartan matrices over extended number fields.
The above projection procedure has thus yielded asymmetric induced Cartan matrices. In the context of Kac-Moody algebras and Coxeter groups, it is often of interest to know if an asymmetric (generalised) Cartan matrix A is symmetrisable:

Definition III.4 (Symmetrisability). An asymmetric Cartan matrix A is symmetrisable if there exist a diagonal matrix D with positive integer entries and a symmetric matrix S such that A = DS.
We have investigated the symmetrisability of the induced non-symmetric Cartan matrices 1 .
They are indeed symmetrisable, but the entries of the resulting symmetric matrices are no longer from Z[τ] (see also the discussion in Section V). Given that the Cartan matrix is defined in terms of the geometry of the roots as A i j = 2(α i |α j )/(α i |α i ), i.e. is given in terms of the angles between root vectors and their length, such matrices would imply a geometry for the root system that is no longer compatible with an (aperiodic) quasilattice, and the corresponding affine groups would therefore lose their distinctive structure. Indeed, it is that relation with quasilattices that makes these affine extended groups mathematically interesting, and distinguishes them from the free group obtained by an extension via a random translation. Therefore, we will not use these symmetric matrices in our context.

B. Invariance of the projections under Dynkin diagram automorphisms
Before we classify the induced affine extensions, we show in this section that no additional induced extensions arise from the Dynkin diagram automorphisms of the simple and affine Lie algebras considered above. Proof. We consider the four cases in turn.

Lemma III.5 (Invariance of the induced extensions). The induced affine extensions H
1. The Dynkin diagram of D 6 has a Z 2 -automorphism that acts by permuting the roots α 5 and α 6 (denoted as 5 ↔ 6 in the following). The projection displayed in Fig. 3, however, is not symmetric in α 5 and α 6 . Therefore, the choice of projection could potentially alter the induced affine extension. However, as can be seen from equations (6), (7) and (8) 2. Similarly, the simply-laced extension D = 6 has an additional D 4 automorphism symmetry (here D n denotes the dihedral group of order n) that allows one to swap the roots labelled by 5 ↔ 6 or 0 ↔ 1 separately, as well as an overall left-right symmetry of the diagram obtained by swapping the pairs of terminal roots (0, 1) ↔ (5, 6) together with 2 ↔ 4, 3 ↔ 3 2 . This symmetry is made manifest in the diagram shown in Fig. 9. Thus, the four terminal roots are equivalent, and one could define four different projections, depending on which terminal root one considered as the affine root. Once one decides on the affine root, the rest of the diagram is fixed by the projection. However, the formula for the affine root is symmetric in (0, 1, 5, 6) as can be seen from (6). Thus, the induced affine extension is again independent of which projection one chooses.
3. Likewise, the A 4 diagram has a Z 2 -symmetry swapping left and right, that is broken by the projection. However, the affine root (9) is again invariant, so that the induced affine extension is not affected.

The extended diagram A =
4 has an enhanced D 5 -automorphism symmetry 2 , under which the affine root can be seen as invariant by rewriting Eq. (9) as α 0 + α 1 + α 2 + α 3 + α 4 = 0. (18) Thus, in this case, one could choose any of the roots of the extended diagram as the affine root, and the others are then fixed by the projection.
The invariance is at the level of the affine roots before projection, so it does not matter into which invariant subspace one projects. Thus, the Dynkin diagram automorphisms do not affect the induced affine extensions.

C. Extending by two nodes
Until now, we have considered extending a diagram by a single root, and we have projected this single affine root onto the single induced affine root. However, as is shown in Fig. 2, other roots are projected in pairs, e.g. α 1 and α 7 project onto a 1 and τa 1 , which results in the single H 4 -root a 1 . In analogy, we now consider affine extensions of the diagrams by two nodes such that the two additional roots project as a pair onto a single affine root. This can be achieved by further extending the above affine extensions by another node, or by extending the initial diagrams by two nodes at once.
We first check whether further extending the above groups A = 4 , D > 6 , D = 6 , D < 6 and E = 8 by another node leads to new induced affine extensions. We show that in such a case, the only possibilities are in fact the above diagrams with a disconnected node. Thus, this type of extension is trivial, and the additional affine root will not be a superposition of the other roots:  The other possibility is to extend by two nodes at once, and to demand that the Cartan matrix of the double-extension be affine, i.e. that it has zero determinant, but that none of the principal minors has this property.

Definition III.7 (Affine double extension). An affine double extension is a Kac-Moody-type extension of a diagram by two nodes.
We analyse here the simply-laced double extensions with a trivial projection kernel, which give 292 such matrices for E 8 , 27 for D 6 , and 6 for A 4 . Note, however, that the number of different Dynkin diagrams is actually lower. For instance, there are only three diagrams that occur for A ++ 4 , which are displayed in Fig. 10. The first corresponds to a single extension of the D-series, c.f. D = 6 above. The second is a single extension of A 5 , and the third a diagram with a trivial disconnected node. The diagrams for D 6 and E 8 have a richer mathematical structure. However, it shall suffice here to consider these matrices from the projection point of view -a more detailed analysis of double extensions will be relegated to future work. Proof. In all the cases mentioned above (292 for E 8 , 27 for D 6 and 6 for A 4 ), it is not possible to express both additional roots simultaneously in terms of linear combinations of the roots of the unextended group, c.f. the diagrams for A 4 in Fig. 10. Hence, the Cartan matrices can be obtained only in terms of higher-dimensional vectors, i.e. the kernel is non-trivial. Earlier, we have demonstrated that the induced affine extensions do not depend on the nontrivial automorphism properties of the simple and extended diagrams. Therefore, in summary, we conclude that the ten cases considered above are actually the only cases that arise in the context of trivial projection kernels.

5
A H af f and H a f f 4 , the unique symmetric affine extensions in 20 . Note that Coxeter angles of 2π/5 lead to labels 5 2 (or, τ ′ in the notation of 20 ). In 1 , we have found infinite families of generalisations of these examples, which are obtained from the symmetric cases via scalings with τ and thus follow a Fibonacci recursion relation. We have also found more general examples, which likewise display this scaling property.

IV. CLASSIFICATION OF INDUCED AFFINE EXTENSIONS
The affine extensions induced via the projection in Section III (right arrow in Fig. 1) are subsets of the infinite families developed purely in a non-crystallographic framework in 1 (bottom arrow in Fig. 1). Therefore, we first summarise the relevant results from this paper in Subsection IV A, and then analyse in Subsection IV B how the induced affine extensions relate to our classification in 1 .

A. Construction of affine extensions in the non-crystallographic case
In the case of non-crystallographic Coxeter groups, which have Cartan matrices given in terms of the extended integer ring Z[τ], the earlier definition of affine extensions from Section II C via introducing affine hyperplanes H α 0 ,i as solutions to the equations (x|α 0 ) = i , where x ∈ E , α 0 ∈ Φ and i ∈ Z, is not possible because the crystallographic restriction 15 implies that the planes cannot be stacked periodically; In contrast, the definition of affine extensions of non-crystallographic Coxeter groups via the Kac-Moody-type extensions of their Cartan matrices in Definition II.7 still works. For instance, affine extensions along a 2-fold axis of icosahedral symmetry T 2 = (1, 0, 0) (i.e. along the simple roots) have the following general form 1 since T 2 is orthogonal to two of the simple roots, and thus there is only one pair of off-diagonal entries that is non-zero. This family of matrices contains H = 3 in Eq. (13)  By choosing an ansatz similar to Eq. (19), one can also obtain translations along 3-and 5-fold axes of icosahedral symmetry, T 3 = (τ, 0, σ ) and T 5 = (τ, −1, 0). These cases correspond to one pair of non-zero entries (x, y) in the Cartan matrix in the fourth row and column, and in the second row and column, respectively, since these symmetry axes are again orthogonal to two of the simple roots. From now on, we will use the notation In summary, one can thus label an affine extension in the following way: A solution is given in terms of an integer quadruplet (a, b; c, d) that is related to a particular reference solution via rescaling with a power k of τ, together with a multiplier pair (γ, δ ).

B. Identification of the induced affine extensions in the Fibonacci classification
We now identify the induced affine extensions derived in Section III within the Fibonacci families from 1  We summarise the relation of all induced affine extensions to the Fibonacci classification 1 in Table II.  Table II,

V. DISCUSSION
We have shown that via affine extensions of the crystallographic root systems E 8 , D 6  (denoted by P). Again, we denote generic extensions E + 8 , D + 6 and A + 4 by their additional row and column vector in the Cartan matrix as follows We begin by lifting the affine extensions H