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Adv. Appl. Cliﬀord Algebras (2021) 31:57

c

The Author(s) 2021

0188-7009/030001-35

published online June 28, 2021

https://doi.org/10.1007/s00006-021-01139-2

Advances in

Applied Cliﬀord Algebras

Cliﬀord Spinors and Root System Induction:

H4and the Grand Antiprism

Pierre-Philippe Dechant∗

Communicated by RafalAblamowicz

To John Horton Conway, Richard Kenneth Guy and Michael Guy.

Abstract. Recent work has shown that every 3D root system allows the

construction of a corresponding 4D root system via an ‘induction theo-

rem’. In this paper, we look at the icosahedral case of H3→H4in detail

and perform the calculations explicitly. Cliﬀord algebra is used to per-

form group theoretic calculations based on the versor theorem and the

Cartan–Dieudonn´e theorem, giving a simple construction of the Pin and

Spin covers. Using this connection with H3via the induction theorem

sheds light on geometric aspects of the H4root system (the 600-cell) as

well as other related polytopes and their symmetries, such as the famous

Grand Antiprism and the snub 24-cell. The uniform construction of root

systems from 3D and the uniform procedure of splitting root systems

with respect to subrootsystems into separate invariant sets allows fur-

ther systematic insight into the underlying geometry. All calculations are

performed in the even subalgebra of Cl(3), including the construction of

the Coxeter plane, which is used for visualising the complementary pairs

of invariant polytopes, and are shared as supplementary computational

work sheets. This approach therefore constitutes a more systematic and

general way of performing calculations concerning groups, in particular

reﬂection groups and root systems, in a Cliﬀord algebraic framework.

Mathematics Subject Classiﬁcation. Primary 52B15; Secondary 52B11,

15A66, 20F55, 17B22, 20G41.

Keywords. Exceptional symmetries, spinors, 600-cell, grand antiprism,

Cliﬀord algebras, Coxeter groups, root systems, Platonic solids.

Supplementary Information The online version contains supplementary material available

at https://doi.org/10.1007/s00006-021-01139-2.

∗Corresponding author.

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57 Page 2 of 35 P. Dechant Adv. Appl. Cliﬀord Algebras

1. Introduction

Like many other mathematical concepts, symmetry groups have a rich struc-

ture including regular families and sporadic or exceptional phenomena [16].

The exceptional symmetry group in 3D is icosahedral symmetry. This fact

has profound implications for the natural world around us, because objects

consisting of identical building blocks that are ‘maximally symmetric’ dis-

play icosahedral symmetry. This includes most viruses and many fullerenes,

as well as artiﬁcial nanocages in bionanotechnology and geodesic domes in

architecture [13]. Even before any of these examples were known, icosahedral

symmetry had inspired Plato to formulate a ‘uniﬁed theory of everything’ in

his dodecahedral ‘ordering principle of the universe’. This pattern of (excep-

tional) symmetries inspiring ‘grand uniﬁed theories’ continues to this day,

with A4=SU(5) in GUTs, and E8in string theory and GUTs, as well as

D4=SO(8) and B4=SO(9) being critical in string and M theory.

Traditionally, people seek to understand symmetries ‘top-down’. For

instance E8includes A4and H4,orH4includes H3, so that people seek

to understand the smaller groups as subgroups of the larger ones. In recent

work [6,8,10,11] the author has shown that instead there is also a ‘bottom-

up’ view, by which e.g. H4and even E8can be constructed from H3. In this

sense the key to the larger exceptional symmetry groups is already contained

in the smaller exceptional group. In particular, the author proved a uniform

theorem that any 3D root system/reﬂection group induces a corresponding

4D root system/reﬂection group in a systematised way. The unusual abun-

dance of exceptional symmetry structures in 4D could thus be based on (the

accidentalness of) this construction, because it gives rise to the 4D excep-

tional objects D4(triality), F4(the largest 4D crystallographic group) and

H4(the largest non-crystallographic group altogether). There is an immediate

connection with Arnold’s Trinities, mysterious connections between diﬀerent

triplets of exceptional objects throughout mathematics [1,2,12].

There is therefore a lot of additional geometric insight to be gained

from understanding this connection between 3D and 4D geometry, rather

than looking at these phenomena from a 4D perspective alone. In fact, the

link with 3D is much wider, including an ADE-type correspondence between

3D and 4D but also between 3D and ADE-type diagrams, in addition to the

famous McKay correspondence [12,27]. The connection between 3D and 4D

geometry arises because 3D reﬂections give rise to rotation groups via spinors.

These 3D spinors themselves behave like 4D objects and can be shown to

satisfy the root system axioms. This paper seeks to provide an example of

concrete calculations performed entirely within the 3D Cliﬀord algebra Cl(3)

and its even subalgebra rather than in R4. There is of course also a connection

with quaternions, because of an isomorphism. A lot of previous work has

been done using quaternions and a purely algebraic description [4,22–26]

with some somewhat haphazard results. However, we would argue that a lot

of the deeper geometric insight, and its universality and systematic approach,

have been lost by following an approach that is algebraically equivalent but

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Vol. 31 (2021) Cliﬀord Spinors and Root System Page 3 of 35 57

is really not well suited for uncovering and understanding the underlying

geometry and the generality of the situation.

In this paper we therefore give a comprehensive and pedagogical expo-

sition of how to perform group theoretic calculations for H3in a Cliﬀord

approach, leading to the detailed construction of H3→H4. We give explicit

results of calculations in the paper, as well as making python Jupyter note-

books available as supplementary information for convenience and repro-

ducibility. These detail the calculations, results and algorithms used based

on the galgebra python package [3] as well as some visualisations also from

SageMath [30]. These notebooks are shared in the interest of open science,

collegiality and reproducibility, and may be useful to the readers for adapting

them for their own calculations. If they are useful please cite this paper along

with the original software provided by e.g. [3] and [30].

After the basic construction of the H4root system (the 600-cell) from H3

we investigate various subrootsystems that arise within H4. These groups and

root systems are of course separately invariant within H4, and the 120 vertices

of the H4root system can be split into two separately invariant sets by

taking the complement of the subrootsystem in H4. Conway and Guy found

the Grand Antiprism computationally 55 years ago this year [5]. Conway

and Michael Guy’s father, Richard Guy, himself a famous mathematician,

populariser of mathematics and collaborator of Conway, have unfortunately

passed away this year (2020). This paper is dedicated to their memory. The

construction and the symmetries of the Grand Antiprism actually beneﬁt

from the construction from 3D as noted in [7]. In this paper we follow in

detail how the H2×H2subgroup arises naturally within H4in the induction

process. This subrootsystem is then used to split the 120 vertices of H4

into the set of 20 roots of H2⊕H2and the 100 vertices of its complement,

which is exactly the Grand Antiprism. A completely analogous construction

works in a uniform way for other subrootsystems of H4that arise via the 3D

construction, either as subrootsystems of H3or even subgroups of 2I.The

analogous cases include D4and the snub 24-cell, A4

1,A2⊕A2and A4.

We organise this paper as follows. We review some basics of Cliﬀord

algebras, reﬂection groups and root systems in Sect. 2, leading to the Versor

Theorem and the Induction Theorem. In Sect. 3we build on the Versor Theo-

rem to set up a framework for explicit group theoretic calculations, including

the construction of the Pin(H3) and Spin(H3) groups, and discussion of their

conjugacy classes. Subrootsystems can arise either as even subgroups of the

spinor group (here the binary icosahedral group 2I), via subrootsystems of

H3or generated via the inversion e1e2e3, which is discussed in Sects. 4and 5,

respectively. The Coxeter plane is a convenient way to visualise root systems

and other polytopes in any dimension, and the calculations in this case can

also be entirely performed within the even subalgebra, as shown in Sect. 6.

The next sections give detailed results for splitting the H4root system with

respect to various subrootsystems, yielding complementary pairs of invariant

polytopes, including H2⊕H2and the Grand Antiprism in Sect. 7,D4and the

snub 24-cell in Sect. 8, and examples from A4

1,A2⊕A2and A4in Sect. 9.In

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57 Page 4 of 35 P. Dechant Adv. Appl. Cliﬀord Algebras

Sect. 10 we conclude with a few words about the insights that can be gained

from this novel approach via induction from 3D.

2. Root System Induction

The setting for reﬂection groups and root systems stipulates the existence of

a vector space with an inner product, so without loss of generality one can

construct the corresponding Cliﬀord algebra. The Cliﬀord algebraic frame-

work used here is very standard, though we broadly follow [14,17]. We deﬁne

an algebra product via the geometric product xy =x·y+x∧y, where

the inner product (given by the symmetric bilinear form) is the symmetric

part x·y=1

2(xy +yx), and the wedge/exterior product the antisymmetric

part x∧y=1

2(xy −yx). These relations also mean that parallel vectors com-

mute whilst orthogonal vectors anticommute. The full 2n-dimensional algebra

is generated via this geometric product, linearity and associativity. For our

purposes we only consider the Cliﬀord algebra of 3D Cl(3) generated by three

orthogonal unit vectors e1,e2and e3, though some of the following statements

hold under very general conditions. This yields an eight-dimensional vector

space generated by the elements

{1}

1 scalar

{e1,e

2,e

3}

3 vectors

{e1e2=Ie3,e

2e3=Ie1,e

3e1=Ie2}

3 bivectors

{I≡e1e2e3}

1 trivector

,

with an even subalgebra consisting of the scalar and bivectors, which is 4-

dimensional. Note that for the orthogonal unit vectors e.g. e1e2=e1∧e2.

We follow the galgebra L

A

T

EX output, which has the wedge version.

Root systems and in particular their simple roots are convenient objects

to characterise reﬂection and Coxeter groups. We therefore brieﬂy introduce

the relevant terminology here:

Deﬁnition 2.1. (Root system)Aroot sy ste m is a collection Φ of non-zero

(root) vectors αthat span an n-dimensional Euclidean vector space V

endowed with a positive deﬁnite bilinear form, that satisﬁes the two axioms:

1. Φ only contains a root αand its negative, but no other scalar multiples:

Φ∩Rα={−α, α}∀α∈Φ.

2. Φ is invariant under all reﬂections corresponding to root vectors in Φ:

sαΦ=Φ∀α∈Φ. The reﬂection sαin the hyperplane with normal

vector αis given by

sα:x→sα(x)=x−2(x·α)

(α·α)α,

where (x·y) denotes the inner product on V.

Unlike other popular conventions here we assume unit normalisation for

all our root vectors for later convenience.

Proposition 2.2. (Reﬂection) In Cliﬀord algebra, the reﬂection formula sim-

pliﬁes to

sα:x→sα(x)=−αxα

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Vol. 31 (2021) Cliﬀord Spinors and Root System Page 5 of 35 57

for normalised root vectors, so that double-sided (‘sandwich’)application of

a root vector generates the corresponding reﬂection.

Proof. Using the Cliﬀord form for the inner product x·y=1

2(xy +yx)inthe

(simple) reﬂection formula in Deﬁnition 2.1 si:x→si(x)=x−2(x·α)

(α·α)αand

assuming unit normalisation of roots αi·αi= 1 yields the much simpliﬁed

version

si:x→si(x)=x−2·1

2(xαi+αix)αi=x−xα2

i−αixαi=−αixαi.

Proposition 2.3. (Double cover) In Cliﬀord algebra, the reﬂections αand −α

doubly cover the same reﬂection sα.

Proof. Straightforward, since due to the double-sided application the signs

cancel out.

A subset Δ of Φ, called simple roots α1,...,α

nis suﬃcient to express

every element of Φ via linear combinations with coeﬃcients of the same sign.

For a crystallographic root system, these are Z-linear combinations, whilst for

the non-crystallographic root systems one needs to consider certain extended

integer rings. For instance for H2,H3and H4one has the extended integer

ring Z[τ]={a+τb|a, b ∈Z}, where τis the golden ratio τ=1

2(1+√5) =

2cos π

5,andσis its Galois conjugate σ=1

2(1 −√5) (the two solutions to

the quadratic equation x2=x+ 1), and linear combinations are with respect

to this Z[τ]. This integrality property of the crystallographic root systems

(types A-G) leads to an associated lattice which acts as a root lattice for Lie

algebras, which are named accordingly. In contrast, no such lattice exists for

the non-crystallographic groups (types Hand I), which accordingly do not

have associated Lie algebras, and are perhaps less familiar as a result.

The reﬂections corresponding to simple roots are also called simple

reﬂections. The geometric structure of the set of simple roots encodes the

properties of the reﬂection group and is summarised in the Cartan matrix and

Coxeter–Dynkin diagrams, which contain the geometrically invariant infor-

mation of the root system as follows:

Deﬁnition 2.4. (Cartan matrix and Coxeter–Dynkin diagram)TheCartan

matrix of a set of simple roots αi∈Δ is deﬁned as the matrix

Aij =2

(αi·αj)

(αi·αi).(2.1)

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

3have a simple link, and other angles π

mhave a

link with a label m.

Example. The Cartan matrices for H3and H4are respectively given by

A(H3)=⎛

⎝

2−10

−12−τ

0−τ2

⎞

⎠,A(H4)=⎛

⎜

⎜

⎝

2−10 0

−12−10

0−12−τ

00−τ2

⎞

⎟

⎟

⎠.

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57 Page 6 of 35 P. Dechant Adv. Appl. Cliﬀord Algebras

Possible choices of simple roots are e.g.

a1=e2,a

2=1

2(−τe1−e2−(τ−1)e3),a

3=e1

and

α1=1

2(τe1−e2+(τ−1)e4),α

2=e2,

α3=−1

2((τ−1)e1+e2+τe3)andα4=e3.

Remark 2.5. What is a slight drawback of the diagrammatic approach is that

it is rather unobvious which subgroups are contained e.g. in H4–atleast

exhaustively. We will see some non-obvious examples later on.

Root systems and simple roots are therefore convenient paradigms for

considering reﬂection groups: each root vector deﬁnes a hyperplane that it is

normal to and thereby acts as a generator of a reﬂection in that hyperplane. In

Cliﬀord algebra this root vector in fact directly acts as a reﬂection generator

via the geometric product. Multiplying together such simple reﬂections si:

x→si(x)=x−2(x|αi)

(αi|αi)αi=−αixαitherefore generates a reﬂection group.

This is in fact a Coxeter group, since the simple reﬂections sisatisfy the

deﬁning relations:

Deﬁnition 2.6. (Coxeter group)ACoxeter group is a group generated by

a set of involutory generators si,s

j∈Ssubject to relations of the form

(sisj)mij = 1 with mij =mji ≥2fori=j.

Deﬁnition 2.7. (Coxeter element and Coxeter number) The product of all the

simple reﬂections in some order is called a Coxeter element. All such elements

are conjugate and as such their order is well-deﬁned and called the Coxeter

number.

These reﬂection groups are built up in Cliﬀord algebra by performing

successive multiplication with the unit vectors deﬁning the reﬂection hyper-

planes via ‘sandwiching’

s1...s

k:x→s1...s

k(x)=(−1)kα1...α

kxαk...α

1=: (−1)kAx ˜

A, (2.2)

where the tilde denotes the reversal of the order of the constituent vectors

in the product A=α1...α

k. In order to study the groups of reﬂections one

therefore only needs to consider products of root vectors in the Cliﬀord alge-

bra, which form a multivector group under the geometric product and yield

a Pin double cover of the corresponding reﬂection group [29]. The inverse of

each group element is of course simply given by the reversal, because of the

assumed normalisation condition. Since αiand −αiencode the same reﬂec-

tion, products of unit vectors are double covers of the respective orthogonal

transformation, as Aand −Aencode the same transformation. We call even

products R, i.e. products of an even number of vectors, ‘spinors’ or ‘rotors’,

and a general product A‘versors’ or ‘pinors’. They form the Pin group and

constitute a double cover of the orthogonal group, whilst the even prod-

ucts form the double cover of the special orthogonal group, called the Spin

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Vol. 31 (2021) Cliﬀord Spinors and Root System Page 7 of 35 57

group. Cliﬀord algebra therefore provides a particularly natural and simple

construction of the Spin groups.

In fact, more general groups can be constructed in this way because

of the fundamental importance of reﬂections according to the Cartan–

Dieudonn´e Theorem [15].

Theorem 2.8. (Cartan–Dieudonn´e Theorem) Every orthogonal transforma-

tion in an n-dimensional symmetric bilinear space can be described as the

composition of at most nreﬂect ion s.

The above approach to group theory via multivector groups (2.2)is

therefore a much more general way of doing group theory.

In Cliﬀord algebra, instead of using matrices to perform linear transfor-

mations one can use spinors/rotors/pinors/versors to perform linear transfor-

mations that leave the inner product invariant i.e. orthogonal transformations

[18–20]. Here the normalisation condition has been dropped as long as the

vectors are non-null since the inverse of multiplication with a non-null vector

xis simply x−1=x

|x|2since xx =x·x=|x|2(in the positive signature spaces

we will consider there are no null vectors anyway). Therefore the multivector

Athat is a product of vectors is invertible and preserves the inner product,

though the inverse is no longer just given by the reverse:

Theorem 2.9. (Versor Theorem) Every orthogonal transformation Acan be

expressed in the canonical form A:x→x=A(x)=±A−1xA where Ais a

versor and the sign is its parity.

A concept familiar from abstract group theory via generators and rela-

tions is that group elements can be written as words in the generators. It

is noteworthy here that in contrast to for instance the Coxeter group ele-

ments as words in the generators sα, in the Cliﬀord algebra approach the

root vectors are directly generators for the Pin double cover under multipli-

cation with the geometric product. To stress this slight distinction we call

these ‘generator paths’ for each versor v, with ±veach versor corresponding

to one Coxeter group word.

Therefore Cliﬀord algebras and root systems are frameworks that per-

fectly complement each other since performing reﬂections in Cliﬀord algebras

is so simple and only assumes the structure of a vector space with an inner

product that is already given in the root system deﬁnition. Therefore Clif-

ford algebras are perhaps the most natural framework for studying reﬂection

groups and root systems, and through the above arguments also more general

groups [9]. In the next section we perform a detailed computation of the Pin

and Spin covers of the icosahedral groups to illustrate the principles. Fol-

lowing the above two frameworks, given (simple) roots in a root system one

can start multiplying these together using the geometric product. General

products will be in Pin whilst even products are in Spin. For now we will

concentrate on the Spin and the even subalgebra.

Proposition 2.10. (O(4)-structure of spinors) The space of Cl(3)-spinors has

a 4D Euclidean structure.

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57 Page 8 of 35 P. Dechant Adv. Appl. Cliﬀord Algebras

Proof. For a spinor R=a0+a1e2e3+a2e3e1+a3e1e2, the norm is given by

R˜

R=a2

0+a2

1+a2

2+a2

3, and the inner product between two spinors R1and

R2is (R1,R

2)=1

2(R1˜

R2+R2˜

R1).

Remark 2.11. For rotors, the inner product 1

2(R1˜

R2+R2˜

R1)isofcourse

invariant under Ri→RRi˜

Rbut also just under Ri→RRi.

From the double cover property of Proposition 2.3 we have the following

corollary:

Corollary 2.12. (Discrete spinor groups) Discrete spinor groups are of even

order since if a spinor Ris contained in the group then so is −Rsince it

encodes the same orthogonal transformation.

Proposition 2.13. (Spin group closure properties) Spin groups are closed

under:

•multiplication using the geometric product

•reversal

•multiplication by −1

Proof. Straightforward:

•by deﬁnition of multivector groups via the geometric product

•by the inverse element group axiom since reversal is equivalent to the

inverse

•by Corollary 2.12 both Rand −Rare contained in the group

Following the formula for fundamental reﬂections from Deﬁnition 2.1

one can likewise deﬁne reﬂections on this spinor space with respect to the

inner product between spinors.

Proposition 2.14. (Spin reﬂections) Reﬂections between spinors using the

spinor inner product are given by

R2→R

2=−R1˜

R2R1.

Proof. In analogy to Proposition 2.2, for normalised spinors R1and R2

and using the deﬁnition of the spinor inner product from Proposition 2.10

this amounts to R2→R

2=R2−2(R1,R

2)/(R1,R

1)R1=R2−(R1˜

R2+

R2˜

R1)R1=R2−R1˜

R2R1−R2˜

R1R1=−R1˜

R2R1.

Proposition 2.15. (3D spinor–4D vector correspondence) Spinor reﬂections

in the spinor R=a0+a1e2∧e3−a2e1∧e3+a3e1∧e2are equivalent to 4D

reﬂections in the 4D vector (a0,a

3,−a2,a

1).

Proof. By direct calculation or see supplementary material.

In practice, the exact mapping of the components is a matter of con-

vention and often irrelevant, since most root systems contain roots up to ±

and (cyclic) permutations anyway.

Theorem 2.16. (Induction Theorem) Any rank-3 root system induces a root

system of rank 4.

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Vol. 31 (2021) Cliﬀord Spinors and Root System Page 9 of 35 57

Table 1. List of the rank 3 root systems and their induced

root systems in four dimensions, as well as the binary poly-

hedral groups that act as the spinor group intermediaries

Start root system Induced root system Binary polyhedral group

A3

1A4

1Q

A1⊕I2(n)I2(n)⊕I2(n)2,2,n

A3D42T

B3F42O

H3H42I

Proof. A root system in three dimensions Φ(3) gives rise to a group of spinors

by taking even products of the root vectors. From Corollary 2.12, this group

contains −Rif it contains R, and therefore satisﬁes the ﬁrst root system

axiom from Deﬁnition 2.1. The set of spinors has a 4D Euclidean structure

by Proposition 2.10 and can thus be treated as a collection of 4D vectors Φ(4)

with the inner product as given in the Proposition. It remains to show that

this collection of vectors Φ(4) is invariant under reﬂections (axiom 2), which

is satisﬁed by Proposition 2.3.

Closure of the root system is thus ensured by closure of the spinor group.

This also has very interesting consequences for the automorphism group of

these spinorial root systems, which contains two factors of the spinor group

acting from the left and the right [7] (in this sense, the above closure under

reﬂections amounts to a certain twisted conjugation).

There is a limited number of cases which we can just enumerate. The

3D root systems are listed in Table 1along with the 4D root systems that

they induce as well as the intermediate spinor groups (the binary polyhedral

groups). In this article, we focus on the case H3→H4.

Deﬁnition 2.17. (Subrootsystem) By a subrootsystem Φ1of a root system Φ2

we mean a subset Φ1of the collection of vectors Φ2that itself satisﬁes the

root system axioms.

From the Induction Theorem 2.16 we immediately get:

Corollary 2.18. (Induced subrootsystems) A subrootsystem Φ(3)

1of a root sys-

tem Φ(3)

2induces a subrootsystem Φ(4)

1of the induced root system Φ(4)

2.

Any subrootsystem of H3therefore induces a subrootsystem of H4.For

instance, the A3

1inside H3induces the rather boring A4

1in H4. Similarly, A2

and H2are contained, if rather boring as 2D root systems. If H3contained

A1⊕A2and A1⊕H2subrootsytems (which it doesn’t), then this would lead

to the doubling A2⊕A2and H2⊕H2inside the H4. This is not quite the

case, but nearly so, which we will return to later. Similarly, it follows from

the Induction Theorem that any even subgroup of a spinor group will also

yield a subrootsystem. We will explore these points in the Sects. 4and 5

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57 Page 10 of 35 P. Dechant Adv. Appl. Cliﬀord Algebras

which follow the next section, where we will discuss the multivector group

calculation framework.

3. Pin Group and Spin Group

In this section we pick up on the practical implications of the versor Theorem

2.9 and the Cartan–Dieudonn´e theorem in order to do explicit calculations in

group theory in the Cliﬀord algebra approach. We use the concrete example of

the H3reﬂection/Coxeter group. The icosahedral rotation group (in SO(3))

is the alternating group A5of order 60, also known simply as I(which we will

avoid due to the pseudoscalar often being denoted by that too). This group

is of course doubly covered in Spin(3) by its spin double cover, with its nice

Cliﬀord algebra construction via the reﬂection formula in Proposition 2.2.We

might denote this group by Spin(H3) but it is also commonly known as the

binary icosahedral group 2I. From the induction theorem 2.16 of the previous

section the elements of this group of course give the 120 roots of the H4root

system. The rotational group is of course also doubly covered in O(3) by its

double cover H3=A5×Z2. Both double covers are of course of order 120.

H3however is itself also doubly covered in Pin(3) by a group of order 240,

which doesn’t have a common name since it is simply H3×Z2=A5×Z2×Z2

but which we might for consistency call Pin(H3).

For the reader’s convenience and for reproducibility in Tables 2,3,4,5

we list the diﬀerent group elements of Pin(H3) in our Cliﬀord approach explic-

itly. The python Jupyter notebooks in the supplementary material contain

the algorithms used, which are based on the galgebra software package [3].

For convenience we group the elements as whole conjugacy classes but we

give the order in which the elements are generated by repeated application

of the generating simple roots. This gives a number for each group element

as a reference for ease of access, as well as the word in the generators (the

‘generator path’) that generates this particular group element in terms of the

H3simple roots/generators a1,a

2,a

3. For the simple roots of H3we pick

a1=e2,a

2=1

2(−τe1−e2−(τ−1)e3),a

3=e1.

Note that the wedge could be omitted since we have picked the orthogo-

nal unit vectors e1,e

2,e

3so the wedge product is synonymous with the full

geometric product. We multiply all group elements by 2 to save clutter.

The spin group Spin(3) = 2Iis given in Tables 2and 3. Its nine conju-

gacy classes lead to irreducible representations of dimensions 1,3,3,4,5which

are shared by A5, as well as the spinorial ones of dimensions 2,2,4,6. (Inter-

esting connections with the binary polyhedral groups and the McKay corre-

spondence [27] are explored elsewhere [12]). Since Pin(H3) = Spin(H3)×Z2,

the remaining Tables 4and 5list the remaining 9 conjugacy classes achieved

by multiplying those of Spin(H3) with the inversion e1e2e3.

Tables 2and 3: The only normal subgroup of 2Iconsists of the ﬁrst two

conjugacy classes i.e. ±1. We note that the conjugacy class of order 4 consists

of thirty pure bivectors, and that they give rise to the 2-fold rotations of the

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Vol. 31 (2021) Cliﬀord Spinors and Ro ot System Page 11 of 35 57

Table 2. The ﬁrst set of the conjugacy classes of Spin(H3),

the ones with orders of ‘crystallographic type’ 1,2,3,4,6.

These conjugacy classes all contain their own reverses i.e.

inverses. The only normal subgroup consists of the ﬁrst two

conjugacy classes and is ±1, and the order 4 conjugacy class

consists of pure bivectors

Order Number Element ×2 Generator path

14 2 11

226 −2 1313

35 −1+τe1∧e2+σe2∧e312

37 −1−τe1∧e2−σe2∧e321

336 −1−τe1∧e2+σe2∧e33123

338 −1+τe1∧e2−σe2∧e33213

382 −1+e1∧e2+e1∧e3−e2∧e3231232

383 −1−e1∧e2−e1∧e3+e2∧e3232132

3 124 −1−e1∧e2+e1∧e3+e2∧e312312321

3 125 −1+e1∧e2−e1∧e3−e2∧e312321321

3 128 −1−e1∧e2+e1∧e3−e2∧e312323123

3 131 −1−e1∧e2−e1∧e3−e2∧e313213232

3 134 −1−σe1∧e2+τe1∧e321321323

3 137 −1+e1∧e2+e1∧e3+e2∧e323213213

3 141 −1+σe1∧e2−τe1∧e332132132

3 143 −1+e1∧e2−e1∧e3+e2∧e332132321

3 170 −1+σe1∧e2+τe1∧e31213213231

3 172 −1−σe1∧e3−τe2∧e31213231232

3 174 −1+σe1∧e3−τe2∧e31232132312

3 177 −1−σe1∧e2−τe1∧e31321321321

3 184 −1−σe1∧e3+τe2∧e32132312321

3 187 −1+σe1∧e3+τe2∧e32321323121

46 −2e1∧e213

49 2e1∧e231

430 τe1∧e2−σe1∧e3−e2∧e32132

431 −τe1∧e2+σe1∧e3+e2∧e32312

466 −τe1∧e2−σe1∧e3+e2∧e3121321

468 τe1∧e2+σe1∧e3−e2∧e3123121

489 −τe1∧e2+σe1∧e3−e2∧e3321323

491 τe1∧e2−σe1∧e3+e2∧e3323123

4 129 −τe1∧e2−σe1∧e3−e2∧e313213213

4 130 τe1∧e2+σe1∧e3+e2∧e313213231

4 139 e1∧e2+τe1∧e3+σe2∧e323213232

4 140 −e1∧e2−τe1∧e3−σe2∧e323231232

4 175 −e1∧e2+τe1∧e3−σe2∧e31232132321

4 176 e1∧e2−τe1∧e3+σe2∧e31232312321

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57 Page 12 of 35 P. Dechant Adv. Appl. Cliﬀord Algebras

Table 2. continued

Order Number Element ×2 Generator path

4 181 −σe1∧e2+e1∧e3−τe2∧e32132132132

4 182 σe1∧e2−e1∧e3+τe2∧e32132132312

4 186 −e1∧e2−τe1∧e3+σe2∧e32321321323

4 188 e1∧e2+τe1∧e3−σe2∧e32321323123

4 209 σe1∧e2+e1∧e3+τe2∧e3121321321321

4 211 −σe1∧e2−e1∧e3−τe2∧e3121321323121

4 213 −e1∧e2+τe1∧e3+σe2∧e3123213213213

4 214 e1∧e2−τe1∧e3−σe2∧e3123213213231

4 221 σe1∧e2−e1∧e3−τe2∧e3321321321323

4 222 −σe1∧e2+e1∧e3+τe2∧e3321321323123

4 233 −2e2∧e312132132132132

4 234 2e2∧e312132132132312

4 235 −σe1∧e2−e1∧e3+τe2∧e312321321321323

4 236 σe1∧e2+e1∧e3−τe2∧e312321321323123

4 237 2e1∧e321321321321323

4 238 −2e1∧e321321321323123

628 1+τe1∧e2−σe2∧e31323

632 1+τe1∧e2+σe2∧e32313

637 1−τe1∧e2−σe2∧e33132

639 1−τe1∧e2+σe2∧e33231

681 1−e1∧e2−e1∧e3+e2∧e3213232

684 1+e1∧e2+e1∧e3−e2∧e3232312

6 123 1 + e1∧e2−e1∧e3−e2∧e312132321

6 126 1 + e1∧e2−e1∧e3+e2∧e312321323

6 127 1 −e1∧e2+e1∧e3+e2∧e312323121

6 132 1 + e1∧e2+e1∧e3+e2∧e313231232

6 136 1 + σe1∧e2−τe1∧e321323123

6 138 1 −e1∧e2−e1∧e3−e2∧e323213231

6 142 1 −σe1∧e2+τe1∧e332132312

6 144 1 −e1∧e2+e1∧e3−e2∧e332312321

6 169 1 −σe1∧e2−τe1∧e31213213213

6 171 1 + σe1∧e3+τe2∧e31213213232

6 173 1 −σe1∧e3+τe2∧e31232132132

6 179 1 + σe1∧e2+τe1∧e31321323121

6 183 1 + σe1∧e3−τe2∧e32132132321

6 185 1 −σe1∧e3−τe2∧e32321321321

icosahedron around its 30 edges. The 20 3-fold rotations around the 20 trian-

gular faces are split into two conjugacy classes of orders 3 and 6, which are

related by multiplication by −1, and each contain their own reverse/inverse.

Similarly, the two sets of twelve 5-fold rotations are doubly covered by 4

conjugacy classes which are related by multiplication by −1. The two classes

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Vol. 31 (2021) Cliﬀord Spinors and Ro ot System Page 13 of 35 57

Table 3. The second set of conjugacy classes of Spin(H3),

the ones with orders of ‘non-crystallographic type’ i.e. the

ones related to 5-fold symmetry. These conjugacy classes

also all contain their inverses. The ﬁrst column denotes the

order of the elements in each conjugacy class. The second

column is the position in the order in which our algorithm

generates this element, for convenience (c.f. the supplemen-

tary material). The ﬁnal column denotes the order in which

the generators with the corresponding labels are applied to

generate this group element, i.e. is eﬀectively the ‘word in

the generators’ that yields this element. To avoid confusion,

pairs of such words doubly cover the rotations of the icosa-

hedral group A5, which are also often considered in terms

of words in the A5generators. The ones meant here are the

root vectors multiplied by using the geometric product

Order Number Element ×2 Generator path

58 −τ+e1∧e2−σe1∧e323

510 −τ−e1∧e2+σe1∧e332

523 −τ−e1∧e2−σe1∧e31231

527 −τ+e1∧e2+σe1∧e31321

565 −τ−σe1∧e2−e2∧e3121312

567 −τ+e1∧e3+σe2∧e3121323

572 −τ+σe1∧e2−e2∧e3123232

575 −τ−e1∧e3+σe2∧e3132312

577 −τ+σe1∧e2+e2∧e3213121

580 −τ+e1∧e3−σe2∧e3213231

585 −τ−σe1∧e2+e2∧e3232321

590 −τ−e1∧e3−σe2∧e3323121

534 −σ−τe1∧e2−e1∧e32323

540 −σ+τe1∧e2+e1∧e33232

571 −σ+τe1∧e2−e1∧e3123231

576 −σ−τe1∧e2+e1∧e3132321

5 122 −σ−e1∧e2+τe2∧e312132312

5 135 −σ+e1∧e2−τe2∧e321323121

5 178 −σ+e1∧e2+τe2∧e31321321323

5 190 −σ−e1∧e2−τe2∧e33213213231

5 212 −σ−τe1∧e3+e2∧e3121321323123

5 215 −σ+τe1∧e3+e2∧e3132132132132

5 217 −σ−τe1∧e3−e2∧e3213213213213

5 220 −σ+τe1∧e3−e2∧e3232132132312

10 22 τ+e1∧e2+σe1∧e31213

10 24 τ+σe1∧e2+e2∧e31232

10 25 τ+e1∧e2−σe1∧e31312

10 29 τ−e1∧e2+σe1∧e32131

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57 Page 14 of 35 P. Dechant Adv. Appl. Cliﬀord Algebras

Table 3. continued

Order Number Element ×2 Generator path

10 33 τ−σe1∧e2−e2∧e32321

10 35 τ−e1∧e2−σe1∧e33121

10 69 τ−e1∧e3−σe2∧e3123123

10 74 τ+e1∧e3−σe2∧e3132132

10 79 τ−e1∧e3+σe2∧e3213213

10 87 τ−σe1∧e2+e2∧e3312323

10 88 τ+e1∧e3+σe2∧e3321321

10 92 τ+σe1∧e2−e2∧e3323213

10 70 σ−τe1∧e2+e1∧e3123213

10 73 σ−τe1∧e2−e1∧e3131232

10 78 σ+τe1∧e2+e1∧e3213123

10 86 σ+τe1∧e2−e1∧e3312321

10 121 σ+e1∧e2−τe2∧e312132132

10 133 σ−e1∧e2+τe2∧e321321321

10 180 σ−e1∧e2−τe2∧e31321323123

10 189 σ+e1∧e2+τe2∧e33213213213

10 210 σ+τe1∧e3−e2∧e3121321321323

10 216 σ−τe1∧e3−e2∧e3132132132312

10 218 σ+τe1∧e3+e2∧e3213213213231

10 219 σ−τe1∧e3+e2∧e3232132132132

describe rotations by ±2π/5and±4π/5, respectively, around the 5-fold axes

of symmetry, the icosahedral vertices.

Tables 4and 5: The ﬁrst two conjugacy classes are the inversion and

its negative. The conjugacy class consisting of pure vectors of course corre-

sponds to the 30 roots of H3which generate the reﬂections, and which are

of course related to the 30 2-fold rotations since the inversion is contained in

the group (so one can dualise a (root) vector to a pure bivector). The con-

jugacy classes of order 12 are the two inversion-related versions of the 3-fold

rotations, and are rotoreﬂections. The four conjugacy classes of order 20 are

both related to the 5-fold rotations, as well as serving as the versor analogues

of the Coxeter elements e.g. w=a1a2a3. These are in one conjugacy class in

the reﬂection/Coxeter group framework where their order gives the Coxeter

number. But in this Cliﬀord double cover setup these ‘Coxeter versors’ are

given in 4 conjugacy classes that are related by reversal and multiplication

by e1e2e3.

Remark 3.1. It has been noted that a1a2and a2a3generate the quaternionic

root system multiplicatively e.g. for H4. This is pretty obvious when thought

of in terms of the 3D simple roots and the Induction Theorem, as they of

course generate Spin(H3), which gives rise to the H4root system.

This example illustrates how one can perform practical computations in

group theory via versors in this Cliﬀord algebra framework, and in galgebra

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Vol. 31 (2021) Cliﬀord Spinors and Ro ot System Page 15 of 35 57

in particular. We will discuss group and representation theoretic aspects in

more detail elsewhere.

4. Subgroups

The Induction Theorem 2.16 from Sect. 2showed that every 3D root system

determines a 4D root system. This proceeded essentially via using the 3D

roots to construct a group of spinors (via multiplication with the geometric

product), which satisﬁes the properties of a root system. Our main example

is of course H3which induces H4in four dimensions via the binary icosa-

hedral group of order 120 as the spinor group intermediary. It is therefore a

straightforward corollary of the Induction Theorem that each even subgroup

of 2Ialso yields a root system.

Corollary 4.1. (Subgroups of 2I)Each even subgroup Gof the binary icosa-

hedral group 2Idetermines a corresponding root system Φthat is a subset of

the H4root system, the 600-cell.

Theorem 4.2. (Induced subrootsystems of H4)The binary icosahedral group

2Ihas the following subgroups that determines the corresponding root sys-

tems:

•The normal subgroup ±1which gives A1.

•The quaternion group Qconsisting of ±1,±e1e2,±e2e3and ±e3e1,

which gives A1×A1×A1×A1.

•The binary dihedral groups of orders 6and 10, which yield A2and H2.

•The binary tetrahedral group, which yields D4.

Remark 4.3. Note that although the A3root system is not contained in H3,

2Tis a subgroup of 2Iand therefore D4is contained in H4. We will revisit

these examples in later sections and in the next section investigate this deli-

cate relationship between subgroups and other subrootsystems further.

Proposition 4.4. (Simple roots of induced subrootsystems) A2and H2are

generated straightforwardly from the H3generators a1,a

2,a

3e.g. via the

‘spinorial simple roots’ a1a1=1and a1a2for A2and a1a1=1and a2a3

for H2.

One possible choice of simple roots for D4contained in H4is given by

(a1a1,a

1a2,a

1a2a3a2a3a1a2a3,a

3a2a1a3a2a1a3a2),

but it is of course not unique.

Explicit versions of these simple roots can be looked up in the earlier

tables via the ‘generator path’, which can be used to explicitly verify the

correct Cartan matrix and closure of the root system (see e.g. supplementary

information).

5. Subrootsystems

There is a subtlety at play here since there are subrootsystems of H4that

are neither induced by 3D subrootsystems nor by even spinor subgroups. We

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57 Page 16 of 35 P. Dechant Adv. Appl. Cliﬀord Algebras

Table 4. The ﬁrst set of the remaining conjugacy classes of

Pin(H3). Note that the ﬁrst two conjugacy classes show that

the inversion e1e2e3is contained in the group. The class of

order 2 consists of pure vectors and are thus their own reverse

i.e. inverse. The other pair are each other’s reverses/inverses

Order Number Element ×2 Generator path

4 240 2e1∧e2∧e3121321321323123

4 239 −2e1∧e2∧e3121321321321323

21 2e21

22 −τe1−e2+σe32

23 2e13

211 τe1−e2−σe3121

213 −2e1131

217 −σe1+τe2+e3232

219 −2e2313

221 −τe1+e2−σe3323

244 −τe1+e2+σe312313

245 σe1+τe2−e312321

248 τe1−e2+σe313123

249 τe1+e2−σe313132

250 −τe1−e2−σe313213

251 τe1+e2+σe313231

253 σe1−τe2−e321312

260 −σe1−τe2−e323232

293 −σe1−τe2+e31213121

2 101 σe1−τe2+e31232321

2 103 σe1+τe2+e31312323

2 108 −σe1+τe2−e31323213

2 109 −e1+σe2−τe32132132

2 110 e1−σe2+τe32132312

2 145 e1+σe2+τe3121321321

2 147 −e1−σe2−τe3121323121

2 166 −e1−σe2+τe3321321323

2 168 e1+σe2−τe3321323123

2 199 −e1+σe2+τe313213213213

2 200 e1−σe2−τe313213213231

2 229 −2e32132132132132

2 230 2e32132132132312

12 52 −τe1−σe2−e1∧e2∧e313232

12 55 −τe2+σe3−e1∧e2∧e321323

12 58 τe1+σe2−e1∧e2∧e323213

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Table 4. continued

Order Number Element ×2 Generator path

12 62 τe2−σe3−e1∧e2∧e332132

12 64 τe1−σe2−e1∧e2∧e332321

12 94 −τe1+σe2−e1∧e2∧e31213123

12 96 −τe2−σe3−e1∧e2∧e31213231

12 98 −e1−e2−e3−e1∧e2∧e31231232

12 100 −e1+e2−e3−e1∧e2∧e31232312

12 104 τe2+σe3−e1∧e2∧e31321321

12 112 e1−e2+e3−e1∧e2∧e32312321

12 115 e1+e2+e3−e1∧e2∧e32323121

12 116 e1−e2−e3−e1∧e2∧e32323123

12 120 −e1+e2+e3−e1∧e2∧e33231232

12 149 −e1−e2+e3−e1∧e2∧e3123213213

12 156 e1+e2−e3−e1∧e2∧e3132312321

12 191 σe1−τe3−e1∧e2∧e312132132132

12 201 −σe1+τe3−e1∧e2∧e321321321321

12 228 σe1+τe3−e1∧e2∧e31321321323123

12 232 −σe1−τe3−e1∧e2∧e32321321323123

12 46 τe1−σe2+e1∧e2∧e312323

12 57 τe2−σe3+e1∧e2∧e323123

12 59 −τe1−σe2+e1∧e2∧e323231

12 61 τe1+σe2+e1∧e2∧e331232

12 63 −τe2+σe3+e1∧e2∧e332312

12 95 τe2+σe3+e1∧e2∧e31213213

12 97 e1+e2+e3+e1∧e2∧e31213232

12 99 e1−e2+e3+e1∧e2∧e31232132

12 102 −τe1+σe2+e1∧e2∧e31312321

12 106 −τe2−σe3+e1∧e2∧e31323121

12 111 −e1+e2−e3+e1∧e2∧e32132321

12 113 −e1−e2−e3+e1∧e2∧e32321321

12 114 −e1+e2+e3+e1∧e2∧e32321323

12 119 e1−e2−e3+e1∧e2∧e33213232

12 150 e1+e2−e3+e1∧e2∧e3123213231

12 155 −e1−e2+e3+e1∧e2∧e3132132321

12 192 −σe1+τe3+e1∧e2∧e312132132312

12 203 σe1−τe3+e1∧e2∧e321321323121

12 227 −σe1−τe3+e1∧e2∧e31321321321323

12 231 σe1+τe3+e1∧e2∧e32321321321323

have observed in the spinor Induction Theorem that A1⊕I2(n) root systems

experience a doubling to I2(n)⊕I2(n) in the induction process. Indeed, such

A2⊕A2and H2⊕H2within H4are induced, but not because they have an

orthogonal A1. Instead, it is because the group H3contains the inversion,

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57 Page 18 of 35 P. Dechant Adv. Appl. Cliﬀord Algebras

Table 5. The four conjugacy classes of order 20 are related

to the 5-fold rotations in Spin(H3) via multiplication by

e1e2e3. They are also the spinor version of the Coxeter ele-

ments e.g. w=a1a2a3. They are related by reversal and

multiplication by e1e2e3

Order Number Element ×2 Generator path

20 14 e1−τe2−σe1∧e2∧e3132

20 15 −e1+τe2−σe1∧e2∧e3213

20 20 −e1−τe2−σe1∧e2∧e3321

20 41 e1+τe2−σe1∧e2∧e312131

20 43 τe1+e3−σe1∧e2∧e312312

20 56 −τe1−e3−σe1∧e2∧e323121

20 105 τe1−e3−σe1∧e2∧e31321323

20 118 −τe1+e3−σe1∧e2∧e33213231

20 159 e2+τe3−σe1∧e2∧e3213213232

20 162 −e2−τe3−σe1∧e2∧e3232132312

20 193 e2−τe3−σe1∧e2∧e312132132321

20 197 −e2+τe3−σe1∧e2∧e312321323121

20 12 −e1−τe2+σe1∧e2∧e3123

20 16 e1−τe2+σe1∧e2∧e3231

20 18 −e1+τe2+σe1∧e2∧e3312

20 42 −τe1−e3+σe1∧e2∧e312132

20 47 e1+τe2+σe1∧e2∧e313121

20 54 τe1+e3+σe1∧e2∧e321321

20 107 −τe1+e3+σe1∧e2∧e31323123

20 117 τe1−e3+σe1∧e2∧e33213213

20 160 −e2−τe3+σe1∧e2∧e3213231232

20 161 e2+τe3+σe1∧e2∧e3232132132

20 194 −e2+τe3+σe1∧e2∧e312132312321

20 195 e2−τe3+σe1∧e2∧e312321321321

20 146 σe1−e2−τe1∧e2∧e3121321323

20 151 −e1+σe3−τe1∧e2∧e3123213232

20 154 σe1+e2−τe1∧e2∧e3132132312

20 158 −σe1−e2−τe1∧e2∧e3213213231

20 163 e1−σe3−τe1∧e2∧e3232132321

20 167 −σe1+e2−τe1∧e2∧e3321323121

20 198 −e1−σe3−τe1∧e2∧e312321323123

20 202 σe2−e3−τe1∧e2∧e321321321323

20 205 e1+σe3−τe1∧e2∧e323213213213

20 207 −σe2+e3−τe1∧e2∧e332132132132

20 224 σe2+e3−τe1∧e2∧e31213213213231

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Table 5. continued

Order Number Element ×2 Generator path

20 226 −σe2−e3−τe1∧e2∧e31232132132312

20 148 −σe1+e2+τe1∧e2∧e3121323123

20 152 e1−σe3+τe1∧e2∧e3123231232

20 153 −σe1−e2+τe1∧e2∧e3132132132

20 157 σe1+e2+τe1∧e2∧e3213213213

20 164 −e1+σe3+τe1∧e2∧e3232312321

20 165 σe1−e2+τe1∧e2∧e3321321321

20 196 e1+σe3+τe1∧e2∧e312321321323

20 204 −σe2+e3+τe1∧e2∧e321321323123

20 206 −e1−σe3+τe1∧e2∧e323213213231

20 208 σe2−e3+τe1∧e2∧e332132132312

20 223 −σe2−e3+τe1∧e2∧e31213213213213

20 225 σe2+e3+τe1∧e2∧e31232132132132

which is manifested at the level of the pin group by virtue of containing the

pseudoscalar e1e2e3from Table 4. These are all contained in H3, and have

the eﬀect of creating a second orthogonal I2(n) in the even subalgebra. We

illustrate the idea and its complexities with some examples.

Example. Take as a ﬁrst example A3

1with simple roots e1,e

2,e

3but think of

it as I2(2) ⊕A1. Now this gets doubled to I2(2) ⊕I2(2) = A4

1via the spinor

group ±1, ±e1e2,±e2e3and ±e3e1.

Example. Now consider the following twist: take A2

1with simple roots e1,e

2

but instead of having e3available as another orthogonal simple root, we just

have the inversion e1e2e3available. So we get ±1and±e1e2from multiplying

the simple roots. But by operating in the whole pin group we can multiply e1

by the pseudoscalar e1e2e3, which yields e2e3,whichis in the spin part. So

similarly we get ±e2e3and ±e3e1, i.e. we get the same spinor group as in the

previous example, without actually having the third simple root e3available.

This therefore induces the same A4

1root system.

Proposition 5.1. (Doubling—even case) For even nthe root system I2(n)

together with the inversion e1e2e3yields the doubling I2(n)⊕I2(n).

Proof. Without loss of generality take e1as the ﬁrst simple root. Since n

is even, the number of roots is a multiple of 4 and therefore e2is also a

root. Therefore having e1,e2and e1e2e3available is equivalent to having

e3available as well, which via the Induction Theorem leads to a doubling

I2(n)⊕I2(n).

A convenient choice of simple roots for I2(n)isα1=e1and α2=

−cos π

ne1+sinπ

ne2.

Proposition 5.2. (Doubling—general case) The root system I2(n)together

with the inversion e1e2e3yields the doubling I2(n)⊕I2(n). A possible choice

of simple roots for I2(n)⊕I2(n)is given by α1α1,α

1α2,α

1e1e2e3,α

2e1e2e3).

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57 Page 20 of 35 P. Dechant Adv. Appl. Cliﬀord Algebras

Proof. Direct calculation conﬁrms that these constitute two orthogonal I2(n)

root systems with respect to the spinor inner product, and the simple roots

give the correct Cartan matrix

A(I2(n)⊕I2(n)) = ⎛

⎜

⎜

⎝

2−2cos π

n00

−2cos π

n200

002−2cos π

n

00−2cos π

n2

⎞

⎟

⎟

⎠.

C.f. also the computational proof in the supplementary information.

Remark 5.3. The inclusion of the A2⊕A2or H2⊕H2subrootsystems there-

fore arises in the more oblique way for odd nroot systems as n=3and

n= 5. The Cartan matrices are given by

A(A2⊕A2)=⎛

⎜

⎜

⎝

2−10 0

−12 0 0

002−1

00−12

⎞

⎟

⎟

⎠,A(H2⊕H2)=⎛

⎜

⎜

⎝

2−τ00

−τ20 0

002−τ

00−τ2

⎞

⎟

⎟

⎠

Proposition 5.4. (Simple roots of other subrootsystems) A2⊕A2and H2⊕H2

are generated straightforwardly from the H3generators a1,a

2,a

3e.g. via the

‘spinorial simple roots’

a1a1=1,a

1a2,a

1e1e2e3,a

2e1e2e3

for A2and

a1a1=1,a

2a3,a

2e1e2e3,a

3e1e2e3

for H2.

One possible choice of simple roots for A4within H4is given by

(a1a1,a

1a2,a

1a3a2a1a3a2a1a3,a

3a2a1a3a2a1a3a2a3a1a2a3),

but other choices are of course possible.

Remark 5.5. The fact that H3can’t have A1⊕A2or A1⊕H2subrootsystems

is clear from the following: the H3root system is the icosidodecahedron with

vertices at the 2-fold axes. It contains decagonal/hexagonal grand circles

which are valid A2or H2subrootsystems. However, a root normal to those

can’t exist because they would be the vertices of the icosahedron (5-fold

axes) or dodecahedron (3-fold axes), which is of course inconsistent with the

vertices being the 2-fold axes of the icosidodecahedron.

Remark 5.6. For o d d nthe 4D root systems induced by A1⊕I2(n) and via

I2(n) in combination with the inversion e1e2e3are subtly diﬀerent (related

via e1↔e2). However, for even nthey of course coincide.

Remark 5.7. The inversion e1e2e3is often contained in Coxeter groups, but

is famously not contained in the Anfamily for odd n. As such A3, the tetrahe-

dral group, is a prime example of where this isn’t the case; it is even obvious

from the tetrahedron itself that it is not inversion invariant. The existence of

the inversion in a group means that one can use this pseudoscalar to dualise

root vectors to pure bivectors. In work on quaternions it was often regarded

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Vol. 31 (2021) Cliﬀord Spinors and Ro ot System Page 21 of 35 57

as deeply meaningful that the pure quaternion roots e.g. of H4are exactly

the H3root system, and analogously for other cases (e.g. [28]). However, a

more useful way of viewing this is that this is pretty obvious from the Induc-

tion Theorem as long as the inversion is contained in the group. And that

rather than it being proof that the ‘top-down’ approach is somehow deeply

signiﬁcant it is rather a sign of the opposite: that the ‘bottom-up’ approach

constructs H4from H3whilst one can dualise the 30 roots of H3directly

to pure bivectors/quaternions using the pseudoscalar/inversion. This is not

possible for A3where the inversion is missing and no pure quaternion repre-

sentation of A3within D4exists; however, the Induction Theorem still holds

and yields A3→D4[6].

Having shown the existence and nature of various H4subrootsystems

we now brieﬂy discuss a nice way of visualising 4D polytopes in the Coxeter

plane, before using the subrootsystems of H4in order to construct pairs of

invariant polytopes which we then visualise in the Coxeter plane.

6. The Coxeter Plane

The Coxeter plane is a convenient way of visualising any root system in

any dimension. The exposition is not necessary for the following sections

but helps with the visualisation. We will brieﬂy summarise the construction

of this plane that is invariant under a corresponding Coxeter element. Its

existence relies on the bipartite nature of the corresponding graphs (a two-

colouring) [21], which means that the simple roots can be partitioned into two

mutually orthogonal sets (e.g. black and white), as can the reciprocal basis,

the basis of fundamental weights. The properties of the Cartan matrix further

mean that a Perron–Frobenius eigenvector with all positive entries exists.

The components of this eigenvector corresponding to the black, respectively

white, roots are used in a linear combination of the black, respectively white,

fundamental weights. This gives a pair of (black and white) vectors which

together determine a plane, which can be shown to be invariant under the

Coxeter element. Since of course several such Coxeter elements exist (that are

conjugate to one another), there are likewise several such planes. However,

they give an equivalent description.

The Cliﬀord view of the Coxeter plane more generally has been inves-

tigated in [11]. Here, we instead perform all calculations in the 3D even

subalgebra. The 4D simple roots can be chosen as follows in terms of the 3D

H3simple roots:

α1=a1a1=1

α2=a1a2=−1

2+1

2τe1∧e2+1

2σe2∧e3

α3=e1e2a2e3=1

2σe1∧e2−1

2e1∧e3+1

2τe2∧e3

α4=a2e1e2e3=1

2σe1∧e2+1

2e1∧e3−1

2τe2∧e3

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57 Page 22 of 35 P. Dechant Adv. Appl. Cliﬀord Algebras

The reciprocal basis in this spinorial setup (with respect to the spinor

inner product) is given by the following basis of fundamental weights:

ω1=1−τe2∧e3−(τ+1)e1∧e3

ω2=−2τe2∧e3−2(τ+1)e1∧e3

ω3=−(2τ+1)e2∧e3−3(τ+1)e1∧e3−τe1∧e2

ω4=−(2τ+1)e2∧e3−(3τ+1)e1∧e3−τe1∧e2

The Perron–Frobenius eigenvector of the H4Cartan matrix is given by

v=

⎛

⎜

⎜

⎜

⎜

⎜

⎝

4+4

√5

21+√5√5+6√5+30+7

6√5+30+8+4

√5+√56√5+30

−1+√5+6√5+30

√5+6√5+30+7

⎞

⎟

⎟

⎟

⎟

⎟

⎠

whilst the eigenvalue is

λ=2−1

27+√5+6√5+30.

Figure 1shows the projection of the 120 vertices and 720 edges of the

H4root system (aka the 600-cell) into its Coxeter plane. The projection is

performed via the respective inner product with the ‘black and white spinors’.

The computations are shown in the supplementary information.

We will use this Coxeter plane as a means to visualise the H4substruc-

tures of the following sections, including the Grand Antiprism and the snub

24-cell with their H2⊕H2and D4(aka 2T) complements, as well as anal-

ogous constructions with A1⊕A1⊕A1⊕A1,A2⊕A2and A4and their

complements in the 600-cell.

7. The Grand Antiprism and H2×H2

It is of course simple to show that H3contains an H2root system (generated

by the a2and a3simple roots), which leads to a corresponding root system H2

in the 4D space of spinors. However, since the inversion e1e2e3is contained in

the group H3this gets doubled to two orthogonal copies H2⊕H2sitting inside

the H4root system, as seen above. A possible set is shown below, which is the

one multiplicatively generated by a2and a3in combination with e1e2e3via

the geometric product. Of course the H3root system, the icosidodecahedron,

contains many such decagonal circles, but for this set of simple roots this set

could be considered preferred:

Of course they are invariant under their own automorphism group

Aut(H2⊕H2) of order 20×20 = 400. In previous work the author has already

argued that the automorphism group of a spinorial/quaternionic root system

is just the group acting on itself by left and right multiplication, leading to

two factors of the same group [7]. This spinorial group multiplication is to

be distinguished from the above ‘spin reﬂections’, which one could consider

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Vol. 31 (2021) Cliﬀord Spinors and Ro ot System Page 23 of 35 57

Figure 1. Projection of the 120 roots of H4into the Coxeter

plane. This is a familiar view—however, all the calculations

have been done in the even subalgebra of 3D

Element ×2 Element ×2

22e2∧e3

−2−2e2∧e3

−τ−e1∧e2+σe1∧e3σe1∧e2+e1∧e3−τe2∧e3

σ−τe1∧e2−e1∧e3−e1∧e2+τe1∧e3+σe2∧e3

τ−e1∧e2+σe1∧e3σe1∧e2+e1∧e3+τe2∧e3

τ+e1∧e2−σe1∧e3−σe1∧e2−e1∧e3+τe2∧e3

−σ+τe1∧e2+e1∧e3e1∧e2−τe1∧e3−σe2∧e3

−σ−τe1∧e2−e1∧e3−e1∧e2+τe1∧e3−σe2∧e3

−τ+e1∧e2−σe1∧e3−σe1∧e2−e1∧e3−τe2∧e3

σ+τe1∧e2+e1∧e3e1∧e2−τe1∧e3+σe2∧e3

a diﬀerent type of multiplication, generating the reﬂection/Coxeter groups,

which was termed ‘conjugal’ in the above paper. The H2⊕H2root system

is invariant under this conjugal group multiplication (i.e. 4D reﬂections) by

virtue of being a spinor group by Proposition 2.13.

But since this H2×H2is a subgroup of H4, its complement in the 600-

cell is separately left invariant. By this we mean the collection of vertices

derived from the 600-cell by subtracting from the 120 vertices the 20 vertices

from the H2⊕H2root system. This orbit / collection of points is separately

invariant and therefore has the same automorphism group of order 400. This

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57 Page 24 of 35 P. Dechant Adv. Appl. Cliﬀord Algebras

Figure 2. The projection of the 20 roots of the H2⊕H2sit-

ting inside the 600-cell/H4root system into the H4Coxeter

plane

Figure 3. Removal of those 20 roots of the H2⊕H2from

the 600-cell/H4root system leads to the Grand Antiprism

with 100 vertices. Their projection into the H4Coxeter plane

is shown above

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Vol. 31 (2021) Cliﬀord Spinors and Ro ot System Page 25 of 35 57

can also be veriﬁed straightforwardly by direct computation (see supplemen-

tary material). This complement of the H2⊕H2root system is a 4D polytope

with 100 vertices and 500 edges which was found in 1965 in [5] by computa-

tional means and is called the ‘Grand Antiprism’. This is also discussed from

a quaternionic perspective in [22–25]. The construction from 3D seems both

simpler in terms of deriving the vertex set and in shedding light on its sym-

metry group, as well as the conceptual and uniform construction that carries

over in the other cases below. The projection of the two orthogonal H2sinto

the H4Coxeter plane is shown in Fig. 2. The corresponding projection of the

Grand Antiprism is shown in Fig. 3. For ease of visualisation the edges are

also computed and projected, and plotted in 3D in a SageMath visualisation

that can be further explored [30] (supplementary information).

Proposition 7.1. (H2×H2split of H4vertices) The sets H2⊕H2and its

complement the Grand Antiprism are separately invariant under H2×H2.

Proof. Straightforward or by straightforward explicit calculation (supplemen-

tary material).

8. Snub 24-Cell

From a 3D perspective it is obvious that although A3is not a subrootsystem

of H3, the binary tetrahedral group 2Tis contained in the binary icosahedral

group 2I(this can also easily been seen since the tetrahedral group is the

alternating group A4whilst the icosahedral group is A5and thus the former

is contained in the latter, and this also holds for their spin double covers) and

the root system D4is thus contained in H4. As a spin/quaternionic group,

D4is of course just 2Tand its automorphism group is just 2T×2Tof order

242= 576. Removing the 24 vertices from the 600-cell leads to a set of 96

points that is separately invariant under D4, in analogy to the construction

of the Grand Antiprism above. This collection of 96 vertices connected by

432 edges is known as the ‘snub 24-cell’. Again we believe the construction

from 3D to be conceptually clearer, more systematic and more economical.

The projection of the 24 2Tspinors (listed below) aka the D4root system

along with its 96 edges into the H4Coxeter plane is shown in Fig. 4.The

corresponding projection of the snub 24-cell and its edges is shown in Fig. 5.

Note that because of the spin double cover property they come at least

in pairs. But if they are not their own reverse, then they even come in quadru-

plets related via reversal and multiplication by −1 (c.f. Proposition 2.13).

Proposition 8.1. (D4split of H4vertices) The sets D4and its complement

the snub 24-cell are separately invariant under D4.

Proof. Straightforward or by straightforward explicit calculation (supplemen-

tary material).

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57 Page 26 of 35 P. Dechant Adv. Appl. Cliﬀord Algebras

Figure 4. The projection of the 24 roots of the D4root

system sitting inside the 600-cell/H4root system into the

H4Coxeter plane. These vertices of course come from the

binary tetrahedral group 2Tinside the 2I

Element ×2 Element ×2

2−2

e1∧e2−τe1∧e3+σe2∧e3−e1∧e2+τe1∧e3−σe2∧e3

σe1∧e2−e1∧e3+τe2∧e3−σe1∧e2+e1∧e3−τe2∧e3

τe1∧e2−σe1∧e3+e2∧e3−τe1∧e2+σe1∧e3−e2∧e3

1+e1∧e2−e1∧e3+e2∧e3−1−e1∧e2+e1∧e3−e2∧e3

1−e1∧e2+e1∧e3−e2∧e3−1+e1∧e2−e1∧e3+e2∧e3

1+τe1∧e2+σe2∧e3−1−τe1∧e2−σe2∧e3

1−τe1∧e2−σe2∧e3−1+τe1∧e2+σe2∧e3

1+σe1∧e2−τe1∧e3−1−σe1∧e2+τe1∧e3

1−σe1∧e2+τe1∧e3−1+σe1∧e2−τe1∧e3

1+σe1∧e3−τe2∧e3−1−σe1∧e3+τe2∧e3

1−σe1∧e3+τe2∧e3−1+σe1∧e3−τe2∧e3

9. A4

1,A2×A2and A4

We follow the above uniform construction of splitting the H4root system

with respect to its subrootsystems, using the remaining examples of A1⊕

A1⊕A1⊕A1,A2⊕A2and A4.

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Vol. 31 (2021) Cliﬀord Spinors and Ro ot System Page 27 of 35 57

Figure 5. The projection of the snub 24-cell into the H4

Coxeter plane. The 96 vertices can be derived by removing

the 24 vertices of D4/the binary tetrahedral group from the

600-cell

The basic unit vectors e1,e2,e3of course generate the quaternion group

consisting of ±1, ±e1e2,±e2e3and ±e3e1. The projection of this 16-cell with

its 8 vertices and 24 edges into the H4Coxeter plane is shown in Fig. 6.Of

course its complement is also invariant under A1×A1×A1×A1:

Proposition 9.1. (A1×A1×A1×A1split of H4vertices) The sets A1⊕A1⊕

A1⊕A1and its complement are separately invariant under A1×A1×A1×A1.

Proof. Straightforward or by straightforward explicit calculation (supplemen-

tary material).

The projection of this complement of this 16-cell with its 112 vertices

and 624 edges into the H4Coxeter plane is shown in Fig. 7.

Similarly to the H2case above, it is simple to show that H3contains

an A2root system (generated via reﬂections in the a1and a2simple roots),

which leads to a corresponding root system A2in the 4D space of spinors.

However, since the inversion e1e2e3is contained in the group H3this A2

again gets doubled to two orthogonal copies A2⊕A2sitting inside the H4

root system, as seen above. A possible set is shown below, which is the one

multiplicatively generated by a1and a2simple roots of H3in combination

with the inversion e1e2e3. But other choices would be possible, since of course

the H3root system, the icosidodecahedron, contains many such hexagonal

circles.

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57 Page 28 of 35 P. Dechant Adv. Appl. Cliﬀord Algebras

Figure 6. The projection of the 8 roots of the A4

1sitting

inside the 600-cell/H4root system into the H4Coxeter plane

Element ×2 Element ×2

2−2e1∧e3

−22e1∧e3

−1+τe1∧e2+σe2∧e3σe1∧e2+1e1∧e3−τe2∧e3

1+τe1∧e2+σe2∧e3σe1∧e2−1e1∧e3−τe2∧e3

1−τe1∧e2−σe2∧e3−σe1∧e2−1e1∧e3+τe2∧e3

−1−τe1∧e2−σe2∧e3−σe1∧e2+1e1∧e3+τe2∧e3

The projection of these 12 points into the H4Coxeter plane as before

is shown in Fig. 8. Its complement consisting of 108 vertices and 576 edges

is shown in Fig. 9.

Proposition 9.2. (A2×A2split of H4vertices) The sets A2⊕A2and its

complement are separately invariant under A2×A2.

Proof. Straightforward or by straightforward explicit calculation (supplemen-

tary material).

Our ﬁnal example is the A4root system contained in H4as we saw

above. Again the A4root system and its complement are both invariant.

Proposition 9.3. (A4split of H4vertices) The sets A4and its complement

are separately invariant under A4.

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Vol. 31 (2021) Cliﬀord Spinors and Ro ot System Page 29 of 35 57

Figure 7. The projection of the 112 vertices into the H4

Coxeter plane that are left from the 600-cell by removing

the above 8 vertices constituting the A4

1root system, along

with its 624 edges

Proof. Straightforward or by straightforward explicit calculation (supplemen-

tary material).

The root system consists of 20 vertices and 60 edges (shown in Fig. 10)

whilst its complement consists of 100 vertices and 480 edges and is shown in

Fig. 11.

A possible choice of A4roots is shown here:

Element ×2 Element ×2

2−2

τe1∧e2+σe1∧e3+e2∧e3−τe1∧e2−σe1∧e3−e2∧e3

e1∧e2−τe1∧e3+σe2∧e3−e1∧e2+τe1∧e3−σe2∧e3

σe1∧e2−e1∧e3−τe2∧e3−σe1∧e2+e1∧e3+τe2∧e3

1+τe1∧e2+σe2∧e3−1−τe1∧e2−σe2∧e3

1−τe1∧e2−σe2∧e3−1+τe1∧e2+σe2∧e3

1+σe1∧e3+τe2∧e3−1−σe1∧e3−τe2∧e3

1−σe1∧e3−τe2∧e3−1+σe1∧e3+τe2∧e3

1+σe1∧e2−τe1∧e3−1−σe1∧e2+τe1∧e3

1−σe1∧e2+τe1∧e3−1+σe1∧e2−τe1∧e3

Note that because of the spin double cover property these elements come

at least in pairs (which satisﬁes the corresponding root system axiom). But

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57 Page 30 of 35 P. Dechant Adv. Appl. Cliﬀord Algebras

Figure 8. The projection of the 12 roots of the A2⊕A2sit-

ting inside the 600-cell/H4root system into the H4Coxeter

plane

Figure 9. The projection of the 108 vertices into the H4

Coxeter plane that are left from the 600-cell by removing

the above 12 vertices from the A2⊕A2root system

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Vol. 31 (2021) Cliﬀord Spinors and Ro ot System Page 31 of 35 57

Figure 10. The projection of the 20 roots of the A4root

system sitting inside the 600-cell/H4root system into the

H4Coxeter plane

Figure 11. The projection of the complement of the above

into the H4Coxeter plane. The 100 vertices can be derived

by removing the 20 vertices of A4from the 600-cell

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57 Page 32 of 35 P. Dechant Adv. Appl. Cliﬀord Algebras

if they are not their own reverse, then they even come in quadruplets related

via reversal and multiplication by −1 (c.f. Proposition 2.13).

This concludes our listed examples, which illustrate the uniform

approach of constructing subrootsystems via the connection with 3D and

splitting the vertices of the 600-cell into two separately invariant sets. The

H4Coxeter plane provides a nice visualisation for each complementary pair

based on H4subrootsystems.

10. Conclusions

The intention for this article was to ﬁrstly shed light on 4D geometry, root

systems and polytopes through the connection with 3D spinors via the uni-

form Induction Theorem. This gives additional insight into 4D root systems

and polytopes along with their symmetries via another uniform construc-

tion that splits the H4root system into a complementary pair of separately

invariant polytopes, consisting of a subrootsystem and its complement in

H4. These can also be consistently visualised via a projection into the Cox-

eter plane. As we have shown in previous work there are many connections

across exceptional objects throughout mathematics, including Trinities and

ADE correspondences, which this work relates to. In particular, the ‘bottom-

up’ view of exceptional objects has led to a very fruitful and insightful line

of research, and is perhaps mirrored by some constructions in ﬁnite group

theory for some of the sporadic groups. We continue to advocate the use of

geometric insight in addition to the purely algebraic manipulation in terms of

quaternions, by viewing quaternions as arising in a geometric guise as spinors

in 3 dimensions with a much clearer and consistent geometric interpretation.

Secondly, we continue to advocate that Cliﬀord algebras and root sys-

tems/reﬂection groups are natural and complementary frameworks - in a

setting with a vector space with an inner product, and a powerful reﬂection

formula - and can therefore be synthesised into one powerful and coherent

framework. Thirdly, this provides a general arena to do group theoretical cal-

culations in as demonstrated with some detailed examples in this paper, and

we will investigate group and representation theoretic topics in more detail

in future work. In particular, in the spirit of open science and reproducibility

I hope that the computational work sheets provided as supplementary mate-

rial are useful and can help further research in this area by adapting my code

and using other free software provided by the community such as galgebra

and Sage. Finally, at the end of 2020, we wish to commemorate and honour

some of the giants in this ﬁeld, Conway and the Guys.

Acknowledgements

This paper is dedicated to the memory of the late John Horton Conway

and Richard Kenneth Guy, as well as Michael Guy. I would like to thank

Yang-Hui He, Yvan Saint-Aubin, Jiˇri Patera, Peter Cameron, John McKay,

Robert Wilson, Alexander Konovalov, Jean-Jacques Dupas, Tony Sudbery,

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Vol. 31 (2021) Cliﬀord Spinors and Ro ot System Page 33 of 35 57

Reidun Twarock, David Hestenes, Anthony Lasenby, Joan Lasenby and Eck-

hard Hitzer for interesting discussions over the years, and Hugo Hadﬁeld and

Eric Wieser for help with setting up the galgebra software package [3].

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Vol. 31 (2021) Cliﬀord Spinors and Ro ot System Page 35 of 35 57

Pierre-Philippe Dechant

School of Science, Technology and Health

York St John University

York YO31 7EX

UK

and

Department of Mathematics

University of York

Heslington YO10 5GE

UK

and

York Cross-disciplinary Centre for Systems Analysis

University of York

Heslington YO10 5GE

UK

e-mail: ppd22@cantab.net

Received: December 29, 2020.

Accepted: April 8, 2021.

Content courtesy of Springer Nature, terms of use apply. Rights reserved.

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