Graph Coverings for Investigating Non Local Structures in Proteins, Music and Poems

We explore the structural similarities in three different languages, first in the protein language whose primary letters are the amino acids, second in the musical language whose primary letters are the notes, and third in the poetry language whose primary letters are the alphabet. For proteins, the non local (secondary) letters are the types of foldings in space (α-helices, β-sheets, etc.); for music, one is dealing with clear-cut repetition units called musical forms and for poems the structure consists of grammatical forms (names, verbs, etc.). We show in this paper that the mathematics of such secondary structures relies on finitely presented groups fp on r letters, where r counts the number of types of such secondary non local segments. The number of conjugacy classes of a given index (also the number of graph coverings over a base graph) of a group fp is found to be close to the number of conjugacy classes of the same index in the free group Fr−1 on r−1 generators. In a concrete way, we explore the group structure of a variant of the SARS-Cov-2 spike protein and the group structure of apolipoprotein-H, passing from the primary code with amino acids to the secondary structure organizing the foldings. Then, we look at the musical forms employed in the classical and contemporary periods. Finally, we investigate in much detail the group structure of a small poem in prose by Charles Baudelaire and that of the Bateau Ivre by Arthur Rimbaud.
About Klee Irwin

Klee Irwin is an author, researcher and entrepreneur who now dedicates the majority of his time to Quantum Gravity Research (QGR), a non-profit research institute that he founded in 2009. The mission of the organization is to discover the geometric first-principles unification of space, time, matter, energy, information, and consciousness.
As the Director of QGR, Klee manages a dedicated team of mathematicians and physicists in developing emergence theory to replace the current disparate and conflicting physics theories. Since 2009, the team has published numerous papers and journal articles analyzing the fundamentals of physics.
Klee is also the founder and owner of Irwin Naturals, an award-winning global natural supplement company providing alternative health and healing products sold in thousands of retailers across the globe including Whole Foods, Vitamin Shoppe, Costco, RiteAid, WalMart, CVS, GNC and many others. Irwin Naturals is a long time supporter of Vitamin Angels, which aims to provide lifesaving vitamins to mothers and children at risk of malnutrition thereby reducing preventable illness, blindness, and death and creating healthier communities.
Outside of his work in physics, Klee is active in supporting students, scientists, educators, and founders in their aim toward discovering solutions to activate positive change in the world. He has supported and invested in a wide range of people, causes and companies including Change.org, Upworthy, Donors Choose, Moon Express, Mayasil, the X PRIZE Foundation, and Singularity University where he is an Associate Founder.
structure group letter protein secondary structure Music
About Klee Irwin

structure group letter protein secondary structure Music
Article
Graph Coverings for Investigating Non Local Structures in
Proteins, Music and Poems
Michel Planat 1,*,†
, Raymond Aschheim 2,†
, Marcelo M. Amaral 2,†
, Fang Fang 2,†
and Klee Irwin 2,†


Citation: Planat, M.; Aschheim, R.;
Amaral, M.M.; Fang, F.; Irwin, K.
Graph Coverings for Investigating
Non Local Structures in Proteins,
Music and Poems. Sci 2021, 3, 39.
https://doi.org/10.3390/sci3040039
Academic Editor: Claus Jacob
Received: 20 August 2021
Accepted: 18 October 2021
Published: 1 November 2021
Publisher’s Note: MDPI stays neutral
with regard to jurisdictional claims in
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Copyright: © 2021 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
1
Institut FEMTO-ST CNRS UMR 6174, Université de Bourgogne/Franche-Comté, 15B Avenue des
Montboucons, F-25044 Besançon, France
2 Quantum Gravity Research, Los Angeles, CA 90290, USA; Raymond@QuantumGravityResearch.org (R.A.);
Marcelo@quantumgravityresearch.org (M.M.A.); Fang@QuantumGravityResearch.org (F.F.);
Klee@quantumgravityresearch.org (K.I.)
* Correspondence: michel.planat@femto-st.fr
†
These authors contributed equally to this work.
Abstract: We explore the structural similarities in three different languages, first in the protein
language whose primary letters are the amino acids, second in the musical language whose primary
letters are the notes, and third in the poetry language whose primary letters are the alphabet. For
proteins, the non local (secondary) letters are the types of foldings in space (α-helices, β-sheets,
etc); for music, one is dealing with clear-cut repetition units called musical forms and for poems
the structure consists of grammatical forms (names, verbs, etc). We show in this paper that the
mathematics of such secondary structures relies on finitely presented groups fp on r letters, where r
counts the number of types of such secondary non local segments. The number of conjugacy classes
of a given index (also the number of graph coverings over a base graph) of a group fp is found to be
close to the number of conjugacy classes of the same index in the free group Fr−1 on r− 1 generators.
In a concrete way, we explore the group structure of a variant of the SARS-Cov-2 spike protein and
the group structure of apolipoprotein-H, passing from the primary code with amino acids to the
secondary structure organizing the foldings. Then, we look at the musical forms employed in the
classical and contemporary periods. Finally, we investigate in much detail the group structure of a
small poem in prose by Charles Baudelaire and that of the Bateau Ivre by Arthur Rimbaud.
Keywords: protein structure; musical forms; poetry; graph coverings; finitely generated groups;
SARS-Cov-2; Arthur Rimbaud
1. Introduction
In this paper, we point out for the first time a remarkable analogy between the pattern
structure of bonds between amino acids in a protein (the protein secondary structure [1])
and the non local structures observed in tonal music and in poems. We explain the origin
of these analogies with finitely generated groups and graph covering theory.
A protein is a long polymeric linear chain encoded with 20 letters (the 20 amino
acids). The surjective mapping of the 43 = 64 codons to the 20 amino acids is the DNA
genetic code. It can be given a mathematical theory with appropriate finite groups [2,3].
In addition, a protein folds in the three dimensional space with structural elements such as
coils, α-helices and β-sheets, or other arrangements that determine its biological function.
The number of proteins encoded in genomes depends on the biological organism (typically
from 1 to 102 proteins in viruses, from 102 to 103 proteins in bacteria and from 103 to 104
proteins in eukaryotes). The protein database (or PDB) contains about 1.8× 105 entries [4].
Proteins ensure the language of life, amino acids are the alphabet, proteins are the words
and the set of proteins in an organism are the phrases.
Analogously in music, a note is a letter encoding a musical sound. In the 12-tone
chromatic scale [5], each of the 12 notes (or letters) has the frequency of the previous note
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multiplied by 21/12 ≈ 1.0595. The form refers to the secondary structure of a musical
composition in terms of clear-cut units of equal length, for example, A-B-A in the sonata
form or A-B-C-B-A in an arch form [6].
Now, we come to human language and the Latin alphabet. There are 26 letters
organized into words of various types such as names, adjectives, verbs, and so on. In the
following, we will show that a verse in a poem or a phrase in prose have distinctive features,
the former being closer to our theory.
Our mathematical theory of the secondary structures in proteins, music and poems
relies on the concept of a finitely generated group and the corresponding graph coverings,
as explained in Section 2.
We will investigate three applications of the graph covering approach. In Section 3, we
look at the secondary structures of two proteins. We take as examples the spike protein of
the SARS-Cov-2 virus and a glycoprotein playing a role in the immune system (see [3] for
our earlier work). In Section 4, the secondary structures are the musical forms of western
music in the classical age and twentieth century music. Then, in Section 5, the secondary
structures in the verses of selected poems are obtained from an encoding of the types of
words (names, verbs, prepositions, etc).
A Brief Review of the Literature
After we received an invitation to contribute to the present special issue of Sci “Math-
ematics and poetry, with a view towards machine learning” we thought that our current
group theoretical approach of protein language [3] could be converted into an understand-
ing of the poetic language, as well as an understanding of some musical structures.
Our goal in this subsection is to point out earlier work in the same direction as ours.
There are many papers attempting to relate group theory to the genetic code, as reviewed
in [2] but we found none of them featuring the secondary structure of proteins along the
chain of amino acids, as we did in [3] and as we do below with the graph coverings.
Poetry inspired mathematics has been the common thread of most papers exploring
the connection between poems and maths [7–10]. However, it is more challenging to explain
what type of structure and beauty occurs in a poem in the language of mathematics [11].
Perhaps mathematical linguistics is the proper frame for making progress [12] and artificial
intelligence (AI) may help in the classification of languages [13].
Although both subjects have been connected for centuries, comparing musical struc-
tures to mathematics is a fairly new research domain [14]. For a different perspective,
the readers may consult Reference [15].
2. Graph Coverings and Conjugacy Classes of a Finitely Generated Group
Let rel(x1, x2, . . . , xr) be the relation defining the finitely presented group f p =
〈x1, x2, . . . , xr|rel(x1, x2, . . . , xr)〉 on r letters (or generators). We are interested in the con-
jugacy classes (cc) of subgroups of f p with respect to the nature of the relation rel. In a
nutshell, one observes that the cardinality structure ηd( f p) of conjugacy classes of sub-
groups of index d of f p is all the closer to that of the free group Fr−1 on r− 1 generators as
the choice of rel contains more non local structure. To arrive at this statement, we experi-
ment on protein foldings, musical forms and poems. The former case was first explored
in [3].
Let X andX̃ be two graphs. A graph epimorphism (an onto or surjective homomor-
phism) π : X →X̃ is called a covering projection if, for every vertexṽ ofX̃, π maps the
neighborhood ofṽ bijectively onto the neighborhood of πṽ. The graph X is referred to as a
base graph (or a quotient graph) andX̃ is called the covering graph. The conjugacy classes
of subgroups of index d in the fundamental group of a base graph X are in one-to-one
correspondence with the connected d-fold coverings of X, as it has been known for some
time [16,17].
Graph coverings and group actions are closely related.
Let us start from an enumeration of integer partitions of d that satisfy:
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l1 + 2l2 + . . . + dld = d,
a famous problem in analytic number theory [18,19]. The number of such partitions is
p(d) = [1, 2, 3, 5, 7, 11, 15, 22 · · · ] when d = [1, 2, 3, 4, 5, 6, 7, 8 · · · ].
The number of d-fold coverings of a graph X of the first Betti number r is ([17], p. 41),
Iso(X; d) =
∑
l1+2l2+...+dld=d
(l1!2l2 l2! . . . + dld ld!)r−1.
Another interpretation of Iso(X; d) is found in ([20], Euqation (12)). Taking a set of
mixed quantum states comprising r + 1 subsystems, Iso(X; d) corresponds to the stable
dimension of degree d local unitary invariants. For two subsystems, r = 1 and such a
stable dimension is Iso(X; d) = p(d). A table for Iso(X, d) with small d’s is in ([17], Table
3.1, p. 82) or ([20], Table 1).
Then, one needs a theorem derived by Hall in 1949 [21] about the number Nd,r of
subgroups of index d in Fr
Nd,r = d(d!)r−1 −
d−1
∑
i=1
[(d− i)!]r−1Ni,r
to establish that the number Isoc(X; d) of connected d-fold coverings of a graph X (alias
the number of conjugacy classes of subgroups in the fundamental group of X) is as follows
([17], Theorem 3.2, p. 84):
Isoc(X; d) =
1
d ∑
m|d
Nm,r ∑
l| dm
µ
(
d
ml
)
l(r−1)m+1,
where µ denotes the number-theoretic Möbius function.
Table 1 provides the values of Isoc(X; d) for small values of r and d ([17], Table 3.2).
Table 1. The number Isoc(X; d) for small values of first Betti number r (alias the number of generators of the free group Fr)
and index d. Thus, the columns correspond to the number of conjugacy classes of subgroups of index d in the free group of
rank r.
r
d = 1
d = 2
d = 3
d = 4
d = 5
d = 6
d = 7
1
1
1
1
1
1
1
1
2
1
3
7
26
97
624
4163
3
1
7
41
604
13,753
504,243
24,824,785
4
1
15
235
14,120
1,712,845
371,515,454
127,635,996,839
5
1
31
1361
334,576
207,009,649
268,530,771,271
644,969,015,852,641
The finitely presented groups G = fp may be characterized in terms of a first Betti
number r. For a group G, r is the rank (the number of generators) of the abelian quotient
G/[G, G]. To some extent, a group fp whose first Betti number is r may be said to be close
to the free group Fr since both of them have the same minimum number of generators.
3. Graph Coverings for Proteins
As a follow up of our previous paper [3] we first apply the above theory to two
proteins of current interest, the spike protein in a variant of SARS-Cov-2 and a protein that
plays an important role in the immune system.
3.1. The D614G Variant (Minus RBD) of the SARS-CoV-2 Spike Protein
As a first example of the application of our approach, let us consider the D614G variant
(minus RBD: the receptor binding domain) of the SARS-CoV-2 spike protein. In the Protein
Data Bank in Europe, the name of the sequence is 6XS6 [22]. D614G is a missense mutation
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(a nonsynonymous substitution where a single nucleotide results in a codon that codes for
a different amino acid). The mutation occurs at position 614 where glycine has replaced
aspartic acid worldwide. Glycine increases the transmission rate and correlates with the
prevalence of loss of smell as a symptom of COVID-19, possibly related to a higher binding
of the RBD to the ACE2 receptor: an enzyme attached to the membrane of heart cells.
A picture of the secondary structures can be found in Figure 1.
Figure 1. A picture of the secondary structure of D614G variant (minus RBD) of the SARS-CoV-2
spike protein found in the protein data bank in Europe [22].
The D614G variant (minus RBD) of the SARS-CoV-2 spike protein contains 786 amino
acids (aa) forming a (long) word as follows:
AYTNSFTRGVYYPDKVFRSSVLHSTQDLFLPFFSNVTWFHAIHDNPVLPF. . .
AYRFNGIGVTQNVLYENQKLIANQFNSAIGKIQDSLSSTASALGKLQDVV.
NTQEVFAQVKQIYKTPPIKDFGGFNFSQILPDPSKPSKRSFIEDLLFNKV. . .
FVTQRNFYEPQIITTDNTFVSGNCDVVIGIVNNTV
Such a protein sequence, comprising 20 amino acids as letters of the primary code,
can be encoded in terms of secondary structures. Most of the time, for proteins, one makes
use of three types of encoding that are segments of α helices (encoded with the symbol H),
segments of β pleated sheets (encoded by the symbol E) and the segment of random coils
(encoded by the segment C) [1,3,23].
A finer structure may be obtained by using methods such as the SST Bayesian method.
A summary of the approach can be found in Reference [23].
We used a software prepared in [24] to obtain the following secondary structure
rel(H,E,C,G,I,T,4) =
CCCCCCCCEEEEEECCCCCCCEEEEECCCCCCCCCCEEEEEECCCCCCCC. . .
HHHHHHHHCC444444CHHHHHHHHHHHHHHHHHHHHCCCGGGGGHHHHH
HHIIIIICCCCCCCCCCCCCCCCCCTTTTTCCCCCCCCCHHHHHHHHHHH. . .
CCCTTTTTCCCCCTTTTTCCCC44444EEEEEECC,
where G means a 310 helix, 4 means α-like turns, I means a right-handed π helix and T
corresponds to unspecified turns.
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For the group analysis, we slightly simplify the problem by taking 4 = H just one
form of α turn so that the sequence is encoded with 6 letters only. Then, we further simplify
by taking T = C to obtain a 5-letter encoding. We further simplify by taking I = H, then by
taking G = H to get 4-letter and 3-letter encodings, respectively. The results are in Table 2.
Table 2. Group analysis of the D614G variant (minus RBD) of the SARS-CoV-2 spike protein. The bold numbers mean that
the cardinality structure of cc of subgroups of G fits that of the free group Fr−1 when the encoding makes use of r letters.
In the last column, r is the first Betti number of the generating group fp.
PDB 6XS6: AYTNSFTRGVYYPDKVFRSSVLHSTQDL . . .
Cardinality Structure of cc of Subgroups
r
6 letters H, E, C, G, I, T
[1,31,1361,334576]
5
5 letters H, E, C, G, I
[1,15,235,14120]
4
4 letters H, E, C, G
[1,7,41 604,14720]
3
3 letters H, E, C
[1,3,7,30,127,926]
2
We observe that the cardinality structure of the cc of subgroups of the finitely presented
groups f p = 〈H, E, C, G, I, T|rel〉, . . . , f p = 〈H, E, C|rel〉 fits the free group Fr−1 when the
encoding makes use of r = 6, 5, 4, 3 letters. This is in line with our results found in [3] on
several kinds of proteins.
3.2. The β-2-Glycoprotein 1 or Apolipoprotein-H
Our second example deals with a protein playing an important role in the immune
system [25]. In the Protein Data Bank, the name of the sequence is 6V06 [26] and it contains
326 aa. All models predict secondary structures mainly comprising β-pleated sheets and
random coils and sometimes short segments of α-helices.
We observe in Table 3 that the cardinality structure of the cc of subgroups of the
finitely presented groups f p = 〈H, E, C|rel〉 approximately fits the free group F2 on two
letters for the first three models but not for the RAPTORX model. In one case (with the
PORTER model [27]), all first six digits fit those of F2 and higher order digits could not be
reached. The reader may refer to our paper [3] where such a good fit could be obtained for
the sequences in the arms of the protein complex Hfq (with 74 aa). This complex with the
6-fold symmetry is known to play a role in DNA replication.
A picture of the secondary structure of the apolipoprotein-H obtained with the soft-
ware of Ref. [24] is displayed in Figure 2.
Table 3. Group analysis of apolipoprotein-H (PDB 6V06). The bold numbers means that the cardinality structure of cc of
subgroups of fp fits that of the free group F3 when the encoding makes use of 2 letters. The first model is the one used in
the previous Section [24] where we took 4 = H and T = C. The other models of secondary structures with segments E, H
and C are from softwares PORTER, PHYRE2 and RAPTORX. The references to these softwares may be found in our recent
paper [3]. The notation r in column 3 means the first Betti number of fp.
PDB 6V06: GRTCPKPDDLPFSTVVPLKTFYEPG. . .
Cardinality Structure of cc of Subgroups
r
Konagurthu
[1,3,7,26,218,2241]
2
PORTER
[1,3,7,26,97,624]
.
PHYRE2
[1,3,7,26,157,1046]
.
RAPTORX
[1,7,17,134,923,13317]
3
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Figure 2. A picture of the secondary structure of the apolipoprotein-H obtained with the software [24].
4. Graph Coverings for Musical Forms
We accept that this structure determines the beauty in art. We provide two examples
of this relationship, first by studying musical forms, then by looking at the structure of
verses in poems. Our approach encompasses the orthodox view of periodicity or quasi-
periodicity inherent to such structures. Instead of that and the non local character of the
structure is investigated thanks to a group with generators given by the allowed generators
x1, x2, · · · , xr and a relation rel, determining the position of such successive generators,
as we did for the secondary structures of proteins.
4.1. The Sequence Isoc(X; 1), the Golden Ratio and More
4.1.1. The Fibonacci Sequence
As shown in Table 1, the sequence Isoc(X; 1) only contains 1 in its entries and it
is tempting to associate this sequence to the most irrational number, the Golden ratio
φ = (
√
5− 1)/2 through the continued fraction expansion φ = 1/(1 + 1/(1 + 1/(1 +
1/(1 + · · · )))) = [0; 1, 1, 1, 1, · · · ).
Let us now take a two-letter alphabet (with letters L and S) and the Fibonacci words
wn defined as w1 = S, w2 = L, wn = wn−1wn−2. The sequence of Fibonacci words wn is
as follows
S, L, LS, LSL, LSLLS, LSLLSLSL, LSLLSLSLLSLLS, LSLLSLSLLSLLSLSLLSLSL, · · ·
and its length corresponds to the Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, 21, · · · .
Then, one can check that the finitely-presented group fp(n) = 〈S, L|wn〉whose relation
is a Fibonacci word wn possesses a cardinality sequence of subgroups [1, 1, 1, 1, 1, 1, 1, 1 · · · )
equal to Isoc(X; 1), up to all computable orders, despite the fact that the groups fp(n)
are not the same. It is straightforward to check that the first Betti number r of fp(n) is 1,
as expected.
4.1.2. The Period Doubling Cascade
Other rules lead to a Betti number r = 1 and the corresponding sequence Isoc(X;1).
Let us consider the period-doubling cascade in the logistic map xl+1 = 1− λx2
l . Period
doubling can be generated by repeated use of the substitutions R→ RL and L→ RR., so
that the sequence of period doubling is [28]
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R, L, RL, RLR2, RLR3LRL, RLR3LRLRLR3LR3, RLR3LRLRLR3LR3LR3LRLRLR3LRLRL, · · ·
and the corresponding finitely presented groups also have first Betti numbers equal to 1.
4.1.3. Musical Forms of the Classical Age
Going into musical forms, the ternary structure L-S-L (most commonly denoted
A − B − A) corresponding to the Fibonacci word w4 is a Western instrumental genre
notably used in sonatas, symphonies and string quartets. The basic elements of sonata
forms are the exposition A, the development B and recapitulation A. While the musical
form A− B− A is symmetric, the Fibonacci word A− B− A− A− B corresponding to
w5 is asymmetric and used in some songs or ballads from the Renaissance.
In a closely related direction, it was shown that the lengths a and b of sections A and
B in all Mozart’s sonata movements are such that the ratio b/(a + b) ≈ φ [29].
4.2. The Sequence Isoc(X; 2) in Twentieth Century Music and Jazz
In the 20th century, musical forms escaped the classical channels that were created.
With the Hungarian composer Béla Bartók, a musical structure known as the arch form
was created. The arch form is a sectional structure for a piece of music based on repetition,
in reverse order, so that the overall form is symmetric, most often around a central move-
ment. Formally, it looks like A− B− C − B− A. A well known composition of Bartok
with this structure is Music for strings, percussion and celesta [30]. In Table 4, it is shown
that the cardinality sequence of cc of subgroups of the group generated with the relation
rel=ABCBA corresponds to Isoc(X; 2) up to the higher index 9 that we could check with
our computer. A similar result is obtained with the symmetrical word ABACABA.
Our second example is a musical form known as twelve-bar blues [31], one of the most
prominent chord progressions in popular music and jazz. In this context, the notation A is
for the tonic, B is for the subdominant and C is for the dominant, each letter representing
one chord. In twelve-bar blues, there are twelve chords arranged as in the first column of
Table 4. We observe that the standard twelve-bar blues are different in structure from the
sequence of Isoc(X; 2). However, variations 1 and 2 have a structure close to Isoc(X; 2).
In the former case, the first 9 orders lead to the same digit in the sequence.
Our third example is the musical form A-A-B-C-C. Notably, it is found in the Slow
movement from Haydn’s ‘Emperor quartet Opus 76, N◦3 [32] (Figure 3), much sooner than the
contemporary period. (See also Ref. [33] for the frequent occurrence of the same musical
form in djanba songs at Wadeye.) As in the aforementioned examples, the cardinality se-
quence of the cc of subgroups of the group built with rel=AABCC corresponds to Isoc(X; 2)
up to the highest index 9 that we could reach in our calculations.
Figure 3. Slow movement from Haydn’s ‘Emperor’ quartet Opus 76, N◦3.
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Table 4. Group analysis of a few musical forms whose structure of subgroups, apart from exceptions, is close to Isoc(X; d)
with d = 2 (at the upper part of the table) or d = 3 (at the lower part of the table). Of course, the forms A-B-C and A-B-C-D
have the cardinality sequence of cc of subgroups exactly equal to Isoc(X; 2) and Isoc(X; 3), respectively.
Musical Form
Ref
Card. Struct. of cc of Subgr.
r
A-B-C-B-A
arch, Belá Bartók
[1,3,7,26,97,624,
2
.
.
4163,34470,314493]
.
A-B-A-C-A-B-A
.
.
.
A-B-A-C-A, A-B-A-C-A-B-A
rondo
.
.
A-B-A-C
.
.
A-A-B-C-C
Haydn [32],
.
.
.
djanba ([33], Figure 9.8)
.
.
A-A-A-A-B-B-A-A-C-C-A-A
twelve-bar blues,
[1,7,14,109,396,3347,
3
.
standard
19758,287340]
.
A-A-A-A-B-B-A-A-C-B-A-A
twelve-bar blues,
[1,3,7,26,97,624,
2
.
variation 1
4163,34470,314493]
.
A-A-A-A-B-B-A-A-B-C-A-C
twelve-bar blues,
[1,3,7,26,127, 799,
.
.
variation 2
5168, 42879]
.
A-B-C
Isoc(X; 2)
[1,3,7,26,97,624,
2
.
.
4163,34470,314493]
.
A-A-B-B-C-C-D-D
pot pourri
[1,15,82,1583,30242]
4
A-B-A-C-A-D-A
rondo
[1,7,41,604,13753,504243]
3
A-B-C-D
Isoc(X; 3)
[1,7,41,604,13753,
3
.
.
504243,24824785]
.
Further musical forms with 4 letters A, B, C, and D and their relationship to Isoc(X; 3)
are provided in the lower part of Table 4.
Not surprisingly, the rank r of the abelian quotient of fp = 〈A, B, C|rel(A, B, C)〉 is
found to be 2 when the cardinality structure fits that Isoc(X; 2) in Table 4. Otherwise, the
rank is 3. Similarly, the rank r of the abelian quotient of fp = 〈A, B, C, D|rel(A, B, C, D)〉 is
found to be 3 when the cardinality structure fits that Isoc(X; 3) in Table 4. Otherwise, the
rank is 4.
5. Graph Coverings for Prose and Poems
5.1. Graph Coverings for Prose
Let us perform a group analysis of a long sentence in prose. We selected a text by
Charles Baudelaire [34]:
Le gamin du céleste Empire hésita d’abord; puis, se ravisant, il répondit: “Je vais vous
le dire ”. Peu d’instants après, il reparut, tenant dans ses bras un fort gros chat, et le
regardant, comme on dit, dans le blanc des yeux, il affirma sans hésiter: “Il n’est pas
encore tout à fait midi.” Ce qui était vrai.
In Table 5, the group analysis is performed with 3, 4 or 5 letters (in the upper part)
and is compared to random sequences with the same number of letters (in the lower part).
The text of the sentence is first encoded with three letters (H for names and adjectives,
E for verbs and C otherwise), we observe that the subgroup structure has cardinality close
to that of a free group F2 on two letters up to index 3. If one adds one letter A for the
prepositions in the sentence (in addition to H, E and C), then the subgroup structure has
cardinality close to that of a free group F3 on three letters. If adverbs B are also selected, then
the subgroup structure is close to that of the free group F4. In all three cases, the similarity
holds up to index 3 and that the cc of subgroups are the same as in the corresponding free
groups. The first Betti numbers of the generating groups are 2, 3 and 4 as expected.
In Table 5, we also computed the cardinality structure of the cc of subgroups of
small indexes obtained from a random sequence of 250 letters (like the number of letters
in the previously studied sentence of the small poem in prose). One took 10 runs with
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random sequences having 3, 4 or 5 letters. We see that the cardinality structure of the cc
of subgroups for the cases with 4 or 5 letters tends to align to that of the free group Fr−2
(not Fr−1). The 3-letter case is the most random one and does not correspond to F1 (or F2),
in most runs.
Our conclusion is that the considered prose sequence contains a structure close to that
of Fr when we select r + 1 letters for the encoding of the sentence, a result that is similar to
that which we found in the group analysis of proteins in Section 3 and musical forms in
Section 4.
Table 5. Group analysis of an excerpt of a small poem in prose Le vieux saltimbanque by Charles Baudelaire. The text is split
into segments encoded by the symbol H (for names and adjectives), E (for verbs), A for prepositions, B for adverbs, or C
(for the other types: conjunctions, punctuation marks and so on). The cardinality structure of the cc of subgroups of a small
index is compared to the one obtained with 10 runs of a sequence of words of a similar length (i.e., the length 250) with the
corresponding number of letters.
Le Gamin du Céleste Empire . . . Ce Qui était Vrai.
Card. Seq. of cc of Subgroups
r
3 letters: rel=C2H5C2H7H6E6C7CC4CC2E8C · · ·
[1,3,7,34,131]
2
4 letters: rel=C2H5 A2H7H6E6C7CC4CC2E8C · · ·
[1,7,41,636,14364]
3
5 letters: rel=C2H5 A2H7H6E6B7CB4CC2E8C · · ·
[1,15,235,14376,.]
4
[Random[1,3]: i in [1..250]]
[1,1,1,2,4,4]
1
(10 runs)
[1,3,2,9,5,20]
2
[1,3,1,6,6,15]
.
[1,3,7,30,124,987]
.
[1,7,17,126,323,2445]
3
etc
Isoc(X;2)
[1,3,7,26,97,624]
2
[Random[1,4]: i in [1..250]]
[1,3,7,30,.] (×3)
2
(10 runs)
[1,3,10,51,.] (×3)
.
[1,3,7,26,457]
.
[1,3,10,39,.]
.
[1,3,13,52,.]
.
[1,7,20,143,.]
3
Isoc(X;3)
[1,7,41,604,13573]
3
[Random[1,5]: i in [1..250]]
[1,7,41,620,.] (×3)
3
(10 runs)
[1,7,41,636,.] (×3)
.
[1,7,41,604,.] (×2)
.
[1,7,41,668,.]
.
[1,7,50,819,.]
.
Isoc(X;4)
[1,15,235,14120,1712845]
4
5.2. Graph Coverings for Poems
In poems, the verses are generally of a smaller length than that for a sentence in prose.
We selected the first strophe of the poem, Le Bateau Ivre, by Arthur Rimbaud. The poem
may be found on a wall in Paris, see Figure 4. The verses in the strophe have about 35
letters. We compare the group structure of the four verses in the first strophe to that of
random sequences of length 35 in Table 6 (when the encoding is with 3 letters H, E and C)
and in Table 7 (when the encoding is with 4 letters H, E, C and A). Adverbs are too rare in
verses of such a small length so that we did not considered the 5-letter case.
Let us first look at the 3-letter case in Table 6. Apart from the first verse in the strophe,
the structure of the poem is very close to that of F2, up to the index 6 (for the second verse)
and up to the index 7 (for verses 3 and 4). Higher order indices could not be reached
in our calculations. For the English translation, the closeness to F2 holds as well but is
not so perfect. It is not so surprising since the poem was originally composed in French.
For a French translation of a poem in English one would have obtained a similar (small)
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discrepancy to the group structure to F2. We looked at the cardinality structure of the cc of
subgroups by taking random sequences of length 35 in 10 runs and we observe that the
closeness to F2 is much less than in the case of the poem.
Figure 4. Part of the poem Le Bateau Ivre of Arthur Rimbaud on a wall (Rue Férou) in Paris.
Table 6. Group structure of the poem Le Bateau Ivre’ (The Drunken Boat) by Arthur Rimbaud. Only the first strophe (that has
four lines) is analyzed, firstly in its original form, then in an English translation. Each line is split into segments encoded
by the symbol H (for names and adjectives), E (for verbs) or C (for the other types: conjunctions, adverbs, prepositions,
punctuation marks and so on). The group relation is displayed for the first line only.) The cardinality structure of cc of
subgroups of a small index is compared to the one obtained with 10 runs of a sequence of random 3-letter words of similar
length (i.e., the length 35).
Comme je descendais des fleuves impassibles,
[1,1,7,17,114, 1395,36973]
1
rel=C4C2E10C3H7H11C
Je ne me sentis plus guidé par les haleurs:
[1,3,7,26,97, 624,4171]
2
Des Peaux-Rouges criards les avaient pris pour cibles
[1,3,7,26,97, 624,4163]
.
Les ayant cloués nus aux poteaux de couleurs.
[1,3,7,26,97, 624,4163]
.
As I was floating down unconcerned rivers
[1,3,7,26,97, 624,4163,34470]
2
rel=C2 ∗ C ∗ E3 ∗ E8 ∗ C4 ∗ E11 ∗ H6
I now longer felt myself steered by the haulers:
[1,3,7,26,101, 656,4227]
2
Gaudy Redskins had taken them for targets
[1,3,7,26,97, 624,4163,324935]
.
Nailing them naked to coloured states.
[1,3,7,42,202, 1682,9204]
.
[Random[1,3]: i in [1..35]]
[1,3,7,30, .] (×3)
2
(10 runs)
[ 1,3,7,26, . ]( ×3)
.
[1,3,7,.,.,]
.
[ 1,3,10,.,. ](×2)
.
[ 1,3,13,.,.]
.
Isoc(X;2)
[1,3,7,26,97, 624,4163,34470]
2
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Table 7. The same as in Table 6, but each line is split into segments encoded by the symbol H (for names and adjectives),
E (for verbs), A for prepositions, or C (for the other types: conjunctions, adverbs, punctuation marks and so on). The
cardinality structure of cc of subgroups of a small index is compared to the one obtained with 10 runs of a sequence of
random 4-letter words of similar length (i.e., the length 35).
Comme je descendais des fleuves impassibles,
[1,7,41,604,13753]
3
rel=C4C2E10 A3H7H11C
Je ne me sentis plus guidé par les haleurs:
[1,7,41,604,13753]
.
Des Peaux-Rouges criards les avaient pris pour cibles
[1,7,41,604,13753]
.
Les ayant cloués nus aux poteaux de couleurs.
[1,7,41,604,13753]
.
As I was floating down unconcerned rivers
[1,7,59,1386,27011]
3
rel=C2CE3E8 A4E11H6
I no longer felt myself steered by the haulers:
[1,7,41,604,13753]
.
Gaudy Redskins had taken them for targets
[1,7,50,1763,51582]
.
Nailing them naked to coloured states.
[1,7,59,1002,18671]
.
[Random[1,4]: i in [1..35]]
[1,7,50,755,.] (×2)
3
(10 runs)
[1,7,41,604,.] (×3)
.
[ 1,7,41,.,.](×2)
.
[ 1,7,50,739,.](×2)
.
[ 1,7,59,.,.]
.
Isoc(X;3)
[1,7,41,604,13753]
3
The group structure with 3 letters can also be obtained for the group structure with 4
letters in Table 7 but the closeness is to F3 (not F2), as expected.
6. Conclusions
The graph covering approach has been shown to be useful for understanding how
complex structures are encoded in nature and in art. For proteins, there exists a primary
encoding with 20 amino acids as letters and the secondary encoding determines the folding
of proteins in the 3-dimensional space. This is useful for recognizing the relationship
between the structure and function of the protein. We took examples based on a present
hot topic: a variant of the SARS-Cov-2 spike protein and the alipoprotein-H. For music,
the secondary structures are called musical forms and the choice of them determines
the type of music. For poems, we took the French (or English) alphabet with 26 letters,
but many other alphabets may be used for the application of our approach. The secondary
structures are defined from the encoding of the words (names, verbs and so on).
It is also interesting to speculate about the possible existence of a primary code and a
secondary code in other fields, for example, in physics at the elementary level like in particle
physics and quantum gravity [35]. According to the experience of the authors of this paper,
the structure has much to do with complete quantum information. The reader may consult
paper [36] about particle mixings or [3,37] about the genetic code in which finite groups
are the players. Here, we are dealing with infinite groups so that the representation theory
of finite groups (with characters) has to be defined on finitely-presented groups (most of
the time of infinite cardinality). This will be explored further in our next paper [38].
Author Contributions: Conceptualization, M.P., F.F. and K.I.; methodology, M.P. and R.A.; software,
M.P.; validation, R.A., F.F. and M.M.A.; formal analysis, M.P. and M.M.A.; investigation, M.P., F.F. and
M.M.A.; writing—original draft preparation, M.P.; writing—review and editing, M.P.; visualization,
F.F. and R.A.; supervision, M.P. and K.I.; project administration, K.I.; funding acquisition, K.I. All
authors have read and agreed to the published version of the manuscript.
Funding: Funding was obtained from Quantum Gravity Research in Los Angeles, CA.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: Not applicable.
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Conflicts of Interest: The authors declare no conflict of interest.
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