Galois Connections: Mathematics, art, and archives

http://dx.doi.org/10.14236/ewic/eva2015.18

176

Galois Connections:

Mathematics, Art, and Archives

Jonathan P. Bowen

Tula Giannini

Birmingham City University

School of Computing, Telecommunications & Networking

Birmingham, UK

http://www.jpbowen.com

Pratt Institute

School of Information & Library Science

New York, USA

http://mysite.pratt.edu/~giannini/

[email protected]

Évariste Galois (1811–1832) has been increasingly recognised as an important mathematician who

despite his short life developed mathematical ideas that today have led to applications in computer

science (such as Galois connections) and elsewhere. Some of Galois’ mathematics can be

visualised in interesting and even artistic ways, aided using software. In addition, a significant

corpus of the historical documentation on Galois and his family (including his brother Alfred

Galois, who was an artist), can now be accessed online as a growing number of institutional

archives digitise their collections. This paper introduces some of the mathematics of Galois, ways

in which it can be visualised, and also considers the issues and new opportunities with respect to

visualising information on Galois and his family (including the connections between them).

Although the story of Galois and his close relations can be seen as one of tragedy with lives cut

short, from a historical viewpoint Évariste Galois’ contribution to humankind has been a triumph.

Computer science. Digital archives. Galois connections. History of mathematics. Social history. Visualisation.

1. INTRODUCTION

“Unfortunately what is little recognized is that the

most worthwhile scientific books are those in

which the author clearly indicates what he does

not know; for an author most hurts his readers

by concealing difficulties.”

– Évariste Galois (1811–1832)

In this paper, we explore various aspects of the

French mathematician Évariste Galois (1811–

1832), who died when only aged 20 but made very

important contributions to his field, producing an

eponymous theory only properly recognised and

developed further after his death (Neumann 2011).

Rigatelli (1996) provides an extended biography of

Galois and there is every a fictionalised version of

his life where Galois is represented by a

mathematical “i” (Petsinis 1995). We consider his

life and work with respect to both visualising his

mathematical ideas as artistically interesting

structures and discovering archival material relating

to Galois and his family that is increasingly

accessible online in digital form (Bowen & Giannini

2014), although still difficult to find in practice.

In Section 2, Galois’ mathematics is considered in

overview, including the concept of a Galois

connection. Section 3 demonstrates how some of

Galois’ mathematics can be visualised as pleasing

and sometimes artistic patterns. In Section 4,

archival investigations on historical documents

relating to Galois and his close relatives, together

with the connections between them, are presented,

some of which are available in digitised form.

Finally a brief conclusion is presented, noting that

there are opportunities for further visualisation and

archival research.

2. MATHEMATICS

The contribution of Évariste Galois to mathematics

is extraordinary considering his early death at the

age of 20 (Neumann 2011, Daintith & Nelson 1989,

p. 141). He is an extreme example of the widely

held notion that a mathematician produces their

best work when young – typically in their 20s, but

Galois had barely entered this decade of his life.

That said, creative breakthroughs need years of

commitment and complete dedication (Robinson

2010). Galois achieved this in his teenage years, a

remarkable feat even for a mathematician.

While still in his teens, Galois determined a

necessary and sufficient condition for a polynomial

to be solvable using radicals, thus resolving a 350

year-old problem. His mathematical investigations

were foundational for two important aspects of

abstract algebra: Galois theory and group theory.

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One of Galois’ most celebrated contributions in

computer science is that of what is now known as a

Galois connection (Melton et al. 1986, Mu &

Oliveira 2011). For an abstract specification level

S(a) and a more concrete design level D(c), there

should be a linking predicate G(c,a) connection the

two levels. Formally (Hoare & He 1998, p. 41):

[(∃c ⦁ D(c) ∧ G(c,a) ⇒ S(a)] iff

[D(c) ⇒ (∀a ⦁ G(c,a) ⇒ S(a))]

This is known as a Galois connection and is very

useful in linking theories in computer science.

Determining a suitable predicate G(c,a) connects

the two levels or theories in a mathematically useful

manner.

A Galois connection can also be modelled as a pair

of functions, L and R, between two complete

lattices, S and T, in each direction, with the

following two inequations holding (Hoare & He

1998, p. 98):

R(L(X)) ⊑ X

Y ⊑ L(R(Y))

where X is in S and Y is in T. Note that X and Y

may be predicates in stronger and weaker theories

respectively. This model of such a Galois

connection can be visualised as in Figure 1 (Hoare

& He 1998, p. 100).

A Galois lattice (or concept lattice) is useful in

formal concept analysis, for example in the study of

ontologies. It is a mathematical structure in which

the relation between the sets of concepts and

attributes is a Galois connection. Such a lattice can

be usefully visualised to find patterns within its

structure. Building a Galois lattice can be

considered as a method for conceptual clustering

since it provides a concept hierarchy (Godin et al.

1995).

Using a Galois lattice helps to display an order

structure, in which dependencies among row and

column objects, together with dependencies

between rows and columns, are both revealed; for

example, in visualisation for social network analysis

(Freeman 2000). Marghoubi et al. (2006) have

investigated the use of a Galois lattice for the

extraction and visualisation of association rules,

applied to spatial data mining, helping to discover

hidden relationships. First the spatial context is

determined and then pattern mining is undertaken

in an efficient manner.

Figure 1: Visualisation of a Galois connection

(Hoare & He 1998, p. 100)

3. ART

The Galois family had an artistic streak running in it

and Évariste’s brother Alfred was an artist. Alfred

Galois produced one of the few images of Évariste

posthumously (see Figure 2).

Mathematical structures can often be beautiful and

complex (Bakshee 1999, Beddard & Dodds 2009,

Wolfram 2002). Some of Évariste Galois’

mathematics can be visualised as interesting and

even artistic patterns. We explore some examples

in this section.

Galois fields have been visualised (Stein 2012). It

is convenient to have Galois fields of size 2

for

various n for software applications to allow them to

fit into computer bytes or words. The Galois field

GF(2

) has 2

elements, each represented as

polynomials of degree less than n with all the

coefficients having values of either 0 or 1. Then to

encode an element of GF(2

) as a number, an n-bit

binary number is required. For example, consider

the Galois field GF(2

) with 2

or 8 elements.

These are 3-bit binary numbers. For example, the

binary element 011 represents the polynomial x + 1

and the element 110 stands for x

+ x. Using 8-bit

representations of elements of GF(2

) for example,

it is possible to create an image where the pixel in

the i

row and j

column is the sum in the Galois

field of i and j (as binary numbers). See Figure 3

for a visualisation of this.

stronger

weaker

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Figure 2: Posthumous portrait of Évariste Galois

by his brother Alfred Galois (1848)

Figure 3: Visualisation of a Galois field (Stein 2012)

An online Galois visualisation tool illustrating

arithmetic symmetries using roots of unity is

available for use by anyone (Balakrishnan &

Venkatachalam 2014). Figures 4 and 5 provide

examples of simple and more complex visual-

isations, generated using this interactive tool

(http://people.maths.ox.ac.uk/balakrishnan/galois/).

Figure 4: Simple visualisation of arithmetic symmetry

of roots using mathematical ideas by Galois

(Balakrishnan & Venkatachalam 2014)

Figure 5: More complex visualisation of arithmetic

symmetry (Balakrishnan & Venkatachalam 2014)

4. ARCHIVES

Archival documents can visualise history and as

seen for example in documentary films since they

provide the necessary detail to create vivid pictures

of points or moments in time. Much archival

material has yet to be discovered, due to the large

amount of it and the difficulty of searching it

effectively.

With new archival documentation from the Archives

Nationales, Archives de Paris, and État Civil of

Bourg-la-Reine, we visualise the family life of

Évariste in Paris, focusing on the Galois residences

at the rue Jean de Beauvais, where his father

committed suicide in 1829, and the rue d’Enfer

Saint Michel 57, where his brother, Alfred Galois, a

painter, died after a prolonged illness on 4 June

1849.

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Documents relating to the life of Évariste Galois’

mother, Adélaïde Marie Demante (born Paris 1788;

died Paris 1871, aged 83), reveal the social milieu

of Évariste’s short life and his mother’s connections

to academic circles in Paris, where her father,

Thomas François Demante, was docteur agrégé à

la Faculté de droit de l'ancienne université de Paris

(see Figure 6), while her brother, Antoine Marie

Demante, and his son, Auguste Gabriel, were

members of the Faculté de droit and also Chevalier

de la Légion d’honneur. His mother’s classical

education and enthusiasm for learning no doubt

influenced Évariste’s passionate approach to life.

Évariste’s father, Nicolas Gabriel, the mayor of

Borg-la-Reine, Évariste’s place of birth, was the

son of Théodore Michel Galois (born 1774, died

1831), who was an Officer of the Royal Order of the

Légion d’honneur (see Figure 7).

Figure 6: Faculté de Droit de Paris document

4.1 Évariste’s family life in Paris at 16 rue Jean

de Beauvais, 1829–1832: four fateful years

Focusing on the critical years of the Galois and

Demante families in Paris from the death of Nicolas

Gabriel Galois to that of Évariste, we present an

archival narrative, beginning with the inventory after

the death of Évariste Galois’ father in 1829, to the

marriage contract of Évariste’s mother, Adélaïde

Marie Demante with Jean Christophe Loyer in

1832, which took place shortly after Évariste’s

death. As an epilogue, we consider Loyer’s 1837

inventory after his death, the marriage contract and

inventory of Alfred Galois, Évariste’s brother, and

the 1851 sale by Mme Demante of the Galois

house in Bourg la Reine, and finally her death there

in 1871.

Figure 7: Copie Certifiée La Garde Impériale (Certified

copy of the Imperial Guard) for Théodore Michel Galois

(http://www.culture.gouv.fr/LH/LH078/PG/FRDAFAN83_

OL1064022v004.htm)

Central to this l’histoire de famille, is the Demante

family house on rue Jean de Beauvais no.16

(formerly, St. Jean de Beauvais), where the

dialogue between Évariste and his parents and

grandparents no doubt took centre stage, and

where the republican views of Évariste Galois’

father held sway, influencing the young mind of

Évariste, and drawing him into his ill-fated political

struggles. Set in the fifth arrondissement in the

quartier of the Sorbonne at the heart of academic

life in Paris, the home on the rue Jean de Beauvais

belonged to Évariste’s maternal grandparents,

Thomas François Demante (lawyer to Parliament,

doctor of the faculty of law of Paris, and former

President of the civil tribunal of Louviers), and his

wife, Marie Elisabeth Thérèse Durand.

Évariste’s sister, Nathalie Théodore Galois, married

Benoit Chantelot in Paris on 28 January 1829, with

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a second ceremony on 5 February 1829 in Bourg

La Reine. Her marriage took place just five months

before her father’s suicide by asphyxiation on 2

July 1829 at the house on rue Jean de Beauvais,

where his inventory after death was taken on 11

November 1829. The inventory establishes that his

three children from his marriage with Mme

Demante, namely Nathalie Théodore, Évariste, and

Alfred (see Figure 8), were the unique inheritors of

his estate, each receiving one third. Significantly,

Évariste Galois’ father’s inventory details the 1825

estate settlement of his father-in-law, Thomas

François Demante, by which Adélaïde Marie

Demante received a one-third share of the house

on the rue Jean de Beauvais. In October 1829, she

moved from Bourg la Reine to Paris, taking up

residence there, a move that coincided with

Évariste’s acceptance at the l’Ecole préparatoire.

Consequently, Évariste and his mother were living

at the family house in Paris just months after his

father’s suicide there and shortly before he

received his baccalauréat des sciences on 14

December 1829, so that mother and son were

living there through the tumultuous years of the

political upheaval of 1830 and 1832. Thus, the

house on the rue Jean de Beauvais became in

some respects an arbiter of history as it brought

together the Galois and Demante families at critical

moments of personal crisis. For a family tree of the

Galois and Demante families, see Figure 8 below.

The July Revolution of 1830 proved highly

disruptive to Évariste’s studies and the intricate

deep thinking of his mathematical work seems to

belie his impulsive and passionate politics. This

was a turning point in his life that distracted him

from his true mathematical vocation. Without the

political upheavals, he could have made even more

interesting mathematical discoveries.

Figure 8: Galois/Demante family tree

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The year of 1831 saw the death of Évariste’s

paternal uncle, Théodore Michel Galois, on 4

February 1831, a member of the Royal Order of the

Legion of Honour as lieutenant colonel of the 6

regiment of infantryman. Évariste enjoyed a close

relationship with him, so that his death added to his

loss of family support. 1831 was also the year of

Évariste’s arrest and imprisonment for political

activities at a time when his mother was dividing

her time between Paris and Bourg la Reine. We

see that at most critical moment in his life, Évariste

lacked parental guidance; one can imagine the

anguish of Mme Demante as she observed his

intensifying anti-monarchy sentiments whilst fearing

that she was on the verge of losing her cherished

son.

31 May 1832, the day after Évariste’s ill-fated pistol

duel, marks the tragic end of the life of a great

mathematician, ironically just a few days before the

Paris uprising of 1832, on 5–6 June. Perhaps

surprisingly, only four months after her son’s death,

on 8 October 1832, Mme Demante married Jean

Christophe Loyer, described in the marriage

contract as “Maître d’Hôtel Garni à Elbeuf” and a

widower with two children from his first marriage to

Victoire Françoise Rouaux. Significantly, Loyer

owned a house in Paris located at 6 rue Jean de

Beauvais, which he brought to the marriage,

providing Mme Demante with a quarter share.

Thus, it seems likely that the newlyweds met on or

around rue Jean de Beauvais. After the marriage

Mme Demante moved from no. 16 to no. 6.

4.2 Epilogue

When Loyer died on 21 August 1837, he and Mme

Demante were still residing at 6 rue Jean de

Beauvais, where his inventory after death was

taken, even though he died in Bourg la Reine

where Mme Demante maintained property. A few

years later, Alfred Galois, born 17 December 1814

in Bourg la Reine, married Pauline Henriette

Alexandrine Elodie Chantelot on 14 December

1841. By that date, Mme Demante was living with

her son, Alfred, on the rue d’Enfer Saint Michel, at

no. 61.

Alfred had two children with Mlle Chatelot:

Elisabeth Julia Galois, born 6 April 1843 and died

25 May 1855, at age 12, and a son named after his

brother, Évariste Galois, born 8 September 1848

and died on 4 April 1850 at only 18 months old.

Sadly Alfred’s death in 1849 at his apartment on

rue d’Enfer Saint Martin 57, which was due to

illness, ended his seemingly happy family life. His

marriage contract and inventory after his death

describe him as a “peintre, artiste,” although little is

known of his painting. These documents further

establish that the life of the Galois family was

marked by turbulent times and tristesse for which

the historical backdrop was the French revolution of

1830.

Mme Demante, now twice a widower, was still

living on the rue d’Enfer when she sold the Galois

family house with “jardins Anglais” in Bourg la

Reine at the Grande Rue no. 20, on 26 March 1851

to Pierre Ravon, mayor of Bourg la Reine, and

Dame Amadine Zoe Deriquebourcq. According to

the contract of sale, the house was sold for 35,000

francs together with an annual rente viagère for

Mme Demante, by which she would receive 850

francs annually. Page 9 of the contract shows that

by 1851 her only surviving heirs were Nathalie

Théodore and Alfred’s daughter, Julie Elisabeth; it

notes the deaths of Évariste, Alfred, and Alfred’s

son Évariste (see Figure 9).

Figure 9: Archives Nationales, Minutier Central, Sale by

Mme Loyer (Mme Demante) to M. and Mme Ravon, 26

March 1851 – Déclaration sur l’état civil

Mme Demante, retired, aged 83, died on 1 August

1871 in Bourg la Reine. No family witnesses were

present, as was also the case when Évariste died,

aged 20, in Paris at the Chochin Hospital in the fifth

arrondissement, 39 years earlier. The declaration

of his death, which incorrectly states his age to be

21 (a mistake that has been perpetuated by others

since then), was made by two hospital employees.

Documenting here the series of emotionally

charged family interactions at the Demante

residence in Paris at 16 rue Jean de Beauvais, it

becomes clear that Évariste’s family life profoundly

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182

influenced his thoughts and actions, leading to his

premature death in 1832.

7. CONCLUSION

“No mathematician should ever allow him to

forget that mathematics, more than any other art

or science, is a young man's game. … Galois

died at twenty...”

– G. H. Hardy (1877–1947)

Figure 10: The duel of Évariste Galois in 1832

Évariste died in Paris on 31 May 1832 from the

gunshot wounds that he suffered in a duel (see

Figure 10). Even his last letter to a friend, Auguste

Chevalieron, on the eve of the duel, was on the

subject of mathematics, including some that can be

visualised as regular solids and a Fano plane (see

Figure 11), an order 2 finite projective plane with

the smallest possible number of points and lines,

namely seven of each, where there are three lines

passing through each point and three points on

every line (Kostant 1995, Le Bruyn 2008). The

portrait of Évariste by his younger brother, Alfred,

remains an iconic image of a great mathematician

(see Figure 2 earlier), celebrated by later

mathematicians to this day (Neumann 2011) and

also theoretical computer scientists (Hoare & He

1998) for his important foundational achievements.

Figure 11: Visualisation of a Fano plane, relating to

Évariste Galois’ last letter in 1832 (Wikimedia Commons)

The death record of Évariste Galois is shown in

Figure 12. Bringing archival documents together as

a visualisation, we have created an archival map of

place (Paris and Borg-la-Reine), time (1811–1871),

family connections, and documents, illustrating the

cascading events of Évariste’s life, leading to his

death in 1832, followed his mother’s remarriage

shortly afterwards in that same year, and beyond

that to her death much later in 1871.

Figure 12: Archives de Paris, Etat Civil, death record,

Évariste Galois, 31 May 1832

This paper has attempted to provide a view of the

French mathematician Évariste Galois (1811–1832)

and his close relatives (including his brother, the

painter Alfred Galois) from several aspects,

including an overview of his mathematical

contribution, the possibilities of visualising his

mathematics ideas, and the issues of searching

historical archives to help understand his family life

and background better. There are still many

possibilities of visualising Galois’ mathematics in

more interesting and artistic ways and discovering

further historic documents with information on his

wider family.

Figure 13: Mention of Évariste and Alfred Galois

in the 1829 inventory of Galois’ father (page 2)

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183

8. ACKNOWLEDGEMENTS

Thank you to Évariste Galois for his influence on

mathematics in general and theoretical computer

science in particular, including the application of

Galois connections to formal program development

(Morgan 1994, Hoare & He 1998), which was the

original inspiration for this paper, together with

connections within mathematical communities

(Bowen & Wilson 2012). Thanks also to Évariste’s

brother Alfred Galois for his artistic input and the

rest of the extended Galois/Demante families for

their archival legacy. (See Figures 13 and 14.)

Jonathan Bowen is also grateful to Museophile

Limited for financial support.

Figure 14: Galois and Demante family signatures

in the 1829 inventory of Nicolas Gabriel

9. REFERENCES

Bakshee, I. (1999) The Pattern of Beauty: The Art

of Igor Bakshee. Wolfram Media, Graphica 2: The

World of Mathematica Graphics.

Balakrishnan, J. & Venkatachalam, V. (2014)

Galois Visualizations: Program Guide. Math-

ematical Institute, University of Oxford, UK.

http://people.maths.ox.ac.uk/balakrishnan/galois/ga

lois_guide.pdf (retrieved 6 January 2015).

Beddard, H. & Dodds, D. (2009). Digital Pioneers.

V&A Pattern, V&A Publishing.

Bowen, J. P. & Giannini, T. (2014) Digitalism: the

new realism? In K. Ng, J. P. Bowen, & S. McDaid

(eds.), EVA London 2014: Electronic Visualisation

and the Arts. British Computer Society, Electronic

Workshops in Computing, pp. 324–331. DOI:

10.14236/ewic/eva2014.38

Bowen, J. P. & Wilson, R. J. (2012) Visualising

virtual communities: from Erdös to the arts. In S.

Dunn, J. P. Bowen, & K. Ng (eds.), EVA London

2012: Electronic Visualisation and the Arts. British

Computer Society, Electronic Workshops in

Computing, pp. 238–244.

http://ewic.bcs.org/content/ConWebDoc/46141

(retrieved 28 May 2015).

Kostant, B. (1995) The graph of the truncated

icosahedron and the last letter of Galois. Notices of

the AMS, 42(9), 960–968.

http://www.ams.org/notices/199509/kostant.pdf

(retrieved 4 June 2015).

Daintith, J. & Nelson, R. D. (eds.) (1989) The

Penguin Dictionary of Mathematics. Penguin

Reference Books.

Freeman, L. C. (2000) Visualizing social networks.

Journal of Social Structure, 1(1).

Godin, R., Missaoui, R., & Alaoui, H. (1995)

Incremental concept formation algorithms based on

Galois (concept) lattices. Computational Intel-

ligence, 11(2), 246–267.

Hoare, C. A. R. & He, J. (1998) Unifying Theories

of Programming. Prentice Hall.

Le Bruyn, L. (2008) Galois’ last letter. NeB blog.

http://www.neverendingbooks.org/galois-last-letter

(retrieved 27 May 2015).

Marghoubi, R., Boulmakoul, A., & Zeitouni, K.

(2006) The use of the Galois lattice for the

extraction and the visualization of the spatial

association rules. IEEE International Symposium

on Signal Processing and Information Technology.

DOI: 10.1109/ISSPIT.2006.270872

Melton, A., Schmidt, D. A., & Strecker, G. E. (1986)

Galois connections and computer science

applications. In Category Theory and Computer

Programming. Springer, Lecture Notes in Computer

Science, vol. 240, pp. 299–312. DOI: 10.1007/3-

540-17162-2_130

Morgan, C. C. (1994) The cuppest capjunctive

capping, and Galois. In A. W. Roscoe (ed.), A

Classical Mind: Essays in Honour of C. A. R.

Hoare, chapter 19, pp. 317–332. Prentice Hall

International Series in Computer Science.

Mu, S.-C. & Oliveira, J. N. (2011) Programming

from Galois connections. In Relational and

Algebraic Methods in Computer Science. Springer,

Lecture Notes in Computer Science, vol. 6663, pp.

294–313. DOI: 10.1007/978-3-642-21070-9_22

Neumann, P. M. (2011) The Mathematical Writings

of Évariste Galois. European Mathematical Society

Publishing House.

Petsinis, T. (1995) i = Galois: A Fictional Biography

of the French Mathematician Évariste Galois

(1811–1832). Masters thesis, Department of

Humanities, Victoria University of Technology,

Australia.

Rigatelli, L. T. (1996) Évariste Galois, 1811–1832.

Birkhäuser.

Robinson, A. (2010) Sudden Genius? Oxford

University Press.

Stein, P. (2012) Visualizing Galois fields. Nklein

Software, 17 May.

http://nklein.com/2012/05/visualizing-galois-fields/

(retrieved 6 January 2015).

Wolfram, S. (2002) A New Kind of Science.

Wolfram Media.