Elli Heesch, Heinrich Heesch and Hilbert’s eighteenth problem: collaborative research between philosophy, mathematics and application

This paper examines the hitherto unknown scientific collaboration between the siblings Elli Heesch (1904–1993) and Heinrich Heesch (1906–1995). Heinrich Heesch, a well-known mathematician, was spearheading the early development of the computer-aided proof of the four-colour theorem. Much less is known about his sister Elli Heesch, a philosopher and logician. Together with her brother she investigated tiling problems and worked out a solution of Hilbert’s 18th problem. In 1944, Elli and Heinrich Heesch wrote a joint treatise on the industrial application of the tessellation method, which was of great interest to the German war and armaments industry. The collaboration of the Heesch siblings illustrates individual, disciplinary, cultural, and political aspects of knowledge production. The common interplay of close family relations and socio-political conditions that we find here underlines the fact that women’s contributions to solving mathematical problems often remained invisible.


2018).
A memorial plaque at Wilhelmsplatz 4 in the city of Kiel reminds us that the mathematician and crystallographer grew up at this place (Liebau 2006).Hans-Günther Bigalke has written a comprehensive biography of Heinrich (Bigalke 1988).
Hardly anything is known about the sister Elli Heesch.So far, no one has taken the two siblings seriously to be a scientific couple.At first glance, this is not surprising.Elli Heesch's academic path led her from Kiel via Münster to Innsbruck and finally to Prague and Warsaw.She pursued a career as a teacher at an early age.Later she became a nun.Heinrich Heesch's path led him via Munich and Zurich to Göttingen and finally to Hanover.Heinrich became professor of mathematics.Despite these different paths, it is also worth taking a closer look at the life of Elli and Heinrich Heesch. 2  Elli Johanna Anna Heesch was born on 4 April 1904 in Schleswig, Germany.She was the second daughter of Peter Heinrich and Bertha Heesch, née Herzer.Heinrich Heesch was born two years later, on 25 June 1906.He was the third child in the family.The oldest daughter was Käthe Heesch.She died after a long illness in 1925.The siblings Elli and Heinrich grew up in a musical home.Their father was a military musician, their mother a trained singer.Siegfried Heller, the siblings' teacher, had an important influence on their academic development.Heller had received his doctorate in mathematics from the University of Kiel under Paul Stäckel in 1904.Later, Heller was a teacher at the 'Oberrealschule', but remained in close contact with the 'History of Mathematics Seminar' at Kiel University, which was organized by Julius Stenzel, Heinrich Scholz and Otto Toeplitz.
In 1922, Elli Heesch began studying mathematics at the Christian-Albrechts University of Kiel.She attended lectures by Ernst Steinitz and Otto Toeplitz.She also attended seminars in philosophy given by Heinrich Scholz and soon developed a close friendship with him.Elli told Scholz about her brother's musical talent.Scholz enthusiastically passed this information on to Steinitz and Toeplitz with whom he shared a passion for music.So, Scholz, together with Steinitz and Toeplitz, soon joined the hospitality of the Heesch family in order to play music together at Wilhelmsplatz 4 in Kiel.Their private relationships were strengthened by common bereavements: Scholz's wife Elisabeth died on 6 August 1924, and Käthe Heesch on 21 February 1925 (Bigalke 1988, 23).In April 1925, shortly after the death of his older sister Käthe, Heinrich Heesch began to study physics and mathematics at the University of Munich.He attended courses by Arnold Sommerfeld and Constantin Carathéodory. In 1928, he Mss. 1995.30).Göttingen received the collection as a gift from Professor Hans-Günther Bigalke in 1995.My own archival research on Elli Heesch would not have been possible without the information that Adelheid-Hamacher-Hermes gathered on Elli Heesch. Hamacher-Hermes' article (2008) is the only one that has been published on Elli Heesch up to the present day.
in Munich in the winter term 1926/27, he prepared a transcript for Arnold Sommerfeld who used it for his Lectures on Theoretical Physics, vol. 1: Mechanics (Sommerfeld 1943).The 1964 English translation is based on the 5th edition in German (Sommerfeld 1964).The names of the Heesch siblings are not mentioned.In the German preface to the 1943 edition, Sommerfeld thanks Heinrich and Elli Heesch by name (Sommerfeld 1943, vii).
One year later, Elli Heesch passed her academic exam for the 'Lehramt für höhere Schulen' at the University of Kiel on 21 August 1928.In the winter semester of 1928/29, she enrolled at the University of Münster in order to get her degree in philosophy.Her advisor was Heinrich Scholz, who had received a chair for philosophy at the University of Münster.Elli postponed her doctoral studies and continued her teacher training, called 'praktischer Vorbereitungsdienst für die Lehramtsprüfung an höheren Schulen'.Elli Heesch began her school employment as an assessor on 1 April 1930, at the girls' school 'Hildegardis-Oberlyzeum' in Bochum, where she continued to teach until the fall of 1931 (Hamacher-Hermes 2008, 267).
Heinrich received his doctorate degree in 1929 at the ETH Zurich in the field of mathematical crystallography after submitting two papers which were published in the Zeitschrift für Kristallographie (Heesch 1929a(Heesch , 1929b). 3The advisor was Gregor Wentzel.In Zurich, Heinrich Heesch met Hermann Weyl.Weyl offered Heesch an assistantship, which Heesch gratefully accepted.In 1930, Heesch went with Hermann Weyl to Göttingen.The same year, Elli moved from Bochum to the 'Oberlyzeum' under the head of the Franciscans of Nonnwenwerth ('Aloisianum') and taught there until Easter 1932 (Figure 1).
During this time, Elli worked on the completion of her doctoral thesis on Bernard Bolzano's Wissenschaftslehre (Bolzano 1837).The disputation took place on 18 and 19 February 1932 (see the protocols of the examinations in Philosophy, sgd.Scholz; mathematics, sgd.Behnke; physics, sgd.Kratzer). 4Elli Heesch received her doctorate on 30 May 1933.Her doctoral thesis, 'Grundzüge der Bolzano'schen Wissenschaftslehre',was officially published two years later in the journal Philosophisches Jahrbuch der Görres-Gesellschaft (Heesch 1935a).5 Elli's doctoral thesis examines in a highly 3 Heinrich Heesch's doctoral thesis is composed of two articles published in 1929 in the Zeitschrift für Kristallographie (Heesch 1929a(Heesch , 1929b)).A third paper followed in 1930 in the same journal (H.Heesch 1930).In these papers, Heesch introduced the so-called black-white groups (a subclass of magnetic space groups).Specifically, Heesch introduced an anti-symmetry operation to the 32 crystallographic point groups which gives a total of 122 magnetic point groups.An example of an anti-symmetry operation is magnetic spin so that each atom side can have one of two possible values: spin up or spin down if the magnetic values are not randomly arranged, but aligned.A further example is the case where one atom is colored in one colour, say black, and the other identical atom symmetrically related in position is coloured in a different colour, say white.The concept was more fully explored by Alexei Vasilievich Shubnikov in terms of 'colour symmetry' (Shubnikov and Belov 1964). 4For these and further information concerning Elli Heesch's    themselves; (ii) Bolzano's system of the interrelation of true propositions; (iii) the possibility of knowing truths (for the human mind). 6n subsequent years, social and political conditions interfered with the plans of the two siblings to pursue their habilitation.In 1933, when Elli completed her doctorate, Heinrich witnessed the racist purges at the University of Göttingen. 7After the National Socialists seized power, many academics were dismissed for racial or political reasons.Mathematics was most severely affected, losing, among others, the highly renowned professors Richard Courant, Hermann Weyl, and Edmund Landau, as well as Emmy Noether (Siegmund-Schultze 2009).The only way Heinrich Heesch could have remained a member of the university and thus could have continued to work toward obtaining a habilitation would have been by joining the 'NS-Dozentenbund' (The National Socialist German Lecturers League).
Since this was not an option for Heesch, he resigned from his university position in 1935 and worked privately at his parents' home in Kiel until 1948 (Mertens 2004, 287).For Elli, too, her hopes of finishing her habilitation were likewise dashed.Elli thus worked from 5 January 1932 to Easter 1933 at the University of Tübingen, under the chair of the Roman Catholic theologian Paul Simon.In 1932, Paul Simon became rector of the University of Tübingen.However, in 1933 he was dismissed from the civil service at his own request (due to his opposition to the National Socialist).Elli thus worked without official employment at the University of Tübingen, receiving only occasional payment.
On a trip to Italy, Elli met the Austrian philosopher Alfred Kastil, an expert on Franz Brentano.Elli knew Kastil from her time as a student at the University of Innsbruck.Kastil invited her to come to Prague and Innsbruck from Easter 1933 to Easter 1934.From May 1933, Elli worked on behalf of the Brentano Society with Professor Kastil and Ernst Foradori at the University of Innsbruck.8Among other things, she prepared a publication on Ernst Mach's Erkenntis und Irrtum (Mach 1906).She also intended to write on the concept of infinity (from Antiquity, Giordano Bruno and Leibniz to modern set theory).Supported by Kastil, Elli moved to Prague in February 1934 with the aim of completing her habilitation under Oskar Kraus, head of the Prague Brentano Society.But all academic assistant positions were already occupied. 9During her time in Prague, Elli learned about the Vienna Circle and established personal contacts with Rudolf Carnap and Otto Neurath (Hamacher-Hermes 2008). 10In a letter to Heinrich Heesch on 30 April 1934 (Figure 2), she reports that she has engaged with the work of Alfred Tarski and Kurt Gödel, but so far no archival records of this project could be found.In the spring of 1935, Elli Heesch worked together with Jan Łukasiewicz in Warsaw for a few weeks.However, she was forced to return to Germany for financial reasons.Shortly after her return to Germany, her paper 'Vom intellektuellen Gewissen ' was published in the Catholic journal Akademische Bonifatius-Korrespondenz in May 1935 (Heesch 1935b).Elli Heesch was not overjoyed about this place for her publication.She would have prefered if the paper had been published in the journal Erkenntnis, the Kant-Studien or Philosophisches Jahrbuch (see Elli Heesch's letters to Heinrich Heesch on March 1935).Like Scholz at the same time, Elli Heesch advocated the new logic called 'Logistik' and defended it against philosophical criticism.She argued that modern, formal logic leads to clear and precise thinking and contributes to intellectual formation and education (Heesch 1935, 23).Following this introduction, she gives a brief but concise overview of the development of recent logic since Leibniz.She mentions Bolzano as a logician influenced by Leibniz, before describing Gottlob Frege and his Begriffsschrift of 1879 as the creator of modern propositional calculus that is developed in axiomatized form.She then turns the reader's attention to the international research landscape of contemporary logic; in addition to the English logicians, e.g.Russell and Whitehead and American logicians, she also refers to the German representatives, particularely those in places such as Münster (Heinrich Scholz), Berlin (Hans Reichenbach, Kurt Grelling, Walter Dubislav), Göttingen (David Hilbert) and the Vienna Circle in Vienna and Prague, not to forget to mention the Polish school, first of all Jan Łukasiewicz.The thesis underlying Heesch's paper is that our intellectual conscience (analogous to the moral conscience in action theory) imposes on us the responsibility to posit freedom from contradiction as a necessary, though not sufficient, criterion that an axiom system must satisfy.
From Easter 1935 to 1937, Elli Heesch lived with her brother in Meldorf (Low German: Möldörp or Meldörp), a small town in the district of Dithmarschen in Schleswig-Holstein.Both were without academic employment at that time and earned their living with private tutoring. 11In retrospect, this period proved to be very productive: both were working on group theory and mathematical tiling problems.
In February 1936, Łukasiewicz gave four lectures in Münster on Heinrich Scholz's invitation, among them 'The Old and the New Logic' (Figure 3).Elli Heesch attended these meetings.In a letter to her brother on 6 February 1936, she writes with a mixture (Also listed is the Polish philosopher and logician Janina Hosiasson-Lindenbaum, a student of Władysław Tatarkiewicz.) 10 There are five letters from Elli Heesch to Otto Neurath and six letters from Otto Neurath to Elli Heesch in the Vienna Circle Archive, Otto Neurath Collection, Inv.nr.243, Noord-Hollands Archief, Haarlem, The Netherlands. 11The mathematician Arnold Scholz wrote to Olga Taussky in a postcard Kiel, 4 July 1935: 'Dear Oli! Thank you very much for your last two cards from Bryn Mawr and Oxford!I hope you are feeling quite well in England!The last time Foradori from Innsbruck, now Berlin, was here, whom you perhaps still know slightly, a very nice, modest person.Miss Heesch also attended his lecture.She and her brother, who live currently in Kiel, have nothing to live on.'Quoted from Lemmermeyer and Roquette 2016, 425).The translation is my own. of hope and skepticism that she might be able to return to Warsaw for the summer semester as a visiting scholar.The research stay never materialized.
A few months later, at a conference of the Deutsche Mathematiker-Vereinigung in Salzbrunn from 14 to 19 September 1936, Elli Heesch presented a talk on 'Gruppen mit vertauschbaren konjugierten Elementen' (E.Heesch, 1936).Elli Heesch's lecture dealt with some of the consequences of the Sylow theorems, a fundamental part of finite group theory with important applications in the classification of finite simple groups.Elli Heesch's intensive engagement with group theory began already during her studies at Kiel University.In a 1928 paper, she investigated the continuity theorems in Victor Eberhard's Zur Morphologie der Polyeder (Eberhard 1891), one of the main research topics of Elli Heesch's teacher Ernst Steinitz.Steinitz spent many years working on the theory of polyhedra.12Elli Heesch's interest in group theory continued after her time in Prague and Warsaw.Her letters to her brother in the early 1930s concerning finite and infinite semi-groups bear witness to this.The period between 1935 and 1945 was a very difficult time for both siblings.Heinrich had no academic position.Elli Heesch had a teaching position since 1937 at school.However, giving up her academic career was not easy for her.Above all, however, she was worried about her brother's financial situation.For this reason, she became a driving force for her brother's contacts with economy and industry.
During the Second World War, Heesch was temporarily employed at Jakob Loef's Steinbock AG machine factory in Moosburg, upper Bavaria, which prevented him from being called up again for military service at the front.After the war, Heinrich resumed his work for Jakob Loef, but only until 1948.In the meantime, he had made the solution of the four-colour problem so much his profession during this time that he turned down a lucrative offer on behalf of the Association of the German Mechanical Engineering Institute in Düsseldorf.For the coming years, from 1951 to 1953, Heesch's former student friend Ernst Peschl, who had in the meantime become director of the Mathematical Institute of the University of Bonn, was able to employ Heesch as a 'wissenschaftliche Hilfskraft' with funds from the 'Notgemeinschaft der Deutschen Wissenschaft', so that Heesch was able to continue his research.In 1955, at the instigation of Otto Kienzle, who had already been interested in Heesch's work during the war, Heesch was initially employed there as a lecturer, or 'Lehrbeauftragter '. In February 1958, Heesch obtained  In March 1972, Elli Heesch had research stays in the United States at Princeton and Washington, among other places (Figure 4).In July 1993, she was awarded the Order of Merit of the Federal Republic of Germany.She died on 18 September 1993, at the age of 89. 13 Two years later, on 26 July 1995, Elli's brother Heinrich died in Hanover.

Working together on Hilbert's eighteenth problem
At first glance, Elli and Heinrich Heesch worked on a variety of topics in separate areas.Elli Heesch was a philosopher and logician, Heinrich Heesch was a mathematician.A closer look, however, reveals that both were exchanging ideas, not least on mathematical research questions.This thesis can be paradigmatically illustrated by the so-called 'regular sphere packing problem', i.e. the solution (for the two-dimensional case) of David Hilbert's 18th problem.Today, Heinrich Heesch is credited with having found a solution for Hilbert's 18th problem.
By definition, a tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called tiles, with no overlaps and no gaps.A periodic tiling has a repeating pattern.Some special kinds include regular tilings with regular polygonal tiles all of the same shape.Mathematically speaking, a tessellation (in the plane) is a (countable) set of tiles (parquet stones).If the symmetry group contains two linearly independent translations, the tiling is said to be periodic (for an introduction see Adams 2022).
At a conference in Paris in 1900, Hilbert presented a list of ten unsolved problems in mathematics (Hilbert 1900).He ultimately put forth 23 problems that to some extent set the research agenda for mathematics in the twentieth century.Hilbert formulated the 18th problem, i.e. 'building up of space from congruent polyhedra', as a twofold question14 : (i) 'Is there in n-dimensional Euclidean space also only a finite number of essentially different kinds of groups of motions with a fundamental region?' (Hilbert 1902, 467) (ii) 'Whether polyhedra also exist which do not appear as fundamental regions of groups of motions, by means of which nevertheless by a suitable juxtaposition of congruent copies a complete filling up of all space is possible' (Hilbert 1902, 467).Hilbert thus asked, first, whether it is possible to generalize a finite classification of crystallographic groups to n-dimensional Euclidean space.Second, he asked whether monohedral but not isohedral tilingss in three dimensions are possible to realize.In 1910, Ludwig Bieberbach answered this part of the question in the affirmative (Bieberbach 1910).The subgroups in question are now called 'Bieberbach groups'.However, the question was open whether a polyhedron exists which tiles 3-dimensional Euclidean space but is not the fundamental region (domain) of any space group.Such tiles are now known as anisohedral.In 1928, Karl Reinhardt found such a tiling (Reinhardt 1928).Heinrich Heesch found a tiling in two-dimensional space, namely a nonconvex anisohedral polygon in the plane that allows for a periodic monohedral tiling.Reinhardt's assumption that for n = 2 monohedral tilings are also isohedral was refuted by Heinrich Heesch.The tile, or parquet stone, presented by Heesch as an example in his 1935 work is a decagon with which the Euclidean plane can be covered simply and without gaps, while it cannot be a fundamental area in a (twodimensional discontinuous) group of cover transformations (see Figure 5).
Heesch's article was published on the recommendation of Helmut Hasse in the Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen.In his monograph on Heinrich Heesch, Hans-Günther Bigalke retrospectively stated: 'For more than twenty years, this was to be Heesch's last mathematical publication.His assistantship expired on March 31, 1935.An extension was not possible' (Bigalke 1988, 114).Bigalke still mentions that Constantin Carathéodory would have liked to see Heesch habilitated with his 1935 paper.
However, that's not the whole story.In fact, Elli and Heinrich were working together on Hilbert's 18th problem.They had already given a proof of completeness for the case of regular tiling in 1932, i.e. a complete classification of 28 types of asymmetric tiles which tessellate (more specifically, of isohedral tilings without reflection symmetry).Elli and Heinrich Heesch submitted their solution to the Mathematische Zeitschrift in 1934.The article was accepted.As they wrote in 1944, they still abstained from publishing it 'for various reasons' (Heesch et al. 1944, 15).According to Bigalke, this was perhaps one of the biggest mistakes Heinrich Heesch made with regard to his academic career (Bigalke 1988, 125).From Bigalke we also learn that Victor Moritz Goldschmidt, a Norwegian and Jewish geologists and chemist, who was appointed the chair of mineralogy in Göttingen in 1929, advised Heinrich Heesch to patent the mathematical invention.In fact, Heinrich Heesch, due to his professional and financial situation, did everything possible to market his work (Bigalke 1988, 126) and Elli Heesch supported him.For almost ten years, with the help of various patent attorney firms, the siblings tried unsuccessfully to obtain a patent from the Reichspatentamt (German Patent Office) as well as abroad.
On 30/31 January 1934, at the invitation of Ludwig Bieberbach, Heinrich Heesch gave two lectures at the Mathematical Institute in Berlin regarding tiling problems: the one lecture was directed to mathematicians, the other lecture was directed to a public audience (see Bigalke 1988, 126).In the auditorium was, among other luminaries, the head of the Dresden branch of the company Villeroy and Boch.Heesch was commissioned to design suitable tile designs for the company.One of these designs would materialize as roof tiles which formed the covering of the Göttingen 'Stadthaus'.
The Heesch siblings sought connections with business and industry, on the one hand with manufacturers of wrapping paper printed with patterns, and on the other with manufacturers of ceramic tiles, such as Villeroy and Boch.Another application arose in the course of the Second World War which came as a surprise to the siblings: the shortage of raw materials made it desirable to minimize waste when stamping sheet metal parts; the mathematical studies of the Heesch siblings provided an economical use of both production material and time.
In the spring of 1938, Heinrich Heesch presented sample patterns at the Leipzig Exhibition.There he met Jakob Loef, the director of the Steinbock AG machine factory in Moosburg.Loef showed strong interest in Heesch's work.In a letter dated 3 February 1938, Heinrich informed his sister that Jakob Loef had sent him some of his own designs and a 'Hilbert article' in which Loef deals with Hilbert's 18th problem.He himself, Heinrich said, found Loef's discussion of Hilbert 'cruelly superficial'.Loef also overestimated the savings achieved by the method of tiling.This motivated Heesch to calculate plate losses more precisely.He sent the results with drafts to Elli Heesch and asked for her opinion.At the same time the Heesch siblings carried out extensive research on tilings and kept discussing Hilbert's 18th problem.For example, when discussing the number of symmetry groups of corresponding tilings in hyperbolic n-space, in a letter dated 13 March 1938, Heinrich remarked that 'you [Elli] did the regular problem for the elliptical plane (=sphere), I did it for the Euclidean plane' (see also Figure 6).
Elli encouraged her brother to cooperate with Loef.In the meantime, Otto Kienzle had also made contact with Heesch.Kienzle had been appointed to the chair of 'Betriebswissenschaften und Werkzeugmaschinen' at the Technical University of Berlin in 1934 and was employed by the German Army Weapons Office during World War II.Kienzle offered Heesch a position at the Technical University Berlin, with the prospect of working towards his habilitation.Heesch declined, mentioning a variety of reasons, primarily political, but also because he had already signed an employment contract with Loef.Loef obtained a leave for Heinrich Heesch from military service ('Wehrdienst') as of 15 August 1944, and for a limited period of time employment with the planning service of the Reich Research Council ('Planungsdienst des Reichsforschungsrats'), which made Heesch available for Moosburg.
With Loef's support, contact was immediately established with Siemens and Halske.The first meeting with 300 specialists of the company under the leadership of Dr. R von Siemens took place on 24 November 1944.In January 1945, it was agreed that the Heesch-procedure would be used on a large scale in telephone production, among other things.On 15 January 1945 Siemens and the Heesch siblings (Elli and Heinrich Heesch) signed a contract.Both together were to receive 300 Reichsmark per month for their consulting work (Bigalke 1988).However, the contract ended up not materializing due to the collapse of the German Reich.In addition to Siemens and Halske, Messerschmitt-Flugzeugwerke had expressed interest in the invention of the siblings.As late as February 1945, Elli and Heinrich Heesch jointly conducted a training course lasting several days at Messerschmitt-Flugzeugwerke's Oberammergau Research Institute in Upper Bavaria.
In 1944, Heinrich Heesch published the work System einer Flächenteilung together with Elli Heesch and Jakob Loef under the Josef Pichlmayr publishing house, Moosburg (Bavaria) (Heesch et al. 1944; see Figure 7).The publication was issued 'in conjunction with the Main Committee for Machines and Apparatuses at the Reich Ministry for Armaments and War Production' ('Hauptausschuß Maschinen und Apparate beim Reichsminister für Rüstung und Kriegsproduktion') with the written note: 'Confidential!Not intended for publication elsewhere, nor for discussion in the press!'In the preface, the authors state that the paper deals with a 'new method of tiling by which remarkable savings can be achieved' (Heesch et al. 1944, preface (without page number)).
The publication is currently achieved as a file ('Sachakte') in the Bundesarchiv under the archive number R 3/653 in Berlin-Lichterfelde.The title of the file in the Bundesarchiv, however, 'System einer Flächenteilung und seine Anwendung zum Werkstoff-und Arbeitsparen nach Heesch und Loef', is misleading.If one opens the file, which can be viewed online15 , the title page reads: 'System einer Thus, the name 'Elli' as co-author is simply flat-out ommited in the file designation.That this is not only misleading but incorrect.
The 1944 publication in the Bundesarchiv consists of two parts: In the first part, Elli and Heinrich Heesch describe the method of tiling the plane isohedrally ('Flächenschluss').In the second part, Josef Loef describes 'the experiences with application in an industrial plant'.Part I starts with an explication of the technical term 'decomposition' ('Zerlegung'), or 'tiling'.It means a simple gapless division and classification of the unlimited plane into congruent, possibly also mirror-inverted congruent single forms, elements, or 'stones' (Heesch et al. 1944, 7).Subsequently, regular tilings are distinguished from irregular ones and illustrated by examples.In this context, Elli and Heinrich Heesch refer to Wilhelm Ostwald's book Harmonie der Formen (Ostwald, 1922), in which many examples of regular tilings can be found (Figure 8).However, Ostwald did not address the question of completeness.It was in fact Hilbert who asked, in his 18th problem, 'whether polyhedra also exist which do not appear as fundamental regions of groups of motions, by means of which, nevertheless, by a suitable juxtaposition of congruent copies a complete filling up of all space is possible' (Hilbert 1902, 467).
Elli and Heinrich Heesch then went on to state that Hilbert's question, for which they presented a solution in 1934/35, was of only minor importance for practical purposes.The exact number of possible tilings would be of interest with respect to questions of application.For the case of regular tiling they had already given a proof of completeness in 1932, i.e. a complete classification of 28 types of asymmetric tiles which tessellate (namely of isohedral tilings without reflection symmetry).Further, they had described the construction rules for each type.This statement is remarkable, because in the vast majority of publications on the regular tiling problem it is claimed that in 1963 Heinrich Heesch and Otto Kienzle proposed a complete classification of 28 types of asymmetric tiles which tessellate (Figure 9).This leaves seven of the 17 plane symmetry groups (called p1, p2, p3, p4, p6, pg, pgg).From these, there are only 28 isohedral types.
The simple idea of the Heesch-Kienzle method runs as follows: Patterns can be made by repeated application of operations.The basis for changes of the initial cells are translations (T), rotations (C) and glide reflections (G).Starting from a polygon, a translation T changes one side and moves it to the opposite side.In a glide reflection G, the changed side is shifted and then mirrored at the centre of the side.In a rotation C, half a side is changed and transferred to the second half by rotation around the centre of the side.The resulting side line is called the C line.By rotating C around a vertex of the polygon, the changed side is transferred to the neighbouring one.Due to the requirement of periodicity of the whole pattern, only rotations of 60°, 90°, 120°and 180°, i.e. point reflections, are admissible.Heesch moved on to prove exactly nine types.Heesch obtained 28 different constructions result by combination with the 17 symmetry groups.Note: The method devises a Figure 9. Heinrich Heesch and Otto Kienzle (1963).Flächenschluß.coding scheme to denote the 28 types.For example (Figure 10) TTTT is the type where each tile has four lines and where these lines are related by translation (T) only. 16The translation (T) operation means that, for example, a concave shape applied to one side of a square requires the same shape applied to the opposite side, now in a convex sense.Thus, translation conserves the area of the original rectangular or hexagonal figure (as do the other operations).
After the war, Elli continued to teach in a Catholic girls' gymnasium and held important leadership positions within the school administration.Heinrich Heesch reestablished his contacts with industry and published a number of articles, for example in the Industrie-Anzeiger Essen (Heesch 1950) and in the Mitteilungen der Forschungsgesellschaft Blechverarbeitung (Heesch 1956(Heesch , 1957)).In 1955, Heesch first became a lecturer, then, after his habilitation in 1958, a salaried private lecturer and finally an associate professor at the Technical University of Hanover.During this period Heesch published two books on the tiling problem, which became very popular (Heesch and Kienzle 1963;Heesch 1968) and did pioneering work in developing methods for a computer-aided proof of the then unproved four colour theorem.

Conclusion
The focus of this paper was the collaboration between the siblings Elli and Heinrich Heesch on the regular tiling problem.Heinrich Heesch is credited for his solution of Hilbert's eighteenth problem and for his classification system of 28 types of tilings.It is virtually unknown that Elli accompanied and supported his research from the very beginning.The purpose of this article was to shed some new light on the sibling's collaboration.The result is instructive for several reasons.First, the Heesch example provides a critical examination of the much-cited Matilda effect.The Matilda effect is a bias against acknowledging the achievements of women scientists.17Nevertheless, one cannot claim that Elli did the work for which Heinrich would have been rewarded.The siblings were close friends throughout their lives.They shared an enthusiasm for mathematics.Many letters attest to their interest in discussing and solving mathematical problems.Both were deeply devout and modest people who had no interest in popularity.(Heinrich became an extremely committed Catholic after his conversion to the Catholic faith in which he followed the example of his sister.)Both shared the ambition of an academic career, and both had to earn their livelihood by teaching and working outside the academia.Seen in this light, the example shows us that academic careers are often marked by precariousness and uncertainty and that the Matilda effect has many faces.
Second, the example provides deep insights into aspects of research ethics concerning the responsibility of mathematicians in times of National Socialism and the Second World War.Heinrich Heesch relinquished his habilitation in Göttingen because he refused to join the National Socialist German Lecturers League (Nationalsozialistischer Deutscher Dozentenbund).Nevertheless, he later worked, together with his sister, for the German Army Weapons Agency.Wasn't that acting against its own ethical principles?In retrospect, one should bear in mind that this job protected Heesch from being drafted as a soldier to the front.Moreover, he was able to earn money in this way in order to survive.Thus, Elli and Heinrich were calculating couples in the sense of pragmatic social strategizingwhen it comes to the specific compromises in order to navigate dilemmas in which there was probably no straightforwardly good choice to be made, at times seemingly collaborating with a government and institutions they oppose.But it would be utterly wrong to call Elli and Heinrich strategists and pragmatists; on the contrary, both were idealists who were deeply convinced of the compatibility of science and faith and guided by an intellectual and moral conscience.Third, the Heesch example reveals an interesting interdependence between foundational research and application.The cooperation between the mathematician and crystallographer Heinrich Heesch on the one side and the logician and philosopher Elli Heesch on the other side was an excellent match, because it bridged the gap between the theory of patterns and formal logic.How does that fit together?Hilbert's 18th problem provides a possible answer.This problem formed an initial point of a research programme concerning the arithmetic theory of crystallographic groups, a powerful tool for studying crystalline structures.At the same time, mathematical logic became formalized and paved the way for the axiomatization and formalization of mathematical concepts concerning tiling structures, which allowed a more detailed analysis of the algebraic, topological and combinatorial properties of tiling problems.

Disclosure statement
No potential conflict of interest was reported by the author(s).
systematic and critical manner three presuppositions of Bolzano's theory of science, namely those concerning (i) Bolzano's theory of ideas, propositions and truths in

Figure 3 .
Figure 3. Announcement to the lecture by Jan Łukasiewicz on 'The Old and the New Logic' in Münster on 4 February 1936.Source: Polkowski (2019, 21).
also received a master class certificate for violin at the State Academy of Music in Munich.Together with his sister, who spent a semester 2 For the following presentation of Elli und Heinrich Heesch's life and carrier I consulted, among other sources: (i) Münster University Archive.Doctoral file Elli Heesch.Faculty of Philosophy 65/2737 (Universitätsarchviv Münster.Promotionsakte Elli Heesch.Philosophische Fakultät.Bestand 65, Nummer 2737).(ii) University and State Library Münster.Heinrich Scholz collection.Sign: N. Scholz (Universitäts-und Landesbibliothek Münster.Nachlass Heinrich Scholz.Sign.: N. Scholz).(iii) Göttingen State and University Library.Heinrich Heesch collection (including Elli Heesch collection).Sign.: Cod.Ms. H. Heesch (Niedersächsische Staats-und Universitätsbibliothek Göttingen.Nachlass Heinrich Heesch.Teilnachlass Elli Heesch.Sign.: Cod.Ms. H. Heesch).The Heinrich and Elli Heesch collection in the Göttingen State and University Library consists of 11 boxes, 16 folders, 7 mpn, 2 rolls (Cod.Ms. H. Heesch; Inventory number: Acc. his habilitation.Thereafter, he taught as an unsalaried 'Privatdozent' at the Technical University Hanover until March 1961, when he was appointed an associate professor and began drawing a monthly salary.No further improvement of his position was granted to Heesch and it was not actively sought by him.He lived with his mother in Kiel until 1960, then as a subtenant in Hanover until 1984.It was not until 1984 that he gave up his parents' apartment in Kiel and moved in his own apartment in Hanover.On 1 July 1946, Elli Heesch entered the novitiate of the Sacred Heart Society in the monastery of St. Adelheid in Bonn-Pützchen.The Society of the Sacred Heart of Jesus (Sacré Coeur) is a religious congregation of women which was founded in 1800 by Sophie Barat in France.Beginning in 1949, Elli Heesch taught there at the order's own school, and on 2 April 1949 at St. Adelheid.With the reestablishment of the order she went to Hamburg in 1951, where she was director of the Catholic Sophie Barat School from 1957 to 1969.At the age of 68, she began studying gerontology.
Table of 28 types of assymmetric isohedral tiles.