Farewell to Alan L. Mackay (1926-2025) - Scientist, Friend

We knew this would come, yet it is not easy to accept that my friend and remote mentor is no longer. He was one of crystallography’s most original thinkers but he was not just a crystallographer; crystallography was his way of life and his crystallography was broad — it was the structure of matter. He was always on guard lest we lose the larger picture of what there is beyond how atoms are arranged in space. In this personal account, I rely on my previous writings about him [1, 2] and our published conversation [3].

Alan L. Mackay had been one of my heroes well before we met in 1981 in Ottawa during the Congress of the International Union of Crystallography. I was anxious to meet him, so I walked up to him and introduced myself. We exchanged a few words, and then he abruptly turned and left. I felt let down and was not even sure that he caught my name. However, in a few weeks I received a gracious letter from him saying that he was glad that we had met and suggesting that we meet again soon. Later, I learned that the Ottawa encounter was not atypical; his hearing difficulties could hinder such social interactions and he did not like small talk. Our children like to refer to Alan’s saying “It comes anyway”, as he declined to talk about the weather or listen to the weather forecast. A decades-long friendship developed with numerous visits in both directions, our wives also taking to each other, and we stayed in each other’s homes. 

Alan came to see us in Budapest in September 1982, and we organized three lectures for him. Two were about fivefold symmetry. My amateurish but deep interest in quasicrystals and buckminsterfullerene dates to his presentations. Until then, fivefold symmetry used to be just one of countless symmetries for me. My research involved the structure of isolated molecules, so I hardly paid any interest to its presumed impossibility in extended structures. As I learned about this fundamental dogma of classical crystallography, the possibility of its fallibility came simultaneously, thanks to Alan. He taught us that if we accept dogmas uncritically, even if we encounter their violations, we may remain blind to them. Alan took pleasure in questioning dogmas. He ascribed the conspicuous success of Jewish scientists in making discoveries to the long history of the Jews suffering from unreasonable rules and regulations and their willingness to question them. He was a rebel himself, coming from fiercely independent ancestors, and he liked to characterize his position in society as an “internal immigrant.” 

Alan Lindsay Mackay was born on September 6, 1926, in Wolverhampton, England. Both his parents were born in Glasgow. They were physicians and lived in Wolverhampton, in the Midlands. Alan’s father served as an infantry officer in World War I and as second-in-charge of a military field hospital in the Middle East in World War II. Alan’s parents ran their own practice in the late 1920s and 1930s, which they sold in 1938. They then became consultants and, especially Alan’s mother, served the community in various capacities dictated by their social conscience. There was always professional talk at their table during their meals, which fascinated Alan. He also understood that what he heard there could not be repeated outside their home. There were also brothers and sisters who eventually dispersed to Australia and America.

Alan started his formal education at the age of five in a small private school and continued at the Wolverhampton Grammar School from 1935 to 1940. He passed an entrance examination at the age of eight to get into it. His school years overlapped with the Second World War. At the age of 13, he was a messenger in the Auxiliary Fire Service. In 1940, he was sent to a boarding school — Oundle School — after he passed another entrance examination. There was talk of a possible German invasion. Alan stayed at Oundle until 1944. Although life was difficult during the war, the education was excellent. His teachers had first-class degrees in science and mathematics — teaching was a sought-after profession during and after the Depression. There was an emphasis on practical applications and experiments. His chemistry teacher demonstrated periodic chemical reactions, today called oscillating or Belousov-Zhabotinsky reactions. The concentrations of the reactants and products undergo periodic changes, and they offer a spectacular view if the participants have colors. However, such reactions can occur far from equilibrium, so, even 20 years later, Belousov found it difficult to get his manuscript describing such reactions accepted for publication.

Alan referred to his developing an independent mind: “I discovered that you should not believe everything that grown-ups tell you nor say what you actually think. … The tradition of my ancestors was to listen to what authority said and keep their doubts to themselves” [4]. His intellectual disposition of being an internal immigrant was strengthened by the predicament of increasing difficulty of hearing, which started to become noticeable in 1955. On the other hand, he developed exceptional reading skills in at least half a dozen languages — he became a voracious reader. He travelled a great deal, especially in Eastern Europe and, from 1961, in Asia, including Japan, China and Korea, as well as India.

Alan earned excellent credentials, and in October 1944, he enrolled at Cambridge with a scholarship for Trinity College. He focused on physics and chemistry, and also studied electronics, mineralogy and mathematics. Sir Lawrence Bragg was one of his professors along with other famous scientists, such as the physical chemist and later Nobel laureate R. G. W. Norrish, the physical chemist Frederick Dainton, and the inorganic chemist H. J. Emeléus. Alan won the Percy Pemberton Prize and graduated in 1947. 

In the summer of 1947, Alan joined a group of students to help build a railway in Yugoslavia. The 'Youth Work Actions' project was a huge contribution to the recovery of the country. This was the beginning of his active interest in politics. In the years 1947‒1949, Alan worked in the crystallography laboratory of Philips Electrical Company Ltd and simultaneously earned his BSc degree in physics. He then decided to study for his PhD and joined Birkbeck College of London University, where he stayed for the rest of his professional life. Soon, he became a member of the crystallography laboratory of J. Desmond Bernal (1901–1971), defended his PhD thesis, and was awarded the degree in 1951.

Alan learned Russian in summer school, and he met Sheila Thorne Hague there, his future wife. They married in 1951, and by 1959, they had three children, two boys and a girl, and moved to their home in North London, where they stayed for their entire life.

Alan’s interests were broad, and he published broadly, which did not help his promotion in the university ranks. He was awarded a DSc degree in crystallography and studies of science in 1986, was appointed Professor of Crystallography in the same year, and became Professor Emeritus in 1991. In 1988, he was elected Fellow of the Royal Society (FRS).

Bernal’s example inspired Alan ever since he chose Bernal’s book The Social Function of Science as his prize for winning a competition in Cambridge. Bernal’s sizzling intellectual environment was superbly conducive to developing Alan’s generalist approach to science. Bernal’s nickname was 'Sage' for he was supposed to know everything worth knowing. In the 1930s, Bernal was a member of the Club for Theoretical Biology, along with Joseph Needham, C. H. Waddington and others. They dealt with such questions as the application of X-ray crystallography and other physical techniques to solving problems in biology. Already in the mid-1930s, Bernal had shone X-rays onto protein crystals, and the fact that he could record interference patterns led him to believe that the structures of such large biological systems could be solved on the atomic level. Bernal liked delegating tasks and delegated the structure determination of large biological molecules to such disciples as the future Nobel laureates Dorothy Hodgkin, Max Perutz and Aaron Klug.

Bernal served as one of the highest-level science advisors during World War II. After the war, his communist politics and uncritical adulation for the Soviet Union were a serious impediment to his obtaining support for building a research center that would have been adequate for implementing his far-reaching ideas. Alan was left-of-center politically, a staunch though not uncritical supporter of the Labour movement, and did not follow Bernal’s communist commitment. In his scientific interactions, apart from his Soviet connections, Bernal preserved objectivity and collected around him an excellent group of scientists in mathematics and computing, in the theory and experiment of X-ray crystallography, and physical chemistry, both inorganic and organic structures, and his laboratory ran a skilled workshop. He had a stream of international visitors, such as the scientists Norbert Wiener, Linus Pauling, André Lwoff and H. S. M. Coxeter, and representatives of world culture, like Maurits Escher, Pablo Picasso and Paul Robeson. Bernal’s associates felt they were “living in the center of the universe” [5]. Alan realized from the start how privileged it was to be part of Bernal’s circle, whose combination of scientific, social and political activities appealed to his own inclinations. 

In 1956, Alan accompanied Bernal to Moscow, where he met such giants of Soviet science as Petr L. Kapitza, Lev D. Landau, Igor E. Tamm and Vladimir A. Fock (of Hartree-Fock fame). Bernal and Mackay visited the Institute of Crystallography (later, Shubnikov Institute of Crystallography) of the Soviet Academy of Sciences and met its director Alexey V. Shubnikov, and Shubnikov’s co-workers, among them the future director Boris K. Vainshtein and Zinovii G. Pinsker. By then, Alan had already begun building up an international network of friends, especially crystallographers, at international meetings, and his interactions with the Moscow crystallographers were especially active. In 1962, he spent five months at the Institute of Crystallography in Moscow. Scientifically, it was not a very fruitful stay, but it was good for getting to know many colleagues and Soviet life in a more realistic way than from propaganda materials. 

Alan’s first research project was at Philips Electrical Ltd on the structural analysis of a particular modification of solid calcium phosphate used in fluorescent tubes. When he moved to Birkbeck College, he joined the section studying the properties of cement, and thus, he continued doing research on inorganic materials. Early on, icosahedral structures became the focus of his interest. He had already met with the structure of beta-tungsten at Philips. Then, he found some old papers at Birkbeck, as there had been interest in these structures at the College before him. Bernal considered the icosahedral arrangement rather early because it would prevent crystallization, and he thought that icosahedral coordination might give some clues to understanding the structures of liquids. Alan was also aware of Pauling’s interest in icosahedral structures. When Bernal was to go to Budapest to give a talk at a meeting honoring Zoltan Gyulai’s 70th birthday, he asked Alan to draw the figures. Bernal’s comprehensive presentation was about the symmetry of solids and liquids [6]. 

The icosahedral arrangement of atoms is interesting because it could also be a step in the progression from the isolated molecule to an extended structure. When a second icosahedral shell surrounds an icosahedron of 12 spheres about a sphere in the center, the size of this second shell is precisely twice the size of the first shell [7]. This second shell contains 42 spheres and lies over the first so that spheres are in contact along the fivefold axis. Further layers can be added in the same fashion. 

The third layer is shown in the figure above. This is known as the Mackay polyhedron or Mackay icosahedron — an example of icosahedral packing of equal spheres. The layers of spheres succeed each other in a cubic close-packing sequence on each triangular face. Each sphere that is not on an edge or vertex touches only six neighbors, three above and three below. Each such sphere is separated by 5% of its radius from its neighbors in the plane of the face of the icosahedron. This assembly can be distorted to cubic close packing in the form of a cuboctahedron. The Mackay icosahedron has “made a tremendous impact on particle, cluster, intermetallics, and quasicrystal researchers…,” [8] according to the late K. H. Kuo, the doyen of Chinese crystallographers. Kuo identified two basic concepts in Mackay’s paper. One was the icosahedral shell structure consisting of concentric icosahedra displaying fivefold rotational symmetry. This structure occurs frequently, not only in clusters but also in intermetallic compounds and quasicrystals. According to Kuo, the other concept was the hierarchic icosahedral structures due to the presence of a stacking fault in the face-centered-cubic packing of the successive triangular faces in the icosahedral shell structure.

Alan questioned dogmas wherever and whenever he met them. This was especially so in the case of crystallography, where the classical rules had worked so well but eventually proved to be limiting the scope of structures the subject embraced. Those rules limited the inclusion of novel kinds of structures that kept emerging as well as structures that had been abandoned by crystallographers; the need arose to include them in a broader system. There was an obvious deficiency when the theoretical constraints of crystal symmetry were confronted with real crystals. The approach to discussing crystal symmetry used to be to think of the formation of a crystal through insertion of individual atoms or groups of atoms into the three-dimensional framework of symmetry elements, whereas in reality — as Alan liked to point out — the symmetry elements emerge as a consequence of the structure being formed through the local interactions between individual atoms or other building elements. 

Above, I have already mentioned fivefold symmetry and the role it played in our interactions. Alan stressed its importance in a broader sense [9]: “The main significance of five-fold symmetry for science is that it furnishes us with an explicit example of frustration, which has proved a most fertile concept in the physics of condensed matter. … Neither we nor nature can have everything simultaneously — not all things are possible … We have only the freedom of necessity. ‘Nature must obey necessity’ as Shakespeare (Julius Caesar IV:iii), Democritus, Monod, Bernal, and many others have also recognized. Science probes the limits of necessity and, in the case of five-fold symmetry, has found a corridor that leads us to a new territory.”

The concept of crystal symmetry itself took center stage in Alan’s inquiry, and he creatively deepened and expanded its meaning. When I asked him which of his papers would be the most representative of his work in this area, he pointed to the one titled Crystal Symmetry [10]. In 1981, he compiled a list of concepts in two versions, classical and modern [11]. This was not the only time he did this, but to me, this shows the changes in a most dramatic way.

It is interesting to follow how several different threads converged in Alan’s research career. He described this [3]: “I used to do science abstracts — for ten years I abstracted all the Russian papers on crystallography — and I remember abstracting a paper on the incommensurate arrangements of spins in iron oxides, in hematite. The period of the helical magnetic spin is not the same as the crystallographic period. So incommensurate structures were current before that time. Even much longer before that I thought of a simple thing about printing wallpaper. Suppose your wallpaper is simply printed from a roller. But suppose you are printing two motifs from two rollers of different diameter. Then you get a non-repeating pattern. I wasn’t able to think of producing an aperiodic two-dimensional pattern in this way. I was only aware of the possibility of one-dimensional incommensurate patterns. I was really interested in hierarchic patterns and not in aperiodicity as such. This came directly from Bernal’s suggestions and the polio virus project. I produced a hierarchic pattern, a hierarchic packing of pentagons. Then, in 1974, I got some help in computing from Judith Daniels at the University College Computing Centre and, incidentally, showed her these patterns. She said that Roger Penrose had something like them. So, I made an appointment with Roger Penrose and Robert, my son, and I went to see Penrose in Oxford, where he showed us the jigsaw puzzle, with the kites and darts and so on. Basically, his concern was with forcing aperiodicity, and my concern was with hierarchic structures. It turned out to be very similar.” 

In his paper about the pentagonal snowflake [11], Alan, à la Penrose, built up a regular but non-periodic (he called it then "non-crystalline") structure from regular pentagons in a plane.

It starts with a regular pentagon of a given size, which we may call the zeroth-order pentagon. Six of these pentagons are combined to form a larger regular pentagon, the first-order pentagon. There are lozenge-shaped gaps where two second-order pentagons abut, and Alan filled these gaps with zeroth-order pentagons. This design is repeated recursively, on an ever-increasing scale.

After the meeting with Penrose, Alan’s son Robert returned to his university at York, where he studied computer science and plotted a tiling on his pen-plotter (see above). He started from pentagons of a certain size, and as he kept going to larger and larger pentagons, he built up a pentagonal snowflake. Alan included Robert’s design in his paper on pentagonal snowflakes to give his considerations added emphasis.

It was at this time that Alan made a significant prediction concerning the possibilities of real three-dimensional structures with fivefold symmetry. He had the idea of producing a simulated diffraction pattern of the Penrose tiling [3]: “First I just drew the Penrose type pattern and sent it to George Harburn in Cardiff who was a colleague of Charles Taylor who had a good optical diffractometer. I had stuck it into a laser beam here, but you need a precise adjustment. You can do many beautiful things with the optical diffractometer that you can’t see on the computer, with very fine detail; it is amazing. Then George Harburn made a second version which instead of consisting of lines, had dots; thus, the diffraction pattern was not dominated by the streaks from the lines” (p. 154).

Alan wrote and published a paper in which he communicated the simulated diffraction pattern [12]. In a broader context, he was considering the characteristics of the pattern and the diffraction it generated [3]: “I had also a theory about collagen and had some patterns bearing on that. The theory was that collagen fibers are connected with the Fibonacci spiral. If you draw a Fibonacci spiral of circles along the spiral, then locally the pattern keeps changing between square packing and hexagonal close packing. This corresponds closely to the diffraction you infer from collagen fibers. Richard Welberry in Canberra, Australia, had a still better optical diffractometer and took some very good diffraction pictures from the Fibonacci spiral. Then [the botanist] Eriksson in Philadelphia showed that the diffraction pattern of the Fibonacci spiral was self-similar to the Fibonacci spiral itself. … This may point to a connection between phyllotaxis — the scattered leaf arrangement about stems — and internal structure on the atomic level” (p. 155).

Alan had predicted the existence of regular but non-periodic structures that Dan Shechtman later observed in his experiments. It would have been a wonderful sequence of events had Shechtman and others known about Mackay’s prediction and embarked on looking for such structures and found them. The search for extended structures with fivefold symmetry had been going on for centuries and involved excellent minds, such as those of Johannes Kepler and Albrecht Dürer. Roger Penrose came up with such a pattern in two dimensions and Alan crucially extended it to the third dimension. He urged experimentalists to be on the lookout for such structures. Nobody took up his challenge, and when Shechtman made his observations he was not aware of Mackay’s predictions. Eventually though, all these lines came together. In 2010, the American Physical Society awarded the Oliver Buckley Prize to Alan Mackay, jointly with Dov Levine and Paul Steinhardt for their contributions to the discovery of quasicrystals. The next year, Shechtman received the Nobel Prize in Chemistry.

Alan’s only reservation in evaluating the importance of the discovery of quasicrystals was that it may have appeared more significant than it was. He thought that the too-restrictive definitions of classical crystallography lent a pivotal character to the discovery. Had the definitions of classical crystallography been broader and more inclusive, there would have been no need to bring about a paradigm change. This sounded as if he were belittling the discovery, both Shechtman’s and his own. My suspicion is that he may have been asked from Stockholm to evaluate the discovery, and his self-deprecating approach may have been taken literally there. As it happened, the discovery of quasicrystals did prove to be pivotal, and it did bring about a paradigm change. Shechtman’s unshared Nobel Prize was met with broad satisfaction, but my feeling is that a Shechtman-Mackay shared prize would not have generated any resentment. However, the value of Alan’s achievements is above expressing it in prizes and awards.

One of Alan’s colleagues called him 'the well-known eclectic' and he did not mind this label at all. Pointedly, he chose this word for the title of a selection of his writings, Eclectica, self-published for personal use in a handsome volume in 2009 [13]. He reproduced many of his published papers and included several unpublished ones. The volume is a rich source of information and ideas and here I merely dip into it to illustrate its scope and depth.

Appropriately, the volume begins with a discussion of copyright — one of Alan’s pet concerns. He was an advocate of protecting the rights of scientist authors to their own intellectual production versus the publishing companies. He thought that professional societies should publish their own electronic journals with open access that would be supported by authors’ fees. Currently, the open-access approach is gaining ground rapidly, but there may be a great divide between authors who can and those who cannot afford the often-hefty fees for having their manuscript published in open-access venues.

Looking at Alan’s oeuvre, the discovery of the Mackay polyhedron and his prediction of quasicrystals did not happen in isolation. He had long been interested in structures that fell beyond the confined system of classical crystallography. He published at least three reviews under the title Generalized Crystallography, the latest in 2002 [14]. Here, he defined the aim of generalized crystallography as “to understand the properties of matter, inert and living, at our human scale, in terms of the arrangement and operation of atoms.” Furthermore, he recognized the pioneering role of X-ray crystal structure analysis in this quest but noted that, in view of the vast array of techniques, it might be advisable to replace the term 'crystallography' by 'structural chemistry'. He also realized though that terms that had long been embedded in the scientific literature would be hard to displace. 

Concerning the pioneering role of X-ray crystallography, Alan called attention to the phenomenon when a pioneering field becomes a brake on further progress. This happened with classical crystallography, whose rigid system hindered the recognition of those structures that fall beyond its classical system. Of course, no blame should be assigned to those who originally worked out the system, but it is our task to overcome the barriers that have emerged due to later development. This kind of success turning into a brake is not unique to classical crystallography. When insulin was discovered for treating diabetes, it was a great triumph of the biomedical sciences. It has since then been gradually recognized that the availability of this successful treatment, which is not a cure, might have diverted efforts and resources from continuing a quest for the cure of diabetes. Another example from the science of structures was the resistance to recognizing other techniques against the background of the enormously successful X-ray diffraction, making it harder for electron crystallography and neutron crystallography to become accepted and spread [15].

Alan’s teachings on generalized crystallography fell onto fertile ground; suffice it to mention a couple of additional contributions to the issue of Structural Chemistry dedicated to his 75th anniversary [16, 17].

Alan’s impact on the structural science community is hard to measure, but the impression is that it will be long-lasting. He has impacted us through his writing and through personal interactions. He adapted himself easily to local conditions on his many visits. When he spent a longer period at the Institute of Crystallography in Moscow, he developed the habit of carrying a shopping bag with him. This was not only because the shops did not give out such bags to carry away their goods but even more because one never knew what purchase might suddenly become available. After his return to London, he did not find it easy to give up the habit of having his shopping bag in readiness. Although his stay at the Institute of Crystallography in Moscow did not produce scientific results, his interactions with the Azerbaijani crystallographer Khudu Mamedov (1927–1988) helped Mamedov to become known in the West (see https://www.iucr.org/news/newsletter/volume-29/number-1/macgillavryeschermamedov-and-periodic-patterns). Mamedov prepared periodic drawings that were reminiscent of Escher’s patterns, but he used historical/cultural motifs from his region, thus creating a unique inter-relationship between art and science. Mamedov, perhaps in Mackay’s style, used the term 'crystallographic' in a broad sense. Mackay dedicated a talk to Mamedov’s memory in 1991, “Form and pattern in Azerbaijani civilization”, and its text is reproduced in Eclectica.

Alan and Bernal co-authored a presentation, “Towards a science of science”, for the 11th International Congress for the History of Science in Warsaw in 1965. They outlined what 'science of science' was, why it was needed, and the methods of their inquiry. Their program included practical recommendations, such as establishing departments of the history of science and the need to look at science as a whole rather than merely taking up its specifics. Furthermore, they called for establishing the profession of science critic similar to that of literary critic and called for international cooperation in the recognition of science as a worldwide activity. They also suggested experimental work to find the best means of science training. They emphasized the importance of learning about non-European cultures where emphases were different from European cultures, as illustrated, for example, by a lower priority for written records but a higher one for master-pupil relationships. This joint Mackay-Bernal presentation has been reproduced in several publications and in a number of languages, yet it is not easily accessible. Hence, it is very useful to have it in Eclectica. The idea of the 'science of science' permeated Alan’s activities throughout his career and he co-edited a volume about it [18]. 

In the early 1980s, Mackay ran a column called “Anecdotal evidence” in The Sciences, and the entries are reproduced in Eclectica. Bringing together seemingly disparate ideas and facts suited him eminently. Even the titles reveal some aspects of his approach, such as “Science and Travel”, “Rhyme and Reason”, “How to write a best-seller”, “Mackay’s Michelin”, “Molecules and Moores” (referring to Henry Moore), “Message in a Bottle” and suchlike. The column served the readers of this unusual periodical well, but its editors liked to smooth over his often unorthodox style of writing.

Eclectica contains a list of Alan’s works, including scientific publications (176 entries), miscellaneous publications (130) and book reviews (46). There is then a list of 30 unpublished papers, 10 entries which he called 'indirect material' and publications by others in which he figured, including the special issue in Structural Chemistry in 2002 dedicated to him [19]. There was another special issue in Structural Chemistry in 2017 dedicated to his 90th birthday [20].

The adjectives 'consistent' and 'rational' are among Alan’s many characteristics, and they shine through the following poem that he composed a few years ago, which sounds like a parting gift:

Atoms and our Vision of the World

There are no gods.
We are alone.
I am thus two-fold alone
but I have the second sight of science.
As my eyes grow dim,
my mind sees the future.
I see a handwriting on the wall -
the wall surrounds a giant alembic -
built to win gas from coal.
The Chinese hand wrote large
the character which stands for entropy.
It questions the solid state of Earth.
Asking my computer, I find the words
“disordered hyperuniformity” -
today’s myopic Vision of the World  
glimpsed in the microcosm of atoms.
Death came to my wife of more than sixty years.
Her flame went out. Her body was cremated -
atoms to atoms - Lucretius saw truth.
But where is past history now?
Information increases locally from time to time -
but Entropy will win.

A. L. M.  30 August 2015 [21].

Alan often expressed his views and sentiments through poetry. His published poems expressed topics in crystallography and the science of structures [21]. He titled his collection of poems published in 1980 The Floating World of Science after the works of Japanese artists who lived in the latter half of the 18th century and the first half of the 19th. According to Alan: “Scientists inhabit a kind of Floating World of their own, a kind of Global Village, in which they have friends, or friends of friends, everywhere. Rather like members of a religious order, they can go to any laboratory dealing with their field of study and be hospitably received” [22]. Alan and Sheila practiced this very hospitable attitude toward many members of the international scientific community.

Even when he was composing prose, it sometimes sounded like poetry. Consider this example: “Amorphous materials may be shapeless, but they are not without order. Order, like beauty, is in the eyes of the beholder. If you look only with X-ray diffraction eyes, then all you see is translational order, to wit crystals. … [T]here is a wide range of structures, between those of crystals and those of gases, … Other structures need not be failed crystals but are sui generis [in a class by itself]” [23].

It is often said that scientific discoveries, however important, are sooner or later overshadowed by new developments, and this could also happen with Alan’s contributions to crystallography and the science of structures. However, his demeanor as a researcher and scientific discoverer will serve as inspiration for a long time.

Acknowledgments

My original writings about Alan were greatly assisted by John Finney, Magdolna Hargittai, Alan Mackay and Robert Mackay. The present article has greatly benefited from the constructive criticism and suggestions from John Finney, Mike Glazer, Robert Mackay and Marjorie Senechal.

References

[1] Hargittai, I. (2017). Generalizing crystallography: a tribute to Alan L. Mackay at 90. Struct. Chem. 28, 1–16.

[2] Hargittai, I. (2020). Mosaic of a Scientific Life, pp. 99–102. Springer.

[3] Hargittai, I. & Hargittai, M. (2000). Our Own Image: Personal Symmetry in Discovery. New York: Kluwer/Plenum. (Excerpts from a conversation with Alan L. Mackay in 1994, London.)

[4] Hargittai, I. (1997). Quasicrystal discovery: a personal account. Chem. Intell. 3, 25–49 (actual quote, p. 26).

[5] Mackay, A. L. (1998). Pre-history: For the 50th Anniversary of the Crystallographic Laboratory of Birkbeck College. Unpublished notes of a presentation on November 26, 1998, in The Clore Lecture Theatre in London.

[6] Bernal, J. D. (1957). The importance of symmetry in solids and liquids. Acta Physica Acad. Sci. Hungaricae, 8, 269–276.

[7] Mackay, A. L. (1962). A dense non-crystallographic packing of equal spheres. Acta Cryst. 15, 916–918.

[8] Kuo, K. H. (2002). Mackay, anti-Mackay, double-Mackay, pseudo-Mackay, and related icosahedral shell clusters. Struct. Chem. 13, 221–230.

[9] Mackay, A. L. (1990). Crystals and five-fold symmetry. Quasicrystals, Networks, and Molecules of Five-fold Symmetry, edited by I. Hargittai, pp. 1–18. New York: VCH.

[10] Mackay, A. L. (1976). Crystal symmetry. Phys. Bull. 27, 495–497.

[11] Mackay, A. L. (1981). De Niva Quinquangula: on the pentagonal snowflake. Kritallografiya (Sov. Phys. Crystallogr.) 26, 910–919 (517–522).

[12] Mackay, A. L. (1982). Crystallography and the Penrose pattern. Physica A, 114, 609–613.

[13] Mackay, A. L. (2009). Eclectica from the writings of Alan Mackay. Un-published (self-published), London.

[14] Mackay, A. L. (2002). Generalized crystallography. Struct. Chem. 13, 215–220.

[15] Finney, J. L. (2002). Crystallography without a lattice. Struct. Chem. 13, 231–246.

[16] Ogawa, T. & Ogawa, T. (2002). Proportional representation system as generalized crystallography and science on form. Struct. Chem. 13, 297–304.

[17] Terrones, M., Terrones, G. & Terrones, H. (2002). Structure, chirality, and formation of giant icosahedral fullerenes and spherical graphitic onions. Struct. Chem. 13, 373–384.

[18] Goldsmith, M. & Mackay, A. L. (1964). The Science of Science. London: Souvenir Press.

[19] Hargittai, I. et al. (2002). Special issue honoring Alan L. Mackay. Struct. Chem. 13, 213–412.

[20] Hargittai, I. et al. (2017). Special issue honoring Alan L. Mackay. Struct. Chem. 28, 1–256.

[21] Mackay, A. L. (2015). Private communication, August 30, by e-mail.

[22] Mackay, A. L. (1980). The Floating World of Science: Poems. London: The RAM Press.

[23] Mackay, A. L. (1987). Quasi-crystals and amorphous materials. J. Non-Cryst. Solids, 97–98, 55–62.

 

Istvan Hargittai is at the Budapest University of Technology and Economics.