Teaching crystallographic symmetry
I remember a particular day in 2010 vividly. Mois I. Aroyo had spent eight hours of that hot Saturday (summertime in Montevideo) talking about Representations of Space Groups, after a full Monday–Friday week of Space Group Symmetry shared between Mois and Massimo Nespolo, with short contributions by Ernesto Estevez Rams, Gustavo Echeverría and Eduardo Granado on Thursday. Mois showed no sign of exhaustion and remained enthusiastic. The surviving audience (~25 of the ~50 school participants), including tutors like myself, were all exhausted but tried to follow the lectures. I had met Mois and Massimo personally only eight days before, on Saturday 28 November, when they landed in Montevideo after a long flight from Madrid. Ernesto Estevez Rams had introduced me to them virtually and pushed me into agreeing to organize this International School on Fundamental Crystallography (ISFC2010). I was a recent Associate Professor at Universidad de la República, and having the chance to organize a regional School sounded like an excellent challenge and a good opportunity to establish a network on teaching fundamentals of crystallography that would complement the already existing School, centered on practical aspects of X-ray diffraction and crystallography.
December 2020 marked the 10th anniversary of that final day of the ISFC2010 MaThCryst School in Latin America  held at Facultad de Química, Universidad de la República, Montevideo, Uruguay. This was the first of a series of International Schools that I’ve since organized in Montevideo or helped to organize in Latin American institutions (before and within LACA’s auspices). Joining the International Organizing Committee of the MaThCryst Schools significantly impacted my career, both in network-building and teaching activity aspects. In this article, I want to share some of the more important matters learned in this period, aiming to push colleagues from all over the world to lead similar initiatives that help build the community and educate our students and future colleagues.
Starting with a brief historical description of the development of the series of MaThCryst Schools in Latin America and the most important details of the organization of the events, a review of the outcome based on the opinions of students will follow. After this, I want to discuss how this activity has impacted my teaching work at the local level in the Crystallography “101” course and at symmetry lectures at applied crystallography schools. Finally, I want to share how I would like my teaching activities at all levels and future MaThCryst Schools to develop. The reason for this is that we need more crystallographers able to understand and use symmetry for their analyses. In addition, they have to profit from the modern tools that have been developed by generous colleagues in freely available web servers and databases, and by the IUCr, through the Teaching Edition of International Tables for Crystallography Volume A and the online version of International Tables for Crystallography.
2. MaThCryst Schools in Latin America. The birth of a periodic event
The International School on Mathematical and Theoretical Crystallography in Havana, Cuba, in 2007, was supported by the IUCr Commissions on Mathematical and Theoretical Crystallography (MaThCryst), Crystallographic Teaching and Inorganic and Mineral Structures, and locally organized by Ernesto Estevez-Rams, Arbelio Penton and Cristy Azanza (Havana University, Cuba) with the participation of Commission members Massimo Nespolo (Chair, Université de Lorraine, France) and Bernd Souvignier (Radboud University, The Netherlands) (among other Lecturers) . After this event, the need for this kind of intensive School on Symmetry and the International Tables for Crystallography in Latin America became evident to the organizers. I was not involved in that School but had met Ernesto at the LNLS (Brazilian Synchrotron Light Laboratory) in the late 1990s and visited him at Havana University in 2003 for the X Summer School on Materials Science and Technology. This was followed by a short stay during my PhD studies to learn more about X-ray diffraction analysis of materials showing stacking faults. Some years later, when I was back in Montevideo after my postdoc, I joined the XVI Summer School of 2009, but this time as a Lecturer. During that School, Ernesto convinced me to organize a School on Mathematical and Theoretical Crystallography, similar to 2007, but in Montevideo, farther south in the Latin American region. I accepted his invitation with only one big but: the name of the School could not contain the words "Mathematical and Theoretical” if we wanted chemistry, structural biology, materials science and other students to attend. He put me in touch with Massimo Nespolo, and in January of 2010, we agreed to organize the International School on Fundamental Crystallography in December of that year . Our deal was that I would help Massimo manage some schools of this kind in South America. I was hesitant, though, to have them too close in time since I ignored how much demand there would be for learning Crystallographic Symmetry and the financial resources for these schools. A plan was made to organize three such schools, in 2010 (Uruguay), 2012 (Brazil) and 2014 (Argentina), to cover the largest crystallographic communities of South America and put the idea and format of the School to the test. With this in mind, the ISFC2010 MaThCryst School I organized with Ernesto Estevez-Rams, Massimo Nespolo and Mois I. Aroyo also counted on the participation of Raimundo Lora-Serrano, Eduardo Granado and Gustavo Echeverría, future organizers of ISFC2012 and ISFC2014 in Brazil and Argentina, respectively (Fig. 1).
The School in Montevideo (with 50 attendees from 6 countries) was followed by Uberlandia (2012)  (with another 50 attendees from 5 countries) and La Plata, Argentina (together with an IUCr/UNESCO OpenLab in 2014)  (with 37 participants from 9 countries) as planned. The success of the school series, measured by the number and diverse nationalities of participants, was demonstrated, so the ISFC MaThCryst School would continue and cover the rest of Latin America. To get back to the north of the region, Havana, Cuba was selected again for 2016 , to celebrate a decade of MaThCryst Schools in Latin America (counting 35 attendees from 9 countries). Bogotá, Colombia in 2018  would follow, with Mario Macías – a participant in Havana – assisted by Elizabeth Jimenez and colleagues, organizing the event. The ISFC2018 School had Arbelio Penton replacing Ernesto Estévez in the International Organizing Committee and the Lecturer’s team and involved 39 participants from 9 countries. The seventh MaThCryst School in Latin America was programmed for November 2020 in Lima, Perú but had to be canceled for reasons every reader knows and is now in the process of being rescheduled for 2022 (see below).
The number of attendees at ISFC Schools was on average 40, with a majority (about half of the participants) from the country where the School was held. From that group, between a third and a half came from the city or region of the university hosting the event. Gender balance is a natural characteristic of crystallography in the area and was easily achieved even before the IUCr policy was established. The number of non-local attendees was always limited by the generous but overall scant resources provided by the IUCr and other sponsors of each event. For example, in the case of Montevideo 2010, USD 4650 was given to 14 IUCr Young Scientist Awardees, and more than USD 7000 was spent to cover lodging and lunch for a total of 30 students from abroad. Still, the number of registered participants who could not attend due to lack of financial support was above 30.
3. ISFC format and outcome
The ISFC MaThCryst Schools developed with a consistent format. Lectures were taught in university classrooms with basic teaching media support and wireless internet. First, on Sunday, optional introductory lectures were given, primarily for local participants, on the basics of Fourier Transforms and vector and matrix algebra. Next, there were five days of lectures (Monday to Friday) on crystallographic symmetry. Finally, a one-day workshop (Saturday) was organized on a topic of interest to the local organizers. From Monday to Wednesday, point groups, 2D and 3D lattices, plane and space groups were covered, using the notation and examples from International Tables for Crystallography Volumes A and A1. Aroyo and Nespolo alternated lectures with the rest of the lecturers/tutors guiding the students with exercises. Thursday was devoted to symmetry of the diffraction pattern and space-group determination from diffraction data (details of the lecturers can be obtained from the events’ websites [1–6]). Friday was the day for transformations, group–subgroup relations, and application of these and other concepts using the tools available by the Bilbao Crystallographic Server. The one-day Workshops on the final day (also of optional attendance) covered Representation Theory of Space Groups (2010) , Twinning (2012/2014) [3–4], Nanocrystallography (2016)  and Phase Transitions (2018) .
In the first three days of lectures, where the fundamental concepts and definitions are outlined, the classes consistently follow a pattern beginning with an introduction of a concept or definition: an example is solved, and exercises are given to the attendees for them to solve with the assistance of the lecturers and tutors. This mechanism is a very useful way to quickly diagnose the level of understanding of the students and promote cooperation among them. It also sets the pace of the lectures and keeps everyone active during long days consisting of seven hours of lectures (with two coffee and one lunch break). Even when the concepts or definitions have not been fully understood by the students, having the tutors around allows one-on-one explanations to be given that usually solve the problems and pushes the less active students to keep trying.
As is usual in IUCr-sponsored events, travel and local expenses support grants for non-local participants were given based on academic merit and the background of the applicants. This implied that, in most schools, non-local attendees were slightly more experienced (Master’s/PhD students or Professors) and educated in crystallography than the locals, where undergraduate students were a majority. This may be a bit less accurate for large countries such as Brazil or Argentina, where the nationals may come from longer distances than some international participants and may require more expensive travel support. Still, after eliminating some applicants based on merit, a significant number of them had little or no knowledge of International Tables for Crystallography Volume A. Fig. 2 shows a collection of answers of the attendees in the post-school evaluation form where their background before the event was recorded. This is a central reason why MaThCryst schools have been so well received and are so needed in Latin America.
Even though all of the participants are exhausted by the end of the week, and in every evaluation of the School a request for a free afternoon is repeated, the students evaluate the School very positively (Fig. 3). The need to profit as much as possible from the Lectures, devoting the maximum amount of time possible in the classroom, justified the non-stop program.
Over the years, I have seen a couple of dozen participants in the ISFC Schools become colleagues. Some of them are currently working at highly respected institutions, doing research, teaching and/or developing tools for data analysis. I have exchanged memories of the Schools with many of them, and they all have mentioned how good the ISFC was for their careers. This may just be a demonstration of politeness and respect, so in the following months, I plan to distribute an anonymous questionnaire to the ~200 participants of the different ISFC Schools. I would like to collect their opinion about the impact of ISFC Schools on their formation as crystallographers and ask them for suggestions on improving future Schools. This will probably justify a future article. It has already happened that former ISFC participants have organized a School. Eventually, they will become the Tutors and Lecturers of ISFC, so why not get them thinking about it now?!
4. Teaching crystallographic symmetry in other Schools or regular courses
Before 2010 I had taught crystallographic symmetry (among other topics) over 10 different years (1996–2004, 2009) within the Crystallography “101” course at Facultad de Química, Udelar, only missing the course during my postdoc years. The course was intended for advanced undergraduate and graduate students in chemistry. The focus of the course was to introduce the participants to the most relevant theoretical and practical aspects of crystallography and X-ray diffraction (both single crystals and powder), giving them tools to analyze small-molecule structures or perform qualitative powder X-ray diffraction. The course was not expected to delve deeply into mathematical or physical aspects of the discipline, so every qualitative simplification of the topics was welcome (not only by the students but also by myself as I was learning on the road). I had based the initial course on some basic books (Stout & Jensen, 1968 ; Woolfson, 1969 ) and IUCr Teaching Pamphlets #13 and #14 [9–10], where I had learned about symmetry myself. I have to say that initially (~1996–1998), the quality of my teaching was poor but evolving in parallel with the knowledge I was acquiring during experimental work. Initially, during my Masters’ studies (1996–1999), I solved small-molecule structures from single-crystal X-ray diffraction data collected on a point-detector diffractometer (twins, incommensurate or pseudosymmetric structures were not frequent problems). Later, during my doctoral studies (1999–2003), I used synchrotron X-ray and neutron powder diffraction data and the Rietveld method for refining perovskite structures with magnetic components, applying Shubnikov’s space groups.
My classroom approach to crystallographic symmetry changed drastically after 2010. Even if I could not reproduce the same content Mois and Massimo taught at ISFC Schools (the audience was significantly different), some central ideas of their method became stepping stones of my classes (Fig. 4). Almost all crystallographic symmetry could be built based on the knowledge of symmetry operations and on the definition of a group. So starting in 2011, my symmetry classes, spanning eight 2-hour lectures, started by introducing symmetry operations that form simple crystallographic point groups and, using group axioms, add more symmetry elements to obtain more complex ones. This is done in a geometrical way first (easier to visualize for chemistry students used to representing molecules in space) and in algebraic form later, using matrices. Later, simple lattices in 2D are obtained from identical periodic rows of nodes and ideas on how to derive Bravais lattices from 2D lattices are introduced (not formally derived for time economy). Finally, symmorphic space groups are introduced by combining simple point groups with lattice translations. Later, symmetry operations with translation components are added, emphasizing that space groups have an infinite number of periodically ordered symmetry elements. International Tables for Crystallography are used to show the progression of complexity of space groups and the standard notation. Feet and hand patterns from Teaching Pamphlet #14 are also used, first to learn how to search for symmetry operations, later to determine the coordinates of equivalent positions and the matrix-column pairs for symmetry operations, and finally to determine the space-group symbol.
The logic behind this approach is simple and straightforward, and it could be used to introduce crystallographic symmetry even in a couple of 2-hour lectures, as is the norm in single-crystal or powder diffraction schools of which I've been an Organizer or Lecturer. I have always devoted one class to point groups with geometrical and algebraic approaches finishing with a quick view of Bravais lattices and the final class to space groups and International Tables for Crystallography Volume A. For time economy, I discuss triclinic, some of the simpler orthorhombic and the monoclinic P21/c space group, which is so frequently encountered (one out of three structures deposited at the CSD)  that the students have heard of it or will need it sooner or later. It may not be possible in such a short time to derive and study the more complex point or space groups or give an exhaustive classification of them. But an idea of how they could work them out remains.
Some may say it is a waste of time to teach what is in International Tables for Crystallography Volume A, and what they are used for, in a couple of 2-hour lectures with at most 2-hours of exercises. I would agree that not much will be of use for an absolute beginner after the class. However, I emphasize the idea that even the most complex space groups could be deduced by adding symmetry operations to simpler space groups in a step-by-step procedure. The idea in the mind of all of the students is, in my humble opinion, that it is possible to understand crystallographic symmetry by patiently going through all steps, point groups, then lattices and then space groups. It will take them time, but it is possible.
With the cancellation of Schools during the COVID-19 pandemic, there will be a gap in the preparation of students regarding all aspects of crystallography, but especially of crystallographic symmetry, rarely covered in depth in local courses in the region. The slow restart of international schools in 2022 will generate increased demand, as has been already observed in online courses (the 3rd LACA School on Small Molecule Crystallography held online in December 2020 had 60 participants, but 120 registered).
The International School of Fundamental Crystallography has not been held for three years already, and we are planning the next for the last trimester of 2022, almost certainly at Anápolis, Goiás, Brazil, organized by MaThCryst member Hamilton B. Napolitano, Professor at the Universidade Estadual de Goiás. I expect that we will have well over 100 applications, but the number of attendees will not exceed 50 or 60. On one side, classroom space (and likely limitations to the occupancy of classrooms are in place) will prevent larger numbers. On the other side, larger audiences will be tough for Lecturers and Tutors in the practical sessions. In addition, financial resources for travel support will also limit the attendance to those students that could pay for most of their expenses or will travel from shorter distances.
Other international and national schools held periodically in Latin America will also resume in 2022 and 2023. Hopefully, the hybrid in-person/virtual mode of attendance will be implemented to widen the audiences and reduce the financial costs of the organization. These will also have an increased demand in any of the modes, as the growth of the synchrotron and neutron users communities develop, with the stepwise opening of beamlines at the Sirius synchrotron source in Campinas, SP, Brazil  and the start of operation of the Laboratorio Argentino de Haces de Neutrones (LAHN) in Ezeiza, Buenos Aires, Argentina  in 2024.
On the bright side, this gap gave time for the significant efforts of Mois I. Aroyo (Editor), contributing authors and the IUCr Editorial Office in Chester to finish successfully the new Teaching Edition of International Tables for Crystallography: Crystallographic symmetry , which was published in April 2021. This book, better adapted for teaching and for individual study , will become an essential addition to the toolbox of Lecturers and Tutors and may play a significant role in the education of Latin American (and all around the world) students during the next decade if its availability (price, web access, etc.) is adequately adapted to the circumstances. I think an effort by the IUCr to give away e-copies of the book for free to school applicants/attendees and allow for access to the online tools of International Tables for Crystallography Volumes A and A1 and the Symmetry Database will be very important to support the efforts of Lecturers in our region.
In conclusion, after a decade of participation as a lecturer of crystallographic symmetry in X-ray diffraction schools and symmetry-focused schools in Latin America, I have witnessed the evolution of demand and progress in the community. The pandemic came with a toll, as reduced access to travel reduced the number of Schools and probably the quality learning for the virtual attendees of online events. A strong effort will be needed to restore the situation to pre-2020 levels in the next couple of years on the side of Lecturers and institutions organizing more and better Schools and on the side of students to adapt to reduced travel funds and mobility. However, the IUCr (accompanied by other institutions and companies that profit from the existence of many good crystallographers in the region) is capable of helping to defray the cost of this situation by promising more financial help and free access to educational resources. The MaThCryst Commission I represent and LACA are looking forward to working hard for more and better crystallography in Latin America and around the world.
 Stout, G. H. & Jensen, L. H. (1970). X-ray structure determination. A practical guide. The Macmillan Company 1968. 4th Printing.
 Woolfson, M. M. (1970). An Introduction to X-ray Crystallography. Cambridge University Press. ISBN 0 521074 40 1.
 Dent Glasser, L. S. (1984). “International Union of Crystallography Teaching Pamphlet #13 – Symmetry”, edited by C. A. Taylor. University College Cardiff Press. ISBN 0 906449 17 0. Available at https://www.iucr.org/__data/assets/pdf_file/0012/14304/13.pdf
 Meier, M. W. (1984). “International Union of Crystallography Teaching Pamphlet #14 – Space Group Patterns”, edited by C. A. Taylor. University College Cardiff Press. ISBN 0 906449 18 9. Available at: https://www.iucr.org/__data/assets/pdf_file/0016/14371/14_annotated.pdf
 “CSD Space Group Statistics – Space Group Frequency Ordering – 1 January 2021” https://www.ccdc.cam.ac.uk/support-and-resources/ccdcresources/84242e1ccbe540f8b816e1f6fee4eedb.pdf
 Sirius, Brazilian Synchrotron Light Source. https://www.lnls.cnpem.br/sirius-en/
 Laboratorio Argentino de Haces de Neutrones. https://www.lahn.cnea.gov.ar/
 Teaching Edition of International Tables for Crystallography: Crystallographic symmetry (2021). Edited by Mois I. Aroyo. IUCr/Wiley. ISBN 978-0-470-97422-3.
|Editor's note: The IUCr is pleased to announce that it plans to put the Teaching Edition online. This resource will be made available free of charge to IUCr-supported schools along with the customary free access to International Tables Online and the Symmetry Database.
The paperback version of the Teaching Edition is available at low cost from Wiley [USD 40.00/EUR 33.90/GBP 29.99 excluding taxes (if applicable) and postage]. Bulk discounts may be available; please contact Wiley's Corporate and Custom Sales team.
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