Resolving the Space-Group-Choice Dilemma: Case-Based Learning Approach in Modern Crystallography Education

Introduction: Why Symmetry Still Matters

Imagine gazing through a microscope at a tiny crystal that might just hold the answer to your latest scientific question. In front of you is a wonder of nature, where symmetry and mathematics intertwine. For many chemistry researchers, crystallography is the ultimate tool, as it reveals the true arrangement of atoms within matter. Yet, as technology has made these measurements easier to obtain, one question remains as tricky as ever: How do we choose the correct space group for a crystal structure?

Advancements in crystallographic equipment and software have significantly expedited structural resolution, yet the cultivation of critical thinking skills among students remains paramount, particularly when confronting complex concepts such as symmetry-related issues in crystallography. Over recent decades, we have incorporated various active-learning methods and diverse learning opportunities into our curriculum (Campbell et al., 2016; Dong & Zheng, 2021; Malbrecht et al., 2016; Zheng & Campbell, 2018; Zheng et al., 2018; Zheng & Campbell, 2021; Zheng et al., 2025). This journey has taught us that crystallography requires more than simply pressing a button. Understanding why a certain symmetry was selected, and how to challenge our own assumptions, is key to inspiring the next generation of chemical crystallographers.

The Beauty and Challenge of Symmetry

The beauty of crystallography centers around symmetry. It enables us to use a simple model with just a few atoms in the basic unit cell to describe the trillions of molecules that comprise a crystal. Symmetry brings elegance and efficiency to how we interpret and communicate structural information; however, while crystallography has long served as the "gold standard" for chemists, errors and pitfalls still occur. One of the most fundamental and often overlooked issues is the complex and challenging space-group-choice dilemma including:

• Assigning symmetry that is too low, as a result of missing symmetry elements.
• Assigning symmetry that is too high, often as a result of unrecognized twinning in a lower-symmetry space group.
• Assigning symmetry that is too high, when pseudosymmetry is mistaken for true higher symmetry, even though a fully ordered structure in a lower-symmetry space group may actually provide a better model.

Historically, students have usually made the mistake of lowering the symmetry of a structure without sufficient justification. Richard Marsh (1922-2017) (see https://tinyurl.com/bknp4ze2) notably corrected hundreds of published crystal structures with missing symmetry elements, highlighting just how widespread this problem was. Thanks to modern crystallographic software, such as Platon and CheckCIF developed by Ton Spek, many of these errors can be detected and have been greatly reduced.

However, technological advances can introduce new challenges. The proliferation of automated indexing tools and education has introduced a tendency to prioritize higher-symmetry solutions, sometimes at the expense of important structural information found in lower-symmetry space groups. This overemphasis on high-symmetry groups can mask important details, leading to incomplete understanding or incorrect structural models (Spek, 2020; Lutz, 2023).

Improvements in hardware have also made a significant impact on the field. Modern diffractometers equipped with area detectors are capable of detecting weak diffraction signals and identifying twinning phenomena, which could be easily missed with point-detector instruments (Müller et al., 2021). While these improvements help crystallographers make more accurate space-group determinations, some modern rapid data collection strategies and fully automated processing may fail to capture weak reflections, resulting in a space group with a smaller unit cell and overlooking the true symmetry of the structure (Zheng et al., 2025).

From Tradition to Transformation: Teaching Crystallography in the Modern Era

For decades, crystallography was the province of specialists, with mastery gained through hands-on trial and error, as well as mentorship. Today, advances in software have democratized small-molecule crystallography. Many students in a chemical laboratory now have access to automated tools. Yet this progress brings its own dangers. Push-button structures can appear correct, even when based on an incorrect symmetry assumption. The differences can be subtle, but such errors often affect the conclusions drawn in chemistry, biology, and materials science.

This forms the heart of what we call the "space-group-choice dilemma". Over the decades, we have responded by integrating active-learning methods into our curriculum, with a case-based learning approach proving particularly successful (Dong & Zheng, 2021). This approach directly addresses the complex and often-overlooked decisions scientists face when selecting the correct space group in small-molecule crystallography.

Bring Symmetry to Life: Key Concepts in the Classroom

To bring abstract concepts to life, we use hands-on, visual exercises to help students think differently about symmetry.

For example, in one activity, a drawing of a two-dimensional lattice (an array of points, Fig. 1) is given. At first glance, it may appear to fit an oblique unit cell, but upon closer inspection, a C-centred unit cell actually has higher symmetry and is the better choice. This demonstrates an essential principle of crystallography: by convention, a unit cell should normally be chosen as the smallest repeating unit with the highest symmetry possible.

However, the process is not always so straightforward. Diffraction spots contain both position and intensity information. During the unit-cell determination process, the indexing software typically focuses on the symmetry of positions in reciprocal space (known as "metric symmetry") without considering the symmetry of their intensities. But sometimes, even if the spots line up symmetrically, differences in their intensity can reveal a lower symmetry (called "Laue symmetry"). Students learn that relying only on software can lead to mistakes, particularly when it comes to resolving subtleties in real data.

To highlight these key concepts, students are asked to examine two-dimensional Escher drawings and assign them to one of the 17 possible plane symmetry groups. What starts as a seemingly playful exercise quickly becomes complex, prompting open discussion about previously unexamined assumptions. When inspecting the possible C-centered rectangular unit cell, students realize that the important distinction here is to take into consideration the content of each lattice point for the true symmetry. When each lattice spot is replaced by distinct motifs (Fig. 2, fish/boat), local symmetry is lost, and overall space-group symmetry is reduced. The exercise becomes even more complex if, in an originally symmetric arrangement (Fig. 4, fish/duck/lizard), certain motifs are selectively recolored (as in Fig. 3, where one-third of the fish display a different color). In these cases, the apparent symmetry of the underlying grid is preserved, but the true symmetry of the structure is markedly lowered. Assigning the correct space group can be particularly confusing when a motif, such as one of three fish with a different color in a specific or random position, further complicates the assessment. These nuanced scenarios require students to confront and clarify what symmetry truly means, challenging their initial interpretations and demonstrating that true crystallographic symmetry depends both on the periodic placement and on the identity of the motifs populating the lattice, not just the points themselves.

Another example involves a simple one-dimensional structure with repeating molecules, such as N-methylcyclohexanimine, where a methyl group flips orientation in every third molecule (Fig. 5). If the chosen unit is too small (one molecule), the structure appears disordered. Expanding such a "subcell" (one molecule) to a "supercell" (three molecules) allows the model to be fully ordered, which both fits the data better and reflects the true chemistry behind the material (Nespolo, 2019; Dong & Zheng, 2021). This case teaches students the importance of revisiting the choice of unit cell to avoid a mistaken impression of disorder.

Visualizing the symmetry of diffraction patterns in reciprocal space further deepens students’ understanding. From inspection of 2D reciprocal space reconstructions (as seen in "precession images"), students learn to identify missing diffraction spots (systematic absences) and similar or varying intensities, and find out whether the symmetry proposed by the crystallography software really matches observation. For example, a pattern that appears orthorhombic from indexing software might actually be monoclinic with β-angle close to 90°, when all intensity information is considered. Automated crystal structure determination can miss weak diffraction spots, leading to incorrect unit cells and space groups, and flawed models (Müller et al., 2021). Carefully examining the precession images can reveal the true symmetry and prevent potential errors.

The Case-Based Learning Approach

Building on these practical insights, our classrooms rely on the case-based learning approach to crystallography. We believe that real-world scientific challenges offer the best learning opportunities (Campbell et al., 2016). By integrating case-based learning into crystallography education, we help students confront space-group-choice dilemmas in small-molecule crystallography. Rather than memorizing tables or following routine steps, our students become investigators who tackle authentic and sometimes messy problems. They work through case studies drawn from published structures and dive into crystallography data from databases. In the process, they debate, make mistakes, and most importantly, learn to justify their choices. Some cases involve ambiguous diffraction patterns, questionable symmetry assignments, or even published papers with conflicting identifications. The students realize that the so-called "messiness" in crystallography is not a flaw, but an opportunity for learning and discovery. Through active learning, students develop the ability to critically analyze real data and gain a deeper understanding of the key concepts.

Another central objective in designing case-based learning was to tackle classic misconceptions in chemical crystallography (Nespolo, 2019; Dong & Zheng, 2021). For example:

• Believing the software will always choose the correct space group.
• Assuming that higher symmetry is always preferable.
• Thinking that space-group selection is disconnected from chemical logic.

Through guided discussions and learning from mistakes, students discover the importance of using software as a support tool, not as a substitute for scientific reasoning.

Our approach also emphasizes peer tutoring and reflections. A small group of students assigned to each case hold an oral presentation to discuss their findings with the class, offering both technical insight and encouragement. Group debriefings and written reflections reinforce these lessons. Students learn that there is rarely one clear-cut answer in experimental science. Instead, scientific progress is a process of discussion, critical review, and learning from mistakes and discoveries alike. The structured “messiness” of these problems helps students gain not just technical skill but true scientific insight.

Lasting Impact: Student Growth Beyond the Classroom

By learning to evaluate and defend their conclusions, our students become more confident in navigating complex crystallographic solutions and responding independently to peer-review comments during publication. Many have even contributed valuable insights to scientific literature by building on their strengthened critical thinking skills. Four of our students have been awarded the Ludo Frevel Crystallography Scholarship, reflecting their interest in incorporating crystallography as a critical component of their research workflow. Additionally, some students have continued their crystallography education by learning and continuing to use crystallography as a tool for their research projects at Harvard and in their independent academic careers, significantly contributing to various fields (Dong & Zheng, 2021; Zheng et al., 2025). This positive ripple effect shows how empowering learners contributes to the advancement of the entire field.

Looking Ahead: Crystallography in the Age of Automation

We live in an era of abundant information, with artificial intelligence and automation playing an ever-increasing role. However, no algorithm can replace scientific judgment, particularly when it comes to interpreting results at the boundaries of knowledge. Understanding when to rely on computational recommendations and when to apply critical thinking is central to scientific accuracy.

Crystallography teaches humility. A crystal structure is always a model, open to challenge and revision. Case-based learning provides students with the resilience and insight they need for a lifetime of scientific inquiry. By rooting education in real cases and critical questioning, we prepare scientists to meet complex challenges in crystallography and beyond.

As our tools grow more powerful, the foundations of the field remain unchanged. Careful observation, critical analysis, and the ability to navigate complexity are skills that automation cannot replicate. Teaching students to appreciate both the elegance and complexity of symmetry empowers them to make lasting scientific contributions. Ultimately, it is human curiosity and judgment that turn data into genuine understanding.

Acknowledgements

We thank the Teaching Assistants and the students of Chem255 at Harvard for helping us develop and improve the case study discussions in our class. We acknowledge support from the Major Research Instrumentation (MRI) Program of the National Science Foundation under NSF award No. 2216066 for the crystallography facility.

References

Campbell, M. G., Powers, T. M. & Zheng, S.-L. (2016). J. Chem. Educ. 93, 270–274.

Dong, Y. & Zheng, S.-L. (2021). J. Chem. Educ. 98, 3180–3188.

Lutz, M. (2023). J. Mol. Struct. 1284, 135362.

Malbrecht, B. J., Campbell, M. G., Chen, Y. & Zheng, S. (2016). J. Chem. Educ. 93, 1671–1675.

Müller, U., Ivlev, S., Schulz, S. & Wölper, C. (2021). Angew. Chem. Int. Ed. 60, 17452–17454.

Nespolo, M. (2019). J. Appl. Cryst. 52, 451–456.

Spek, A. L. (2020). Acta Cryst. E76, 1–11.

Zheng, S.-L. & Campbell, M. G. (2018). J. Chem. Educ. 95, 2279–2283.

Zheng, S.-L. & Campbell, M. G. (2021). Acta Cryst. E77, 864–866.

Zheng, S.-L., Chen, Y.-S., Wang, X., Hoffmann, C. & Volkov, A. (2018). J. Appl. Cryst. 51, 909–914. 

Zheng, S.-L., Litak, N. P., Campbell, M. G., Handford, R. C., Dogutan, D. K., Carsch, K. M. & Betley, T. A. (2025). J. Appl. Cryst. 58, 269–275.