Virtual and Printed 3D Models for Teaching Crystal Symmetry and

Article pubs.acs.org/jchemeduc

Virtual and Printed 3D Models for Teaching Crystal Symmetry and Point Groups Lluís Casas* and Eugènia Estop Departament de Geologia, Facultat de Ciències, Universitat Autònoma de Barcelona, Campus de la UAB, 08193 Bellaterra, Catalonia, Spain S Supporting Information *

ABSTRACT: Both, virtual and printed 3D crystal models can help students and teachers deal with chemical education topics such as symmetry and point groups. In the present paper, two freely downloadable tools (interactive PDF files and a mobile app) are presented as examples of the application of 3D design to study point-symmetry. The use of 3D printing to produce tangible crystal models is also explored. A series of dissection puzzles that will be especially useful for teaching crystallographic concepts such as asymmetric unit and general/special positions is presented. Educators are encouraged to use the presented tools in their classes, and we expect our work to inspire other college educators to design and share similar tools.

KEYWORDS: First-Year Undergraduate/General, General Public, Chemoinformatics, Computer-Based Learning, Hands-On Learning/Manipulatives, Humor/Puzzles/Games, Crystals/Crystallography, Group Theory/Symmetry, Materials Science

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INTRODUCTION Very popular tools for teaching and illustrating chemical structures are molecular models based on spheres representing atoms either directly connected or linked by rods representing bonds. These models are used both as tangible kits, usually made of plastic, and in form of 3D designs in molecular viewing software. More recently, taking advantage of the multiple possibilities of 3D printing, very complex 3D molecular designs have been used to produce complex tangible objects based on the same kind of models.1,2 Besides molecule structures, crystal structures can also be represented using spheres and rods and extra rods to delimit the unit cell. Crystal-structure models are not as standardized as molecular models, and many creative and cheap ways to represent them have been presented.3 Like molecular models, there are many viewing programs used both in teaching and research that produce 3D designs of structures and these have also been used to generate prints of tangible objects to represent complex structures.4 Finally, some specifically designed 3D printed models have been tested to teach the concept of unit cell.5 Crystallography courses are not only part of chemistry graduate programs; for instance they can be often found in geology and material science programs.6 Paradoxically, less attention has been devoted to innovate on another interesting teaching tool, the crystal models representing crystal morphologies, which are used in basic crystallography courses. In addition to crystals structures, morphological crystal models are used to teach and practice crystal symmetry concepts © XXXX American Chemical Society and Division of Chemical Education, Inc.

related to the crystallographic point groups. Idealized crystal models (polyhedra), classically made of wood, were first developed by pioneering personalities of modern crystallography, back in the 18th century,7 and they have been used for many decades in crystallography courses. Wood models continue to be the most used in the classroom. However, when these have to be replaced due to wear and tear, an expensive new copy has to be ordered from a skilled carpenter. Commercial wooden kits are available, but they are rare and expensive; the German company Krantz is one of the few suppliers. Low cost alternatives have been presented using paper,8,9 though these have a lower durability. Both 3D design and 3D printing could contribute to modernize this classical tool. Regarding 3D design, only few Web resources deal with crystal morphology and symmetry.10,11 Some commercial packages exist to generate and view morphological crystal models,12,13 though these are not designed for educational purposes. However, some printable 3D models for teaching Geosciences (including Mineralogy and Crystallography) are freely available,14 and the list of downloadable items will possibly grow quickly. Most visualization software packages recognize the existing standardized Crystallographic Information Framework (CIF) as an input/output format. Recently, some efforts have been made to link the CIF format with the stl (stereolithography) format used in 3D printing.15,16 With the use of existing crystallo-

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DOI: 10.1021/acs.jchemed.5b00147 J. Chem. Educ. XXXX, XXX, XXX−XXX

Journal of Chemical Education

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graphic databases, a number of 3D printed resources intended for Crystallography education are starting to be available.17 The morphological crystal models presented here are 3D designs conceived for teaching purposes. These have been used to create entertaining and useful educational tools that will be presented as well. Finally, some possibilities of the 3D-printed designs are also explored.

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3D VIRTUAL MODELS

Selection of Polyhedral Models and Design Protocol

As a consequence of the crystallographic restriction theorem, there are only 32 possible combinations of point symmetry elements (the so-called 32 Point Symmetry Groups). However, there is an infinite number of possible crystal morphologies to illustrate any point group. A minimum of one possible morphology was designed to represent every crystallographic point group. For holohedrial groups (i.e., the maximum symmetry point groups within each system), more than one morphology were designed. Finally, for non-holohedrial groups but bearing high symmetry, several models were designed as well. For instance, up to 10 polyhedra were designed for the cubic holohedry (m3̅m). Altogether, 82 polyhedra were obtained including basic forms like prisms with pedion, bipyramids and some of their combinations. These morphologies came from known wooden models, drawings18 and variations of them. Every polyhedral model was designed parametrically as a 3D object using SolidWorks or Google SketchUp, from the coordinates of their vertices (point cloud). The methodology described in our previous paper19 was used. Succinctly, the point cloud was obtained applying the symmetry operators of a given point group to a set of generator vertices (i.e., the nonsymmetry-equivalent vertices). Once imported into the 3D design software, the points were linked to form edges and faces (Figure 1). The final result is a SLDPRT file (SolidWorks) or a SKP file (Google SketchUp). These files can be exported and used in several ways.

Figure 2. Screenshot of one of the interactive PDF files displaying all the symmetry elements of a cube (point group m3m ̅ ). Check marks indicate the options selected for the 3D design currently on display.

produce an interactive file. Apart from controlling the position of the embedded 3D objects, the user of the file can control which layers are shown or hidden by clicking on the buttons of the control console. The final result consists in 32 PDF files to work interactively with the crystallographic point groups and some of their possible crystal morphologies. Color conventions are used to correlate the Hermann-Mauguin notation of the groups with their standard stereographic projection and the actual symmetry elements. The files are freely downloadable at http://departaments.uab.cat/geologia/PSG. They can be used both by teachers and students in the classroom and at home as they do not require specialized software, not even an Internet connection. The files were designed under version 9 of Adobe Acrobat, and the best performance is achieved using version 9 or above on a computer though they can also be viewed using versions 7 and 8 and even PDF viewers for smartphones. Detailed features of the PDF files can be found in our previous paper,15 though usage is the best way to learn their possibilities. Additionally, a User’s Guide is incorporated as a second page within each PDF file. Quiztallography App

Smartphones are widespread among high school and college students, and education-related mobile apps are becoming increasingly used to promote learning among the students.20 In the field of chemistry, a large amount of apps dedicated to provide chemical data, mainly linked to the Periodic Table scheme21 exist. Taking advantage of the progressive improvement of graphics and CPU computing power, a number of powerful molecular viewer and structure drawing apps are also available.21 The presented 3D crystal models have been used as the main feature of a quiz app named with the pun Quiztallography. The app is currently available for Android and Windows Phone platforms. The app was developed using Java and following a model-view-controller pattern.22 The present version of the app contains more than 1000 questions on point symmetry, including time trial and infinite time playing modes. The best performance of the present version is achieved using version Android 4.0 or above, though the app can already run under version 2.3. Concerning the Windows Phone platform, version 8 is required. Despite the large number of questions there is a quite limited number of

Figure 1. Production of a 3D polyhedron using SolidWorks: (a) point cloud with the generator vertices highlighted in red; (b) point cloud with edges; (c) final polyhedron with faces.

Interactive PDF Files

The free and widespread Adobe Acrobat Reader software supports embedding and viewing 3D objects since version 7. The 3D designs were used as the main feature of interactive PDF files containing a control console to interact with the 3D crystal models (Figure 2), their symmetry and their associated representation conventions. For every point group, the corresponding 3D designs along with their symmetry elements (mirror planes and rotation axes also designed as 3D objects) were embedded in a PDF file in separate layers. Other layers containing basic textual information, icons and the stereographic projection were also integrated into the PDF file to B



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(Figure 4a). Both the core and the overgrowth exhibit crystal facets, but a change of crystal habit is apparent: the edges of the

different question typologies. Most of them present an interactive 3D design of a crystal model that can be smoothly rotated using one-finger swiping on the screen. Some of the questions show, besides the 3D model, highlighted planes or directions to illustrate a question wording on mirror planes or rotation axes (Figure 3). The app’s users can develop learning

Figure 3. Mobile displaying one question using the app Quiztallography (left) and a zoomed screenshot of it (right). Figure 4. Morphology analysis of a garnet overgrowth texture. (a) Microphotograph of a petrographic thin section of a garnet crystal exhibiting a core and an overgrowth with different crystal habits (analyzer out). (b) Overlapping of a rhombic dodecahedron and a trapezohedron (larger) sharing the same symmetry orientation; the images were obtained using the corresponding interactive PDF file; the link with the microphotograph is apparent. (c) Individual views of the rhombic dodecahedron (left) and the trapezohedron (right) displaying the 3-fold axes; check marks indicate the options selected the 3D designs on display.

without reading textbooks, using approaches based on trial and error and reproductive learning. However, a disciplined user would like to read the learning section included within the app before playing. The learning section contains basic information on crystallography, symmetry and crystals. The Quiztallography app is freely downloadable from the Google play and Windows Phone app stores easily accessible just typing Quiztallography into the corresponding app search engine.23,24 Example of Application

Both the interactive PDF files and the mobile app have been presented to the students in the classroom and they were encouraged to use them as part of their self-learning activities. Besides that, the interactive PDF files have been used in a computer-room session. The session typically lasts 2 h and in this case was divided in two parts; in the first one, the students got acquainted with the visualization options of both the polyhedral models and their symmetry elements, and in the second one, they focused on the orientation relationships between the 3D ensemble (model + symmetry elements) and the corresponding stereographic projection. At the end of the session, the students were asked to fill in a questionnaire including questions on the full Hermann-Mauguin notation, the number of symmetry elements of a given polyhedron and their orientation relationships. Regarding the developed app, application to face-to-face sessions was also achieved in form of a competition between students playing together, each with their own mobile device. This was actually part of the commemorative activities for 2014, International Year of Crystallography. However, the PDF files can also be useful for research and introduction to research activities. One example of this appeared casually during a final year project that dealt with the mineralogical characterization of an abandoned iron ore mine in the Montseny massif (near Barcelona). Among the different minerals of the site there were euhedral crystals of grossular-andradite garnet. An interesting feature of these crystals is the presence of zoning and birefringence25 and a distinct overgrowth of isotropic iron-rich garnet (andradite)

core crystals do not coincide with those of the overgrowth. Garnets are known to belong to the crystal class m3̅m, developing different habits (12-sided rhombic and 24-sided trapezoidal crystals). These two forms can be identified using the corresponding PDF file, and more interestingly, the orientation relationship between both forms and their common symmetry elements can be easily tracked. Figure 4 shows how the m3̅m PDF file can help to understand the thin section of these garnet crystals.

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3D PRINTED MODELS

Printing Process

The 3D virtual models are SLDPRT or SKP files that can be easily modified using the corresponding 3D software (SolidWorks and Google SketchUp, respectively). Among the exportation possibilities for these files, there is the stl (stereolithography) extension. This file format describes the surface geometry as a mesh of triangular surfaces using the 3D coordinates of the vertices. The STL files can be easily imported into the 3D printer-specific software and printed. Two types of printing processes have been tested. On one hand, Selective Laser Sintering (SLS)26 with an EOS Formiga P100 system was used. The SLS process involves a laser that selectively hardens a pattern in a layer containing polyamide powder (Fine Polyamide PA 2200); the process acts progressively layer-by-layer. The final result is a solid model within a block of powder. On the other hand, Fused Deposition Modeling (FDM)27 using an UP 3D Printer has been also C



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3D-printed crystal morphologies are already commercially available through the Internet,11,28 but some basic polyhedral models are freely available from 3D-printable-things sharing communities.14,29 However, the possibilities of the application of 3D printing to point symmetry are not restricted to the reproduction of crystal models. The edition of the 3D designs can be used to produce the model split in several parts to illustrate how a given symmetry element generates the whole model by applying its symmetry operations. If the split of the model is applied systematically, taking into account all the symmetry elements of a given point group, a kind of dissection puzzle is obtained (see Figures 6 and 7). The pieces can be stuck using Blu-Tack or a

tested. In the FDM process a heated extrusion nozzle melts a thermopolymer (acrylonitrile butadiene styrene, ABS) filament and deposits it following patterns on successive layers. The models produced by SLS are white, solid and they feel powdery to the touch; the layer thickness can be 0.1 mm (Figure 5b).

Figure 5. A variety of crystal models produced by 3D printing using Fused Deposition Modeling (a) and Selective Laser Sintering (b) and a classical wooden model (c).

The models produced by FDM can be directly produced in any color using colored filaments; they are light (they interior is almost empty) and slightly flexible, and the layer thickness can be 0.15 mm (see Figure 5a). Printings by FDM are cheaper to produce, but some models require support material and slight deformations can occur during the printing process. Both SLS and FDM processes allow the user to scale the models and produce them in the desired size.

Figure 7. 3D-printed dissection puzzles of a rhombic dodecahedron and a trapezohedron. In (a) the mounted puzzle shows that every face in a rhombic dodecahedron is shared by 4 pieces. In (b) the subgroup of their faces has been highlighted. In (c) one piece has been detached to show its internal mirror plane (dashed line). Finally, (d) shows a trapezohedron with two enantiomorphic pieces.

Printing Examples and Applications

similar product. These puzzles are particularly useful to deal with high-symmetry crystal models. The shape of all the pieces of the puzzle will be exactly the same for models representing chiral point groups, whereas they will be divided into two enantiomorphic versions of the same piece for achiral point groups (Figure 7d). A single piece of the puzzle is actually

By printing virtual models of crystals, the virtual object become tangible. 3D prints of models are a cheap and durable alternative to the classical wood models (Figure 5c). In spite of virtual tools, the value of the tangible models is still in facilitating communication of symmetry notions in the classroom and between students, even at home. A number of

Figure 6. 3D-printed dissection puzzles of a cube (a) and an octahedron (b) where the symmetry axes have been highlighted using needles and a trapezohedron (c) where a mirror plane is highlighted using a sheet of paper. D



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ACKNOWLEDGMENTS Financial support from Generalitat de Catalunya (project 2010MQD00113) and the Spanish Ministerio de Economı ́a y Competitividad (project FCT-13-5966). We are grateful to Vı ́ctor Arribas, Anı̈s Khoury, Xavier Moreno and Aaron Negrı ́n for their help developing the virtual tools. Mercè Corbella and Gerard Casado are also acknowledged for providing the garnet samples.

analogous to the asymmetric unit of a space group and it could be used to explain this space symmetry concept. The number of pieces of the puzzle indicates the multiplicity of the pointgroup, and in the assembled dissection puzzle, the symmetry axes can be easily highlighted using, for instance, decorated needles and similarly the mirror planes using a sheet of paper (Figure 6). Furthermore, the position of the faces of a given polyhedral model can be easily identified; a face in a general position will be formed by a single puzzle piece and the face will be shared between several puzzle pieces if placed at a special position (Figure 7a). The number of puzzle pieces sharing a given face indicates the multiplicity of the subgroup of the corresponding special position and that subgroup can be easily highlighted using a needle and sheets of paper (Figure 7b). Despite the resemblance between a single puzzle piece and the concept of asymmetric unit, it is worth noting that, unlike asymmetric units, the piece of a point-symmetry puzzle could actually contain symmetry. In a rhombic dodecahedron puzzle, the single puzzle pieces have a mirror plane though this plane does not define collectively a mirror plane in the mounted dodecahedron (Figure 7c), and in any case, it is not a pointsymmetry element because the plane does not pass through the point where all the point-symmetry elements intersect. In the case of symmetric puzzle pieces, there will not be any difference between the two enantiomorphic versions of the piece.

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CONCLUSIONS Different applications of 3D design and 3D printing to produce support materials to study point-symmetry groups have been explored. The availability of tools related to both virtual and tangible crystal models afford crystallography students and teachers with additional resources to overcome the difficulties from learning/teaching symmetry concepts that require capacity for abstraction and spatial vision. An asset of virtual tools is the no need for gathering a large set of tangible models. Besides that, users can work with virtual models on a variety of technological platforms that are particularly attractive to the digital generation students. The strong points of 3D printed models are obviously the addition of the sense of touch to the interaction process. To assemble the kind of dissection puzzles presented here is challenging and once they are assembled possibly give the greatest aid in point symmetry study as the user looks for the right insertion position of the needles that indicate symmetry axes. The 32 interactive PDF files and the Quiztallography app are freely downloadable and several examples of printable .stl files of crystal models are included in the Supporting Information. Educators are encouraged to use these tools in their classes and to develop their own ones inspired by those presented here. ASSOCIATED CONTENT

S Supporting Information *

Several examples of printable .stl files of crystal models (one per crystal system) (zip file). This material is available via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest. E



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Virtual and Printed 3D Models for Teaching Crystal Symmetry and

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