Abstract
Scientific observation has led to the discovery of recurring patterns in nature. Symmetry is the property of an object showing regularity in parts on a plane or around an axis. There are several types of symmetries observed in the natural world and the most common are mirror symmetry, radial symmetry, and translational symmetry. Symmetries can be continuous or discrete. A discrete symmetry is a symmetry that describes non-continuous changes in an object. A continuous symmetry is a repetition of an object an infinite number of times. A consequence of continuous symmetry is the existence of conservation laws. A natural system with discrete and continuous symmetries displays several physical properties, such as the existence of long-range order. Geometric shapes exhibit symmetry as mirror reflections of each other. Symmetry in nature has been a model of beauty since the beginning of civilization. Since the earliest times, nature itself has manifestly been a model, evincing regularity in sundry forms and occurrences–from minerals and plants to the anatomy of living beings, to the regularly recurring stellar constellations. For Ancient Greek philosophers, proportion, symmetry, and harmony, were the basics to determine whether something is beautiful or not. Ancient Greek philosophers were known for bringing logic and rational thinking to phenomena that were previously explained by mythology and Gods. They observed the natural world around them and used their knowledge to answer questions about the proportion in observable objects and the origin of Earth. The Greek words summetria and summetros appear frequently in the Timaeus, defined as parts with each other and with the whole. Plato has a very rough concept of symmetry, and when he uses “beauty” to characterize the so-called Platonic Solids in the Timaeus, he seems to be emphasizing their regularity and indirectly their symmetry. Plato believes the four elements (earth, air, fire, water) have been constructed by the Demiurge, or a divine craftsman who appoints order in an otherwise chaotic universe. Minerals are representative examples of beauty, order, and symmetry in inorganic materials. Well-formed minerals (crystals) are a collection of equivalent faces related by symmetry. The goal of this paper is to relate the perspective of symmetry in ancient Greek philosophy with the modern scientific evidence of the geometric description of crystal structures. The five Platonic solids are ideal models of geometrical patterns that occur throughout the world of minerals. In this paper, minerals serve as a model of connecting the symmetry theory in Plato’s philosophy and modern advances in mineralogy.
This is a preview of subscription content, log in via an institution to check access.
Access this article
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
Similar content being viewed by others
References
Asimow, P. D., Lin, C., Bindi, L., Ma, C., Tschauner, O., Hollister, L. S., & Steinhardt, P. J. (2016). Shock synthesis of quasicrystals with implications for their origin in asteroid collisions. Proceeding National Acadamy Science, 113(26), 7077–81.
Baskar, A. V., Benzigar, M. R., Talapaneni, S. N., Singh, G., Karakoti, A. S., Yi, J., et al. (2022). Self-assembled Fullerene Nanostructures: synthesis and applications. Advanced Functional Materials, 32(6), 2106924. https://doi.org/10.1002/adfm.202106924.
Bindi, L., Kolb, W., Eby, G. N., Asimow, P. D., Wallace, T. C., & Steinhardt, P. J. (2021). Accidental synthesis of a previously unknown quasicrystal in the first atomic bomb test. Proceedings of the National Academy of Sciences, 118(22), e2101350.
Bindi, L., Lin, C., Ma, C., & Steinhardt, P. J. (2016). Collisions in outer space produced an icosahedral phase in the Khatyrka meteorite never observed previously in the laboratory. Scientific Reports, 6(1), 38117. https://doi.org/10.1038/srep38117.
Bindi, L., Steinhardt, P. J., Yao, N., & Lu, P. J. (2009).Natural Quasicrystals. Science, 324(5932),1306–1309. https://doi.org/10.1126/science.1170827
Bindi, L., Steinhardt, P. J., Yao, N., & Lu, P. J. (2011). Icosahedrite, Al63Cu24Fe13, the first natural quasicrystal, 96(5–6),928–931. https://doi.org/10.2138/am.2011.3758
Bindi, L., Yao, N., Lin, C., Hollister, L. S., Andronicos, C. L., Distler, V. V., et al. (2015). Decagonite, Al 71 Ni 24 Fe 5, a quasicrystal with decagonal symmetry from the Khatyrka CV3 carbonaceous chondrite. American Mineralogist, 100(10), 2340–2343. https://doi.org/10.2138/am-2015-5423.
Blacker, S., Polanski, J., & Schwach, M. (1997). The Golden Ratio. http://www.geom.uiuc.edu/~demo5337/s97b/art.htm. Accessed 15 June 2022
Brading, K., & Castellani, E. (2021). Symmetry and Symmetry Breaking. Stanford Encyclopedia of Philosophy. https://plato.stanford.edu/entries/symmetry-breaking/. Accessed 9 March 2022
Burns, G., & Glazer, A. M. (2013). Chapter 2-Crystal Systems. In G. Burns & A. M. Glazer (Eds.), Space Groups for Solid State Scientists (Third Edition) (pp. 25–44). Oxford: Academic Press. https://doi.org/10.1016/B978-0-12-394400-9.00002-2
Buseck, P. R. (2002). Geological fullerenes: review and analysis. Earth and Planetary Science Letters, 203(3–4), 781–792. https://doi.org/10.1016/S0012-821X(02)00819-1.
Castellani, E. (2002). Symmetry, Quantum mechanics, and Beyond. Foundations of Science, 7, 181–196.
Caulton, A. (2015). The role of symmetry in the interpretation of physical theories. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 52, 153–162. https://doi.org/10.1016/j.shpsb.2015.08.002.
Darvas, G. (1997). Mathematical Symmetry Principles in the Scientific World View. In E. Agazzi, & G. Darvas (Eds.), Philosophy of Mathematics Today (pp. 319–334). Dordrecht: Springer Netherlands. https://doi.org/10.1007/978-94-011-5690-5_19.
Darvas, G. (Ed.). (2007). Symmetry, invariance, harmonySymmetry: Cultural-historical and ontological aspects of science-arts relations the natural and man-made world in an interdisciplinary approach (pp. 1–34). Basel: Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-7555-3_1.
Dress, A., Giegerich, R., Grünewald, S., & Wagner, H. (2003). Fibonacci-cayley numbers and repetition patterns in genomic DNA. Annals of Combinatorics, 7(3), 259–279. https://doi.org/10.1007/s00026-003-0185-z.
Fletcher, R. (2006). The Golden Section. Nexus Network Journal, 8(1), 67–89. https://doi.org/10.1007/s00004-006-0004-z.
Giomini, C. (2003). Timaeus’s insight on the shape of the Universe. Nature, 425(6961), 899–899. https://doi.org/10.1038/425899c.
Goldman, A. I., & Widom, M. (1991). Quasicrystal structure and Properties. Annual Review of Physical Chemistry, 42, 685–729.
Greuel, G. M. (2021). Crystals and symmetry–an historical outline. Symmetry: Culture and Science, 32(1), 41–57. https://doi.org/10.26830/symmetry_2021_1_041.
Grimm, U., & Schreiber, M. (2016). Electronic structure of quasicrystals. Reference Module in materials Science and Materials Engineering. Elsevier. https://doi.org/10.1016/B978-0-12-803581-8.01062-6.
Hahn, T., & Klapper, H. (2005). Crystal Symmetry. In F. Bassani, G. L. Liedl, & P. Wyder (Eds.), Encyclopedia of condensed Matter Physics (pp. 312–323). Oxford: Elsevier. https://doi.org/10.1016/B0-12-369401-9/00421-6.
Haldar, S. K. (2020). Chapter 3-Basic mineralogy. In S. K. Haldar (Ed.), Introduction to Mineralogy and Petrology (Second Edition) (pp. 109–143). Oxford: Elsevier. https://doi.org/10.1016/B978-0-12-820585-3.00003-X
Hazen, R. M. (2004). Chiral Crystal Faces of Common Rock-Forming Minerals. In Progress in Biological Chirality (pp. 137–151). Elsevier. https://doi.org/10.1016/B978-008044396-6/50013-7
Heyrovská, R. (2009). The golden ratio in the Creations of Nature arises in the Architecture of Atoms and Ions. In B. Şener (Ed.), Innovations in Chemical Biology (pp. 133–139). Dordrecht: Springer Netherlands. https://doi.org/10.1007/978-1-4020-6955-0_12.
Jennings, B. J., Tseng, T. W. J., Ouhnana, M., & Kingdom, F. A. A. (2022). Different symmetries, different mechanisms. Attention, Perception, & Psychophysics. https://doi.org/10.3758/s13414-022-02599-9
Kotrč, R. F. (1981). The dodecahedron in Plato’s “Timaeus. Rheinisches Museum Für Philologie, 124(3/4), 212–222.
Kroto, H. W., Heath, J. R., O’Brien, S. C., Curl, R. F., & Smalley, R. E. (1985). C60: Buckminsterfullerene. Nature, 318(6042), 162–163. https://doi.org/10.1038/318162a0.
Lalena, J. N., Cleary, D. A., & Hardouin Duparc, O. B. M. (2020). Principles of Inorganic Materials Design, 3rd Edition. Hoboken, NJ, USA: John Wiley & Sons, Inc.
Levine, D., & Steinhardt, P. J. (1984). Quasicrystals: a new class of ordered structures. Physical Review Letters, 53(26), 2477–2480. https://doi.org/10.1103/PhysRevLett.53.2477.
Lima-De-Faria, J. (1994). The stability of crystal structures. In J. Lima-De-Faria (Ed.), Structural mineralogy: an introduction (pp. 33–37). Dordrecht: Springer Netherlands. https://doi.org/10.1007/978-94-015-8392-3_4.
Lloyd, D. R. (2006). Symmetry and asymmetry in the construction of “elements” in the Timaeus. The Classical Quarterly, 56(2), 459–471. https://doi.org/10.1017/S0009838806000462.
Lloyd, D. R. (2010). Symmetry and beauty in Plato. Symmetry, 2(2), 455–465. https://doi.org/10.3390/sym2020455.
Lomas, J. D., & Xue, H. (2022). Harmony in design: a synthesis of literature from classical philosophy, the Sciences, Economics, and design. She Ji: The Journal of Design Economics and Innovation, 8(1), 5–64. https://doi.org/10.1016/j.sheji.2022.01.001.
Lorenz, K. (2005). Reflections on the concept of symmetry. European Review, 13(S2), 3–11. https://doi.org/10.1017/S1062798705000621.
Lower, S. (2017). Matter under the microscope. States of matter-Introduction. chem1 virtual textbook a reference text for General Chemistry. http://www.chem1.com/acad/webtext/states/states.html#Sect. 4. Accessed 15 March 2022
Mainzer, K. (2013). Symmetries of Nature: a Handbook for Philosophy of Nature and Science. https://www.degruyter.com/doi/book/10.1515/9783110886931. Accessed 29 January 2022
Malgrange, C., Ricolleau, C., & Schlenker, M. (2014). Introduction. In C. Malgrange, C. Ricolleau, & M. Schlenker (Eds.), Symmetry and Physical Properties of crystals (pp. 1–11). Dordrecht: Springer Netherlands. https://doi.org/10.1007/978-94-017-8993-6_1.
Marek-Crnjac, L. (2011). Platonic solids and thier connection to garnets. Garnet: metamorphic history, composition, crystallization (pp. 1–14). Nova Science Publishers.
Mehaffy, M. W. (2020). The impacts of Symmetry in Architecture and Urbanism: toward a New Research Agenda. Buildings, 10(12), 249. https://doi.org/10.3390/buildings10120249.
Mortley, R. J. (1969). Plato’s choice of the sphere. Revue des Études Grecques, 82(391), 342–345. https://doi.org/10.3406/reg.1969.1085.
Murayama, H., Tomonoh, S., Alford, J. M., & Karpuk, M. E. (2005). Fullerene production in tons and more: from science to industry. Fullerenes Nanotubes and Carbon Nanostructures, 12(1–2), 1–9. https://doi.org/10.1081/FST-120027125.
Nickel, E. H. (1995). The definition of a mineral. Mineralogical Journal, 17(7), 346–349. https://doi.org/10.2465/minerj.17.346.
Niu, W., Zhang, W., Firdoz, S., & Lu, X. (2014). Dodecahedral Gold Nanocrystals: the missing platonic shape. Journal of the American Chemical Society, 136(8), 3010–3012. https://doi.org/10.1021/ja500045s.
Prato, M. (1999). Fullerene materials. In A. Hirsch (Ed.), Fullerenes and related structures (pp. 173–187). Berlin, Heidelberg: Springer Berlin Heidelberg. https://doi.org/10.1007/3-540-68117-5_5.
Richeson, D. S. (2019). Platonic solids, Golf Balls, Fullerenes, and geodesic domes. Euler’s gem: the polyhedron formula and the birth of topology (pp. 75–80). Princeton, NJ: David S. Richeson: Princeton University Press.
Schwerdtfeger, P., Wirz, L. N., & Avery, J. (2015). The topology of fullerenes. Wiley Interdisciplinary Reviews: Computational Molecular Science, 5(1), 96–145. https://doi.org/10.1002/wcms.1207.
Tanaka, Y., Takeuchi, T., Lovesey, S. W., Knight, K. S., Chainani, A., Takata, Y., et al. (2008). Right handed or left handed? Forbidden X-Ray diffraction reveals chirality. Physical Review Letters, 100(14), 145502. https://doi.org/10.1103/PhysRevLett.100.145502.
Tapp, K. (2012a). Introduction to Symmetry. Symmetry (pp. 1–16). New York, NY: Springer New York. https://doi.org/10.1007/978-1-4614-0299-2_1.
Tapp, K. (2012b). The five platonic solids. In K. Tapp (Ed.), Symmetry: a Mathematical Exploration (pp. 115–137). New York, NY: Springer New York. https://doi.org/10.1007/978-1-4614-0299-2_8.
Verba, E. (2012). The Golden Ratio in Time-based Media. Journal of Arts and Humanities, (1).
Viazminsky, C. P. (2018). Framing Noether’s Theorem. Applied Mathematics, 09(03), 274–286. https://doi.org/10.4236/am.2018.93021.
Wenk, H. R., & Bulakh, A. (2016). Minerals Their Constitution and Origin 2nd edition. Cambridge, UK: Cambridge University Press.
Xiao, Y., Borgh, M. O., Blinova, A., Ollikainen, T., Ruostekoski, J., & Hall, D. S. (2022). Topological superfluid defects with discrete point group symmetries. Nature Communications, 13(1), 4635. https://doi.org/10.1038/s41467-022-32362-5.
Yanofsky, N. S., & Zelcer, M. (2017). The role of Symmetry in Mathematics. Foundations of Science, 22(3), 495–515. https://doi.org/10.1007/s10699-016-9486-7.
Zeyl, D., & Sattler, B. (2019). Plato’s Timaeus. In The Stanford Encyclopedia of Philosophy (Summer 2019 Edition.). Edward N. Zalta. <https://plato.stanford.edu/archives/sum2019/entries/plato-timaeus/
Zoltai, T., & Stout, J. H. (1984). Mineralogy: concepts and principles. Minneapolis, Minn.: Burgess Publishing Company. (Burgess Publishing Company.).
Zschornak, M., Leisegang, T., Meutzner, F., Stöcker, H., Lemser, T., Tauscher, T., et al. (2018). Harmonic principles of elemental crystals—from Atomic Interaction to Fundamental Symmetry. Symmetry, 10(6), 228. https://doi.org/10.3390/sym10060228.
Acknowledgments
We are grateful to Prof. Luca Bindi for the critical review and his thoughtful and constructive feedback that contribute to significantly improve the manuscript. An anonymous reviewer is kindly acknowledged for their comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
On behalf of all authors, the corresponding author states that there is no conflict of interest.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Elmi, C., Goodman, D.L. Beauty of Order and Symmetry in Minerals: Bridging Ancient Greek Philosophy with Modern Science. Found Sci (2023). https://doi.org/10.1007/s10699-022-09897-3
Accepted:
Published:
DOI: https://doi.org/10.1007/s10699-022-09897-3