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From Friezes to Quasicrystals: A History of Symmetry Groups

Handbook of the History and Philosophy of Mathematical Practice

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Abstract

Even if, since many thousands of years, humans were fascinated by symmetry, which is reflected in many preserved ornaments on buildings, paintings, pottery, etc., the mathematical study of symmetrical patterns is just a few hundred years old. In fact, until the nineteenth century, when it received an impulse from crystallography, no systematical approaches were developed. In this chapter, we follow the history of (Euclidean) symmetry from its first appearances and connections with art, over geometric classifications by nineteenth-century crystallographers, the coalescence of this crystallographic-geometrical approach with theory of groups in the second part of the nineteenth century, until various twentieth-century generalizations to symmetry groups in higher dimensions, magnetic, and colored groups.

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Notes

  1. 1.

    J’appelerai polyèdres symétriques deux polyèdres qui, ayant une base commune, sont construits semblablement, l’un au-dessus du plan de celte base, l’autre au-dessous avec celte condition que les sommets des angles solides homologues soient situes a egales distances du plan de la base, sur une meme droite perpendiculaire à ce plan.

  2. 2.

    Eine wissenschaftliche Kenntniss von den Formen der Krystalle, die allen Anforderungen genügt, setzt voraus, dass man mit den allgemeinen Principien vertraut sey, welche die Gestaltenlehre bei Untersuchungen der verschiedenen möglichen Gestalten überhaupt zu befolgen hat. Da diese Principien jedoch in Werken über reine Mathematik noch nicht so bestimmt aus gesprochen sind, indem die allgemeine Gestaltenlehre in der Ausdehnung, wie sie jetzt vorliegt, eine noch jugendliche Wissenschaft ist, so muß sie hier erst entwickelt werden, um so dann die Anwendung derselben auf die Krystallgestalt folgen zu lassen.

  3. 3.

    Some of the groups listed above are not only isomorphic, but the same. For example, C1h = C1v and D1h = C2v.

  4. 4.

    Le groupe forme par les mouvements cher de cette propriete earacteristique, quo si M’ et M” sont deux mouvements quelconques faisant partie de ce groupe, M’M” en fera partie.

  5. 5.

    Eine besonders einfache Aufgabe ist die folgende: es sind alle verschiedenen Strukturen der Symmetrie einer Bordüre (Band, Fries) aufzuzählen; es gibt deren sieben.

  6. 6.

    Es ist dies das Problem der Erweiterung der 17 Nigglischen zweidimensionalen Ebenengruppen zu den 80 dreidimensionalen, so, wie es sich jetzt darum handelt, die 230 dreidimensionalen R3-Gruppen zur Gesamtheit der vierdimensionalen auszubauen.

  7. 7.

    Les éléments de symétrie des causes doivent se retrouver dans les effets produits.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, University of Zagreb, Zagreb, Croatia

    Franka Miriam Brückler

  2. Department of Chemistry, Faculty of Science, University of Zagreb, Zagreb, Croatia

    Vladimir Stilinović

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  2. Vladimir Stilinović
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Correspondence to Franka Miriam Brückler .

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  1. Department of Mathematical Sciences, The University of Montana, Missoula, MT, USA

    Bharath Sriraman

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  1. Department of Mathematical Sciences, The University of Montana, Missoula, MT, USA

    Bharath Sriraman

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Brückler, F.M., Stilinović, V. (2023). From Friezes to Quasicrystals: A History of Symmetry Groups. In: Sriraman, B. (eds) Handbook of the History and Philosophy of Mathematical Practice. Springer, Cham. https://doi.org/10.1007/978-3-030-19071-2_132-1

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  • DOI: https://doi.org/10.1007/978-3-030-19071-2_132-1

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-19071-2

  • Online ISBN: 978-3-030-19071-2

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  1. Latest

    From Friezes to Quasicrystals: A History of Symmetry Groups
    Published:
    21 November 2023

    DOI: https://doi.org/10.1007/978-3-030-19071-2_132-2

  2. Original

    From Friezes to Quasicrystals: A History of Symmetry Groups
    Published:
    12 September 2023

    DOI: https://doi.org/10.1007/978-3-030-19071-2_132-1

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