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  2. Crystallographic restriction theorem - Wikipedia

    en.wikipedia.org/wiki/Crystallographic...

    The crystallographic restriction theorem in its basic form was based on the observation that the rotational symmetries of a crystal are usually limited to 2-fold, 3-fold, 4-fold, and 6-fold. However, quasicrystals can occur with other diffraction pattern symmetries, such as 5-fold; these were not discovered until 1982 by Dan Shechtman .

  3. Penrose tiling - Wikipedia

    en.wikipedia.org/wiki/Penrose_tiling

    The pattern represented by every finite patch of tiles in a Penrose tiling occurs infinitely many times throughout the tiling. They are quasicrystals: implemented as a physical structure a Penrose tiling will produce diffraction patterns with Bragg peaks and five-fold symmetry, revealing the repeated patterns and fixed orientations of its tiles ...

  4. Pentagonal tiling - Wikipedia

    en.wikipedia.org/wiki/Pentagonal_tiling

    In 2016 it could be shown by Bernhard Klaassen that every discrete rotational symmetry type can be represented by a monohedral pentagonal tiling from the same class of pentagons. [15] Examples for 5-fold and 7-fold symmetry are shown below. Such tilings are possible for any type of n-fold rotational symmetry with n>2.

  5. Point groups in three dimensions - Wikipedia

    en.wikipedia.org/wiki/Point_groups_in_three...

    The two groups are obtained from it by changing 2-fold rotational symmetry to 4-fold, and adding 5-fold symmetry, respectively. There are two crystallographic point groups with the property that no crystallographic point group has it as proper subgroup: O h and D 6h. Their maximal common subgroups, depending on orientation, are D 3d and D 2h.

  6. Pyramid (geometry) - Wikipedia

    en.wikipedia.org/wiki/Pyramid_(geometry)

    The base regularity of a pyramid's base may be classified based on the type of polygon: one example is the star pyramid in which its base is the regular star polygon. [24] The truncated pyramid is a pyramid cut off by a plane; if the truncation plane is parallel to the base of a pyramid, it is called a frustum.

  7. Cyclic symmetry in three dimensions - Wikipedia

    en.wikipedia.org/wiki/Cyclic_symmetry_in_three...

    It has reflection symmetry with respect to a plane perpendicular to the n-fold rotation axis. C nv, [n], (*nn) of order 2n - pyramidal symmetry or full acro-n-gonal group (abstract group Dih n); in biology C 2v is called biradial symmetry. For n=1 we have again C s (1*). It has vertical mirror planes. This is the symmetry group for a regular n ...

  8. Alan Lindsay Mackay - Wikipedia

    en.wikipedia.org/wiki/Alan_Lindsay_Mackay

    Mackay has made important scientific contributions related to the structure of materials: In 1962 he published a manuscript that showed how to pack atoms in an icosahedral fashion; a first step towards five-fold symmetry in materials science. [2] These arrangements are now called Mackay icosahedra.

  9. Quasicrystal - Wikipedia

    en.wikipedia.org/wiki/Quasicrystal

    A further study of Khatyrka meteorites revealed micron-sized grains of another natural quasicrystal, which has a ten-fold symmetry and a chemical formula of Al 71 Ni 24 Fe 5. This quasicrystal is stable in a narrow temperature range, from 1120 to 1200 K at ambient pressure, which suggests that natural quasicrystals are formed by rapid quenching ...

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