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  2. Wang tile - Wikipedia

    en.wikipedia.org/wiki/Wang_tile

    Example of Wang tessellation with 13 tiles. In 1961, Wang conjectured that if a finite set of Wang tiles can tile the plane, then there also exists a periodic tiling, which, mathematically, is a tiling that is invariant under translations by vectors in a 2-dimensional lattice. This can be likened to the periodic tiling in a wallpaper pattern ...

  3. Quasicrystal - Wikipedia

    en.wikipedia.org/wiki/Quasicrystal

    The more precise mathematical definition is that there is never translational symmetry in more than n – 1 linearly independent directions, where n is the dimension of the space filled, e.g., the three-dimensional tiling displayed in a quasicrystal may have translational symmetry in two directions.

  4. Fourth dimension in literature - Wikipedia

    en.wikipedia.org/wiki/Fourth_dimension_in_literature

    The idea of a fourth dimension has been a factor in the evolution of modern art, but use of concepts relating to higher dimensions has been little discussed by academics in the literary world. [1] From the late 19th century onwards, many writers began to make use of possibilities opened up by the exploration of such concepts as hypercube ...

  5. Fourth dimension in art - Wikipedia

    en.wikipedia.org/wiki/Fourth_dimension_in_art

    Literature leaving the line and entering the plane. Painting leaving the plane and entering space. Sculpture stepping out of closed, immobile forms. The artistic conquest of four-dimensional space, which to date has been completely art-free. The manifesto was signed by many prominent modern artists worldwide.

  6. Aperiodic crystal - Wikipedia

    en.wikipedia.org/wiki/Aperiodic_crystal

    The history of aperiodic crystals can be traced back to the early 20th century, when the science of X-ray crystallography was in its infancy. At that time, it was generally accepted that the ground state of matter was always an ideal crystal with three-dimensional space group symmetry, or lattice periodicity.

  7. Lattice (group) - Wikipedia

    en.wikipedia.org/wiki/Lattice_(group)

    A lattice in the sense of a 3-dimensional array of regularly spaced points coinciding with e.g. the atom or molecule positions in a crystal, or more generally, the orbit of a group action under translational symmetry, is a translation of the translation lattice: a coset, which need not contain the origin, and therefore need not be a lattice in ...

  8. Bravais lattice - Wikipedia

    en.wikipedia.org/wiki/Bravais_lattice

    For example, a crystal, viewed as a lattice with a single kind of atom located at every lattice point (the simplest basis form), may also be viewed as a lattice with a basis of two atoms. In this case, a primitive unit cell is a unit cell having only one lattice point in the first way of describing the crystal in order to ensure the smallest ...

  9. Unimodular lattice - Wikipedia

    en.wikipedia.org/wiki/Unimodular_lattice

    In geometry and mathematical group theory, a unimodular lattice is an integral lattice of determinant 1 or −1. For a lattice in n-dimensional Euclidean space, this is equivalent to requiring that the volume of any fundamental domain for the lattice be 1. The E 8 lattice and the Leech lattice are two famous examples.