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2. Hexagon - Wikipedia

   en.wikipedia.org/wiki/Hexagon

   The six lines of reflection of a regular hexagon, with Dih 6 or r12 symmetry, order 12. The dihedral symmetries are divided depending on whether they pass through vertices ( d for diagonal) or edges ( p for perpendiculars) Cyclic symmetries in the middle column are labeled as g for their central gyration orders.

3. Frieze group - Wikipedia

   en.wikipedia.org/wiki/Frieze_group

   A symmetry group in frieze group 1, 2, 3, or 5 is a subgroup of a symmetry group in the last frieze group with the same translational distance. A symmetry group in frieze group 4 or 6 is a subgroup of a symmetry group in the last frieze group with half the translational distance. This last frieze group contains the symmetry groups of the ...

4. Hexadecagon - Wikipedia

   en.wikipedia.org/wiki/Hexadecagon

   On the regular hexadecagon, there are 14 distinct symmetries. John Conway labels full symmetry as r32 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars) Cyclic symmetries in the middle column are labeled as g for their central ...

5. Wallpaper group - Wikipedia

   en.wikipedia.org/wiki/Wallpaper_group

   A wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art, especially in textiles, tiles, and wallpaper. The simplest wallpaper group, Group p 1 ...

6. List of planar symmetry groups - Wikipedia

   en.wikipedia.org/wiki/List_of_planar_symmetry_groups

   The symmetry groups are named here by three naming schemes: International notation, orbifold notation, and Coxeter notation. There are three kinds of symmetry groups of the plane: 2 families of rosette groups – 2D point groups. 7 frieze groups – 2D line groups. 17 wallpaper groups – 2D space groups.

7. Octagon - Wikipedia

   en.wikipedia.org/wiki/Octagon

   In geometry, an octagon (from Ancient Greek ὀκτάγωνον (oktágōnon) 'eight angles') is an eight-sided polygon or 8-gon. A regular octagon has Schläfli symbol {8} [ 1 ] and can also be constructed as a quasiregular truncated square, t {4}, which alternates two types of edges. A truncated octagon, t {8} is a hexadecagon, {16}.

8. Line group - Wikipedia

   en.wikipedia.org/wiki/Line_group

   Line group. A line group is a mathematical way of describing symmetries associated with moving along a line. These symmetries include repeating along that line, making that line a one-dimensional lattice. However, line groups may have more than one dimension, and they may involve those dimensions in its isometries or symmetry transformations.

9. Pentagon - Wikipedia

   en.wikipedia.org/wiki/Pentagon

   The regular pentagon has Dih 5 symmetry, order 10. Since 5 is a prime number there is one subgroup with dihedral symmetry: Dih 1, and 2 cyclic group symmetries: Z 5, and Z 1. These 4 symmetries can be seen in 4 distinct symmetries on the pentagon. John Conway labels these by a letter and group order. [10]