When an object is invariant under a specific combination of translation, reflection, rotation and scaling, it produces a new kind of pattern called a fractal. Concentric circles of geometrically progressing diameter are invariant under scaling. FractalsĪlso important is invariance under a fourth kind of transformation: scaling. 3-D objects can also be repeated along 1-D or 2-D lattices to produce rod groups or layer groups, respectively. Solve the equations for x (mentally, if possible). Ralph rolls a number less than 7 on a standard number cube. The various 3-D point groups repeated along the various 3-D lattices form 230 varieties of space group. Tanisha buys a new pair of shoes and the first shoe she pulls out of the box is for the left foot. ģ-D patterns are more complicated, and are rarely found outside of crystallography. A 2-D object repeated along a 2-D lattice forms one of 17 wallpaper groups. A 2-D object repeated along a 1-D lattice forms one of seven frieze groups. To make a pattern, a 2-D object (which will have one of the 10 crystallographic point groups assigned to it) is repeated along a 1-D or 2-D lattice. In 1-D there’s just one lattice, in 2-D there are five, and in 3-D there are 14. The number indicates what-fold rotational symmetry they have as well as the number of lines of symmetry.Ī lattice is a repeating pattern of points in space where an object can be repeated (or more precisely, translated, glide reflected, or screw rotated).
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