Beautiful patterns can be created by drawing sets of nested polygons such that the incircle of the th polygon is the circumcircle of the st and successive polygons are rotated one half-turn at each iteration. The results are shown above for nested triangles through heptagons in alternating black and white and in a cyclic rainbow coloring.The animation above shows successive iterations of a nested octagon.The black region of a nested square has areaif the initial square has unit edge length.
In 1704, Sebastien Truchet considered all possible patterns formed by tilings of right triangles oriented at the four corners of a square (Wolfram 2002, p. 875).Truchet's tiles produce beautiful patterns when laid out on a grid, as illustrated by the arrangement of random tiles illustrated above.A modification of Truchet's tiles leads to a single tile consisting of two circular arcs of radius equal to half the tile edge length centered at opposed corners (Pickover 1989). There are two possible orientations of this tile, and tiling the plane using tiles with random orientations gives visually interesting patterns. In fact, these tiles have been used in the construction of various games, including the "black path game" and "meander" (Berlekamp et al. 1982, pp. 682-684).The illustration above shows a Truchet tiling. For random orientations, the fraction of closed circles is approximately 0.054 and the..
The perspective image of an infinite checkerboard. It can be constructed starting from any triangle , where and form the near corner of the floor, and is the horizon (left figure). If is the corner tile, the lines and must be parallel to and respectively. This means that in the drawing they will meet and at the horizon, i.e., at point and point respectively (right figure). This property, of course, extends to the two bunches of perpendicular lines forming the grid.The adjacent tile (left figure) can then be determined by the following conditions: 1. The new vertices and lie on lines and respectively. 2. The diagonal meets the parallel line at the horizon . 3. The line passes through . Similarly, the corner-neighbor of (right figure) can be easily constructed requiring that: 1. Point lie on . 2. Point lie on the common diagonal of the two tiles. 3. Line pass through . Iterating the above procedures will yield the complete picture. This construction shows..
The fractal-like two-dimensional functionThe function is named for the appearance of a butterfly-like pattern centered around the origin (left figure). In the above illustration, the left plot runs from to 5 and the right plot runs from to 20.
Strang's strange figures are the figures produced by plotting a periodic function as a function of an integer argument for , 2, .... Unexpected patterns and periodicities result from near-commensurabilities of certain rational numbers with the period (Richert 1992). Strang figures are shown above for a number of common functions.