This is a tessellation of Penrose tiles. Penrose tiles can be arranged, as has been done in this image, such that the tiling never repeats, no matter how many tiles are used.The two different tile shapes are a wider rhombus and a thinner rhombus. Additionally, each tile has been filled with four pursuit curves: the dark curves from each corner to a point neat the centre of the tile.
This image was created by iterating systems of Möbius transformations. A Möbius transformation in the complex plane is a rational function of the form
where z is a complex variable, and a, b, c, d are complex numbers satisfying ad − bc ≠ 0.
This double knit scarf brings together two mathematical ideas: a recursive construction of a fractal – the dragon fractal, and the recursive construction of an integer sequence – the Fibonacci sequence. The main panels of the scarf are based on a pattern which arises from the 11th iteration of the dragon fractal, whilst the striping between the main panels illustrates the Fibonnaci sequence (colour changes after 1 row, after 1 row, after 2 rows, after 3 rows, after 5 rows etc).
This image shows 8,000 ellipses. For each k=1,2,3,…,8000 the foci of the kth ellipse are:
The eccentricity of the kth ellipse is D(k), where
This is part of the generating tile of a planar repeating pattern that have no reflection symmetries but do have many glide reflection symmetries as well as translational symmetries and two-fold centres of rotation. “The absence of reflectional symmetries often leads to very fluid and dynamic patterns.”