Another approach that captures the complexity of the world around us is to use fractal tiles. For example, starting with a tessellation by hexagons, replace each line segment by three line segments as shown on the left. This is a version of midpoint modification, so at each stage the resulting shapes tessellate the plane.

Purpose To illustrate three simple methods for making fractals that tile the plane. Materials Drawing software with scale, rotate, and reflect options; IFS software. Conclusion The first method can generate fractal tiles from many starting shapes, but the growth of the core area limits the variety of fractal tiles produced.

The original shape tiles the plane, so also will each iterate, being made up of ever smaller copies of the original tile. To see the limiting tile has a fractal boundary, recall the area-perimeter relationship, by which we found the limiting curve has dimension d = 3/2.

The result is a fractal island, that we shall see is also a fractal tile. For familiar geometric shapes such as a square or circle, there is a simple relation between the area and the perimeter. If A 1 and A 2 denote the areas of two squares, for example, and P 1 and P 2 their perimeters, then: P 2 /P 1 = (A 2 /A 1) 1/2

A tutorial on basic fractal geometry and a panorama of uses of fractals, with supporting software, laboratory exercises, and resources for teachers

A tiling is a collection of shapes (tiles) that fit together to cover the plane, without overlapping or leaving gaps. A fractal tiling or f-tiling is a tiling in which the arrangement of tiles is fractal. A prototile is a tile to which all other tiles in the tiling are similar; i.e., of the same shape, though not necessarily the same size. At ...

In this paper, we demonstrate how our research findings regarding fractal tilings arose through the teaching of a geometry course, and describe the mathematical relationships between the content of the geometry course and our research results.

The mathematician Niels Fabian Helge von Koch (1870-1924) invented a famous curve by iteratively breaking up a line segment into three line segments of equal length and by replacing the central one by an equilateral triangle and removing the base line of the triangle. This produces a fractal geometry (Fig. K1) Fig.K1 Four iterates of the Koch ...

What is a fractal? Fractals have self-similar properties. If a portion of a fractal object is enlarged, the magnified portion always resembles the original figure. E.g. coastal line of...

If all the tiles are regular and similar, only equilateral triangles, squares, and hexagons can be used, because each of their interior angles evenly divides 360. Any point in the plane can be surrounded by six equilateral triangles, four squares, or three hexagons.