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Fractals

A set of five laminated posters with background notes, from The Beauty of Mathematics poster collection.

The Fractals poster set

The posters illustrate different forms of fractals:

• The Sierpinski Triangle
• The Snowflake Curve
• The Dragon Curve
• Polyhedron Fractals
• Julia Sets

Each poster describes the fractals and also gives brief mathematical or historical information, such as the fractal dimension.

See below for more information about the background notes as well as samples from the notes. Larger images of the posters may be seen by going to the Poster Gallery.

• Poster Gallery
• Order
• Other poster sets:
• Curves from Circles
• Tilings
• Polyhedra
• Curves from Lines

“Not everything that can be counted counts, and not everything that counts can be counted.”—Albert Einstein

The Fractals background notes

A descriptive booklet accompanies the posters, explaining the mathematics behind the images in the posters and related topics.

The four pages of background notes give more information about the fractals on the five posters and about fractals in general. The topics covered are:

• Self-similarity
• Dimension
• Fractals
• Cantor Sets
• Space-filling Curves
• Description and properties of the fractals on each poster, and related fractals
• Glossary
• References

The notes help interested students and teachers explore further, and provide ideas for activities connected to the theme of the posters.

Extracts from the background notes

Self-similarity

A shape is said to be self-similar if it has a similar appearance when viewed at different scales. For example, enlarging the shaded section of the figure alongside gives an exact copy of the original, known as the quadratic von Koch curve. The similarity need not be exact; it may be approximate, or even statistical.

Dimension

One way to define dimension is to consider scaling properties—how many copies of a shape are obtained when it is enlarged? For example, enlarging a triangle by a factor of 3 produces a shape which is equivalent to 9 copies of the original, as shown. The number of copies and the scale factor are related by 9 = 3² and the power of 2 in that relationship corresponds to the dimension of the triangle.

The Dragon Curve

Since the dragon curve is space-filling it has dimension 2. The boundary of the region filled by the curve has dimension 1.5236.

Though it is possible to fit copies of the dragon curve together in various ways—an example is shown alongside—the dragon curve does not tile the plane.

The curve on The Beauty of Mathematics poster is coloured so that nearby points on the original unfolded strip of paper have similar colours. The actual colours used are arbitrary.

© A K Jobbings 2004-2008