# FRACTALS

John Robinson's sculpture "Temple of Fire" is analogous to a construction already found in mathematics by W.Sierpinski in 1915.
The Polish mathematician Wraclaw Sierpinski died in 1969. His grave bears only these words:

Explorer of the infinite.

 Sierpinski 'Cheese' 1st iteration

Note that, unlike Robinson's sculpture, the right hand picture does not involve any upside down tetrahedra, and that regular tetrahedra are used instead of elongated tetrahedra.

Fractals: at the infinite limit of a process of iterations.

Sierpinski's aim was to consider iteration, i.e. repetition, of the above process, so that each small tetrahedron of the first iteration is replaced by the first iteration:

The fractal is the "limit" of this process. It is an extraordinarily complicated object with a strong property of self-similarity. These ideas are easier to explain in the 2-dimensional analogue, the Serpinski gasket, which was found first.

 Self-similarity. Note that the whole figure can be found again in its own details: this is true at all scales; this feature is easily understood from the process of construction. The Sierpinski Gasket

 (Click to magnify) Robinson, without knowing of Sierpinski's work, was aware of the idea of fractals and has perceived the importance of this iterative process, as shown in his words: " I also see it being like a Gene, because you can go on adding tetrahedra to the sculpture for ever, so that it becomes like a Family Tree. If you go back 1000 years, we each have millions of ancestors. We are the Genetic melting pots of survival Genes.". On the left is another iteration of the process giving Robinson's sculpture :

Fractals: symbolism and life.

• Robinson has long been fascinated by the origin and development of life, and has expressed the ideas of birth and replication in several sculptures:
"Chain of Life", "Life", "Womb".

• The notion of fractal found in mathematics has been shown to give a model of many processes which govern development of organisms.

The analogy shows how objects which look and indeed are very complicated may be generated by processes which may be defined in a simple way. They can explain how complicated structures, for example, of the kidney, lung, blood vessels, and so on, can be generated.

(Click on the fern to see the replication process that leads to this picture.)

 Fractal ammonite Fractal fern

The bone below is a real one. Fractals are used in the University of Washington to show how the geometrical structure of a bone is important for its strength.

 Trabecular bone

## Going further with fractals

The notion of Iterated Function System (IFS) for constructing fractals: High level account given by Boston University.

The Center for Polymer Studies, a scientific visualisation research center in the Physics department at Boston University.

Fractint:
A public domain software package which enables you to specify an iterative process and then draws the corresponding fractal. It also draws fractals constructed by other methods than iterative processes and provides the possibility of magnifying every detail of the constructed fractal.

Fractamina:
A link to an iteractive program which draws the Sierpinski Gasket by playing dice.

There is a great deal of information about fractals available through a net search.

- by Genevieve Cance -

## THE MATHEMATICAL THEMES

• Borromean Rings - what they are and why they don't exist!
• The Möbius Band - what one looks like, experiments to try, and a beautiful rotating golden one enabling you to see what one really looks like in 3D (this is optional as 90Kb).
• Bernard Morin and the Brehm Model - how Bernard Morin showed John Robinson the Brehm Model of the Möbius Band and how to make one!
• The Projective Plane - how to create and understand the projective plane when it is not possible to physically construct (this relates to the Brehm Model in the Projective Plane). Also the Dirac String Trick.
• Fibre Bundles - what they are, how to make them, and examples of them in John Robinson's Work.
• Knots and Links - Introduction to the subject of Knot Theory, includes history of the subject, and a rotating mathematically constructed Immortality (the trefoil being one of the most basic knots).
• Torus Knots - two pages explaining the basics about torus knots with the help of excellent colourful graphics. There are also 3D moving images of John Robinson's sculptures of the Gordian Knot and the Rhythm of Life.

• Exhibition "Mathematics and Knots"
• Ronnie Brown's Homepage
• John Robinson's Symbolic Sculpture
• Brochure giving a guided Tour around the Symbolic Sculptures at the University of Wales, Bangor.
• About the Centre for the Popularisation of Mathematics