FRACTAL FRAGMENTS

‘Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightening travel in a straight line.”
Benoit Mandelbrot

The resulting forms often exhibit an enigmatic ambiguity, hovering between two adjacent dimensions. Taken to its infinite conclusion the Sierpinski Carpet pictured here dissolves into a foam whose final structure has no area whatever yet possesses a perimeter that is infinitely long. Like the skeleton of a beast whose flesh has vanished, the concluding form is without substance – it occupies a planar surface, but no longer fills it.

This filamentous remnant of a once solid polygon now hovers between a line and a plane: Where a line has one dimension and a plane has two, the Sierpinski carpet has a “fractional” dimension of 1.89. All fractal structures, including coastlines and clouds, possess a fractional dimension, which in the case of the west coast of Britain has been measured at 1.25. Nature is not alone in finding useful applications for fractal forms, fractal antennae based on the Sierpinski Carpet can replace the usual dendritic stalks that clutter our skylines, gathering electromagnetic signals more efficiently and with a fewer number of elements.

In the1920’s a young Austrian named Karl Menger extended the work begun by his mathematical predecessor Sierpinski. Menger attended a course of lectures by Professor Hans Hahn at the University of Vienna entitled What’s New Concerning the Concept of the Curve; under Hahn’s encouragement he embarked on an exploration of the concept of dimension that him to an expanded definition of this seemingly obvious term. Several years later Menger reported his discovery of a three-dimensional version of Sierpinski’s Carpet, which came to be known as the Menger Sponge. Where the Carpet is poised between a line and a plane, the Sponge hovers of the boundary of the plane and the solid - its fractional dimension is 2.73.

Though it manifestly occupies a volumetric space, the Menger sponge is essentially a linear object – it possesses a topological dimension of 1. Menger proved that it is indeed the universal curve - that is, any possible one-dimensional curve is mathematically identical to some part of its infinitely complex internal morphology. Though the classical Menger sponge is constructed in three-dimensions, it can be embodied in any number of higher dimensions; consequently any geometry of loop quantum gravity can be embedded in a Menger Sponge. Interestingly then, the structure of spacetime may be allied with this foam-like form.

A mathematician with an innate interest in form and structure, Menger contributed to many branches of geometry, including probabilistic and hyperbolic geometry. After the end of World War II, however, the new Austrian regime saw little need for such talents and in 1948 Menger accepted a position at the Illinois Institute of Technology where he was to remain for the rest of his life. In a reminiscence on the Vienna Circle and the mathematical colloquium of which Karl Menger was such an integral part, this agile mathematician was described as follows:

He had a great love of music…. He built up a notable collection of decorative tiles from all over the world …. He ate meat sparingly, particularly in his last years. But he was always glad to sample cuisines, from Cuban to Ethiopian, that were new to him. He liked baked apples.

In 1995, Dr Jeannine Mosely set out to construct a level three Menger Sponge from business cards. After 9 years of effort involving hundreds of folders all over America, the Business Card Sponge was completed. In August/September 2006 the Institute For Figuring curated an exhibition of the Business Card Sponge at Machine Project [Link] in Los Angeles.

Dr. Jeannine Mosely is one of the pioneers in the emerging field of computational origami, a branch of mathematics that explores the formal properties and potentialities of folded paper. An expert on the subfields of business card origami and minimalist origami (in which the practitioner is limited to only four folds), she also conducts research on curved crease origami, investigating the forms that can result from non-linear foldings. Dr Moseley was trained as an electrical engineer at MIT and works in the computer graphics industry writing three-dimensional modeling software.