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.* |