Monster curve

The mathematicians of the late 19th and early 20th centuries called the monster curve or teragon (from Greek : teras = dragon, monster) the geometrical curves with extremely strange properties that were discovered at that time.

Examples

Examples of monster curves are:

• The Koch curve , introduced in 1904, is constant everywhere , but nowhere differentiable .
• The Hilbert curve and the Peano curve consist entirely of one-dimensional lines, but fill a two-dimensional area. They are therefore called space-filling curves . Like the Koch curve, both are constant everywhere, but nowhere differentiable.

construction

The monster curves are mainly created by repeated geometric replacement systems: An initial line, the so-called initiator, is replaced by another geometric figure, also called a generator. The resulting new routes can now be seen as initiators and replaced by generators, and this process, if repeated an infinite number of times, leads to curves with the aforementioned strange properties.

Many of these curves can also be generated by Lindenmayer systems .

meaning

Since these properties seemed so strange to mathematicians, these curves were relegated to the realm of mathematical curiosities and not dealt with any further. Only gradually did they delve into the questions they raised, such as the problem of dimensions . These questions often led to decisive advances in mathematics.

Most of the monster curves are fractals .