::* Exact self-similarity: identical at all scales ; e. g. Koch snowflake

::* Statistical self-similarity: repeats a pattern stochastically so numerical or statistical measures are preserved across scales ; e. g., randomly generated fractals ; the well-known example of the coastline of Britain, for which one would not expect to find a segment scaled and repeated as neatly as the repeated unit that defines, for example, the Koch snowflake

:* Iterated function systems – use fixed geometric replacement rules ; may be stochastic or deterministic ; e. g., Koch snowflake, Cantor set, Sierpinski carpet, Sierpinski gasket, Peano curve, Harter-Heighway dragon curve, T-Square, Menger sponge

For instance, a side of the Koch snowflake is both symmetrical and scale-invariant ; it can be continually magnified 3x without changing shape.

Moore compares the multitude of increasingly outlandish Ripper theories to a Koch snowflake, where a finite, fixed location, event and era ( London, in late 1888 ) can have an infinite number of nooks and crannies.

The first four iteration s of the Koch snowflake

The Koch snowflake ( also known as the Koch star and Koch island ) is a mathematical curve and one of the earliest fractal curves to have been described.

The Koch snowflake can be constructed by starting with an equilateral triangle, then recursively altering each line segment as follows:

The Koch snowflake is the limit approached as the above steps are followed over and over again.

In other words, three Koch curves make a Koch snowflake.

So the area of a Koch snowflake is 8 / 5 of the area of the original triangle, or.

Tesselation by two sizes of Koch snowflake

Since each Koch snowflake in the tesselation can be subdivided into seven smaller snowflakes of two different sizes, it is also possible to find tessellations that use more than two sizes at once.

the resulting curve converges to the Koch snowflake.

The image of simple plane curve can have Hausdorff dimension bigger than one ( see Koch snowflake ) and even positive Lebesgue measure ( the last example can be obtained by small variation of the Peano curve construction ).

Niels Fabian Helge von Koch ( Stockholm, January 25, 1870 – ibidem, March 11, 1924 ) was a Swedish mathematician who gave his name to the famous fractal known as the Koch snowflake, one of the earliest fractal curves to be described.

* Koch snowflake

One non-trivial example is the fractal dimension of a Koch snowflake.

* Koch snowflake

* Helge von Koch describes the " Koch snowflake ", one of the earliest fractal curves described.

* Tune Smithy, developed by Robert Walker, for Windows generates music real time using a musical construction similar to the Koch snowflake fractal.

In mathematics, the Lévy C curve is a self-similar fractal that was first described and whose differentiability properties were analysed by Ernesto Cesàro in 1906 and G. Farber in 1910, but now bears the name of French mathematician Paul Pierre Lévy, who was the first to describe its self-similarity properties, as well as to provide a geometrical construction showing it as a representative curve in the same class as the Koch curve.

* Julius Ludwig August Koch begins publication of Die psychopathischen Minderwertigkeiten in Ravensburg, introducing the concept of psychopathology.

In 1883, Robert Koch identified V. cholerae with a microscope as the bacillus causing the disease.

One of the next milestones came in 1904, when Helge von Koch, extending ideas of Poincaré and dissatisfied with Weierstrass's abstract and analytic definition, gave a more geometric definition including hand drawn images of a similar function, which is now called the Koch curve ( see Figure 2 ).

Koch published his findings in 1876, and was rewarded with a job at the Imperial Health Office in Berlin in 1880.

In 1883, Koch worked with a French research team in Alexandria, Egypt, studying cholera.

Because the construction efforts on Veliki Brijun Island were jeopardized by malaria outbreaks which occurred during summer months, and even Austrian steel industrialist Paul Kupelweiser himself fell ill with the disease, at the turn of the century Kupelweiser invited in Koch, who at the time was studying different forms of malaria and quinine-based treatments.

Beside this, Pasteur, along with Robert Koch ( who was awarded the Nobel Prize in 1905 ), founded bacteriology.

He is regarded as one of the three main founders of microbiology, together with Ferdinand Cohn and Robert Koch.

Today, he is often regarded as the father of germ theory and bacteriology, together with Robert Koch.

Hess negotiated with New York mayor Ed Koch.

In 1970, a manifesto authored by Meinhof used the name " RAF " and the red star logo with a Heckler & Koch MP5 submachine gun for the first time.

Variants of the Sten and modern-day Heckler & Koch MP5 have been manufactured with integral suppressors, and such weapons are on occasion used by special forces and police units.

European painters who worked with tempera include Giorgio de Chirico, Otto Dix, Eliot Hodgkin, and Pyke Koch ;.

** Dutch painter Han van Meegeren is arrested for collaboration with the Nazis, but the paintings he had sold to Hermann Göring ( Koch ) are later found to be his fakes.

When the lift started returning to the ground floor, ASIS officers emerged wearing masks and openly brandishing 9mm Browning pistols and Heckler & Koch MP5 submachine guns, two of them with silencers.

The Koch curve originally described by Koch is constructed with only one of the three sides of the original triangle.

Koch had a zoo built by the prisoners in the camp, with a bear pit ( Bärenzwinger ) facing the Appellplatz, the assembly square where prisoner " roll-calls " were conducted.

* New York State Theater ( renamed David H. Koch Theater ) at Lincoln Center, New York City, New York ( with Richard Foster ; 1964 )

* 2004: Stauffenberg, by Jo Baier, with Sebastian Koch as Stauffenberg

Weir contributed to the screenplay along with its original author Christopher Koch, and playwright David Williamson.

In 1948, a film version was produced with a screenplay adaptation by Howard Koch.