Koch snowflake, a fractal that begins with an equilateral triangle and then replaces the middle third of every line segment with a pair of line segments that form an equilateral " bump "

::* 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 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.

The now-defunct Robert Koch Hospital at Koch, Missouri ( south of St. Louis, Missouri ), was also named in his honor.

Koch was also famous for the discovery of the tubercle bacillus ( 1882 ) and the cholera bacillus ( 1883 ) and for his development of Koch's postulates.

One way of proving that a given disease is " infectious ", is to satisfy Koch's postulates ( first proposed by Robert Koch ), which demands that the infectious agent be identified only in patients and not in healthy controls, and that patients who contract the agent also develop the disease.

Likewise, the German Heckler & Koch MP5K ( a weapon small enough to be concealed on one's person or in a briefcase ), is also classed as a compact submachine gun.

They also sponsored a letter-writing campaign to Mayor Lindsay in which the Greenwich Village Democratic Party and Congressman Ed Koch sent pleas to end raids on gay bars in the city.

In 1994 James Sedares conducted a re-recording of the score performed by The Phoenix Symphony Orchestra ( which also included a suite from Bernstein's score for The Hallelujah Trail, issued by Koch Records ; Bernstein himself conducted the Royal Scottish National Orchestra for a performance released by RCA in 1997, but the original film soundtrack was not released until the following year by Rykodisc ( Varèse Sarabande reissued this album in 2004 ).

* September 1 – Ilse Koch, also known as the " Witch of Buchenwald ", commits suicide in the Bavarian prison of Aichach.

While Koch Bihar Kingdom in the North, flourished during the period of 16th and the 17th centuries as well as weathered the Mughals also and survived till the advent of the British.

Further influence of A. M. Cassandre and Rudolf Koch, whose work greatly inspired the young Zapf, can also be seen in Optima.

* Carl Koch ( director ) ( 1892 – 1963 ), also spelled Karl Koch, German film director, writer

Koch Legacy has also released Strauss's recordings of overtures by Gluck, Carl Maria von Weber, Peter Cornelius, and Wagner.

Karl-Otto Koch ( August 2, 1897 – April 5, 1945 ), a Standartenführer ( Colonel ) in the German Schutzstaffel ( SS ), was the first commandant of the Nazi concentration camps at Buchenwald and Sachsenhausen, and later also served as a commander at the Majdanek concentration camp.

Following World War I, Koch worked as an accounting supervisor in a bank and later also in the same role in an insurance company.

He had also helped treat Koch for syphilis.

Exhibit # 254, also introduced by Dodd, was a shrunken head, allegedly used by Ilse Koch as a paperweight, which Dodd claimed was the head of a Polish prisoner at Buchenwald.

The darts of German ( Heckler & Koch P11 ) and Russian underwater firearms, and other similar weapons are also supercavitating.

It is also known as the William A. Koch Memorial Highway, named for the local developer who pushed much of the town's development, as assisted the creation of nearby Lincoln Boyhood National Memorial.

Wichita was also a significant entrepreneurial business center during the pre and post-war period, with Coleman, Mentholatum, Pizza Hut, White Castle, Taco Tico and Koch Industries having all been founded in Wichita.

Quincy is also home to the United States ' longest running Flag Day parade, a tradition that began in 1952 under Richard Koch, a former director of Parks and Recreation, who started the " Koch Club " sports organization for kids and had an annual parade with flags.

V. cholerae was first isolated as the cause of cholera by Italian anatomist Filippo Pacini in 1854, but his discovery was not widely known until Robert Koch, working independently 30 years later, publicized the knowledge and the means of fighting the disease.

Walter Karl Koch ( 3 May 1880 in Dortmund, Germany-1962 ) was a German surgeon best known for the discovery of Koch's triangle, a triangular shaped area in the right atrium of the heart, and of Tawara's node, the atrioventricular node which is the beginning of the auricular-ventricular bundle.

Koch became known for his work on the motor centres of the human heart, and coined the term " sinus node ".

Koch is perhaps best known as assistant to Jean Renoir, who helped get Koch and Reiniger exit visas from Germany in 1936.

He sold thirty or forty lots near his home in west Bloomington, and this neighborhood soon became known as Kochsville, giving Koch the honor of having two McLean County places named in his honor.

In 1940, Koch joined new partners to create a new firm, the Wood River Oil and Refining Company, which is today known as Koch Industries.

* Philip Johnson New York State Theater, now known as the David H. Koch Theater

In Canada, Koch Entertainment ( now known as ( Entertainment One ) released Canadian seasons 1 and 3 ( which correspond to US Seasons 1-3 and 5, respectively ) on DVD in 2004.

M. tuberculosis, then known as the " tubercle bacillus ", was first described on 24 March 1882 by Robert Koch, who subsequently received the Nobel Prize in physiology or medicine for this discovery in 1905 ; the bacterium is also known as " Koch's bacillus ".

* Koch Entertainment LP, now known as E1 Entertainment

* Koch Records, now known as E1 Music

She was one of the first members of Congress to support gay rights, introducing the first federal gay rights bill, known as the Equality Act of 1974, with fellow Democratic New York City Representative, Ed Koch, a future mayor of New York City.

Ilse Koch, known as " the witch of Buchenwald ", was the chief female guard at the Buchenwald camp, and at the same time married to the camp commandant, Karl Koch.

Familiarly known to all as Max, Koch is a retired policeman who saved Moser from a life of crime and became something of a mentor to him.