Karl August Reinhardt (27 January 1895 – 27 April 1941) was a German mathematician whose research concerned geometry, including

polygon In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...

s and

tessellation A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety ...

s. He solved one of the parts of Hilbert's eighteenth problem, and is the namesake of the

Reinhardt domain The theory of functions of several complex variables is the branch of mathematics dealing with functions defined on #The complex coordinate space, the complex coordinate space \mathbb C^n, that is, -tuples of complex numbers. The name of the fiel ...

s in

several complex variables The theory of functions of several complex variables is the branch of mathematics dealing with functions defined on the complex coordinate space \mathbb C^n, that is, -tuples of complex numbers. The name of the field dealing with the properties ...

, and Reinhardt polygons and the Reinhardt conjecture on packing density.

Life

Reinhardt was born on January 27, 1895, in

Frankfurt Frankfurt am Main () is the most populous city in the States of Germany, German state of Hesse. Its 773,068 inhabitants as of 2022 make it the List of cities in Germany by population, fifth-most populous city in Germany. Located in the forela ...

, the descendant of farming stock. One of his childhood friends was mathematician Wilhelm Süss. After studying at the gymnasium there, he became a student at the

University of Marburg The Philipps University of Marburg () is a public research university located in Marburg, Germany. It was founded in 1527 by Philip I, Landgrave of Hesse, which makes it one of Germany's oldest universities and the oldest still operating Prote ...

in 1913 before his studies were interrupted by

World War I World War I or the First World War (28 July 1914 – 11 November 1918), also known as the Great War, was a World war, global conflict between two coalitions: the Allies of World War I, Allies (or Entente) and the Central Powers. Fighting to ...

. During the war, he became a soldier, a high school teacher, and an assistant to mathematician

David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and philosopher of mathematics and one of the most influential mathematicians of his time. Hilbert discovered and developed a broad range of fundamental idea ...

at the

University of Göttingen The University of Göttingen, officially the Georg August University of Göttingen (, commonly referred to as Georgia Augusta), is a Public university, public research university in the city of Göttingen, Lower Saxony, Germany. Founded in 1734 ...

. Reinhardt completed his Ph.D. at

Goethe University Frankfurt Goethe University Frankfurt () is a public research university located in Frankfurt am Main, Germany. It was founded in 1914 as a citizens' university, which means it was founded and funded by the wealthy and active liberal citizenry of Frankfurt ...

in 1918. His dissertation, ''Über die Zerlegung der Ebene in Polygone'', concerned

s of the plane, and was supervised by

Ludwig Bieberbach Ludwig Georg Elias Moses Bieberbach (; 4 December 1886 – 1 September 1982) was a German mathematician and leading representative of National Socialist German mathematics (" Deutsche Mathematik"). Biography Born in Goddelau, near Darmstadt, ...

. He began working as a secondary school teacher while working on his

habilitation Habilitation is the highest university degree, or the procedure by which it is achieved, in Germany, France, Italy, Poland and some other European and non-English-speaking countries. The candidate fulfills a university's set criteria of excelle ...

with Bieberbach, which he completed in 1921; titled ''Über Abbildungen durch analytische Funktionen zweier Veränderlicher'', it concerned

. Bieberbach moved to Berlin in 1921, taking Süss as an assistant. They left Reinhardt in Frankfurt, working two jobs as a high school teacher and junior faculty at the university. In 1924, Reinhardt moved to the

University of Greifswald The University of Greifswald (; ), formerly known as Ernst-Moritz-Arndt University of Greifswald, is a public research university located in Greifswald, Germany, in the state of Mecklenburg-Western Pomerania. Founded in 1456, it is one of th ...

as an extraordinary professor, under the leadership of

Johann Radon Johann Karl August Radon (; 16 December 1887 – 25 May 1956) was an Austrian mathematician. His doctoral dissertation was on the calculus of variations (in 1910, at the University of Vienna). Life RadonBrigitte Bukovics: ''Biography of Johan ...

; this gave him an income sufficient to support himself without a second job, and afforded him more time for research. He became an ordinary professor at Greifswald in 1928. He remained in Greifswald for the rest of his career, "with an outstanding research record and a reputation as a fine, thoughtful teacher". However, despite his now-comfortable position, his health was poor, and he died in Berlin on April 27, 1941, aged 46.

Contributions

In his doctoral dissertation, Reinhardt discovered the five tile-transitive pentagon tilings. In a 1922 paper, ''Extremale Polygone gegebenen Durchmessers'', he solved the odd case of the biggest little polygon problem, and found the Reinhardt polygons,

equilateral polygon In geometry, an equilateral polygon is a polygon which has all sides of the same length. Except in the triangle case, an equilateral polygon does not need to also be equiangular (have all angles equal), but if it does then it is a regular polygon ...

s inscribed in Reuleaux polygons that solve several related optimization problems. He had long been interested in Hilbert's eighteenth problem, a shared interest with Bieberbach, who in 1911 had solved a part of the problem asking for the classification of space groups. A second part of the problem asked for a

of Euclidean space by a tile that is not the fundamental region of any group. In a 1928 paper, ''Zur Zerlegung der euklidischen Räume in kongeuente Polytope'' Reinhardt solved this part by finding an example of such a tessellation. In a later development, Heinrich Heesch showed in 1935 that tilings with this property exist even in the two-dimensional

Euclidean plane In mathematics, a Euclidean plane is a Euclidean space of Two-dimensional space, dimension two, denoted \textbf^2 or \mathbb^2. It is a geometric space in which two real numbers are required to determine the position (geometry), position of eac ...

. Another of his works, ''Über die dichteste gitterförmige Lagerung kongruenter Bereiche in der Ebene und eine besondere Art konvexer Kurven'' from 1934, constructed the

smoothed octagon The smoothed octagon is a region in the plane found by Karl Reinhardt in 1934 and conjectured by him to have the ''lowest'' maximum packing density of the plane of all centrally symmetric convex shapes. It was also independently discovered by ...

and conjectured that, among all centrally-symmetric convex shapes in the plane, it is the one with the lowest maximum

packing density A packing density or packing fraction of a packing in some space is the fraction of the space filled by the figures making up the packing. In simplest terms, this is the ratio of the volume of bodies in a space to the volume of the space itself. ...

. Although the packing density of this shape is worse than the density of

circle packing In geometry, circle packing is the study of the arrangement of circles (of equal or varying sizes) on a given surface such that no overlapping occurs and so that no circle can be enlarged without creating an overlap. The associated ''packing den ...

s, Reinhardt's conjecture that it is the worst possible remains unsolved. Reinhardt also published a textbook, ''Methodische Einfuhrung in die Hohere Mathematik'' (1934). In it he presented

calculus Calculus is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. Originally called infinitesimal calculus or "the ...

in a format reversed from the usual presentation, with areas under curves (integrals) earlier than slopes of curves (derivatives), based on his theory that the material would be easier to learn in this order.

References

External links

* {{DEFAULTSORT:Reinhardt, Karl August 20th-century German mathematicians 1895 births 1941 deaths Goethe University Frankfurt alumni Academic staff of Goethe University Frankfurt Academic staff of the University of Greifswald