Karl August Reinhardt | |
---|---|
Born | January 27, 1895 |
Died | April 27, 1941 46) | (aged
Nationality | German |
Education | Goethe Univ. Frankfurt |
Known for | |
Scientific career | |
Fields | Geometry |
Institutions | |
Thesis | Über die Zerlegung der Ebene in Polygone (1918) |
Doctoral advisor | Ludwig Bieberbach |
Karl August Reinhardt (27 January 1895 – 27 April 1941) was a German mathematician whose research concerned geometry, including polygons and tessellations. He solved one of the parts of Hilbert's eighteenth problem, and is the namesake of the Reinhardt domains in several complex variables, and Reinhardt polygons and the Reinhardt conjecture on packing density.
Reinhardt was born on January 27, 1895, in Frankfurt, 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 in 1913 before his studies were interrupted by World War I. During the war, he became a soldier, a high school teacher, and an assistant to mathematician David Hilbert at the University of Göttingen. [1] [2]
Reinhardt completed his Ph.D. at Goethe University Frankfurt in 1918. His dissertation, Über die Zerlegung der Ebene in Polygone, concerned tessellations of the plane, and was supervised by Ludwig Bieberbach. [1] [3] He began working as a secondary school teacher while working on his habilitation with Bieberbach, which he completed in 1921; titled Über Abbildungen durch analytische Funktionen zweier Veränderlicher, it concerned several complex variables. [1] [2]
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 as an extraordinary professor, under the leadership of Johann Radon; 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. [1] [2]
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. [1] [2]
In his doctoral dissertation, Reinhardt discovered the five tile-transitive pentagon tilings. [2] In a 1922 paper, Extremale Polygone gegebenen Durchmessers, he solved the odd case of the biggest little polygon problem, [4] and found the Reinhardt polygons, equilateral polygons inscribed in Reuleaux polygons that solve several related optimization problems. [5] [6]
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 tessellation 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 [7] 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. [8]
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 and conjectured that, among all centrally-symmetric convex shapes in the plane, it is the one with the lowest maximum packing density. Although the packing density of this shape is worse than the density of circle packings, Reinhardt's conjecture that it is the worst possible remains unsolved. [9]
Reinhardt also published a textbook, Methodische Einfuhrung in die Hohere Mathematik (1934). In it he presented calculus 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. [2]
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 of geometries.
The Kepler conjecture, named after the 17th-century mathematician and astronomer Johannes Kepler, is a mathematical theorem about sphere packing in three-dimensional Euclidean space. It states that no arrangement of equally sized spheres filling space has a greater average density than that of the cubic close packing and hexagonal close packing arrangements. The density of these arrangements is around 74.05%.
Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles, spheres, polygons, and so forth. The subject focuses on the combinatorial properties of these objects, such as how they intersect one another, or how they may be arranged to cover a larger object.
Ludwig Georg Elias Moses Bieberbach was a German mathematician and leading representative of National Socialist German mathematics.
Hugo Hadwiger was a Swiss mathematician, known for his work in geometry, combinatorics, and cryptography.
Euclidean plane tilings by convex regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Kepler in his Harmonices Mundi.
Hilbert's eighteenth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by mathematician David Hilbert. It asks three separate questions about lattices and sphere packing in Euclidean space.
László Fejes Tóth was a Hungarian mathematician who specialized in geometry. He proved that a lattice pattern is the most efficient way to pack centrally symmetric convex sets on the Euclidean plane. He also investigated the sphere packing problem. He was the first to show, in 1953, that proof of the Kepler conjecture can be reduced to a finite case analysis and, later, that the problem might be solved using a computer.
In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which exactly three hexagons meet at each vertex. It has Schläfli symbol of {6,3} or t{3,6} .
In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane. It has Schläfli symbol of {4,4}, meaning it has 4 squares around every vertex. Conway called it a quadrille.
In geometry, a Cairo pentagonal tiling is a tessellation of the Euclidean plane by congruent convex pentagons, formed by overlaying two tessellations of the plane by hexagons and named for its use as a paving design in Cairo. It is also called MacMahon's net after Percy Alexander MacMahon, who depicted it in his 1921 publication New Mathematical Pastimes. John Horton Conway called it a 4-fold pentille.
In geometry, a pentagonal tiling is a tiling of the plane where each individual piece is in the shape of a pentagon.
In geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions. Its dimension can be clarified as n-honeycomb for a honeycomb of n-dimensional space.
Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. Geometry is one of the oldest mathematical sciences.
In geometry, a shape is said to be anisohedral if it admits a tiling, but no such tiling is isohedral (tile-transitive); that is, in any tiling by that shape there are two tiles that are not equivalent under any symmetry of the tiling. A tiling by an anisohedral tile is referred to as an anisohedral tiling.
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 Kurt Mahler in 1947. It is constructed by replacing the corners of a regular octagon with a section of a hyperbola that is tangent to the two sides adjacent to the corner and asymptotic to the sides adjacent to these.
A set of prototiles is aperiodic if copies of the prototiles can be assembled to create tilings, such that all possible tessellation patterns are non-periodic. The aperiodicity referred to is a property of the particular set of prototiles; the various resulting tilings themselves are just non-periodic.
The Voderberg tiling is a mathematical spiral tiling, invented in 1936 by mathematician Heinz Voderberg (1911-1945). Karl August Reinhardt asked the question of whether there is a tile such that two copies can completely enclose a third copy. Voderberg, his student, answered in the affirmative with Form eines Neunecks eine Lösung zu einem Problem von Reinhardt ["On a nonagon as a solution to a problem of Reinhardt"].