**Karl August Reinhardt** was a German mathematician who worked on various tiling problems He gave a partial solution to Hilbert's eighteenth problem.

- Reinhardt attended school in Frankfurt am Main, studying at the Gymnasium there.
- This caused severe disruption of Reinhardt's university education; he fought briefly in the war, served for a while as an assistant to David Hilbert at Göttingen, and spent some time teaching in secondary schools as part of his war service.
- Not only was this important in developing Reinhardt's character, but their discussions also encouraged Reinhardt to look at various problems that Hilbert deemed the most important.
- Reinhardt worked on these problems for his doctorate, advised by Bieberbach.
- In his thesis Reinhardt found five convex pentagons that tile the plane in such a way that the automorphism group acts transitively on the tiles.
- Bieberbach and Süss had both been at the University of Frankfurt up to 1921, and Bieberbach had assisted Reinhardt with his habilitation thesis but, shortly after Reinhardt was appointed, Bieberbach moved to take up the Chair of Geometry at the University of Berlin and Süss went with him as his assistant.
- At this stage Reinhardt was teaching both at secondary level and also at the University of Frankfurt, something which he struggled to do due to the poor state of his health.
- In 1924 Reinhardt left Frankfurt when he was appointed as an extraordinary professor of Pure and Applied Mathematics at the Ernst-Moritz-Arndt University of Greifswald.
- Over the next few years, Reinhardt undertook research on tiling 2-dimensional space and also tilings in higher dimensional space.
- It was in the last of these papers that Reinhardt completed the solution of Hilbert's Eighteenth Problem by finding a polyhedron which, although it is not the fundamental region of any space group, tiles 3-dimensional Euclidean space.
- Reinhardt was made an ordinary professor at Greifswald in 1928.
- Reinhardt told Süss that if he submitted his papers to Greifswald he could habilitate there and obtain a position.
- Süss accepted Reinhardt's suggestion and returned from Japan in 1928 to take up the lecturing post at Greifswald.
- Reinhardt supervised several doctoral students at the Ernst-Moritz-Arndt University of Greifswald, perhaps the best known being Theodor Schmidt who submitted his thesis Über die Zerlegung des n-dimensionalen Raumes in gitterförmig angeordnete Würfel Ⓣ(On the decomposition of n-dimensional space into cubes arranged in a lattice) in 1933.
- Among Reinhardt's other doctoral students we mention Heinrich Engelhardt who submitted the thesis Über die Zerlegung der euklidischen Ebene und des euklidischen Raums in kongruente Bereiche Ⓣ(On the decomposition of Euclidean plane and Euclidean space into congruent areas) in 1933, and Heinrich Voderberg who also studied tiling problems in Form eines Neunecks eine Lösung zu einem Problem von Reinhardt Ⓣ(On a nonagon as a solution to a problem of Reinhardt) (1934).
- In addition to his innovative research career, Reinhardt was interested in teaching, particularly looking at ways to make mathematics more understandable to students at the beginning of their university careers.
- On account of this belief Professor Reinhardt has presented us with a book in which integral calculus is developed first, completely independent of differential calculus; in fact the latter appears simply as the inverse operation of the former.
- By this stage in his career, Reinhardt was in an excellent position with an outstanding research record and a reputation as a fine, thoughtful teacher.
- However, Reinhardt's health had always been delicate and this prevented his continuing a successful career.

Born 27 January 1895, Frankfurt am Main, Germany. Died 27 April 1941, Berlin, Germany.

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Origin Germany

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive