Hugo Hadwiger (23 December 1908 in

Karlsruhe, Germany Karlsruhe ( , , ; South Franconian: ''Kallsruh'') is the third-largest city of the German state (''Land'') of Baden-Württemberg after its capital of Stuttgart and Mannheim, and the 22nd-largest city in the nation, with 308,436 inhabitants. I ...

– 29 October 1981 in

Bern, Switzerland german: Berner(in)french: Bernois(e) it, bernese , neighboring_municipalities = Bremgarten bei Bern, Frauenkappelen, Ittigen, Kirchlindach, Köniz, Mühleberg, Muri bei Bern, Neuenegg, Ostermundigen, Wohlen bei Bern, Zollikofen , website ...

) was a Swiss

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

, known for his work in

geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

, combinatorics, and

cryptography Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or ''-logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of adver ...

Biography

Although born in

, Hadwiger grew up in

.. He did his undergraduate studies at the

University of Bern The University of Bern (german: Universität Bern, french: Université de Berne, la, Universitas Bernensis) is a university in the Swiss capital of Bern and was founded in 1834. It is regulated and financed by the Canton of Bern. It is a compreh ...

, where he majored in mathematics but also studied physics and actuarial science. He continued at Bern for his graduate studies, and received his Ph.D. in 1936 under the supervision of Willy Scherrer. He was for more than forty years a professor of mathematics at Bern.

Mathematical concepts named after Hadwiger

Hadwiger's theorem In integral geometry (otherwise called geometric probability theory), Hadwiger's theorem characterises the valuations on convex bodies in \R^n. It was proved by Hugo Hadwiger. Introduction Valuations Let \mathbb^n be the collection of all c ...

integral geometry In mathematics, integral geometry is the theory of measures on a geometrical space invariant under the symmetry group of that space. In more recent times, the meaning has been broadened to include a view of invariant (or equivariant) transformati ...

classifies the isometry-invariant valuations on

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...

convex set In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex ...

s in ''d''-dimensional Euclidean space. According to this theorem, any such valuation can be expressed as a linear combination of the intrinsic volumes; for instance, in two dimensions, the intrinsic volumes are the

area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an ope ...

, the

perimeter A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference. Calculating the perimeter has several pr ...

, and the Euler characteristic. The Hadwiger–Finsler inequality, proven by Hadwiger with Paul Finsler, is an inequality relating the side lengths and area of any

triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colline ...

in the Euclidean plane. It generalizes

Weitzenböck's inequality In mathematics, Weitzenböck's inequality, named after Roland Weitzenböck, states that for a triangle of side lengths a, b, c, and area \Delta, the following inequality holds: : a^2 + b^2 + c^2 \geq 4\sqrt\, \Delta. Equality occurs if and on ...

and was generalized in turn by Pedoe's inequality. In the same 1937 paper in which Hadwiger and Finsler published this inequality, they also published the Finsler–Hadwiger theorem on a square derived from two other squares that share a vertex. Hadwiger's name is also associated with several important unsolved problems in mathematics: *The Hadwiger conjecture in graph theory, posed by Hadwiger in 1943 and called by “one of the deepest unsolved problems in graph theory,” describes a conjectured connection between graph coloring and

graph minor In graph theory, an undirected graph is called a minor of the graph if can be formed from by deleting edges and vertices and by contracting edges. The theory of graph minors began with Wagner's theorem that a graph is planar if and only if ...

s. The

Hadwiger number In graph theory, the Hadwiger number of an undirected graph is the size of the largest complete graph that can be obtained by contracting edges of . Equivalently, the Hadwiger number is the largest number for which the complete graph is a ...

of a graph is the number of vertices in the largest

clique A clique ( AusE, CanE, or ), in the social sciences, is a group of individuals who interact with one another and share similar interests. Interacting with cliques is part of normative social development regardless of gender, ethnicity, or popular ...

that can be formed as a minor in the graph; the Hadwiger conjecture states that this is always at least as large as the

chromatic number In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices ...

. *The Hadwiger conjecture in combinatorial geometry concerns the minimum number of smaller copies of a convex body needed to cover the body, or equivalently the minimum number of light sources needed to illuminate the surface of the body; for instance, in three dimensions, it is known that any convex body can be illuminated by 16 light sources, but Hadwiger's conjecture implies that only eight light sources are always sufficient. *The Hadwiger–Kneser–Poulsen conjecture states that, if the centers of a system of balls in Euclidean space are moved closer together, then the volume of the union of the balls cannot increase. It has been proven in the plane, but remains open in higher dimensions. *The

Hadwiger–Nelson problem In geometric graph theory, the Hadwiger–Nelson problem, named after Hugo Hadwiger and Edward Nelson, asks for the minimum number of colors required to color the plane such that no two points at distance 1 from each other have the same color. ...

concerns the minimum number of colors needed to color the points of the Euclidean plane so that no two points at unit distance from each other are given the same color. It was first proposed by Edward Nelson in 1950. Hadwiger popularized it by including it in a problem collection in 1961; already in 1945 he had published a related result, showing that any cover of the plane by five congruent closed sets contains a unit distance in one of the sets.

Other mathematical contributions

Hadwiger proved a theorem characterizing eutactic stars, systems of points in Euclidean space formed by

orthogonal projection In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if it wer ...

of higher-dimensional

cross polytope In geometry, a cross-polytope, hyperoctahedron, orthoplex, or cocube is a regular, convex polytope that exists in ''n''- dimensional Euclidean space. A 2-dimensional cross-polytope is a square, a 3-dimensional cross-polytope is a regular octahed ...

s. He found a higher-dimensional generalization of the space-filling Hill tetrahedra. And his 1957 book ''Vorlesungen über Inhalt, Oberfläche und Isoperimetrie'' was foundational for the theory of Minkowski functionals, used in

mathematical morphology Mathematical morphology (MM) is a theory and technique for the analysis and processing of geometrical structures, based on set theory, lattice theory, topology, and random functions. MM is most commonly applied to digital images, but it can be empl ...

Cryptographic work

Hadwiger was one of the principal developers of a Swiss

rotor machine In cryptography, a rotor machine is an electro-mechanical stream cipher device used for encrypting and decrypting messages. Rotor machines were the cryptographic state-of-the-art for much of the 20th century; they were in widespread use in the 1 ...

for encrypting military communications, known as

NEMA The National Electrical Manufacturers Association (NEMA) is the largest trade association of electrical equipment manufacturers in the United States. Founded in 1926, it advocates for the industry, and publishes standards for electrical product ...

. The Swiss, fearing that the Germans and Allies could read messages transmitted on their Enigma cipher machines, enhanced the system by using ten rotors instead of five. The system was used by the Swiss army and air force between 1947 and 1992.

Awards and honors

Asteroid 2151 Hadwiger, discovered in 1977 by Paul Wild, is named after Hadwiger.. The first article in the "Research Problems" section of the '' American Mathematical Monthly'' was dedicated by

Victor Klee Victor LaRue Klee, Jr. (September 18, 1925 – August 17, 2007) was a mathematician specialising in convex sets, functional analysis, analysis of algorithms, optimization, and combinatorics. He spent almost his entire career at the University of ...

to Hadwiger, on the occasion of his 60th birthday, in honor of Hadwiger's work editing a column on unsolved problems in the journal ''Elemente der Mathematik''.

Selected works

Books

*''Altes und Neues über konvexe Körper'', Birkhäuser 1955 *''Vorlesungen über Inhalt, Oberfläche und Isoperimetrie'', Springer, Grundlehren der mathematischen Wissenschaften, 1957 *with H. Debrunner, V. Klee '' Combinatorial Geometry in the Plane'', Holt, Rinehart and Winston, New York 1964
Dover reprint 2015

Articles

*"Über eine Klassifikation der Streckenkomplexe", Vierteljahresschrift der Naturforschenden Gesellschaft Zürich, vol. 88, 1943, pp. 133–143 (Hadwiger's conjecture in graph theory)
with Paul Glur ''Zerlegungsgleichheit ebener Polygone, Elemente der Math, vol. 6, 1951, pp. 97-106''Ergänzungsgleichheit k-dimensionaler Polyeder'', Math. Zeitschrift, vol. 55, 1952, pp. 292-298''Lineare additive Polyederfunktionale und Zerlegungsgleichheit, Math. Z., vol. 58, 1953, pp. 4-14''''Zum Problem der Zerlegungsgleichheit k-dimensionaler Polyeder'', Mathematische Annalen vol. 127, 1954, pp. 170–174

References

{{DEFAULTSORT:Hadwiger, Hugo 1908 births 1981 deaths Modern cryptographers 20th-century Swiss mathematicians Scientists from Bern University of Bern alumni Combinatorialists Geometers German emigrants to Switzerland