Hugo Hadwiger was a Swiss mathematician

Mathematician

A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

, known for his work in geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, combinatorics

Combinatorics

Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...

, and cryptography

Cryptography

Cryptography is the practice and study of techniques for secure communication in the presence of third parties...

.

Biography

Although born in Karlsruhe, Germany, Hadwiger grew up in Bern, Switzerland. He did his undergraduate studies at the University of Bern, where he majored in mathematics but also studied physics and actuarial science

Actuarial science

Actuarial science is the discipline that applies mathematical and statistical methods to assess risk in the insurance and finance industries. Actuaries are professionals who are qualified in this field through education and experience...

. 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

Hadwiger's theorem

In integral geometry , Hadwiger's theorem characterises the valuations on convex bodies in Rn. It was proved by Hugo Hadwiger.-Valuations:...

 in integral geometry

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 transformations from the space of functions on one geometrical space to the...

 classifies the possible isotropic measures

Measure (mathematics)

In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...

 on compact convex set

Convex set

In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object...

s in d-dimensional Euclidean space. According to this theorem, any such measure can be expressed as a linear combination of d + 1 fundamental measures; for instance, in two dimensions, there are three possible measures of this type, one corresponding to area

Area

Area is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat...

, a second corresponding to perimeter

Perimeter

A perimeter is a path that surrounds an area. The word comes from the Greek peri and meter . The term may be used either for the path or its length - it can be thought of as the length of the outline of a shape. The perimeter of a circular area is called circumference.- Practical uses :Calculating...

, and a third corresponding to the Euler characteristic

Euler characteristic

In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent...

.

The Hadwiger–Finsler inequality

Hadwiger–Finsler inequality

In mathematics, the Hadwiger–Finsler inequality is a result on the geometry of triangles in the Euclidean plane, named after the mathematicians Hugo Hadwiger and Paul Finsler...

, proven by Hadwiger with Paul Finsler

Paul Finsler

Paul Finsler was a German and Swiss mathematician.Finsler did his undergraduate studies at the Technische Hochschule Stuttgart, and his graduate studies at the University of Göttingen, where he received his Ph.D. in 1919 under the supervision of Constantin Carathéodory...

, is an inequality relating the side lengths and area of any triangle

Triangle

A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....

 in the Euclidean plane. It generalizes Weitzenböck's inequality

Weitzenböck's inequality

In mathematics, Weitzenböck's inequality states that for a triangle of side lengths a, b, c, and area \Delta, the following inequality holds:Equality occurs if and only if the triangle is equilateral...

 and was generalized in turn by Pedoe's inequality

Pedoe's inequality

In geometry, Pedoe's inequality, named after Daniel Pedoe, states that if a, b, and c are the lengths of the sides of a triangle with area ƒ, and A, B, and C are the lengths of the sides of a triangle with area F, then...

.

Hadwiger's name is also associated with several important unsolved problems in mathematics:

• The Hadwiger conjecture in graph theory
  Hadwiger conjecture (graph theory)
  In graph theory, the Hadwiger conjecture states that, if all proper colorings of an undirected graph G use k or more colors, then one can find k disjoint connected subgraphs of G such that each subgraph is connected by an edge to each other subgraph...
  
  , posed by Hadwiger in 1943 and called by “one of the deepest unsolved problems in graph theory,” describes a conjectured connection between graph coloring
  Graph coloring
  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 of a graph such that no two adjacent vertices share the...
  
   and graph minors. The Hadwiger number of a graph is the number of vertices in the largest clique
  Clique (graph theory)
  In the mathematical area of graph theory, a clique in an undirected graph is a subset of its vertices such that every two vertices in the subset are connected by an edge. Cliques are one of the basic concepts of graph theory and are used in many other mathematical problems and constructions on graphs...
  
   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.
• The Hadwiger conjecture in combinatorial geometry
  Hadwiger conjecture (combinatorial geometry)
  In combinatorial geometry, the Hadwiger conjecture states that any convex body in n-dimensional Euclidean space can be covered by 2n or fewer smaller bodies homothetic with the original body, and that furthermore, the upper bound of 2n is necessary iff the body is a parallelpiped...
  
   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
  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 one from each other have the same color. The answer is unknown, but has been narrowed down to...
  
   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 E. Nelson in 1950; Hadwiger popularized it by including it in a problem collection in 1961. However, in 1945, Hadwiger had published a closely 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 star

Eutactic star

In Euclidean geometry, a eutactic star is a geometrical figure in a Euclidean space. A star is a figure consisting of any number of opposing pairs of vectors issuing from a central origin. A star is eutactic if it is the orthogonal projection of a cross-polytope from a higher-dimensional space...

s, systems of points in Euclidean space formed by orthogonal projection of higher-dimensional cross polytopes. He found a higher-dimensional generalization of the space-filling Hill tetrahedra

Hill tetrahedron

In geometry, the Hill tetrahedra are a family of space-filling tetrahedra. They were discovered in 1896 by M.J.M. Hill, a professor of mathematics at the University College London, who showed that they are scissor-congruent to a cube.- Construction :...

. And his 1957 book Vorlesungen über Inhalt, Oberfläche und Isoperimetrie was foundational for the theory of Minkowski functional

Minkowski functional

In functional analysis, given a linear space X, a Minkowski functional is a device that uses the linear structure to introduce a topology on X.-Example 1:...

s, used in mathematical morphology

Mathematical morphology

Mathematical morphology is a theory and technique for the analysis and processing of geometrical structures, based on set theory, lattice theory, topology, and random functions...

.

Cryptographic work

Hadwiger was one of the principal developers of a Swiss rotor machine

Rotor machine

In cryptography, a rotor machine is an electro-mechanical device used for encrypting and decrypting secret messages. Rotor machines were the cryptographic state-of-the-art for a prominent period of history; they were in widespread use in the 1920s–1970s...

 for encrypting military communications, known as NEMA. 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

Asteroid

Asteroids are a class of small Solar System bodies in orbit around the Sun. They have also been called planetoids, especially the larger ones...

 2151 Hadwiger

2151 Hadwiger

2151 Hadwiger is a main-belt asteroid discovered on November 3, 1977 by Wild, P. at Zimmerwald. It is named after Swiss mathematician Hugo Hadwiger.- External links :*...

, discovered in 1977 by Paul Wild

Paul Wild (Swiss astronomer)

Professor Paul Wild of Berne, Switzerland, is an astronomer who discovered numerous comets and asteroids.- Career :Professor Wild was director of the Astronomical Institute of the University of Berne from 1980 to 1991...

, is named after Hadwiger.

The first article in the "Research Problems" section of the American Mathematical Monthly

American Mathematical Monthly

The American Mathematical Monthly is a mathematical journal founded by Benjamin Finkel in 1894. It is currently published 10 times each year by the Mathematical Association of America....

was dedicated by Victor Klee

Victor Klee

Victor L. Klee, Jr. 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 Washington in Seattle.Born in San Francisco, Vic Klee earned his B.A...

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.