Hugo Hadwiger (23 December 1908 in Karlsruhe, Germany – 29 October 1981 in

Bern, Switzerland Bern () or Berne; in other Swiss languages, gsw, Bärn ; frp, Bèrna ; it, Berna ; rm, Berna is the ''de facto'' capital of Switzerland, referred to as the " federal city" (in german: Bundesstadt, link=no, french: ville fédérale, link=no, i ...

) was a

Swiss Swiss may refer to: * the adjectival form of Switzerland * Swiss people Places * Swiss, Missouri * Swiss, North Carolina * Swiss, West Virginia *Swiss, Wisconsin Other uses * Swiss-system tournament, in various games and sports *Swiss Internatio ...

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 ...

, 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 c ...

combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ap ...

, 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 adve ...

Biography

Although born in Karlsruhe, Germany, 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 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) transformatio ...

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 Briti ...

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 r ...

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 open ...

, 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 prac ...

, and the

Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space' ...

. 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 In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...

. 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 o ...

and was generalized in turn by

Pedoe's inequality In geometry, Pedoe's inequality (also Neuberg–Pedoe inequality), named after Daniel Pedoe (1910–1998) and Joseph Jean Baptiste Neuberg (1840–1926), states that if ''a'', ''b'', and ''c'' are the lengths of the sides of a triangle with area '' ...

. In the same 1937 paper in which Hadwiger and Finsler published this inequality, they also published the

Finsler–Hadwiger theorem The Finsler–Hadwiger theorem is statement in Euclidean plane geometry that describes a third square derived from any two squares that share a vertex. The theorem is named after Paul Finsler and Hugo Hadwiger, who published it in 1937 as part of ...

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 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 o ...

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 i ...

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 popul ...

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 o ...

. *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 Edward Nelson (May 4, 1932 – September 10, 2014) was an American mathematician. He was professor in the Mathematics Department at Princeton University. He was known for his work on mathematical physics and mathematical logic. In mathematical ...

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 ...

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 functional In mathematics, in the field of functional analysis, a Minkowski functional (after Hermann Minkowski) or gauge function is a function that recovers a notion of distance on a linear space. If K is a subset of a real or complex vector space X, th ...

s, 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 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 An asteroid is a minor planet of the inner Solar System. Sizes and shapes of asteroids vary significantly, ranging from 1-meter rocks to a dwarf planet almost 1000 km in diameter; they are rocky, metallic or icy bodies with no atmosphere ...

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 ''The American Mathematical Monthly'' is a mathematical journal founded by Benjamin Finkel in 1894. It is published ten times each year by Taylor & Francis for the Mathematical Association of America. The ''American Mathematical Monthly'' is an ...

'' was dedicated by Victor Klee 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