In geometry, it is an unsolved

conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 19 ...

Hugo Hadwiger Hugo Hadwiger (23 December 1908 in Karlsruhe, Germany – 29 October 1981 in Bern, Switzerland) was a Swiss mathematician, known for his work in geometry, combinatorics, and cryptography. Biography Although born in Karlsruhe, Germany, Hadwige ...

that every

simplex In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...

can be

dissected Dissection (from Latin ' "to cut to pieces"; also called anatomization) is the dismembering of the body of a deceased animal or plant to study its anatomical structure. Autopsy is used in pathology and forensic medicine to determine the cause ...

into orthoschemes, using a number of orthoschemes bounded by a function of the dimension of the simplex. If true, then more generally every

convex polytope A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the wo ...

could be dissected into orthoschemes.

Definitions and statement

In this context, a simplex in

d

-dimensional

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...

is the

convex hull In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...

d+1

points that do not all lie in a common

hyperplane In geometry, a hyperplane is a subspace whose dimension is one less than that of its ''ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyper ...

. For example, a 2-dimensional simplex is just a

triangle A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), 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, an ...

(the convex hull of three points in the plane) and a 3-dimensional simplex is a

tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...

(the convex of four points in three-dimensional space). The points that form the simplex in this way are called its vertices. An orthoscheme, also called a path simplex, is a special kind of simplex. In it, the vertices can be connected by a

path A path is a route for physical travel – see Trail. Path or PATH may also refer to: Physical paths of different types * Bicycle path * Bridle path, used by people on horseback * Course (navigation), the intended path of a vehicle * Desire p ...

, such that every two edges in the path are at right angles to each other. A two-dimensional orthoscheme is a

right triangle A right triangle (American English) or right-angled triangle (British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one angle is a right an ...

. A three-dimensional orthoscheme can be constructed from a

cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only r ...

by finding a path of three edges of the cube that do not all lie on the same square face, and forming the convex hull of the four points on this path. Triangulated cube

A dissection of a shape

S

(which may be any

closed set In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a cl ...

in Euclidean space) is a representation of

S

as a union of other shapes whose

interiors ''Interiors'' is a 1978 American drama film written and directed by Woody Allen. It stars Kristin Griffith, Mary Beth Hurt, Richard Jordan, Diane Keaton, E. G. Marshall, Geraldine Page, Maureen Stapleton, and Sam Waterston. Allen's first full ...

are disjoint from each other. That is, intuitively, the shapes in the union do not overlap, although they may share points on their boundaries. For instance, a

can be dissected into six three-dimensional orthoschemes. A similar result applies more generally: every

hypercube In geometry, a hypercube is an ''n''-dimensional analogue of a square () and a cube (). It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, ...

hyperrectangle In geometry, an orthotopeCoxeter, 1973 (also called a hyperrectangle or a box) is the generalization of a rectangle to higher dimensions. A necessary and sufficient condition is that it is congruent to the Cartesian product of intervals. If all of ...

d

dimensions can be dissected into

d!

orthoschemes. Hadwiger's conjecture is that there is a function

f

such that every

d

-dimensional simplex can be dissected into at most

f(d)

orthoschemes. Hadwiger posed this problem in 1956; it remains unsolved in general, although special cases for small values of

d

are known.

In small dimensions

In two dimensions, every triangle can be dissected into at most two right triangles, by dropping an

altitude Altitude or height (also sometimes known as depth) is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The exact definition and reference datum varies according to the context ...

from its widest angle onto its longest edge. In three dimensions, some tetrahedra can be dissected in a similar way, by dropping an altitude perpendicularly from a vertex

v

to a point

p

in an opposite face, connecting

p

perpendicularly to the sides of the face, and using the three-edge perpendicular paths through

v

and

p

to a side and then to a vertex of the face. However, this does not always work. In particular, there exist tetrahedra for which none of the vertices have altitudes with a foot inside the opposite face. Using a more complicated construction, proved that every tetrahedron can be dissected into at most 12 orthoschemes. proved that this is optimal: there exist tetrahedra that cannot be dissected into fewer than 12 orthoschemes. In the same paper, Böhm also generalized Lenhard's result to three-dimensional

spherical geometry 300px, A sphere with a spherical triangle on it. Spherical geometry is the geometry of the two-dimensional surface of a sphere. In this context the word "sphere" refers only to the 2-dimensional surface and other terms like "ball" or "solid sp ...

and three-dimensional

hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P'' ...

. In four dimensions, at most 500 orthoschemes are needed. In five dimensions, a finite number of orthoschemes is again needed, roughly bounded as at most 12.5 million. Again, this applies to spherical geometry and hyperbolic geometry as well as to Euclidean geometry. Hadwiger's conjecture remains unproven for all dimensions greater than five.

Consequences

Every

may be dissected into simplexes. Therefore, if Hadwiger's conjecture is true, every convex polytope would also have a dissection into orthoschemes. A related result is that every orthoscheme can itself be dissected into

d

d+1

smaller orthoschemes. Therefore, for simplexes that can be partitioned into orthoschemes, their dissections can have arbitrarily large numbers of orthoschemes.

References

{{reflist, refs= {{citation , last1 = Brandts , first1 = Jan , last2 = Korotov , first2 = Sergey , last3 = Křížek , first3 = Michal , doi = 10.1016/j.laa.2006.10.010 , issue = 2-3 , journal = Linear Algebra and its Applications , mr = 2294350 , pages = 382–393 , title = Dissection of the path-simplex in

\mathbb R^n

into

n

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Geometriae Dedicata ''Geometriae Dedicata'' is a mathematical journal, founded in 1972, concentrating on geometry and its relationship to topology, group theory and the theory of dynamical systems. It was created on the initiative of Hans Freudenthal in Utrecht, the N ...

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, mr = 1220122 , pages = 313–329 , title = On the dissection of simplices into orthoschemes , volume = 46 , year = 1993 {{citation , last = Tschirpke , first = Katrin , issue = 1 , journal = Beiträge zur Algebra und Geometrie , mr = 1287191 , pages = 1–11 , title = The dissection of five-dimensional simplices into orthoschemes , volume = 35 , year = 1994 , url = https://www.emis.de/journals/BAG/vol.35/no.1/b35h1tsc.ps.gz Conjectures Unsolved problems in geometry Geometric dissection Multi-dimensional geometry