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 ...

, the Reeve tetrahedra are a family of

polyhedra In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on t ...

three-dimensional space Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position (geometry), position of an element (i.e., Point (m ...

with vertices at , , and where is a positive integer. They are named after John Reeve, who in 1957 used them to show that higher-dimensional generalizations of

Pick's theorem In geometry, Pick's theorem provides a formula for the area of a simple polygon with integer vertex coordinates, in terms of the number of integer points within it and on its boundary. The result was first described by Georg Alexander Pick in 18 ...

do not exist.

Counterexample to generalizations of Pick's theorem

All vertices of a Reeve tetrahedron are

lattice point In geometry and group theory, a lattice in the real coordinate space \mathbb^n is an infinite set of points in this space with the properties that coordinate wise addition or subtraction of two points in the lattice produces another lattice poi ...

s (points whose coordinates are all

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

s). No other lattice points lie on the surface or in the interior of the

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

volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The de ...

of the Reeve tetrahedron with vertex is . In 1957 Reeve used this tetrahedron to show that there exist tetrahedra with four lattice points as vertices, and containing no other lattice points, but with arbitrarily large volume. In two dimensions, the area of every polyhedron with lattice vertices is determined as a formula of the number of lattice points at its vertices, on its boundary, and in its interior, according to

. The Reeve tetrahedra imply that there can be no corresponding formula for the volume in three or more dimensions. Any such formula would be unable to distinguish the Reeve tetrahedra with different choices of from each other, but their volumes are all different. Despite this negative result, it is possible (as Reeve showed) to devise a more complicated formula for lattice polyhedron volume that combines the number of lattice points in the polyhedron, the number of points of a finer lattice in the polyhedron, 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 ...

of the polyhedron.

Ehrhart polynomial

The

Ehrhart polynomial In mathematics, an integral polytope has an associated Ehrhart polynomial that encodes the relationship between the volume of a polytope and the number of integer points the polytope contains. The theory of Ehrhart polynomials can be seen as a highe ...

of any lattice polyhedron counts the number of lattice points that it contains when scaled up by an integer factor. The Ehrhart polynomial of the Reeve tetrahedron of height is

L(\mathcal_r, t) = \fract^3 + t^2 + \left(2 - \frac\right)t + 1.

Thus, for , the coefficient of in the Ehrhart polynomial of is negative. This example shows that Ehrhart polynomials can sometimes have negative coefficients.

References

{{reflist, 30em, refs= {{cite magazine , last = Kiradjiev , first = Kristian , title = Connecting the Dots with Pick's Theorem , date = December 2018 , url = https://cdn.ima.org.uk/wp/wp-content/uploads/2018/12/Connecting-the-Dots-with-Pick-Theore-from-MT-December-2018.pdf , magazine = Mathematics Today , publisher = Institute of mathematics and its applications , access-date = January 6, 2023 {{cite book , last1 = Beck , first1 = Matthias , last2 = Robins , first2 = Sinai , doi = 10.1007/978-1-4939-2969-6 , edition = Second , isbn = 978-1-4939-2968-9 , location = New York , mr = 3410115 , publisher = Springer , series = Undergraduate Texts in Mathematics , title = Computing the Continuous Discretely: Integer-Point Enumeration in Polyhedra , title-link = Computing the Continuous Discretely , at
pp. 78–79, 82
, year = 2015 {{cite journal , last = Kołodziejczyk , first = Krzysztof , doi = 10.1007/BF00150027 , issue = 3 , journal = Geometriae Dedicata , mr = 1397808 , pages = 271–278 , title = An “odd” formula for the volume of three-dimensional lattice polyhedra , volume = 61 , year = 1996 {{cite journal , last = Reeve , first = J. E. , doi = 10.1112/plms/s3-7.1.378 , journal =

Proceedings of the London Mathematical Society The London Mathematical Society (LMS) is one of the United Kingdom's learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh Mathematical S ...

, mr = 0095452 , pages = 378–395 , series = Third Series , title = On the volume of lattice polyhedra , volume = 7 , year = 1957 Digital geometry Lattice points