Reeve tetrahedron
Encyclopedia
In geometry
, the Reeve tetrahedron is a polyhedron
, named after John Reeve, in R3 with vertices at (0, 0, 0), (1, 0, 0), (0, 1, 0) and (1, 1, r) where r is a positive integer. Each vertex lies on a fundamental lattice point (a point in Z3). No other fundamental lattice points lie on the surface or in the interior of the tetrahedron
. In 1957 Reeve used this tetrahedron as a counterexample to show that there is no simple equivalent of Pick's theorem
in R3 or higher-dimensional spaces. This is seen by noticing that Reeve tetrahedra have the same number of interior and boundary points for any value of r, but different volumes.
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 ....
, the Reeve tetrahedron is a polyhedron
Polyhedron
In elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...
, named after John Reeve, in R3 with vertices at (0, 0, 0), (1, 0, 0), (0, 1, 0) and (1, 1, r) where r is a positive integer. Each vertex lies on a fundamental lattice point (a point in Z3). No other fundamental lattice points lie on the surface or in the interior of the tetrahedron
Tetrahedron
In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...
. In 1957 Reeve used this tetrahedron as a counterexample to show that there is no simple equivalent of Pick's theorem
Pick's theorem
Given a simple polygon constructed on a grid of equal-distanced points such that all the polygon's vertices are grid points, Pick's theorem provides a simple formula for calculating the area A of this polygon in terms of the number i of lattice points in the interior located in the polygon and the...
in R3 or higher-dimensional spaces. This is seen by noticing that Reeve tetrahedra have the same number of interior and boundary points for any value of r, but different volumes.