Exploring Reeve Tetrahedra

1. What are Reeve Tetrahedra?

In geometry, the Reeve tetrahedra are a family of polyhedra (3D shapes with flat faces) defined by four specific corner points, called vertices. These vertices have coordinates:, , , and , where is any positive whole number.

They are named after John Reeve, who studied them in 1957. These simple-looking shapes have some surprising properties, especially when considering points with whole number coordinates (lattice points).

The parameter controls the "height" or "sharpness" of the tetrahedron. As you change , the shape changes, and importantly, its volume changes too. The volume of a Reeve tetrahedron is given by a simple formula: .

Interactive Visualisation

Use the slider below to change the value of . Observe how the tetrahedron stretches vertically. The visualisation below shows the shape and calculates its volume based on your chosen .

This 3D model shows the tetrahedron with vertices (0,0,0), (1,0,0), (0,1,0), and (1,1,1).

Notice that the tetrahedron avoids passing through any lattice points, no matter how large you make .

2. Pick's Theorem (2D Analogy)

Pick's Theorem provides a simple formula for the area of a polygon whose vertices are points on a grid (lattice points).

This theorem is elegant because it connects the geometric concept of area with the arithmetic concept of counting points. However, Pick's theorem famously *does not* generalise directly to three dimensions. Reeve tetrahedra are a key example demonstrating this failure.

Interactive 2D Example

Click grid points to define a polygon and see Pick's Theorem calculate its area.

3. Failure of a Simple 3D Pick's Theorem

John Reeve used the tetrahedra to show that there's no simple formula like Pick's Theorem for volume in 3D based solely on the number of interior () and boundary () lattice points.

No matter how large you make in the Reeve tetrahedron, it never passes through any lattice points.

Consider the Reeve tetrahedron with vertices at , , , and . Let's examine its properties:

Crucially, notice that and are the *same* for all values of (), but the volume changes with .

Since we can have different volumes for the same number of interior () and boundary () points using the standard integer lattice ( or ), no simple formula analogous to Pick's can exist for 3D polyhedra.

This failure prompted Reeve to investigate further, leading to a more sophisticated formula involving points from finer lattices.

4. Reeve's Formula using Finer Lattices

While a direct 3D Pick's theorem using only standard integer lattice points () fails, John Reeve discovered a more sophisticated way to calculate the volume of a 3D lattice polyhedron (). His approach involved considering points from "finer" lattices, specifically the lattice which includes points with half-integer coordinates.

For a 3D lattice polyhedron (like the Reeve tetrahedra), Reeve's formula relates the volume to the counts of points from and on the boundary and interior:

The visualisation below shows the Reeve tetrahedron , highlighting the points from and (those not in ) within and on its boundary. It calculates the values for and verifies Reeve's formula against the known volume .

Observe how incorporating points from the finer lattice allows for a consistent volume calculation.

5. Ehrhart Polynomials: A Generalisation

Reeve's formula provides a way to calculate volume using specific lattice counts. Ehrhart theory offers a more general framework that connects volume, lattice points, and topology through polynomials. Crucially, this framework applies this approach to shapes (lattice polytopes) in any number of dimensions.

For a -dimensional lattice polytope , the Ehrhart polynomial counts the number of standard integer lattice points () in the scaled polytope (where is a positive integer).

For the Reeve tetrahedron (assuming is even for this simplified example), the Ehrhart polynomial is:

Let's check for and :

Since , and we know for , we expect . Our calculation confirms this (allowing for floating point representation). The polynomial correctly captures the number of points.

Connection to Reeve's Formula: The Ehrhart polynomial encodes the volume directly as its leading coefficient. Furthermore, a property called Ehrhart reciprocity relates the values of at negative integers to the number of *interior* lattice points in scaled versions of . This reciprocity provides the theoretical underpinning that connects the overall scaling behavior captured by the Ehrhart polynomial to the specific counts of interior and boundary points (like ) used in formulas such as Reeve's.

Thus, the Ehrhart polynomial can be seen as a more general structure from which specific volume formulas involving lattice point counts can be derived.

Visualisation: Scaled Tetrahedra and Point Counting

The visualisation below helps understand Ehrhart polynomials. It shows the Reeve tetrahedron scaled by different integer factors . You can adjust and see how the number of lattice points inside the scaled tetrahedron changes, following the polynomial .

Ehrhart theory also provides a way to relate volume and lattice points in higher dimensions.