# Nullitope

The nullitope, or nulloid, is the simplest polytope possible. It has no elements, and it may be considered an element of every other polytope. By convention, it is said to have a rank of −1.[1] While a point has a location but no other properties, a nullitope does not even have location.

Nullitope
Rank−1
TypeRegular
SpaceNone
Topological properties
OrientableYes
Properties
ConvexYes

A nullitope is a −1-simplex, though it doesn't neatly fit the pattern of the hypercubes and orthoplices.

Other names for it have been null polytope, nought, wessian, essence, namon, nullon[2] or simply empty element.

The nullitope's significance results from the definition of abstract polytopes. It describes the bottom node of the Hasse diagram of any polytope. Though most of the time it makes no sense to consider the nullitope on its own, it can be mathematically convenient in some situations, such as when calculating the pyramid product of two polytopes.

The nullitope is the only polytope without a corresponding hypertope.

## References

1. Johnson, Norman W. Geometries and transformations. pp. 224–225.
2. Inchbald, Guy. http://www.steelpillow.com/polyhedra/ditela.html