# Nullitope

Nullitope
Rank−1
TypeRegular
SpaceNone
Abstract & topological properties
OrientableYes
Properties
Flag orbits1
ConvexYes
NatureTame

The nullitope, or nulloid, is the simplest polytope possible. It has no proper 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.

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.

As a convex polytope, the nullitope corresponds exactly to the empty set. In this sense, the nullitope can be said to exist in any space.

## Properties

### Properties unique to the nullitope

• The nullitope is the only polytope without a corresponding hypertope.
• Since a regular polytope has as many distinguished generators as its rank, the nullitope would have to -1 distinguished generators, a clear impossibility. Thus it is the only regular polytope without a set of distinguished generators. The empty set generates the point.
• Although the nullitope is trivially convex when realized as the empty set, the alternating sum of its element counts (c.f. Euler characteristic) is 1. For all other convex polytopes this alternating sum is 0.
• It is the identity of the pyramid product.

### Properties shared with the point

• The nullitope and the point are the only regular polytopes without a Schläfli type.
• They are the only polytopes with an odd number of flags.

## References

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