1-polytope

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1-polytope
About
Rank1
PluralPolytela
Counts
Regular convex polytela1
Regular star polytela0
Regular tesselations0
Regular finite skew polytela of full rank0
Regular skew apeirotopes0
Hierarchy
← 0-polytope 1-polytope Polygon →
0 1 2

A 1-polytope (also called an edge or sometimes a polytelon) is a polytope of rank 1. By most definitions there is only a single 1-polytope, the dyad. This is due to the diamond property, which is a common component of most definitions.

Proof of uniqueness[edit | edit source]

Abstract polytopes are a combinatorial definition of polytope. The basic properties of abstract polytopes are held by most ideas of polytope in common use (e.g. convex polytopes, Grünbaum's definition of skew polytopes, etc.), so they make a good basis for a proof. The following proves that there is exactly one abstract 1-polytope.

Proof —

By the definition of rank, a rank 1 polytope must have flags of length 3. It must also have unique maximum element of rank 1, and a unique minimum element of rank -1, which we will call 𝓤 and 𝓞 respectively. By the diamond property, there must be exactly two elements x  such that 𝓤 > x  > 𝓞. Thus any 1-polytope must have exactly two proper elements, which are rank 0. This completely determines the 1-polytope to be exactly the dyad.

Maniplexes are a generalization of abstract polytopes, with a different notion of dyadicity. They include a number of things that may sometimes be thought of as polytopes but do not meet the requirements of an abstract polytope. A proof can be used to show that there is exactly one 1-maniplex, although the language of maniplexes may be unfamiliar to some:

Proof —

A 1-maniplex is a non-empty set Ω  and a permutation ρ  on Ω . We will highlight the following properties which are required by the definition:

  1. ρ 2  must be the identity mapping.
  2. ρ  must be transitive on Ω 
  3. ρ  must have no fixed points.

By 1. ρ  can be transitive on at most 2 elements, x  and ρ (x ), so by 2. Ω  can be at most size 2. However Ω  cannot be size 1 otherwise ρ  violates 3. Thus Ω  has exactly two elements and ρ  maps between them. This is the dyad.

We can note that the above proof only actually used properties true of a complex, so dyad is also the the only complex.

Non-dyadic 1-polytopoids[edit | edit source]

Complex 1-polytope
About
Rank1
Counts
Regular complex 1-polytopes

Some definitions such as complex polytopes or quaternionic polytopes, do not require the diamond condition, and thus allow for non-dyadic structures. A complex 1-polytope may have any number of vertices greater than 1.