---
tag: M6YBMXKJYD
kind: concept
name: Cyclic polytope
msc: [52B12, 52B05, 52B11]
specializes: [YVJ15PGKDM]
sources:
  - url: https://en.wikipedia.org/wiki/Cyclic_polytope
same_as:
  wikidata: Q245469
  wikipedia: Cyclic polytope
---

## Summary

A cyclic polytope is the [[convex hull|0FQSDC2EQ2]] of $n \ge d + 1$ distinct points on the moment curve $t \mapsto (t, t^2, \dots, t^d)$ in $\R^d$.
Its combinatorial type depends only on the number $n$ of points and the dimension $d$, and it is denoted $C(n, d)$.
Cyclic polytopes are [[simplicial|YVJ15PGKDM]] and [[neighborly|9V4PR1RVXW]].
They have the largest number of faces of each dimension among all $d$-polytopes with $n$ vertices, which is the content of the [[upper bound theorem|BH8R1M1KVF]].

## Explanations

### L1 human

Pick points on a curve that bends in every direction of space, such as $(t, t^2, t^3)$, and take the solid they span.
The curve bends so much that many sets of points become faces.
In four dimensions, any two of the points are joined by an edge of the solid.
That is far more edges than a three-dimensional solid could have with the same number of corners.