---
tag: V051BD0CZG
kind: concept
name: Unital
msc: ["51E05", "51E15", 05B05]
sources:
  - url: https://en.wikipedia.org/wiki/Unital_(geometry)
same_as:
  wikidata: Q17148807
  wikipedia: Unital (geometry)
---

## Summary

A unital of order $n$ is a [[block design|4JCSS2Z1BA]] with $n^3 + 1$ points and blocks of $n + 1$ points.
Any two points lie in exactly one block.
The main examples sit inside a [[projective plane|DANFNN8C4Y]] of order $n^2$, where they meet every line in $1$ or $n + 1$ points.
The classical example is the Hermitian curve, defined by a [[Hermitian form|48P4A1CX85]] over a field with $n^2$ elements.

## Explanations

### L1 human

A unital is a finite geometry of points and blocks in which two points always share exactly one block.
Its numbers are fixed by one parameter $n$: there are $n^3 + 1$ points and each block has $n + 1$ points.
The standard picture is a curve inside a finite plane whose lines cut it either in one point or in $n + 1$ points.
For $n = 2$ one gets $9$ points with blocks of size $3$, sitting inside the plane of order $4$.