---
tag: SSNFTHZ6YJ
kind: concept
name: Hausdorff dimension
msc: [28A78, 28A80]
sources:
  - url: https://en.wikipedia.org/wiki/Hausdorff_dimension
same_as:
  wikidata: Q565186
  wikipedia: Hausdorff dimension
---

## Summary

Let $A$ be a subset of a [[metric space|0VKD1TP7F3]].
Its Hausdorff dimension is the exponent $s$ at which the $s$-dimensional [[Hausdorff measure|Q1TGZK8RN6]] of $A$ jumps from infinity to zero.
It is a real number, not always an integer.
It agrees with the usual dimension for smooth curves, surfaces and open sets, and it gives fractals such as the [[Cantor set|0BXNDZFQME]] non-integer dimensions.

## Explanations

### L1 human

A curve has length but no area, and a square has area but infinite length.
The Hausdorff dimension finds the exact exponent where a set switches from infinite size to zero size.
For a segment this happens at $1$, and for a filled square at $2$.
The middle-thirds Cantor set switches at $\log 2 / \log 3 \approx 0.63$: it is too thin to have length but too big to be counted as a few points.