---
type: Fractal Concept
title: Complex Dimensions
description: Generalization of the real Hausdorff/box-counting dimension to complex values, used by Lapidus et al. to describe oscillatory geometric and spectral asymptotics of fractal sets.
resource: https://tyson-swetnam.github.io/fractal-notebooks/okf/concepts/dimension-theory/complex-dimensions/
tags:
- dimension-theory
- complex-analysis
- advanced
aliases:
- complex fractal dimension
sources:
- id: glossary
  resource: https://tyson-swetnam.github.io/fractal-notebooks/foundations/glossary/
  title: Fractal Notebooks glossary (Chapter 4)
- id: seed
  resource: https://github.com/tyson-swetnam/fractal-notebooks/blob/main/fractal_rag/src/fractal_rag/concepts/seed.yaml
  title: fractal_rag concept seed (seed.yaml)
generated:
  by: process:fractal-rag-okf-export
  at: "2026-10-24T00:00:00Z"
status: stable
---

# Definition

Generalization of the real Hausdorff/box-counting dimension to complex values, used by Lapidus et al. to describe oscillatory geometric and spectral asymptotics of fractal sets.

# Aliases

`complex fractal dimension`

# Related concepts

* [Box-Counting Dimension](box-counting-dimension.md)
* [Fractal](../geometry/fractal.md)

# Citations

[1] [Fractal Notebooks glossary (Chapter 4)](https://tyson-swetnam.github.io/fractal-notebooks/foundations/glossary/) — `docs/foundations/glossary.md`

[2] [fractal_rag concept seed (seed.yaml)](https://github.com/tyson-swetnam/fractal-notebooks/blob/main/fractal_rag/src/fractal_rag/concepts/seed.yaml) — `fractal_rag/src/fractal_rag/concepts/seed.yaml`