Concept
How many numbers do you need? — From 0D to ∞D and even fractional dimensions in fractals.
Understand it in one breath
"The number of independent coordinates needed to specify a position." A line is 1, a plane 2, our space 3, spacetime 4 — but fractal dimension can be a fraction. Mandelbrot's findings: the Koch snowflake ≈ 1.26, the British coastline ≈ 1.25, the bronchial tree of the lung ≈ 2.8 (Weibel measurement).
At a glance
Object | Dimension (Hausdorff) | Note |
|---|---|---|
Point | 0 | Has no length, however many points are gathered |
Line | 1 | One coordinate determines position |
Cantor set | ≈ 0.6309 (log 2 / log 3) | Smaller than one-dimensional, but larger than a point |
Coastline of Britain | ≈ 1.25 | Richardson, 1961 — measured length changes with the unit of measurement |
Koch snowflake | ≈ 1.2619 | Infinite perimeter, finite area |
Boundary of the Mandelbrot set | 2 (Shishikura 1998) | A fractal curve of dimension 2 — a one-dimensional boundary that nearly fills the plane |
Plane | 2 | Two coordinates |
Bronchial tree in the lungs | ≈ 2.7–2.84 | Fills space almost three-dimensionally, maximizing oxygen uptake (Weibel; box counting) |
The space we inhabit | 3 | |
Spacetime | 4 | Minkowski, 1908 (4D formulation of Einstein’s 1905 special relativity) |
String-theory spacetime | 10 or 11 | 6 or 7 dimensions are curled up and invisible |
A fractal dimension can be fractional — as in nature’s fragmented, self-similar structures.
Key formula
Key moments
Descartes — coordinates give dimension a language
Coordinate geometry suggested that n numbers specify a point in n-dimensional space, even though geometry beyond three dimensions was not yet accepted.
Hamilton — four-dimensional quaternions
William Rowan Hamilton discovered the quaternion number system, giving four-dimensional algebra a formal place in mathematics.
Riemann — n-dimensional manifolds
Riemann established a general theory of spaces with arbitrary dimension, making four, five, or infinitely many dimensions mathematically natural.
Hausdorff dimension — fractional dimension
Hausdorff introduced a way to measure dimension beyond whole numbers. The Koch curve, for example, has dimension log 4/log 3, about 1.26.
Modern applications
High-dimensional data and PCA, four-dimensional spacetime in general relativity, ten- or eleven-dimensional string theory, and embedding spaces in machine learning.
Beyond MathVoyage
Loading…