Hausdorff dimension
Non-integer dimensional metric
Hausdorff dimension is a mathematical concept that extends the notion of dimension to fractals and other irregular shapes that don't fit neatly into whole-number dimensions.
A jagged coastline, for instance, might have a Hausdorff dimension between 1 and 2—more complex than a simple line but not quite filling a plane.
The concept allows mathematicians to quantify how thoroughly an object fills the space it occupies, capturing degrees of roughness or fragmentation that traditional geometry cannot express.
Identical at every magnification
- strange attractorFractal pattern in chaos
- recursiveDefined in terms of itself
- iterationRepeated procedural pass
Collection with nothing inside
Standard of measurement
- compactTightly packed together
- continuityUninterrupted progression or coherence
- embeddingPlacing within surrounding mass
- separableCapable of being divided
Based on exponents
- exponentSuperscript showing power
- dilationWidening of an opening
- contractionReduction in size
- magnificationDegree of visual enlargement
ratio · proportionality · epsilon · limit · resolution
Framework of intersecting lines
- algorithmFinite problem-solving procedure
- approximationNear but inexact value
- regressionReturn to former state
- softwarePrograms directing computer operations
self-similarity
Identical at every magnification
- strange attractorFractal pattern in chaos
- recursiveDefined in terms of itself
- iterationRepeated procedural pass
null set
Collection with nothing inside
metric
Standard of measurement
- compactTightly packed together
- continuityUninterrupted progression or coherence
- embeddingPlacing within surrounding mass
logarithmic
Based on exponents
- exponentSuperscript showing power
- dilationWidening of an opening
- contractionReduction in size
magnification · ratio · proportionality · epsilon · limit · resolution
grid
Framework of intersecting lines
- algorithmFinite problem-solving procedure
- approximationNear but inexact value
- regressionReturn to former state
- dimensions2D, 3D, 4D, height, width
- Fractalsfractal, fractals, self-similar, recursion, complex plane
- Invariantsgenus, homology, signature, connectedness, compactness
Hausdorff dimensionlength
Measure of extent
area · volume · arc length · surface area · cardinality
dimension, dimensions, dimensional, multidimensional, interdimensional, dimensionless, dimensionality, dimensionally, extradimensional