Mandelbrot set
E695943
The Mandelbrot set is a famous complex-plane fractal defined by iterating quadratic polynomials, known for its infinitely intricate boundary and iconic role in chaos theory and complex dynamics.
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
Julia set parameter space
ⓘ
fractal ⓘ mathematical object ⓘ subset of the complex plane ⓘ |
| conjecture | Mandelbrot local connectivity conjecture NERFINISHED ⓘ |
| contains |
infinitely many circular bulbs attached to the cardioid
ⓘ
main cardioid region ⓘ real interval from -2 to 0.25 on the real axis ⓘ |
| definedAs | set of complex numbers c for which the sequence z_{n+1} = z_n^2 + c with z_0 = 0 remains bounded ⓘ |
| discoveredBy | Benoit B. Mandelbrot NERFINISHED ⓘ |
| field |
chaos theory
ⓘ
complex dynamics ⓘ fractal geometry ⓘ |
| hasBoundaryPoint | c = 0.25 ⓘ |
| hasCenter | c = 0 is in the interior ⓘ |
| hasComplementCondition | c is not in the Mandelbrot set if the orbit of 0 under z ↦ z^2 + c escapes to infinity ⓘ |
| hasElementCondition | c is in the Mandelbrot set if the orbit of 0 under z ↦ z^2 + c is bounded ⓘ |
| hasExtremePoint |
leftmost point at c = -2
ⓘ
rightmost point at c = 0.25 ⓘ |
| hasFeature |
elephant valley region
ⓘ
filamentary tendrils extending from the main body ⓘ miniature copies of the whole set (baby Mandelbrots) ⓘ seahorse valley region ⓘ spiral structures near the boundary ⓘ |
| hasParameter | complex parameter c ⓘ |
| hasProperty |
area is finite
ⓘ
boundary has Hausdorff dimension 2 ⓘ boundary has infinite length ⓘ boundary is a fractal ⓘ boundary is infinitely intricate ⓘ compact ⓘ connected ⓘ exhibits approximate scale invariance ⓘ exhibits chaotic dynamics on its boundary ⓘ full subset of the complex plane ⓘ locally connected is conjectured but not yet proven ⓘ self-similar at various scales ⓘ |
| namedAfter | Benoit B. Mandelbrot NERFINISHED ⓘ |
| popularizedIn | The Fractal Geometry of Nature NERFINISHED ⓘ |
| relatedTo |
Julia sets
ⓘ
bifurcation diagrams of dynamical systems ⓘ quadratic polynomials z^2 + c ⓘ |
| symbol | M ⓘ |
| usedIn |
computer graphics
ⓘ
mathematics education ⓘ visual demonstrations of chaos and fractals ⓘ |
| visualizedBy |
coloring points by iteration count before escape
ⓘ
escape-time algorithm ⓘ |
| yearOfFirstComputerVisualization | late 1970s ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.