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The Fractal Geometry of Nature. The Mathematical Intelligencer. Statistical Self-Similarity and Fractional Dimension. Fractal canopy Space-filling curve H tree. Droste effect Mathematical beauty Patterns in nature Sacred geometry.

Encyclopedia of Complexity and Systems Science. Slime mold Brefeldia maxima growing fractally on wood. Self-similarity illustrated by image enlargements.

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The history of fractals traces a path from chiefly theoretical studies to modern applications in computer graphics, with several notable people contributing canonical fractal forms along the way. University of New South Wales.


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Fractal geometry lies within the mathematical branch of topology. For other uses, see Fractal disambiguation. International Journal of Arts and Technology. When two-dimensional fractals are iterated many times, the perimeter of the fractal increases up to infinity, but the area may never exceed a certain value.

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The outputs of the modelling process may be highly artistic renderings, outputs for investigation, or benchmarks for fractal analysis. Clinical Ophthalmology Auckland, N.

Fractals in biology and medicine. This number is what mathematicians call the fractal dimension of the Koch curve.

In other projects Wikimedia Commons. Frost crystals occurring naturally on cold glass form fractal patterns. Fractal defrosting patterns, polar Mars. Circular houses appear in circles of circles, rectangular houses in rectangles of rectangles, and so on.

Manchester University Press. The Fractal Geometry of the Brain. Analytically, fractals are usually nowhere differentiable. Pattern recognition Emergence Mathematics and art. Form, Function, and Fractal Dimension.

Now, consider the Koch curve. Physics and fractal structures. Introducing fractal geometry.

Fractals in the physical sciences. The fractal geometry of nature.


Pattern Recognition Letters. Pattern Formation in Biology, Vision and Dynamics, pp. Because of the trouble involved in finding one definition for fractals, some argue that fractals should not be strictly defined at all. The fact that the Koch curve has a non-integer fractal dimension is what makes it a fractal. Likewise, if the radius of a sphere is doubled, its volume scales by eight, which is two the ratio of the new to the old radius to the power of three the dimension that the sphere resides in.

The same fractal as above, magnified fold, where the Mandelbrot set fine detail resembles the detail at low magnification. Images of fractals can be created by fractal generating programs. Nonlinear Dynamics, Psychology, and Life Sciences. The Journal of Physiology.

Width of image is about a kilometer. Pattern formation in biology, vision and dynamics. Buddhabrot Orbit trap Pickover stalk.

Wikimedia Commons has media related to Fractal. Springer Series in Computational Neuroscience. The connection between fractals and leaves, for instance, is currently being used to determine how much carbon is contained in trees. The result is that one must need infinite tape to perfectly cover the entire curve, i.


Earth and Planetary Science Letters. Same patterns reappear, rubik cube fridrich method pdf making the exact scale being examined difficult to determine.

Ethnomathematician Ron Eglash has discussed the planned layout of Benin city using fractals as the basis, not only in the city itself and the villages but even in the rooms of houses. There is some disagreement among mathematicians about how the concept of a fractal should be formally defined.