User:Maschen/Fractals in Clifford and geometric algebra

In applied computer graphics, fractals in geometric algebra and Clifford algebra are fractals generated using the tools of geometric algebra, an example of the more general formalisms of Clifford algebra.

Motivation

Surfaces and boundaries of real physical objects can be approximated accurately by fractals. Fractals are characterized by a self-similarly repeating geometric shape, which can be described using recurrence relations of numbers, for example the Cantor set is generated by real numbers, the infamous Mandelbrot set is generated by complex numbers, the 3d analogue called the Mandelbulb is generated by 3d Euclidean vectors, and other 4d fractals can be iterated by quaternions.

In geometric algebra, scalars, vectors, complex numbers and quaternions are all part of the same number system. The most general geometric object is a multivector, a sum of scalars and pseudoscalars, vectors and pseudovectors, and so on, and other numbers can be construed as certain cases of multivectors. Fractals can be generated by iterating multivectors which comprise scalars and pseudoscalars. Since multivectors add together objects of different dimensions (technically in GA, the term grade is used), multidimensional fractals can be generated by multivectors. An interesting feature of GA is that, like complex numbers and quaternions but unlike Euclidean vectors, functions (such as powers, sine, cosine, exponential, logarithm, etc) of multivectors can be defined, which easily allows non-linear maps to be defined.

Generation of fractals

Since fractals can be generated by a simple rule, the rule could remain constant throughout the generation of the fractal, i.e. the rule is deterministic. More interesting results occur when if the rule is changed randomly each time it's applied.

Applications

Computer graphics continues to experiment with applying GA. Fractals have been used to approximate realistic geometric shapes, as well as fractal interpolation to approximate plots and graphs where statistical methods alone are insufficient.

Multivector maps take the form:

User:Maschen/Fractals in Clifford and geometric algebra

Motivation

Generation of fractals

Applications

See also

References

External links