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Fractal Geometry and Landscapes (Diamond Square ((This means one formula…
Fractal Geometry and Landscapes
origin:
Originally coined by mathematician Benoit Mandelbrot, and is derived from the Latin word fractus (broken).
fractal is a geometric shape is generated using a series of
recursive rules.
self similarity
fractal has a pattern that repeats at different scales
Exact (strongest)
The fractal appears identical at
different scales
usually found in geometry, not in nature
• Sierpinski triangle, Koch snowflake
Quasi- SS
appears approximately (but not
exactly) identical at different scales
contain small copies of the entire fractal in distorted and degenerate forms
Statistical
numerical or statistical measures preserved across scales.
Random Fractals e.g. fractal landscapes.
Diamond Square
Complex pictures generated by formula and using several
iterations.
This means one formula is repeated with slightly
different values over and over again, taking into account the
results from the previous iteration
Possible to create realistic textured landscapes, such as mountain ranges, coastlines or even destruction patterns
Generate heightmaps from a 2D grid.
Diamond step
for each square, midpoint = avg (all intersecting points) + random()
Start by setting the initial values for the 4 corner points of the grid.
Square step
for each diamond, midpoint = avg (all intersecting points) + random()
Brownian Motion
Robert Brown discovered that matter consists of tiny particles in motion (1827) while observing pollen move in water
Describes the stochastic (random) movement of small particles of matter in a fluid (liquid or gas)
Used by Benoit Mandelbrot for the modelling of natural
phenomena
Mathematical examination by Einstein and Wiener - Dynamics of molecular collisions
Hurst param - The larger the parameter, the smoother the surface.