Fractal Geometry and Landscapes (Diamond Square ( (This means one formula…
Fractal Geometry and Landscapes
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
fractal has a pattern that repeats at different scales
The fractal appears identical at
usually found in geometry, not in nature
• Sierpinski triangle, Koch snowflake
appears approximately (but not
exactly) identical at different scales
contain small copies of the entire fractal in distorted and degenerate forms
numerical or statistical measures preserved across scales.
Random Fractals e.g. fractal landscapes.
Complex pictures generated by formula and using several
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.
for each square, midpoint = avg (all intersecting points) + random()
Start by setting the initial values for the 4 corner points of the grid.
for each diamond, midpoint = avg (all intersecting points) + random()
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
Mathematical examination by Einstein and Wiener - Dynamics of molecular collisions
Hurst param - The larger the parameter, the smoother the surface.