3-D Fractals

This page shows a different 3-D fractal for each day of the week. It is possible to download individual images.

John Conway's game of life can provide three- dimensional cellular automata, too. Application of the algorithm has already been explained extensively, in past fractal automata sections.

It is worth repeating, maybe, that exploiting symmetry will form the fractal cube 8 (and possibly 32, or more) times faster. Also, there are plenty of ways to map the 81 possible output values (or more, if a 5x5x5 mask is used) back to the available colour range.

The perspective transformation for displaying a planar view of 3-D objects therefore may as well get a mention:

X= x.f/(f-z)

Y= y.f/(f-z)

where

x, y, z are the spatial coordinates of any point on the 3-D object
X, Y are the coordinates of the projection of the 3-D space point on a screen perpendicular to the z- axis and placed at the origin
f is the distance of the viewer along the z- axis

The object is contained wholly between the screen and observer.

And the 3-D rotational transform:

X= c_x,xx+ c_x,yy+ c_x,zz

Y= c_y,xx+ c_y,yy+ c_y,zz

Z= c_z,xx+ c_z,yy+ c_z,zz

where

x, y, z are the old coordinates
X, Y, Z are the coordinates after rotation
each of the coefficients is a product of two cosines

The fractal cube is shown at a rotation and elevation (yaw and pitch) of 45^o. The inevitable symmetry on its faces suggests the page title is cheating. It is not! The 3-D algorithm did form the cube, but admittedly cuts along the main diagonals would look much more impressive. They will follow (sometime.)