Solid Texture Mapping
A form of 3d-texture-mapping which [see page 4, applies] a function to each 3D coordinate on the surface of the object (in the world-coordinates).
This [see page 7, differs] from 2D texture mapping by defining the pattern of color variation mathematically, rather than substituting it from an existing texture.
A good function can produce endless patterns.
See an [see page 8, example] of defining a function for a wooden pattern. Note the transformations we apply as we move through the centre axis (u) all [see page 11, contribute] to the final pattern.
Noisy
A classic functional approach produces clean regular patterns... however we often want to apply some Noise to make things appear more natural. Given a noise map we [see page 15, translate] a texture (x,y,z) coordinate to a noise (u,v,w) vector and then linearly interpolate each vertex around the (u,v,w) coordinate to get a noise value.
We [see page 17, apply] this noise to the color output of our function to get a noisier texture surface.
We can also [see page 18, offset] our color lookup using noise to get a disturbance in the pattern that can resemble marble.
We can also use noise with 2D textures (using the \nth{3} (w) axis as time) to add turbulence to the 2D texture over time causing a [see page 19, flame-pattern]. (or I dunno, play a recorded animation of a billboard).