Sphere volumes In ordinary flat Euclidean space the area of a 2D circle is π r 2 , and the volume of a 3D sphere 4/3 π r 3 . … If instead of flat space one considers a space defined by the surface of a 3D sphere—say with radius a —one can ask about areas of circles in this space. Such circles are no longer flat, but instead are like caps on the sphere—with a circle of radius r containing all points that are geodesic (great circle) distance less than r from its center.

On a clear night over flat terrain, air flow can be laminar near the ground.

General associative [cellular automaton] rules With a cellular automaton rule in which the new color of a cell is given by f[a 1 , a 2 ] (compare page 886 ) it turns out that the pattern generated by evolution from a single non-white cell is always nested if the function f has the property of being associative or Flat . … In general, the pattern produced by evolution for t steps is given by NestList[ Inner[f, Prepend[#, 0], Append[#, 0], List] &, {a}, t] so that the first few steps yield {a}, {f[0, a], f[a, 0]}, {f[0, f[0, a]], f[f[0, a], f[a, 0]], f[f[a, 0], 0]}, {f[0, f[0, f[0, a]]], f[f[0, f[0, a]], f[f[0, a], f[a, 0]]], f[f[f[0, a], f[a, 0]], f[f[a, 0], 0]], f[f[f[a, 0], 0], 0]} If f is Flat , however, then the last two lines here become {f[0, 0, a], f[0, a, a, 0], f[a, 0, 0]}, {f[0, 0, 0, a], f[0, 0, a, 0, a, a, 0], f[0, a, a, 0, a, 0, 0], f[a, 0, 0, 0]} and in general the number of a 's that appear in a particular element is given as in Pascal's triangle by a binomial coefficient. … The result can also be generalized to cellular automata with basic rules involving more than two elements—since if f is Flat , f[a 1 , a 2 , a 3 ] is always just f[f[a 1 , a 2 ], a 3 ] .

A perfectly flat surface will reflect light like a mirror.

The shapes produced in each case are very simple, and ultimately consist just of flat facets arranged in a way that reflects directly the structure of the underlying lattice of cells.

Systematic studies of the symmetries of crystals with flat facets began in the 1700s, and the relationship to internal structure was confirmed by X-ray crystallography in the 1920s.

With material where parts can locally expand, but cannot change their shape, page 1007 showed that a 2D surface will remain flat if the growth rate is a harmonic function.

Starting off with a network that is planar—so that it can be drawn flat on a page without any lines crossing—such rules can certainly give all sorts of complex and apparently random behavior.

For if the underlying rule for a network is going to maintain to a certain approximation the same average number of nodes as flat space, then it follows that wherever there are more nodes corresponding to energy and momentum, this must be balanced by something reducing the number of nodes.

For folding is not only involved in producing shapes such as teeth surfaces and human ear lobes, but is also critical in allowing flat sheets of tissue to form the kinds of pockets and tubes that are so common inside animals.