If one has a simple array of hexagons—as in the first picture below—then this can readily be laid out flat on a two-dimensional plane. … A hexagonal array corresponding to flat two-dimensional space. … If every region in the network is in effect a hexagon—as in the picture at the top of the page—then the network will behave as if it is flat.

[Sequences with] flat spectra Any impulse sequence Join[{1}, Table[0, {n}]] will yield a flat spectrum. … It is also flat for maximal length LFSR sequences (see page 1084 ) and for sequences JacobiSymbol[Range[0, p - 1], p] with prime p (see page 870 ). By adding a suitable constant to each element one can then arrange in such cases for the whole spectrum to be flat.

Having made the definition Attributes[s] = Flat the state of a sequential substitution system at a particular step can be represented by a symbolic expression such as s[1, 0, 1, 0] . … (With s being Flat , s[s[1, 0], 1, s[0]] is equivalent to s[1, 0, 1, 0] and so on. A Flat function has the mathematical property of being associative.)

And the pictures below show examples of what can happen if one starts with a flat disk and then adds different amounts of material in different places. … In the top row the disks are always flat, forcing the cells of material to vary in size and shape.

In order for the surface to stay flat its growth rate Log[h[x, y]] must therefore solve Laplace's equation, and hence must be a harmonic function Re[f[x +  y]] . … Note that if the elements of a surface are allowed to change shape, then the surface can always remain flat, as in the top row of pictures on page 412 . … And this may perhaps be related to the flatness observed in the growth of leaves.

When looked at as a waveform over time, shot noise has a flat frequency spectrum. … Like shot noise, thermal noise has a flat frequency spectrum. … Almost all electronic devices also exhibit a third kind of noise, whose main characteristic is that its spectrum is not flat, but instead goes roughly like 1/f over a wide range of frequencies.

The evaluation of functions with attribute Flat in Mathematica provides an example of confluence. If f is Flat , then in evaluating f[a, b, c] one can equally well start with f[f[a, b], c] or f[a, f[b, c]] . Showing only the arguments to f , the pictures below illustrate how the flat functions Xor and And are confluent, while the non-flat function Implies is not.

If one starts, say, from an ordinary continuous surface, then it is straightforward to approximate it as in the picture below by a collection of flat faces. … A surface approximated by flat faces whose edges form a trivalent network.

Given a flat interface, the layer of cells immediately on either side of this interface behaves like the rule 150 1D cellular automaton. On an infinitely long interface, protrusions of cells with one color into a domain of the opposite color get progressively smaller, eventually leaving only a certain pattern of cells in the layer immediately on one side of the interface. 90° corners in an otherwise flat interface effectively act like reflective boundary conditions for the layer of cells on top of the interface.

Properties [of logical primitives] Page 813 lists theorems satisfied by each function. {0, 1, 6, 7, 8, 9, 14, 15} are commutative (orderless) so that a ∘ b ＝ b ∘ a , while {0, 6, 8, 9, 10, 12, 14, 15} are associative (flat), so that a ∘ (b ∘ c) ＝ (a ∘ b) ∘ c .