But the notion of abstract functions in mathematics reached its modern form only near the end of the 1800s.

Rough analogies between the forms and functions of biological and non-biological systems were fairly common among both artists and scientists, but were rarely thought to have much scientific significance.

In 1917 D'Arcy Thompson mentioned that leaves might have growth rates that are simple functions of angle, and drew the first of the pictures shown below.

(The function s[d] has a maximum around d = 5.26 , then decreases rapidly with d .)

. + 0.32 α - 0.067 α 2 + 0.076 α 3 - 0.029 α 4 + … The comparative simplicity of the symbolic forms here (which might get still simpler in terms of suitable generalized polylogarithm functions) may be a hint that methods much more efficient than explicit Feynman diagram evaluation could be used.

Nevertheless, the approach was used with some success, particularly in proving that various mechanical and other engineering systems would behave as intended—although by the mid-1980s such verification was more often done by systematic Boolean function methods (see page 1097 ).

Note that as on page 1007 packings can be constructed in which the sizes of circles vary smoothly with position according to a harmonic function.

Then assume that a polarizer oriented at 0° will measure the spin of such a photon to have value f[ ϕ ] for some fixed function f .

Discretizing yields lattice gauge theories with energy functions involving for example Cos[ θ i - θ j ] for color directions at adjacent sites.

Within say a surface whose points {x 1 , x 2 , … } are obtained by evaluating an expression e as a function of parameters p (so that for example e = {x, y, f[x, y]} , p = {x, y} for a Plot3D surface) the metric turns out to be given by (Transpose[#] . # &) [Outer[D, e, p]] In ordinary Euclidean space a defining feature of geometry is that the shortest path between two points is a straight line.