FOURIER TRANSFORMS

Transform of a set of planes is a row of points.Transform of a lattice is a lattice.

Transform of a single object is a continuous distribution.

A convolution is a thing repeated by the rule of another thing.

e.g. a crystal of molecules is the molecule repeated at all points of the crystal lattice.

Transform of a convolution of two functions is the product of their individual transforms.

A product (multiplying one function by another) is each point of a function multiplied by the equivalent point of the other function.

Transform of a lattice populated by objects is the transform of the object
sampled at the points of the transformed lattice.

Here the transformed "object" function is a continuous distribution and the transformed "lattice" is a lattice with finite values only at the lattice points (zero everywhere else).

Getting it on the blackboard or on “film”

Transform of a projection of a 3D object is a planar section through the 3D transform of the object. (If stay in plane for transform: third dimension position of features of the object don’t matter.)

Transforms for Dummies:

Transform of Short and Fat is Tall and Thin.

All these are reversible.

fftoys.html: Ian's java fourier transformer featuring 63beetle along with other objects

(20121005: Place holder for suggestions of things to do in fftoys.

Also need better selection of examples for fftoys' preloaded images...)

DNA double helix structure deduced from fiber diffraction

fftoys: show solidhelix, segmentedhelix, doubleis2dot1pixel, double helix

solid helix transforms to X

X convoluted with row of wide 1/d spaced (1pixel)dots becomes 1/d repeated X pattern

row of wide 1/d spaced dots transforms to stack of narrow d spaced planes

solid helix multiplied by stack of narrow d spaced planes becomes segmented helix

segmented helix transforms to repeated X pattern

segmented helix convoluted with 2 (3/8)P spaced dots becomes double helix

2 (3/8)P spaced 1 pixel dots transforms to stack of (8/3)(1/P) spaced fat planes

repeated X pattern multiplied by stack of fat planes becomes same repeated X pattern except at the null point between the fat planes, which zero the 4th spot

double helix transforms to the final X pattern with missing 4th spot

Final X pattern matched to actual xray pattern from B-form DNA shows missing 4th spot

The concept of building up the image (of the molecule(s) in the unit cell) is to first uniformly fill the box with average electron density, then one-by-one add in standing waves where density is either added or subtracted according to the position and height of each wave. The VW Beetle figure shows starting this process using an example suggested in “Introduction to Macromolecular Crystallography” by Alexander McPherson, who also once had a ‘63 beetle. (His was yellow, mine was red.)

The first 3 components and their sum are sketched along the side. Note that the 0 (unmeasureable straight forward ray) is needed to make the resulting real space density all positive.

bottom 5, Fourier Transforms