I have been lurking here, watching this thread. Here is some pertinent material I abstracted from a paper by Kazuo Sayanagi.

This information is from a paper titled: Pinhole Imagery, by Kazuo Sayanagi, published in September 1967 by the Journal Of The Optical Society Of America, Vol. 57, No. 9 1091-1099. In this paper, Sayanagi presents a modulation transfer function approach to calculating optimum pinhole diameters for various different conditions.

From Sayangis paper:

The pinhole has been used as an imaging device for centuries. The oldest description of pinhole imagery I could find was written by the Arabian scholar Ibn Al-Haithan (A.D. 965-1038). He described the use of a pinhole to observe the projected image of a solar eclipse. He pointed out that the finest imaging with the pinhole can be obtained by using a very small hole and discussed the image quality when the size and shape of the hole are changed, as follows:

The image of the sun only shows this (crescent shape) property when the hole is very small. If the hole is larger the image changes, and the change is more marked with increasing size of the hole. If the hole is very large, the crescent shape of the image disappears altogether, and the light becomes round if the hole is round, quadrangular if it is quadrangular, and with any shaped opening you like, the image takes the same shape, always provided the hole is large and the receiving surface parallel to it.

The first physical consideration of pinhole imagery, based on a mixed treatment of geometrical and physical optics, was written by Petzval (1857, 1859). He expressed the diameter D of the image point made by the pinhole as the sum of the geometrical diameter of the aperture d and the diameter of the diffraction pattern caused by the aperture d,

D = d + k l lambda / d

Where k is an optimal condition constant (Petzval chose k=2), l is the distance between the pinhole and the receiving plane and lambda is the wavelength of the incident light. The optimum diameter of the hole is defined so as to give the minimum diameter of the point image.

By solving the resulting partial differential equations, Petzval obtained:

d squared = k l lambda, with k = 2, where d is the optimum diameter of the pinhole.

Based on Fraunhofer diffraction in the presence of defocus, Lord Rayleighs analysis (1891) arrived at a k value of 3.6

Sayanagi (1967) used a modulation transfer function approach to calculate a general-purpose k value of 3.8

These approaches all assume an infinitely thin pinhole.

Equations don't work well in this text editor, Sean.