Hi Charles: Your explanation makes a great deal of sense, and it is easy to understand! There is so much e-mail coming in on this question, that I think I'll wait to hear what everybody has to say until I make up my mind as to what is going on.
Because nobody seems really clear as to what the exact mathematical explanation is, (or an easy-to-use formula that predicts bead and hole diameters based on desired focal length), then maybe the best way to design bead-in-a-hole styles is by experimentation, not math. (at least for mathematically challenged people like me!) Thanks, John Carmichael >>The design which worked the best was a 1/8 inch spherical bead, suspended by >>thin brass crosswires, in the exact center of a 1/4 inch round hole. (The >>style was about 24 inches from the analemma). >> >>A very curious thing happens with this type of style. The bead alone, by >>itself, casts a shadow that was twice as big as the bead; but when the 1/8th >>in. bead is in the center of a 1/4" hole, with a space of 1/16th of an inch >>between the bead's edge and the hole edge, the bead's shadow miraculously >>sharpens into a tight, dark shadow that is only 1/16th of an inch in >>diameter, smaller than the bead itself!!!! The wires which keep the bead >>suspended in the middle of the hole are so thin that they don't cast a >>visible shadow onto the analemma. > ><snip> > >>I don't know why this works, but it does. Can any of you explain this? >> >>John Carmichael > >Although there may be some slight effect due to diffraction, it is >negligible. The open ring between the bead and the edge of the hole can be >described another way, as follows: > >The open ring is comprised of an infinite number of round holes, 1/16th >inch in diameter, each with it's center on the circumference of a circle >3/16ths inch in diameter which is coaxial with the bead and the hole. Each >of these 1/16th inch holes acts as a shadow sharpener (pinhole) which >projects an image of the sun onto a surface. On the surface, there are >then projected an infinite number of images of the sun arranged in a >circle. The images overlap each other to form a bright ring with a dark >center. As the bead/hole aperture is moved further away from the surface, >the images of the sun will grow larger, overlapping more and more which >causes the dark center to diminish in size. If the aperture is moved a >large enough distance away, the dark center will disappear and the infinite >number of sun images will almost merge into a single image of the sun. I >say almost, because the centers (indeed any point on the sun) will still >form a ring 3/16ths inch in diameter. > >Does that make any sense? > >Charles > >
