Indeed after a patient search I have it now in my hands, (August 1991, pages 104-106) Mathematical Recreation Column

What in the Heaven is a Digital Sundial? by Ian Stewart,

At the end as further reading indicates the book of Kenneth Falconer, you have mentioned

In retrospect, rather than on the sundial I could not get finished, I remember being more interested on the three different shadows in form of letter (G, E, and B) cast by the very same object, which looks like a particular cube adequately carved. I believe there is a cover of a math book about Godel Incompleteness Theorem with the same picture.

In same way it is like a old familiar puzzle or toy, in which you must find plug for or put it through three holes each one bored in a board with the shapes of triangle, a circle and square, the solution or the special piece of the toy has the shape of a conoid.

So given a body which equation is U=U(x,y,z) to get its projection on plane XY, we remember first we must get grad U and its scalar multiplication by vector k normal to XY must be zero what means partial differential of U with respect of z must be null. Now in order to obtain a F1(x,y) = 0 the contour equation of the shadow on plane XY, we should eliminate variable z between the two equations: U = U(x,y,z) and Du/Dz = 0.

Obviously we will do the same for the other to plane, and we shall obtain the other two contours F2(x,z) = 0 and F3(y,z) = 0.

Once it is said the above, really we are to confront with the reciprocal problem: given F1, F2 and F3 find U, and this problem admits infinite solutions.

Imagine a truncated cone which can cast a shadow of the bigger circle at the base and two trapezes, we can substitute the top smaller circle for an square which sides parallel to the axis and equal to the diameter. This body will cast the same shadows, but equation for the lateral surface it can be arbitrary chosen with the only restrictions of the bases and the outer contours.

However the problem will has an easy solution if instead of a surface we can consider that our set of points belong to a curve the solution it is immediate if the equation of the curve is expressed in parametric for x=x (t); y=y (t) z=z (t) taking by pairs will render at once the shadow on each plane.

Otherwise if the curve is expressed like W1(x,y,z) and W2(x,y,z) and we want the projection on plane XY we will eliminate z from the two equation an we will get W(x,y)=0 and z=0 .

If the contours are given as F1(x,y)= 0; F2(x,z)=0 and F3(y,z)=0. The more convinient way to go is express each of the above in parametric form, using the same parameter, by doing so also will give us the advantage to built a wire model of the desired curve.

Probably you knew all the procedure above, but I got so excited when my memory was able to kick back to the article of Scientific American of August 1991, that I need to communicate my finding.

Just for giving you an idea of how I tred to built my digital sundial. I intended this two techniques none related to fractals in which actually I am just a layman.

The main problem in my opinion is that shadow is a continuous process, but you must discretize to make it digital. In that sense I kind to understand why the fractals technique has been put in place, since its periodicity and the big sudden change in the output variable just in the narrow vicinity of those critical values of the input variable, what it'should work like a sort of escapement mechanism in a mechanical clock, but for a minutes, not for seconds.

You probably are aware that once you have obtained the correct latitude and longitude of the place where you plan to set the sundial. Time-equation is the main problem for get a decent accuracy besides the size of gnomon.

To take care of the time equation I planned to use an analemmatic sundial, so a very narrow beam like a laser will hit a plastic board like a transparency on which a special grid with the time hour and minute were printed, so the beam will illuminate just ideally one cell, of course since the process is continuous, you have an overlapping what indeed is not truly digital,besides size is the oder problem.

My second approach was like setting series of vertical slats which angle is variable between two consecutive ones, with the purpose that the sun as it is changing its azimuth only just one of the slit between two consecutive slats let the sunlight beam to pass. It does work.

As you can imagine you need a lot of slats to take care of the minutes and since you do not have correction for the time equation is rather impractical,beside being a cumbersone artifact, paradoxically you intend your display be accurate to the minute, however you can be off for more than 10 minutes because of the needed correction to consider the time equation.

Being knowledgeable that former Dean and Professor emeritus Dale Corson at Cornell University built a sundial with 30 second of maximum difference with respect to the real time, and sundial was at the campus, I tried to see how it is work, taking advantage of my son being college student there for the last four year . Impossible task during this period the sundial has been in storage due to a major rebuilding project in the campus and Mr. Corson an elderly man was not available either.

Thank you for the link and for your patience reading the above, Englih is my second language.

>On his personal page at St. Andrews University,
>he describes the idea as "a set with essentially any desired
>projections in all directions."

I haven't looked at this in detail, but there certainly must be some constraints on this in the case of 2d projections. For example, consider a 3d set lying inside the cube -1 < x,y,z < 1, and having a projection onto the yz plane that is disconnected, so that one part has strictly x > 0 and the other strictly x < 0. In that case, the original 3d set cannot have any points with coordinate x=0, and hence every projection onto a plane containing the z axis must also be disconnected.

This reminds me of the moving hologram I saw many years ago now at the Seattle Science Center. It was a reflection hologram in the shape of a cylinder, and held the apparantly 3d head of a young woman. As you walked by it, the changing angle of the light caused her to blow a kiss and wink at you. You could walk all the way around it and follow the action. At the time I saw this, I thought that if you made a long strip hologram like this, you could make a continuous movie with no individual frames. Of course, this would probably not be any better than ordinary movies because of that, but it would be genuinely 3d.

It'seems to me that the same technique could be used to make a holographic digital sundial, where the changing angle of the sun's rays would control the image you would see.

I am also reminded of an article a few years ago in Scientific American about holographic computer memory. In this case, there is a solid cube which is written to by lasers, and changing the angle of the writing laser allows you to record many items of data in the same physical space. To read the information back out, you illuminate the crystal with a weak laser at the same angle as the one that wrote the information.

This method works because the way the information is encoded in the crystal is as the Fourier transform of the information specifying the beam angle. In much the same way that a spectrum associates an amplitude with each frequency of light, this method associates a number with each beam angle. (It is a standard theorem in optics that an arbitrary traveling wave of a given frequency can be decomposed into a superposition of plane waves at the same frequency. This is closely related to the Fourier transform, but transforms the spatial frequency, not the temporal frequency of the waves.) The orthogonality property of the Fourier transform then ensures that when a laser illuminates the crystal, it picks up no information that was written by beams at other angles, and only sees "its own" information. One could also of course make a digital sundial this way, provided you filtered the sunlight to a narrow range of frequencies first.

Both of these methods do not involve fractals in any way, because they take advantage of the wave nature of light, instead of treating it as rays. The cylinder hologram (or crystal) is not of course at all related to the point set with arbitrary projections you mentioned earlier. It is essentially a set with arbitrary Fourier projections, instead of arbitrary geometric projections.

--Stuart Anderson

"one part has strictly x > 0 and the other strictly x < 0. In that case, the original 3d set cannot have any points with coordinate x=0,"

should read

'one part has strictly z > 0 and the other strictly z < 0. In that case, the original 3d set cannot have any points with coordinate z=0,"

Sorry for any confusion!

--Stuart Anderson