Dear John, That is a splendid experiment...
> http://advanceassociates.com/WallDial/NodusShadowExperiment.pdf It illustrates all kinds of interesting aspects of nodus design The Purpose, Setup and Execution all earn top marks. The Conclusion, though, is subject to a little caveat... Let's concentrate on just three of your designs: the disc with the 0.25" hole at the top (or leftmost), the cone at the bottom (or rightmost) and the 1" ball next to the cone. Now look at the two sets of shadows: 1. When the shadows are short... (a) the centre of the anti-shadow of the disc with a hole is about 6.1" along your board. (b) the centre of the shadow of the ball is just a little shorter. It seems to be almost spot on the 6" mark. [This is possibly because the supporting stick is not quite vertical. This is not important.] (c) the shadow of the tip of the cone is almost exactly in line with the anti-shadow of the disc with a hole, about 6.1" along your board. 2. When the shadows are long... (a) the anti-shadow of the disc with a hole is no longer clear (as you say) but because there is an equal amount of fuzz at the extremities of the shadow of the disc as a whole you can fairly easily estimate the centre. It seems to be about 23.3". (b) the centre of the shadow of the 1" ball is just a little less easy to estimate because the supporting stick disturbs the fuzz at one of the extremities but one can see that the centre is about the 23" mark. This, as expected, is shorter than the shadow to the centre of the disc and is consistent with 1(b). So far everything ties up. (c) the shadow of the tip of the cone though has now fallen behind the shadow of the centre of the disc. The shadow may be easier to read but IT IS GIVING A FALSE RESULT. The big big trouble with any asymmetric nodus is that you cannot cancel out the fuzz. You have to decide just where in the fuzz is the point of interest. This is difficult. Different people will estimate different points. As noted at 2(b), each of your ball nodi is slightly asymmetric because of the supporting sticks. If you had mounted the balls sideways (as you have the disc with the hole) it would be easier to estimate the centre of the shadow. To my mind, the disc with the hole gives the most accurate result even if its shadow isn't the prettiest! Incidentally, it is worth analysing the hole in your disc in the long shadow case... Diameter of hole 0.25" Height of hole above the board 4" Approximate length of shadow 23.3" Distance of centre of anti-shadow from centre of hole 23.6" Angle of incidence arctan(23.3/4) = 80.3 degrees Now consider the hole viewed from the centre of the anti-shadow. Given that the disc is parallel to the board, the hole will appear as an ellipse whose angular dimensions in radians are: Major axis 0.25/23.6 approx 1/94.4 radians Minor axis 0.25 x cos(80.3) / 23.6 approx 1/558 radians This last figure should be compared with the angular diameter of the sun which (by a diallist's rule of thumb) is about 1/107.5 radians. Now imagine an insect (wearing eye protection) at the point where the centre of the anti-shadow should be. As seen by the insect, the minor-axis of the hole appears to be less than one-fifth the diameter of the sun. The anti-shadow is entirely penumbra and impossible to detect. In my opinion this is NOT a design error. My eccentric view is that a disc with a hole IS the best form of nodus, especially for big sundials (well ALMOST the best) because... when the angle of incidence is small (short shadows) you observe the centre of the anti-shadow and... when the angle of incidence is high (long shadows) you observe the centre of the shadow of the disc as a whole. I say ALMOST the best because there is a special case of a disc with a hole, and that is the great camera obscura sundials which one comes across in Europe. There the angular diameters of the holes are even smaller than yours. [According to Gianni Ferrari, Cassini took the view that the hole should be 1/1000th of its distance to the floor, half the size of your hole at 23.6".]. The disc though is effectively of infinite diameter because the entire building surrounds the hole and you DO see a splodge of light. This is not anti-shadow though; it is actually an image of the sun and you can estimate its centre VERY precisely. Try making your disc of infinite size and you will see how this works :-) MORAL: Sticks and cones are seductive but should be resisted! I hope this hasn't been too tiresome a message for this list! Frank H. King Cambridge, U.K. --------------------------------------------------- https://lists.uni-koeln.de/mailman/listinfo/sundial
