Dear John (and Edley),
Your experiment continues to fascinate me and I have some fresh
comments which include an experiment that everyone can carry our
very easily and which amplify Edley's remarks.
First, many thanks for your dimensions:
> the cardboard thickness: 3/32"
> the hole diameter: 1/4"
> the disk diameter: 2")
I got the external diameter wrong but that's not too important. My
guess at 0.1" for the internal diameter was almost exactly right!
Accordingly, my figures don't really need amending.
THICK VERSUS THIN
I have some comments on your practical points...
> ...if I were to make one for a real sundial, I'd use strong, thin
> metal instead of cardboard!
Interestingly, you don't have to use thin material for a disc nodus.
You could use quite thick material PROVIDED you taper the internal
and external rims to knife edges. Even a thick disc then works as
though it were paper thin!
CONES AND CLOCK HANDS
> On the practical and artistic level, I love the cone gnomons'
> shadows because they look like clock hands.
Yes, I very much accept this. The shape of the long shadow of
your cone is very elegant. It is such a shame that such a shadow
gets foreshortened when the shadows are long.
One thing that hasn't been suggested is to use TWO cones arranged
so that they meet tip to tip. Approximations to this arrangement
are not uncommon. I am thinking of statue sundials where perhaps
two fingers meet almost tip to tip.
AN EXPERIMENT ALL CAN TRY
This isn't what you have in mind when you seek a shape that looks
like a clock hand but it prompts me to suggest a simple experiment
that anyone can do anytime the sun is shining without any equipment
at all. Here's what you do...
1. Stand with your back to the sun about 6 to 10 feet from a
plane surface which is approximately facing the sun and
look hard at this surface. [The experiment doesn't work
well if the sun is shining through glass, especially
double-glazing, so do this outside or, at least, open the
window!]
2. Point your two forefingers at each other so that there is
about a 1" gap between them and arrange that the shadows
of the fingers fall on the plane surface.
3. Now, very slowly, bring the fingers close together. You will
find that, sometime before they actually touch, a mysterious
blob appears between the shadow fingers. The result is that
the shadow fingers appear to touch before the real fingers do.
This effect is, of course, because the sun is not a point source of
light. The critical moment comes when the angular separation of the
fingers becomes less than the angular diameter of the sun. The gap
between the shadows stops receiving full sunlight and becomes penumbra
instead.
You will get something of the same effect if you bring two of your
cones together tip to tip.
I mention all this to demonstrate that curious effects occur in
the vicinity of the shadows of tips. If you have a PAIR of tips
this doesn't matter too much. You can look at the symmetry and
estimate fairly accurately where the mid-point is. If you have
just ONE tip, estimating gets much harder. Edley's message
alludes to this difficulty.
BALL NODI
There is something else your experiments showed up that I hadn't
really appreciated before...
If you are going to use a ball nodus, then the supporting
stick should go RIGHT THROUGH THE BALL so that it sticks
out a little bit, perhaps half a ball diameter.
This thought struck me when I tried estimating the centres of
the shadows in printouts of your photographs. The point on
the shadow where the stick meets the ball is not matched by a
corresponding point on the far side. Once again, the lack of
symmetry makes estimation a little harder.
Often a ball nodus is mounted on a regular gnomon, perhaps
half-way along, so you get the symmetry for free.
Amazingly, I have somehow missed out analysing the shadows of
balls at the ends of sticks (rather than in the middle) so I
am most grateful to you for thrusting these images my way.
I have also become very impressed by the quality of PDF format.
I found I could enlarge your images many times without serious
degradation of quality.
All the best
Frank
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