Hi Frank (cc. SML)
Glad to be of help.
(For your calculations, here are the dimensions of the hole-in-disk nodus.
the cardboard thickness: 3/32" , the hole diameter: 1/4", the disk diameter:
2"). Of course, if I were to make one for a real sundial, I'd use strong,
thin metal instead of cardboard!
On the practical and artistic level, I love the cone gnomons' shadows
because they look like clock hands. Look at a traditional clock hand. It's
basically shaped like a long skinny triangle, wide at the center of rotation
and pointing to the time. A cone's shadow has the same shape does the same
thing.
In fact, you could even make the cone a pointed teardrop or spindle shape or
with indentations that would produce more interesting shadow shapes that
look like differently shaped clock hands.
----- Original Message -----
From: "Frank King" <[EMAIL PROTECTED]>
To: "John Carmichael" <[EMAIL PROTECTED]>
Cc: <[EMAIL PROTECTED]>
Sent: Monday, August 07, 2006 2:30 AM
Subject: Re: Expanded Nodi Shadow Experiment
Dear John
Many thanks for your response.
I'm still pondering all of the information you gleaned from
my experiment.
Well, I am still pondering the experiment! I have often thought
of setting up almost exactly that experiment but never quite got
round to it! You have certainly done it very elegantly. What
is amazing serendipity is that I have recently been doodling with
a dial design which requires a nodus about 4" from the dial plane
so you saved me a whole lot of work!
I need to point out something important about the light
projection (or antishadow) cast by the hole-in-disk nodus:
As you can clearly see in the long shadow photo, the spot of
light (the antishadow) is not there at all.
This is true. As you will see in a moment, the place where the
spot of light should be is getting no sun at all through the hole
but it IS getting some sun from round the edge of the nodus disc!
This I think was due to the thickness of the cardboard that I
used to make the disk.
Indeed it is. For completeness, there are three factors which
determine whether a hole-in-a-disc nodus will project a spot
of light:
1. The diameter of the hole
2. The thickness of the material (you are right here)
3. The diameter of the surrounding disc (if this is too
small it won't block out all the sun which is one
factor in your long-shadow case)
As a designer, you have to chose both the internal and external
diameters of the disc-with-a-hole carefully. You can easily
sketch what's going on. Just draw a right-angled triangle
4" high and 23.3" long and sketch in the cross-section of your
nodus at the top of the 4" side. You will soon see why no
light is getting through!
The golden rule in the analysis is to consider how the nodus
appears when viewed from the centre of the shadow (or anti-shadow).
You don't give the diameter of your disc or its thickness so I
cannot give a precise analysis. If you could let me have these
dimensions I could easily explain more! Here is a first stab...
Let's guess that the disc diameter is 1.25"
We CAN assume the hole diameter is 0.25"
Let's assume that the thickness is zero (just for the moment)
Using my previous analysis, the appearance of the NODUS to an insect
sitting at the centre of the anti-shadow will be an elliptical disc
with an elliptical hole. In your LONG shadow experiment, when the
shadow is 23.3" from the foot of the nodus support and 23.6" from
the centre of the nodus, we get the following angular dimensions:
1. Apparent major axis of disc 1/19 radians
2. Apparent minor axis of disc 1/112 radians
3. Apparent major axis of hole 1/94 radians
4. Apparent minor axis of hole 1/558 radians
5. Diameter of circular solar disc 1/107.5 radians
If you draw a scaled picture so that the solar disc is (say) 1" in
diameter, you will find that the minor axis of the nodus disc isn't
quite big enough to hide the sun (1/112 is smaller than 1/107.5).
Every point in your shadow can actually see a little bit of the sun.
Nowhere is in full shadow.
In short, the disc isn't big enough to hide the sun completely and
the hole isn't big enough to expose the whole sun. In fact the hole
exposes less than one-fifth of the sun even in the zero-thickness
case.
If the disc is made of thick material the insect sees even less of
the sun through the hole. Instead of appearing as an ellipse, the
hole appears as a lens shape. [Try holding a tube up to your eye
and angling it slightly so you don't look straight through it.]
The angular dimension of the minor axis is now given by the formula:
d cos(i) - t sin(i)
------------------------
nodus-to-shadow distance
d is the diameter of the hole
t is the thickness of the cardboard
i is the angle of incidence
In the case of your long shadows, the nodus-to-shadow distance is
23.6" and the angle of incidence is 80.3 degrees.
Clearly a critical value of t is when d cos(i) = t sin(i) or:
critical-t = d/tan(i)
With i = 80.3 degrees the critical value is a mere 0.04". Any
thicker than that and you get no light through at all! And your
cardboard is MUCH thicker than that!!
In fact, when I was tilting the board, it disappeared when the
shadow was 3 times longer than the gnomon.
Well tan(i) is then 3 and 0.25/3 = 0.08" which all adds up. If
your cardboard is just 0.08" thick no light will get through
the hole when the shadow is 3 times longer than the gnomon.
If I had used a paper thin metal to make the disk, then I think
the antishadow would not have disappeared so quickly.
Correct.
My conclusion about this is that the hole-in-disk works fine and
is extremely precise at high solar angles when you can see the
anti-shadow, but is completely useless at low solar angles when
the antishadow disappears.
Do you agree?
Not entirely!!! I have two reasons:
The anti-shadow doesn't have to disappear. If you make your
hole 1.25" in diameter and the disc about 3.5" in diameter you
will find the anti-shadow is just fine even at 23" from the
foot of the 4" gnomon.
You will now say, quite rightly, that this is a huge hole and will
give an enormous anti-shadow when the shadows are short. OK, make
the hole smaller and...
Be happy about the anti-shadow disappearing! It doesn't
matter! You use the anti-shadow at low angles of incidence
when you can see it precisely and you use the shadow of the
disc as a whole at high angles of incidence when the anti-shadow
disappears!
Of course you have to chose both the internal and external diameters
carefully as I said earlier.
I love your term "antishadow" to describe the projected sunlight
through an aperture nodus. Do you invent this term?
I would love to claim that I did but it wouldn't be true! I first
came across the term anti-shadow on this list three years ago but
I cannot remember who wrote the message.
Noting that Italian seems to have a richer dialling vocabulary than
English, I asked Gianni Ferrari whether there was an Italian
equivalent of anti-shadow.
Sadly his answer was no. I hoped that anti-ombra might be used
but this was wishful thinking.
All the best
Frank
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