----- Original Message -----
Sent: Saturday, September 24, 2005 3:53
PM
Subject: Numb3rs
List members who can view U.S. television may want to watch the CBS show
Numb3rs on Friday night, October 14 (10pm Eastern) for a sundial/shadows
tie-in. The title of the episode is "Obsession".
As you may know, the Numb3rs series deals with a mathematics professor
who helps his FBI agent brother solve crimes through the miracles of
mathematics. The producers make a real effort to stick to reality and
the solutions they present are intended to be possible in the real world (or
at least what Hollywood views as the real world!)
A few months ago, while planning was underway for the Oct. 14 episode, I
was contacted by one of the show's researchers. He wanted to know if it
were possible to use the shadow of a building in a date/time stamped photo as
a poor man's GPS system - using mathematics to locate the building
itself.
We had the requisite discussions about the fact that the shadow of a
building might be rather diffuse near its end point, that the slope of the
ground on which the shadow is cast could be a problem, that the time stamp
placed on a photo by a camera might be significantly different from the real
time, that the level of precision needed for their plot just might not be
realistic, etc., etc.
All of that came under the rubric of
suspension of disbelief - and we then had to address the basic theoretical
question.
I told him I didn't see how it could work with a single photo but that if
the script could be arranged to involve 2 photos taken at different times -
then we could do something. So we began working things out.
I suggested that they use a shadowcasting object shorter than a building
- initially they mentioned a skyscraper but I noted that it would be hard to
get the skyscraper and its shadow in one photo - and that the FBI would
probably be able to identify any such building in the Los Angeles area without
resorting to these measures. They would need something of known height
and they would need some way in the photo to measure the length of the
shadow. For the latter problem, I suggested that they include some
brickwork on the ground - knowing the standard size of the bricks, they should
be able to determine the length of the shadow falling across the bricks by
using the simple Pythagorean theorem. To cast the shadow, they
settled on a basketball hoop - assuming that it was set at the regulation 10
foot height, that the brick patio was level, that they could determine what
point on the hoop cast what point on the shadow, and how far that point was
from the pole supporting the hoop etc. etc.
With this done, Charlie (the math prof.) can calculate the sun's altitude
at the date and time of the photo(s); it's simply the arctangent of the
(height divided by the shadow length).
Since Charlie occasionally scribbles his equations on a blackboard as he
explains the math behind his solution, I provided the following:
sin(alt)=sin(lat)*sin(dec)+cos(lat)*cos(dec)*cos(tau-eot-long+merid)
(longitude increasing to the west)
Charlie calculates the solar altitude (alt) from the shadow and known
height of the hoop in the photo. The photo gives the standard time (tau)
and the date, which together give the solar declination (dec) and (eot).
Knowing that the photo is taken somewhere in the general area of Los Angeles
tells us the time zone of the location - therefore giving us the central
meridian (merid) of the time zone.
Having supplied these values, we now have an equation in two unknowns:
latitude (lat) and longitude (long). Graphically, we can represent this
equation by using the magic of television to draw a curve in the sky above all
the places whose latitude and longitude satisfy the equation. If we do
the same thing for the second photo we obtain a slightly different
curve. The two curves will intersect at a single point - directly above
the location at which the photos were taken.
The show is likely to come up with some graphic to illustrate these
curves in the sky. They went through several versions of dialog and
descriptions of the graphics - right up to the day they were shooting the
relevant scene, I was trying to talk them out of the idea that the curves
would somehow be the links from the shadow to the sun and that they would pass
through the sun. I hope they got it right - it was difficult doing
everything by word descriptions over the phone - and they really were enamored
with an incorrect image and had a difficult time understanding
the correct
one. But, hey, they tried - let's see what they came up with.
While we were going through all this, one of the show's creators
mentioned that if I wanted to donate a sundial that would fit nicely in
Charlie's office (or rather in the set used as Charlie's office), they would
be glad to add it to the collection of permanent props for the set. I
decided to send a brass/marble paperweight dial that is a reduced size
reproduction of an 18th century Richard Glynne sundial - I believe that the
original artwork for this reproduction was done by Fer de Vries.
They were thrilled to have the dial and I am told that it was placed in
the office on the day they were shooting a scene for the October 7 episode -
so sharp eyes might be able to spot the dial in the episode before
"Obsession". It will supposedly be a permanent fixture for the
series.
The script was also changed so that Charlie would contemplate the
solution to the problem while picking the sundial up from his desk - so it
should be obvious in this episode - and making it clear that it was the
sundial that gave him the inspiration to solve the problem.
I suggested that if they really wanted to show their appreciation for
coming up with a solution to their dilemma, they could have Charlie mention
NASS in the program. This idea didn't fly with the original writer, but
it evidently was put in when the producers took over responsibility for the
episode. As of the day of shooting, the script called for Don (Charlie's
brother) to ask how Charlie and Larry (Charlie's cosmologist friend) came up
with this solution. Charlie and Larry both turn to each other and
observe that they are card-carrying members of the North American
Sundial
Society!
Whether or not that bit of dialog made it onto the film and through the
editing department remains to be seen. But it just might have made
it. Watch on October 14 to find out.
Fred Sawyer
