Fun in the Sun - Using Mathematics to Tell Time on the Beach
Christmas is a week away, temperatures have been hovering in the
teens, and I'm writing about fun in the sun and something about
using mathematics to tell time on the beach! Good gracious. You
must be thinking how weird I am. Well maybe, but for those who
have been following some of my other articles, you know then
that I seek to show how mathematics--yes even basic
mathematics--functions so universally throughout our everyday
lives. Even the layman can use this beautiful tool do so many
common and ordinary things! Perhaps while I sit here clacking
at my keyboard, the chill running through my body forces me to
hark back to the bright halcyon July and August days when I
frolicked in the sand and listened attentively to the rhythmic
crashing of the waves--and of course, contemplated the hour of
the day by observing the position of the sun in the sky. Now
some might be thinking, "Wait. Navigators and sailors have used
the stars and celestial bodies for time immemorial to do such
things as tell time, pinpoint their position on the globe, tell
their course of direction, and to make sundry other
determinations." I know this.
Yet once again, while spending some time at the beach one day, I
wanted to see whether I could, without any prior research or
study in the field, nor prior knowledge of such, somehow devise
a method of telling the time using the sun. Now why would any
sane person wearing a watch on his wrist do this? Well for one,
to see what it might have felt like before we had such
conveniences as the wrist watch, and for two, to show that with
a little thought and some basic knowledge, man is quite a
formidable thinker! Passing this knowledge and experience on to
you will serve the purpose of aligning your thoughts more with
nature and to show you that you can do extraordinary things--if
only--you start thinking a little.
So there I am on the beach looking at the cloudless azure sky
and feeling somewhat bored, perhaps because the simplicity of
beach life was just too overwhelming that particular day. I mean
with all that sand between your toes, the constant beach chair
repositioning to afford a better tan, the incessant requests
from family and friends for water (having power of attorney over
the beach cooler, I was de facto water boy), beach life can be
somewhat enervating. Not to complain, mind you. Just that
sometimes beach time makes you realize that there are petty
annoyances no matter how good life can be. Hey, but what about
the time thing?
Oh, yes pardon the diversion. So I was looking up at the sky,
luxuriating in the warm caresses of the summer breezes that
wisped intermittently by, and I challenged myself by saying, "If
you're so smart, find a way to tell time using the sun. After
all, you're always preaching to students, friends, and family
about how useful mathematics is and how it can be used in
everyday life." Thus I took the challenge and began quickly to
make the usual assumptions, do some quick calculations, test the
hypotheses...and...lo, I came up with something. I reasoned
thus. Facing the water, I was looking due east. The sun rises in
the east and sets in the west, dropping off the horizon.
>From previous observations, I knew the approximate position of
the sun at the end of the day, which for us, was about 5-5:30
PM, when we would undertake the arduous task of packing up and
heading home. Using approximately 6:00 AM as rising time for the
sun and its approximate due east position, I constructed, using
my finger or a seashell, a circle--albeit crude--in the sand. I
reasoned that the sun should carve out approximately equal arcs
in equal periods of time while traversing the sky from east to
west, from rising, to its final setting position, before
dropping off the horizon. I sliced the circle in half with a
diameter, and this was done so that one end of the diameter
would point to the sun's rising, or due east, position, and the
other would point to the sun's setting, or due west, position.
Using the assumption about the sun sweeping out equal arcs in
equal time intervals, I then divided the semicircle into twelve
equal segments to allow for the time period during which it
would go from due east to due west position. Not having a
compass or other instrument to segment the semicircle equally, I
did my best to divide the 180 degrees in half into two equal
ninety degree segments. I then divided the two ninety degree
segments in half into two equal forty-five degree segments.
Finally, I trisected (divided in three) each of the four
forty-five degree segments to obtain twelve equal fifteen degree
segments. Having done this, I was left with thirteen radii (one
more than the number of segments) or "spokes"on my semicircle,
starting with 6:00 AM and ending with 6:00 PM. Each of these
radial spokes would represent an hour on the face of the
semicircle. At last, I had my crude sundial.
While basking in the sun, there I am looking up at this star
and lining up its position with one of the spokes on my sundial.
Ah, the sun is pointing right at the spoke corresponding to 1:00
PM. I look at my watch, and lo and behold, it's 1:05 PM! (Of
course this has to be approximate because of the method used.
You didn't think I could construct a perpetual motion Rolex in
the sand, did you?)
Yet after testing time and time again, this crude construct
works consistently. To be able to tell time based on nothing
other than a crude construct in the sand, and to be able to do
this within plus or minus ten minutes of the actual time, is, I
would think, quite a feat--especially in this day of jaded
excessiveness. Try it out and let me know your experience. Bear
in mind, you might have to tweak the location of the spokes, or
hour hands on the sundial depending on your exact location;
however, using the basic assumptions made here, you should prove
a marvel to your friends and family at the beach. And what a
great way to spend real quality time with the kids on the sand.
Until next time......