Math is everywhere one chooses to look for it: As a mathematician and
mathematics teacher, I take that phrase very much to heart. I am happy
whenever I encounter new mathematical issues, particularly when they arise
in the world beyond mathematics and the sciences, and I take pleasure in
sharing these observations with others. It is especially gratifying when
mathematical ideas that I have talked about in class crop up in connection
with running.
This past summer I taught a course in elementary number theory for the first
time, and for a while I despaired of finding any connection at all between
our sport and the so called Queen of Mathematics.
But never fear! Let's say I'm jogging on our 200m indoor track
and I notice that I lap a group of walkers about every second lap. Can I
explain this using some arithmetic ( beginning number theory) and
some reasonable assumptions about our respective paces? (Perhaps I should
define the verb "to lap" for those who are unfamiliar with it: Whenever a
runner overtakes another runner on a track with both heading in the same
direction, the overtaking runner is said to "lap" the other.)
The key point here is that between my lappings of
the group of walkers I have run exactly one more lap than they have walked.
Since I've run two laps in that interval, they have walked one. Accordingly,
I'm going twice as fast as they are. Now my jogging pace, if I'm honest about
it, is between 8 and 9 minutes per mile (increasingly more toward the latter,)
making their walking pace 16 to 18 minutes per mile -- quite reasonable for
a group of fitness walkers. (Exercise: did the size of the track matter at
all in the reasoning?)
A few simple questions about runners moving at the same speed lead to more
interesting issues.
1) Suppose two runners start out together and travel at equal speed but in
different lanes. Experience shows the runner in the inside lane will
gradually pull ahead and eventually lap the outside runner, because it is
further around the track in the outer lane. How far will the runners
have gone when this lapping occurs? (Assume the lane numbers of the runners
to be given.)
2) Same question, but now we require that the lapping take place exactly
at the point on the track where the runners started. That is, if the lapping
from question 1 occurs somewhere on the far side of the track, as it may well
do, then the runners continue on, perhaps lapping many times before meeting
at the starting line.
The dual role of the word "lap" as noun and verb tends to get confusing, so
let's coin some other words to define situations where one runner
overtakes another on the track. When this occurs anywhere on the track, I
shall term it a "crossing." When it happens at the start line, as in
the second problem, I can't resist borrowing the colorful Greek word
"syzygy", a technical term from Astronomy for the situation where
3 or more moons or planets line up.
To discuss the problems, let c = the distance per lap in the inner lane
(for example, c = 400m is common for the innermost lane,) and let d denote
the additional distance per lap in the outer lane. Then the inside runner
will lap the outside runner on his lap number x (which need not be an integer,)
provided dx = c, making the distance run cx, or c^2/d. (c^2 = "c squared", or
c times c.)
Simple as it is, this reasoning overlooks one minor subtlety having to do
with the shape of most tracks. The reasoning is correct for exactly
circular tracks, but most tracks have semicircular curves joined by 100m
straightaways. Let's imagine the overtaking runner is 5 meters behind at
the beginning of the final lap before crossing. Then d > 5, and if the
track is circular the gap will be made up at constant rate, so the crossing
will occur 5/d fraction of the way through the last lap. But if 100m
straightaways are present, and if d > 10, then a gap of d/2 - 5 will remain
at the end of the first curve, and no further reduction will occur
until 100m later at the start of the second curve.
Taking these observations into account, a correct result for such tracks is
c^2/d - 200{c/d} if {c/d} <= 1/2 (where {} denotes fractional part,) and
c^2/d -200{c/d} + 100 in the contrary case.
Using some common measurements for outdoor tracks (c = 400m, lanes 1.07 meters
wide,) runners in lanes 1 and 5 will cross at distance 5,874.86m, or 14 laps
plus another 274.86 meters.
It might make an interesting pacing workout for runners of equal ability
to test these numbers on the track. Start at the beginning of a straightaway
in order to match paces and then attempt to continue on at exactly the same
pace until you cross again.
I don't recommend a workout similarly based on the second problem. Here's why:
Syzygy occurs when the runners cross, having each gone an exact integer
number of laps, n and m, say. This requires that nc = m(c+d) for natural
numbers n and m, which can only happen when d/c is a rational
number. Unfortunately, assuming exact measurements and standard tracks, d/c
is normally a rational multiple of pi, and therefore irrational. Exact syzygy
will NEVER occur!
If we only require that a crossing take place within e meters of the start
(think: e small,) then we enter a subfield of number theory called
Diophantine approximation. (After Diophantus of Alexandria, a 3rd century
mathematician.) With x = d/c, the issue boils down to
approximating x to within e/c amount of error using a fraction of minimum
denominator, i.e., to find natural numbers m and n so that
| m/n - x | < e/(nc),
with n as small as possible. See, e.g., chapter 11 of Hardy and Wright, The
Theory of Numbers, Oxford University Press, London, 1938, for more information
on such problems.