It turns out The Simpsons has little math and science jokes making their way into the show - usually as very short sequences that fans can stop and study. In a favorite episode Homer scribbles on the blackboard leaving four items.1 398712 + 436512 - 447212 immediately caught my eye .. it smelled like they were hinting at Fermat's Last Theorem.
There are an infinite number of equations of the form x2 + y2 = z2 .. like 32 + 42 = 52. Fermat hunted in vain for x3 + y3 = z3 and later claimed xn + yn = zn has no solution for whole numbers n > 2. In the margin of a book he stated there are no whole number solutions for the infinite number of equations of that form and then added Cuius rei demonstrationem mirabilem sane detexi, hanc marginis exiguitas non ca- peret - 'I have discovered a truly marvelous proof of this, which this margin is too narrow to contain'. Nasty. Frustrated mathematicians toiled for hundreds of years until it was finally proved in a very non-trivial fashion in the mid 1990s .. indeed it wouldn't fit in the margin of a book:)
I pulled out my old HP calculator and tapped in the numbers. Sure enough it worked. It appeared to be a counterexample .. enough to disprove the proposition. You suspect a near miss .. something lost as a rounding error. The next step was to go beyond the ten digit accuracy of the calculator. I reached for the laptop and fired up Mathematica
the first suggests the first two terms equal the third, but the second shows they aren't in the twelfth digit .. looking at the first fifteen digits gives a bit more. For most purposes it isn't necessary to have that many digits of accuracy .. but there are times when the question requires a bit of deeper knowledge - in this case the precision of a calculator. But swimming pools?
There was a bit of a kerfuffle in Rio trying to explain ties in swimming events. Tightly spaced finishes - often to within a few hundredths of a second are not uncommon and every once and awhile you get two and even three way ties (Phelps, Le Clos, and Cseh in the 100 meter Butterfly). Identical right down to the hundredth of a second. That can't be right can it? Someone must have touched first. The media mostly butchered the explanation. Going to thousandths of a second like speed skating and a few other events - seems to be the way to go, but closer examination turns up a flaw.
It turns out swimming pools aren't rigid. They can't be or they'd break given what they're made of. The ground around them shifts as does the water. Even a degree change of water temperature causes a change of size. In a 50 meter pool the time of the fastest event has a swimmer moving at about 2.39 meters per second. In a thousandth of a second - a millisecond - the swimmer would travel about 2.4 millimeters. It turns out the allowable tolerance of the length of a lane is 3 centimeters - something that has to be re-established several times a day during an important event. The pool can vary by more than ten times the distance the swimmer travels in a millisecond .. timing at that level becomes meaningless. Records couldn't be compared with each other at that precision and even the lengths of lanes in the pool can't be guaranteed. 2
Which brings up another problem. How do you have a fair start? A starting gun goes off and runners or swimmers start out. The bang of a gun travels at the speed of sound - you probably know the rule of thumb of counting seconds between the flash of lightning and its thunder and multiplying by one thousand feet. The speed of sound varies with temperature, pressure and humidity, with the standardized value of about 343 meters per second. Sound travels a about meter in three milliseconds. Imagine a starting gun at the side of a track. The path from the gun to a runner can vary by up to fifteen meters in some situations - that's 45 milliseconds .. over four hundredths of a second and a big deal in some of the shorter events.
You might get around that by flashing a light, but the problem is we're wired in such a way that we respond to sound faster than to light - and by a lot ... like 50 milliseconds. The solution is to put a speaker under each starting block and connect it to an electronic gun. But another interesting question comes up. How do you judge a misstart? That one works out nicely. Elite runners have a reaction to the starting gun of about 145 milliseconds and remarkably similar to each other. A misstart is judged at something shorter than what reaction time would allow .. usually 100 milliseconds. Some runners have very slow starts - reaction time and just getting the muscles going .. Usain Bolt is one of the slowest, so go figure...
And for fun consider a problem I've given to students... we all know refrigeration is inefficient and that opening the door of a refrigerator is a way to heat a room. What would it take to detect the effect of a refrigerator in an average house? What would it take to detect the impact of a single refrigerator on the power grid? The first case isn't too difficult .. the second case demands some very specialized knowledge.
A thread connects these examples. If you want to make an accurate measurement you need to understand something about what you're measuring and how you're measuring it. The measuring tool, what you're measuring, its environment and even how they interact over time. There are limits and when you bump into them it is useful to realize something is amiss so you know if you need to fix it or not - or if the situation needs a complete rethinking.
1 I suspect this was aimed at physicists as the two physics jokes are a bit deeper than the math jokes. two math and two physics. The first has terms you'd see in particle physics - and suggest they predict the mass of the Higgs boson (this was fifteen years before it was confirmed). The terms may look normal, but the equation don't make sense. If you plug in numbers it does give about 775 GeV which isn't terribly far from the real figure of 125 GeV. (admission - I did just that immediately) The third, Ω(t0) > 1, suggests the universe will implode .. when things in the room start rushing in on each other Homer replaces the > with a < which causes a major explosions. The fourth is a topology joke. A circle and a triangle are the same, technically homeomorphic, in topology. Consider a circle made of a perfectly pliable rubber. You could easily stretch it into a square, pentagon and almost anything else without holes (that's a bit of a stretch, but you get the idea:-) An object with a hole in it, a doughnut for example, can be stretched into a coffee cup - one hole that goes all the way through is the distinguishing characteristic. A doughnut is not homeomorphic with a sphere. Homer suggests that taking a bite out of a doughnut leaves it a doughnut, but you're removed the hole and destroyed its doughtnutness in the process.
For those who love math there are other gems scattered through the shows.
In one a screen comes up in the Springfield stadium:
d. no way to tell
(a perfect number, a narcissistic number and an Mersenne prime)
at one point the spine of one of Lisa's books reads eiπ + 1 = 0
great stuff... to see this on tv is astonishing.
Futurama has some cultural and writing overlap and tends to leave techie trailings..
2 There are other problems with swimming that get a bit complex and pool dependent, but suffice it to say some lanes are better than others and the faster swimmers usually are given the best.
Grilled Sweet Potatoes
° 1 tbl cumin seeds
° about 2 pounds or a kilo of sweet potatoes cut into inchish wedges
° 2 tbl olive oil
° 1 tbl maple syrup
° salt and pepper
° 1/4 cup tahini
° 1 tbl maple syrup
° 1 tbl lemon juice
° 2 tbl water
° fresh cilantro
° get the grill to a good medium high heat
° toast cumin seeds to light brown in a small pan while the grill is heating. Then move them to a chopping board and crush them until fine
° toss the sweet potatoes and everything else up to dip
° make a half dozen aluminum foil packets with two large pieces of foil each (cut it off roughly square) brush the inside with a bit of olive oil
° lay the wedges in a line in each packet and fold tightly
° grill till tender .. about 15 minutes for me, but my grill isn't great. check in a packet and look for the beginnings of charring. Flip the packets a couple of times during the process
° remove and carefully open .. let them cool.
° whisk the dip ingredients and drizzle..
° garnish with cilantro