## Posts Tagged ‘Bones’

### How tall was the murderer?

December 13, 2008

In a recent episode (“The Bone that Blew”) of the TV series Bones — and yes, I am thinking of creating an entire category on this blog just for brief mentions of math in Bones — a body was discovered and it turned out to have been dragged by a choke chain. Temperance Brennan points out that the angle of the fracture was 18°, indicating that the leash was being pulled at that angle. The blond assistant guy then added “Assuming a standard 4 foot leash, the person who dragged the victim is at most 5’5″.” They showed a picture like the following, only it had a skull and broken neck at the bottom. I spared you that.

How did they come up with the 5’5″ height? Since the leash was 4 feet long and being held at 18°, the vertical height from the body to the hand that was dragging it must have been 4·sin(18°)=1.24 feet. The leash probably didn’t touch the ground, though: it would have been attached to the neck, and therefore something like 6″ above the ground. Let’s assume the vertical height is therefore 1.74 feet above the ground.

Presumably Blond Guy knows some formula that relates how tall one’s hand is above the ground with their total height, although my extensive searching (i.e. Google) didn’t reveal any such formula. Never one to be deterred by a lack of actual facts, I forged ahead and found that my height is 2.9 times as big as the distance from the floor to my hand. Assuming that ratio holds constant for everyone [which is certainly doesn't], this would mean that the person dragging the body was (1.74·2.9=5.05) just over 5 feet tall.

So I still have no idea where the 5’5″ came from, assuming it wasn’t just made up out of thin air (which it wasn’t, right? Because that would make me really sad.) Maybe the person was shorter. Maybe my adding 6″ was an underestimate and it should have been closer to 9″ (which would mean I’d add 0.75 to 1.24, getting 1.99, which yields 5’9″ when multiplied by 2.9). Maybe the 2.9 really varies quite a bit person by person. Or maybe the person was tall, but leaning over, and Blond Guy took that into account.

So alas, I really don’t have an answer. But fortunately for the folk at the Jeffersonian, they found additional evidence (identifying the trajectory of the bullets that killed the man, and used that to get a different height that led them to the killer). If you’re dying to know who did it — so to speak — you can see the episode at least for the time being here on Fox. It’s Season 4, Episode 10, and will probably only be up for a few more weeks.

### Math in Bones

December 1, 2008

So I’ve been watching DVDs of Bones, as you might have gathered from an earlier post. And one of the episodes from Season 2, “Spaceman in the Crater”, has three separate spots with some math! Not a lot of math, but it doesn’t take much to make me happy.

The first bit comes early on, when Special Agent Booth and Forensic Anthropologist Dr. “Bones” Brennan examine a body that’s fallen from the sky and formed a crater.

Brennan (immediately after commenting on the man’s loafers): He hit the ground at approximately 200 kilometers per hour.
Booth: How can you tell that by his shoes?
Brennan: 124 miles per hour is terminal velocity for a falling human.

See that fancy unit conversion? It’s right, too. (OK, not hard math, but for all we know Brennan used Fibonacci numbers to figure it out, since 200=144+55+1 [i.e. F12 + F10 + F2] so the miles would be approximately 89+34+1=124 [i.e. F11 + F9 + F1].

Then a little later, Dr. Zack Addy took it a bit further:

A human being reaches terminal velocity after falling 200 to 220 meters, depending upon air resistance. Velocity would be achieved between 5 and 8 seconds, depending upon atmospheric conditions, body position, and clothing. He fell from a minimum of 1200 feet. I can run through the math if you like.

Sadly, no one wanted Addy to run through the math so I can’t see what played into this. Presumably if air resistance were negligible (which it isn’t) and if the body was dropped rather than pushed, then it would fall ½(9.8)t2 meters after t seconds; that’s 122.5 meters per second after 5 seconds and 313.5 meters per second after 8 seconds. The upper bound of 8 seconds makes sense to me, since air resistance would slow the speed down, but I’m a little surprised by the lower bound of 5.  Shouldn’t it be at least 6 seconds, and probably closer to 7?  Maybe Addy is thinking the body might have been pushed.

In the final math segment there’s a bit of exponential decay. Addy shares the following observation a bit later, after they’ve determined that the victim was an astronaut:

Astronauts lose 2% of their bone mass for each month spent in space. Our victim’s legs, hips, and lower vertebrae have demineralized over 20%, indicating 10 months in space.

Using that 2% per year, the amount of bone mass left should be (0.98)n after n months. Solving (0.98)n=0.8, in order to find out when the boned demineralized 20%, leads to n=log(.8)/log(.98) [where the log is to your favorite base], or n≈11.04 months. That doesn’t fit with the 10 months mentioned above. In this case, though, rounding could be the culprit: if it was really 2.4% of bone decay per month and that number was rounded down to 2% for convenience, the formula becomes (0.976)n=0.8, giving n=log(.8)/log(.976)≈9.2, which fits with the data. I’ll grant Addy this one.

(And yes, the rest of the episode was good too!)

### Perfection

November 26, 2008

In the episode “The Boneless Bride in the River” in Season 2 of the TV show Bones, the body of a young woman is found in a river, and it’s discovered that she was likely brought over to the US on a fiancée visa. About 14 ½ minutes into the episode, two of the main characters (Special Agent Seeley Booth and Forensic Anthropologist Dr. Temperance “Bones” Brennan) have this conversation:

Booth: Homeland Security says the fiancée visa was expedited by a lawyer on retainer into a smaller bride agency here in town called “The Perfect Wife”.
Brennan: Oh that sounds archaic.
Booth:
No, you know, in therapy I learned that superlatives like perfect are meaningless.
Brennan:
Not in science. A perfect number is a number whose divisors add up to itself, as in one plus two plus three equals six.
Booth:
Well, in therapy I learned that definitive statements are by their very nature, wrong.
Brennan:
Isn’t the statement “definitive statements are by their very nature wrong”, definitive, and thus wrong?

Speaking of wrong, Brennan was a little bit wrong in her definition: a perfect number is one whose proper divisors add up to itself. But still, neat math in a neat show is always worth a mention. And perfect numbers are pretty neat, because like so much in number theory they’re simple but there are still open problems about them.

A bit of history [where "a bit" apparently means "a lot because I don't know how to edit today"]: perfect numbers were studied by Pythagoras, which makes the concept at least 2500 years old. Euclid also talked about perfect numbers a few hundred years later in Book IX of The Elements. In particular, Thomas Heath’s translation of Proposition 36 states:

If as many numbers as we please beginning from a unit are set out continuously in double proportion until the sum of all becomes prime, and if the sum multiplied into the last makes some number, then the product is perfect.

As a translation of this translation, this is saying that if 1+2+22+…+2k is prime, then that sum times the last number in the sum (2k) must be perfect. For example, 1+2=3 is prime so 3·2=6 is perfect. Likewise, 1+2+4=7 is prime, and 7·4=28 is perfect. [As a note, the formula is is sometime written out algebraically with 2k+1-1 used instead of 1+2+22+...+2k; in addition, when it's written that way it's sometimes reindexed so that k is used in place of k+1, and the statement becomes "If 2k-1 is prime, then (2k-1)·2k-1 is perfect."]

A few hundred years after Euclid, Nicomachus wrote some more information about perfect numbers. According to the St. Andrew’s web site, he made
five claims:

(1) The nth perfect number has n digits.
(2) All perfect numbers are even.
(3) All perfect numbers end in 6 and 8 alternately.
(4) Euclid’s algorithm [described above] to generate perfect numbers will give all perfect numbers
(5) There are infinitely many perfect numbers.

Nicomachus’s word was law as far as perfect numbers were concerned, and his claims, while unproven, were believed for decades centuries a really really long time. Of course, there were only four perfect numbers known at that time (6, 28, 496, and 8128), so they really didn’t have much to go on. In reality, the 5th and 6th perfect numbers (33,550,336 and 8,589,869,056 respectively) disprove claims (1) and the “alternately” portion of (3), but it took a while for someone to discover those.

(Which leads to a little tangent: who did discover those numbers? Ismail ibn Ibrahim ibn Fallus (1194-1239) made a list of ten numbers he thought were perfect. Three of them weren’t, but the other seven were. Sadly, a lot of other mathematicians had no idea about this list: mathematicians in Western Europe had to wait another 350 years for those numbers to enter their collective psyche, during which time a few other mathematicians found the 5th and 6th perfect numbers and were equally ignored.)

But even after it was known that Nicomachus’s claims weren’t themselves perfect (ba DUM!), mathematicians continued to study the numbers. In the 1600s Pierre Fermat tried to find patterns, and ended up discovering his Little Theorem as a consequence. Marin Mersenne also spent some time on it, and in fact his exploration of when 2k-1 is prime, as a part of that theorem of Euclid’s mentioned above, led to the notion of Mersenne primes (primes of the form 2k-1 where k itself is prime).

In the 1700s Leonhard Euler entered the fray. He couldn’t prove that Euclid’s formula generated all perfect numbers, but he did show that it generated all even perfect numbers. And a bunch of other mathematicians spent a lot of time trying to show that numbers were or were not perfect (which was related to showing that specific numbers of the form 2k-1 were or were not prime), a challenging task in the pre-computer days.

Not that we’re doing much better now. As of this moment, we still only know of 46 perfect numbers, and they’re pretty big. We do know a few cool things about perfect numbers in general:

• Even perfect numbers end in 6 or 8.
• Even perfect numbers are triangular numbers (e.g. 6=1+2+3 and 28=1+2+3+4+5+6+7) where the ending digit is one less than a power of 2.
• The reciprocals of the divisors of perfect numbers all add up to 2:
$\frac{1}{1} + \frac{1}{2}+\frac{1}{3} + \frac{1}{6}=2$
$\frac{1}{1} + \frac{1}{2}+\frac{1}{4} + \frac{1}{7}+\frac{1}{14} + \frac{1}{28}=2$

But there’s a lot we still don’t know:

• We don’t know if all perfect numbers are even.
• We don’t know if there are a finite or infinite number of perfect numbers.

In other words, perfection continues to eludes us.