Thursday, January 27, 2011

Igloos Rock!

A good portion of the nation is digging out from under the snow right now.  It seems like the perfect time for an igloo problem.  How warm is it in an igloo? 

Human bodies undergo various chemical reactions and in the process give off heat.  By "heat", I should be clear that I mean thermal energy not temperature.  If you only have your body to keep warm (i.e. no hand warmers or space heaters), then the amount of heat produced in a day should be equal to about 2000 Calories since this is the amount of chemical energy in food that one eats in a day.  This 2000 Calories per day is equivalent to about 97 W of power.  A good portion of this energy will eventually turn into heat, but where does this heat go? 

If heat doesn't somehow escape the igloo, then the temperature inside would keep increasing and the Inuit would burn themselves to death.  Fortunately, heat escapes igloos in many different ways.  For example, as the air inside heats up, the air molecules begin to bounce around very fast.  When they collide with the walls of the igloo, some of their energy gets absorbed and the igloo molecules start to wiggle around a little bit.  As the igloo molecules wiggle around they hit other igloo molecules and this process continues until eventually some of the igloo molecules on the outside surface start jiggling too and bumping the cold air outside.  This whole process ultimately results in heat being transferred out through the walls of the igloo.  Now if the room is going to stay the same temperature (i.e. have the same thermal energy), then the extra energy being produced by thermal processes has to be equal to the energy lost through the walls.1   This means a one person igloo will have 96 W of power flowing through its walls.  There's a nice equation that describes how energy flows through a material, 

P = k · A  · (Thot − Tcold) / x, 

where P is the power passing through, k is called the thermal conductivity, A is the area of the material, x is the thickness, Thot and Tcold are the temperature of the hot and cold ends, respectively. The thermal conductivity of snow is about 0.16 W/m·°C.  On a cold day, the outside temperature can be as low as −45 °C (−49 °F).  The walls might be 0.3 m (~1 ft) thick.  If we make our igloo 2 m (~6.5 ft) in diameter, then the total area of the walls would be 3.1 m2.2   Solving for Thot we find that the temperature inside the igloo is

Thot = · x / [k · A] + Tcold
= (96 W) · (0.3 m) / [ (0.16 W/m·°C) · (3.1 m2)] − 45 °C
= 13 °C.

That's a toasty 13 °C or 55 °F.  According to Wikipedia, igloos can be anywhere from −7 °C (19 °F) to 16 °C (61 °F) when warmed by body heat alone.  So if you're stuck in the cold, warm up by building yourself a nice little igloo hut.

[1] I'm assuming the energy lost through the floor and through a small door is negligible.
[2] This assumes the area of the walls is roughly that of a hemisphere of diameter 3 m.

Tuesday, January 25, 2011

Time Lapse Aging

There's something cool about time lapse photography.  It seems several folks on YouTube have put together time lapse photographs of themselves aging over time.  By taking 1 photo per day every day for years, you can put together a little animation that shows you aging.  From what I've found, the longest stretch of time over which someone took photos like these is 17 yearsIf you took one photo of yourself per day for your whole life, how long would the video last?


Films generally run around 30 frames per second.  A person can live over 100 years or 36,000  days. This means you would show a total of 36,000 pictures.  If each picture being shown for 1/30th of a second, the video would last 20 minutes. 

Sadly, unless someone started doing this at least 30 years ago, I won't be able to see a full life progression sped up in my lifetime.

Wednesday, January 19, 2011

Split Ended

I hate split ends.  They make me sad.  Fortunately, my wife Anna doesn't mind cutting them off for me.  How long will it take to cut off all my split ends?

Sadly, there are no hair care products that can remove splits ends.  I could just shave my head and eliminate them rather quickly, but I'd like to keep my rather luxuriant flowing hair so I'm just going to consider how long it will take to go through each hair individually.  In How Many Licks?, I estimated that there were about 250,000 hairs on a (non-bald) human head.  Roughly 5% or 12,000 of my hairs have splits ends.  If my wife can sift through 10 hairs per minute and cut all the split ends off, then it would take about 17 straight days of sifting to remove all the split ends.  

We could go further with this.  There's some question about what to do with the hair she's already looked at so that she doesn't end up checking it twice.  Moreover, my computer science friends might suggest that a more efficient search algorithm could decrease the time requirement immensely, but I'll leave that to the nice people at Google and Bing.

Monday, January 17, 2011

Astronomical BS

 A few years back, a friend and I got into an argument about astrology. She started telling me how much of a Gemini I am and how I very clearly match all the personality traits. As an experiment, I decided to screw with her and see how she reacted:  

Me: Actually, due to the precession of the equinoxes in the year I was born, I'm actually a Taurus. It's a very rare phenom. Most astrologists don't even know about it.  My birth year was the only time it's happened in the last 153 years.1

Her: Oh my God!  That's so amazing!  I've always thought you acted like a Taurus.  Now I know why!

She then proceeded telling me how much of a Taurus I am and how I very clearly match all the personality traits.   I've repeated this experiment several times with different people, always with similar results.  No matter what sign I give, cognitive dissonance sets in, then there's rationalization, then I'm told how much of a [fill in the blank] I am and how I very clearly match all the personality traits.

Apparently, my little experiment was coincidentally prescient.  In light of the latest astrological buzz, it turns out I actually am a Taurus.2  Unbeknownst to people who follow astrology, the zodiac calendar has been wrong for years.  In addition to the various dates being off, there's also a 13th astrological sign that was left out.  How many astrology-believing Americans just had their signs switched?

Despite the absence of evidence or even a plausible mechanism3, many Americans still believe the stars control our personalities.  According to Michael Shermer's Why People Believe Weird Things, a 1990 Gallup poll showed that 52% of Americans believe in astrology.  There's very little overlap between the zodiac calendar and the actual positions of constellations in the sky. For this reason, about 90% of people's zodiac signs changed. This means that

(3.0×108 Americans) · (0.52 believers per American) · (0.9 zodiac changes)
= 1.4×10
8 zodiac sign changes.

That's 140 million people and a whole lot of cognitive dissonance.  Thank you, Laurie for suggesting this one.

[1] At this point, I was basically making things up and throwing in ad-libed technical jargon.  The actual "precession of equinoxes" has nothing to do with astrology.
[2] But, no, this does not justify my astrologist friend's assertion.  As mentioned earlier, it doesn't matter what sign I give, astrologers will always say my personality matches.

[3]  And, no, the fact that the moon controls the tides does not constitute a mechanism.  Tidal forces are due to gravity, and as I show in How Many Licks?, the tidal force on you at birth due to the obstetrician is about 40 times larger than the largest tidal force of any heavenly body.

Thursday, January 13, 2011

Seth MacFarlane: Family Guy Creator or Math Genius?

On December 26, 2010, Family Guy creator and voice actor extraordinaire Seth MacFarlane tweeted, "If the T1000 had to kill everyone named Aiden Connor or Dylan Connor, he’d have tons of killing to do."  How accurate is his estimate?

According to Book Of Odds, each name has the following probabilities:
  • The odds a male is named Dylan are 1 in 6,250 (US, 5/1990 - 9/1990). 
  • The odds a male born in 2000 is named Aiden are 1 in 2,370 (US, 2000). 
  • The odds a person's last name is Conner are 1 in 4,442 (US, 2000).
Assuming there are no weird correlations, this means the probability that a person is named "Dylan Conner" is 1 in 28 million and the probability that a person is named "Aiden Conner" is 1 in 11 million.  Just considering the U.S. population, there are 150 million males roughly 5 and 12 of which would be named "Dylan Conner" and "Aiden Conner", respectively.  If each Conner weighed 150 lbs, the total weight would be 1.3 tons.

Well done Mr. MacFarlane!  [In poorly attempted Stewie voice] Victory is yours!

Wednesday, January 12, 2011

Tuesday, January 11, 2011

Plague Gauge

"This is what the LORD says: By this you will know that I am the LORD: With the staff that is in my hand I will strike the water of the Nile, and it will be changed into blood. The fish in the Nile will die, and the river will stink; the Egyptians will not be able to drink its water."
— Exodus 7:17–18

Eeeew!  How many people would it take to make a river of blood? 

According to the Wikipedia, the Nile discharges 2830 m3 of water each second.  You'd need an equivalent flow rate of blood to produce a Nile-sized river.  According to the American Red Cross, you can give 1 pint of blood once every 56 days. To produce a river worth of blood, you would need a total of

# of people = (flow rate required) / (flow rate per person)
= ( 2830 m3/s ) / ( 1 pint / 56 days )
= 2.9 ×1013 people.

That's 4000 times more people than there are in the world today.


Unlike plants, cats, or small children, this numbers blog doesn't complain when I don't feed it for two months.  Still, I've been feeling bad about not posting, so my New Years resolution is to start writing regularly again.  More to come in a moment...