Category Archive

As a continuation of yesterday’s discussion, I created a simple graphic to illustrate the relative timescales of significant (to us) events in our cosmic history. The two inner squares show the Cretaceous-Tertiary Extinction (65.5 Ma) and Emergence of Homo (2.5 Ma) (Ga = gigaannum = 1 billion years; Ma = megaannum = 1 million years). Any events more recent would be smaller than a pixel.

I wonder how many galactic empires rose and fell in the blue area, before our own place in the galaxy even began to take shape.

Dear Friends / Family,

It’s that time of year again! What a wonderful year it has been for all of us. The whole family has had a great year, full of memories and adventure. Where should we begin to recount how very blessed we have been? We have never done a Christmas letter before, but fortunately there are letter-writing tools available for people like us.

To start off, ______Deb___________ has been very involved in:
     ___ the mafia as a money launderer
     ___ restoring peace to Iraq
     _X_ starting a homeless shelter for abandoned and battered cats
     _X_ playing extreme Scrabble
     ___ empathizing with the warming globe
     _X_ scratch and sniff lottery, but not for the money

Meanwhile, in ___Pennsylvania_____, ______Jacob_________ has been pursing:
     ___ world conquest
     ___ perfecting sleek and discreet flossing technique
     _X_ an enriching career in hydrotherapy
     _X_ rehabilitation of glue-sniffing ants
     ___ moderate success in minimal endeavors occasionally
     ___ study at the Palestinian glass-blowing institute

And last, but certainly not least, _______Luke_________ has enjoyed living in _Scotland & Minnesota_ while succeeding in:
     _X_ teaching mute people how to walk
     ___ further developing fungus taxonomy
     _X_ acquiring a wide variety of superpowers including, but not limited to:
          ___ infrared vision
          ___ super strength
          ___ the ability to melt
          _X_ an uncanny ability to talk to fish, although they don’t usually talk
                 back
     ___ continued development of Opti-Grab™ technology
     ___ being macho
     _X_ selling glue to ants

In short, we could not have possibly asked for a better year. Some people wish for a perfect life; we lived it, and we are so blessed. We could go on, but we are sure your lives are so much fuller than ours—and so busy too. We only wish that we could have shared each special moment with every one of you.

Sincerely / Love / Affectionately / Yours,

_______Deb__________     _______Jacob________     _______Luke_________

For some reason, my friend Seth Schwartzhoff Boyd and I came up with this in middle school.

The perfect response to the question: Are you hungry?

Oh, I’m Chad. Why don’t you Russia on over to visit U.S. sometime. We’ll go in the Palestine, sit on the Afghanistan, and have Turkey dipped in Greece served on our best China, followed by a big Bolivia of Chile and Tunisia on the Sandwich Islands; but make sure you wash your Honduras before you eat or you’ll have Germanies all over them.

My brother sent me a recursive problem yesterday, and I worked out this solution last night.

Assume a population begins with resources at 100 (some unitless quantity). Half of these resources are spent, while the other half are passed to the next generation. The next generation begins with 100, plus 20 to account for progress, plus the half passed down to them. Likewise, the following generation begins with 100, plus 40 to account for progress, plus the half passed down to them. What is the resource value of the nth generation?

We can induct a recursive function by simply looking at the first few terms. Let f(x) == “resource level for generation x”, a = 100, and b = 20:

f(1) = a
f(2) = (a + b) + f(1)/2
f(3) = (a + 2*b) + f(2)/2
f(4) = (a + 3*b) + f(3)/2
f(5) = (a + 4*b) + f(4)/2

Following the pattern, we can write the general case:

f(n) = a + b*(n-1) + f(n-1)/2

This recursive solution can correctly predict the value of the nth generation, but it is O(n) in complexity (that is, it requires n calculations to find the nth generation). Using a computer to solve the equation is a common solution, but we can also solve this recurrence relation to remove the recursion.

Back-substitution for the recurrence relation reveals that f(n) can be written in the form:

f(n) = λ₁(n)*a + λ₂(n)*b

From the first few values of λ₁ (1, 3/2, 7/4, 15/8, 31/16, 63/32) we can induct:

λ₁(n) = (2ⁿ - 1)/(2^{n - 1})

This is not as easily done with λ₂, though (0, 1, 5/2, 17/4, 49/8, 129/16), since λ₂ has the form

λ₂(n) = (n - 1) + λ₂(n - 1)/2

yet another recurrence relation. This equation can be solved, though, using the method of undetermined coefficients to give:

λ₂(n) = 4*(1/2)ⁿ + 2*n - 4

Therefore, we can write the final solution to our initial recurrence relation:

f(n) = [(2ⁿ - 1)/2^n-1]*a + [4*(1/2)ⁿ + 2*n - 4]*b

I saw a banner add for a site called “Tickle Your Brain” (at least according to the ad; I didn’t click the link). It featured the puzzle below, for which I cached the image as well, but I didn’t want an annoying animated advertisement in the post. The obvious answer they are going for is even part of the filename(!), but as a point of contention I present justifications for any of the solutions as valid.

Which does not belong in the group?

U E A S O

U, because none of the other characters can be approximated by a function of a single variable.

E, because it is the only character with all of its lines parallel or perpendicular to one another.

A, since none of the others are used in the standard U.S. letter-grade system.

S, because it is the only one with no axis of symmetry.

O, since it is the only character with just two edges.

Below, I have categorized several (but not all) forms of communication based on four properties:

Simultaneity (boolean) - Does the mode of communication require simultaneous participation by all parties?

Location Dependent (boolean) - Does the method of communication require being in a specific location?

Logging (boolean) - Does the method of communication typically include archival features?

Preparation (time) - How much time can reasonably be spent crafting, revising, and modifying a response?

Face to Face communication shares the same benefits and limitations as a Land Line on this chart (aside from logging), since I did not include physical distance as a factor. Snail Mail and Email both have the benefit of thoughtfulness; that is, you can spend time composing a letter or email, with the opportunity to analyze your writing and rework ideas. Instant messaging provides this benefit as well, although the active participation of both parties in Instant Messaging places a modest upper limit on composition time. The mobile phone, by definition, is the most portable of these communication modes–although there are an increasing number of portable email and Instant Messaging devices (often embedded with phones).

Any mode of communication with a true value of simultaneity will be interrupted if a similar form of communication is attempted at the same time. For example, Face to Face conversation is interrupted when a participant elects to answer their Mobile Phone. If, on the other hand, the participant received an email (or a voice message), they can ignore it until the Face to Face communication is complete.

In general, I like my available communication modes to change with my physical location, so unless I’m travelling I use my Mobile Phone as if it were a Land Line. But many people disagree, preferring communicative homogeny independent of location.

Why is xor not part of everyday speech? It would make certain situations much less ambiguous:

A) Would you like cream or sugar?
B) Would you like soup or salad?

In case A, it is generally acceptable to take cream, sugar, or both cream and sugar. In case B, though, the chose is exclusionary between soup and salad. But with xor in the lexicon, we can remove the degeneracy:

A’) Would you like cream or sugar? (inclusive or)
B’) Would you like soup xor salad? (exclusive or)

See how much easier this would be?

From a brief discussion on KarpAcrossAmerica.com.

The terms pop, soda, Coke, and other names for carbonated beverages are a source of great conflict among people of different regions. Soft drink is arguably the most neutral and politically correct term for these beverages, particularly because it fits into a nice beverage trifecta:

1) The Soft Drink
2) The Mixed Drink
3) The Hard Drink

With these three drink types, we can define some basic mixing relations (assume addition is commutative):

Soft + Soft == Soft
Soft + Mixed == Mixed
Soft + Hard == Mixed
Mixed + Mixed == Mixed
Mixed + Hard == Mixed
Hard + Hard == Hard

If a friendly game of Kings involves a community cup containing a Jack & Coke, wine, and water, the beverage can be expressed as: Soft + (Hard + (Hard + Soft)) = Soft + (Hard + Mixed) = Soft + Mixed = Mixed. For some reason, I think my 9th grade algebra teacher should be proud of this.

Yesterday I came home to the following voice mail message: “Hi Luke, this is Dr. Rutter’s office calling. We were wondering if we could switch your 11pm appointment tomorrow for 3pm. I’ll assume this switch works for you, but please call us back to let us know.” This confused me a bit, since my brother’s name is Luke, but he is studying overseas and shouldn’t have any appointments in the U.S. Nevertheless, I called him in Scotland and my mom in Minnesota just to make sure, but they were as clueless as I. This morning I called the office, just in case they really meant an appointment for myself; it turns out it was a completely wrong number, and by sheer coincidence the patient’s name was Luke.

Although this is a retrospective view, it seems unlikely for a wrong number to request the name of one of my immediate family members. How unlikely? I checked the U.S. Census Bureau’s name database from the 1990 census (the database is not yet available for 2000) to find out the probabilities for my famliy:

The “coincidence factor” is much less in the case of extended family names, since I have many aunts, uncles, and first cousins. Even so, 11% is a fairly high probability (although this was not unexpected). Fortunately, I do not receive enough mistaken phone calls to collect data and compare against this prediction.

For quick reference, below is a list of all holidays in the United States celebrated after Thanksgiving through New Year’s Day. The list includes all holidays from the Earth Calendar listed as “United States” holidays. Hopefully this clears up something.

Day of the Ninja (December 5)
Pearl Harbor Day (December 7)
Nobel Prize Day (December 10)
Poinsettia Day (December 12)
Bill of Rights Day (December 15)
Boston Tea Party (December 16)
Louisiana Purchase Day (December 20)
Forefather’s Day (December 21)
Festivus (December 23)
Christmas Day (December 25)
Kwanzaa (December 26-31)
New Year’s Eve (December 31)
New Year’s Day (January 1)

To settle regional disputes in our department (e.g., which states comprise the South?), several of us decided to make our own colored regional maps. It’s late and I’m a bit tired, so I’ll just defer to Jared for the rules:

1. The lower 48 must be divided into exactly 11 regions. Why 11? No good reason really, other than that it’s prime and Jacob gravitates toward primes. That, and when Jacob & I arbitrarily tried to do this with exactly 7 regions (since 7 is prime, God’s number, and because this is God’s country), that turned out to be quite obnoxious. Seven might actually potentially be the worst possible number of regions into which to divide the country.

2. Each state must belong to one region and only to one region. This forces people to make the tough decisions, such as what in the world to do with Kentucky or Pennsylvania. Moyer apparently decided to be a bit cavalier with this rule, splitting California and Florida in two (the southern half of both states he called the “Vapid Whores” region, haha). If people don’t like the whole-state rule, then they’re invited to create an additional map of regions that ignores all state boundaries.

3. Single-state regions are allowed. Multi-state regions must be contiguous. This ideally would’ve prevented Moyer from making southern Cali & Florida into the same region, but whatever.

With that in mind, I present my own eleven-region USA map:

Okay, so I guess that doesn’t help settle any regional disputes. But you have to admit it feels right to leave Utah and New Jersey as their own regions.

Others participating in “One Nation, Eleven Regions”:
- Jared Lee
- Adam Moyer
- Chris Allen
- Walter Kolczynski
- Jeff Frame

Many people have plotted world domination to at least a minor degree, although most of these plots are never realized. Part of the problem with world domination schemes is the desire to be recognized by all/most of the world’s human population. As I see it, there are two primary categories of world dominators:

The visible tyrant - Here, the desire is to maintain control (often by force) of the population. Benevolence may work for some world dominators, fear for others. But in either case, the population is knowingly coerced into following the self-appointed leader. This often leads to dissension and historically has led to the fall of many dictators.

The invisible tyrant - Recognition is less important for this type of world dominator. Using more subversive means the population could be manipulated in such a way that they are actually under the control of this dominator, although they may never be fully aware of this fact. If the dominator were to appear in public to try and forcibly exert authority in person, this would likely result in failure. Because the presence of this dominator is hard to detect, subversion by the population is difficult.

Of course, I cannot divulge my plan for world domination (if in fact I have such a plan), but I will admit that it falls under the “invisible tyrant” category. And that should make you wonder: have I already succeeded? You may never know.

Yesterday I posted the results of my stochastic Senate predictions. But that got me thinking: what is the purpose of doing a weighted coin flip to aid in prediction of a random event?

Consider two events, A and B. The probability of event A is p(A) and the probability of event B is p(B) = 1 - p(A). Assume p(A) >= p(B). If I want to use a random device to decide whether I should predict the outcome A or B, I could select a weighted coin that chooses event A with probability f. The probability of correctly choosing outcome A is then p(A)*f, and the probability of correctly choosing outcome B is p(B)*(1 - f) = (1 - p(A))*(1 - f). The total probability of success is therefore T = p(A)*[2*f - 1] - f + 1. In other words, my predictive probability is maximized for f = 1 (which gives T = p(A) and is equivalent to not flipping a coin and always selecting A). This probability T is minimized for f = 0 (which gives T = p(B) and is equivalent to not flipping a coin and always selecting B). Thus, the most accurate outcome is gained without the use of a weighted decision-making coin. Furthermore, if such a coin is necessary, it would be beneficial to use a coin that has a higher weight than the actual probabilities (i.e., f > p(A)). What, then, is the advantage of the weighted coin?

If I were to select event A consistently, I would have a higher probability of greater total accuracy, but I would never correctly select the occurrence of event B. Thus, the use of a weighted coin allows for the correct prediction of either A or B as an outcome. How should this weighting be done, though? If we assume that f = p(A), then the probability of successfully predicting A is [p(A)]² and that of predicting B is [p(B)]². If f > p(A) there is a greater probability of correctly predicting A but a lesser probability of correctly predicting B.

Therefore, the weighted coin flip has its use in solving the coupled requirements that both p(A) and p(B) be nonzero and maximized. This does not yield the highest probability of total accuracy, but it allows the prediction of both A and B events while maintaining as much accuracy as possible. Such a technique may prove useful in a March Madness bracket (especially for someone who doesn’t know too much about college basketball).

Yesterday I suggested that Mole Day be celebrated by the congregation of a mole (quantity) of moles (small burrowing mammals). Some people might have a hard time visualizing this, though, so here are some helpful numbers. First of all, it helps to know that Avagadro’s Number N_A is a quantity approximately equal to 6.022×10²³. We will also assume that the mean mass of a mole (small burrowing mammal) is 0.10 kg, and the mean length is 0.15 m. We can then estimate that a mole of moles would…

…have a mass equivalent to that of our moon.
…have the same volume as Mars.
…reach the Andromeda galaxy and back, twice, if stacked end to end.
…cover the surface of the Sun, all the planets in the Solar System, and 1000 other similar stellar systems.
…feed the present world population for 100 billion years.
…require 10 quadrillion years to assign names or numbers to all of them.
…really annoy a lot of people.

Maybe it’s better if we just let moles be moles.

Do you have a fear of aging or a bit of anxiety when your birthday comes around? Here’s a great way to have fun with your age: whenever you have to write your age, do so in a way that is a bit more mathematically interesting. Here are some examples for 20-29. While I am sad to be leaving one of the prime years of my life, it is nice to know that I will be turning the ripe age of 4!.

20 –> ₆C₃ (number of ways you can choose 3 items from a set of 6)

21 –> 10101₂ (10101 base 2 == 21 base 10)

22 –> Ti (atomic number of titanium)

23 –> no change (23 is prime)

24 –> 4! ( = 4*3*2*1 = 24, also an excited 4-year-old)

25 –> 5²

26 –> 0×1A (1A in hexidecimal == 26 base 10)

27 –> 3³

28 –> no change (28 is a perfect number; i.e., it is equal to the sum of its multiples: 1 + 2 + 4 + 7 + 14 = 28)

29 –> no change (29 is prime)

The Science Creative Quarterly today published my article “Predictability in the Game of War” that I posted here about four months ago. You can read the full text of the article on the SCQ: http://www.scq.ubc.ca/?p=563

The card game of war is typically considered a children’s game, as it requires no skill to play and minimal understanding of playing card relationships in a standard deck. With skill eliminated as a factor in determining the outcome, it is often assumed that war is simply a random game of chance. Yet the game cannot be purely determined by chance, as the initial conditions of the game must have some bearing on the final state. The existence of random factors in the game do not allow for the claim that war is a deterministic game, yet it still possible to quantify properties of the initial state that are indicative of a victory probability. [Continue reading]

Here’s a rare sports-related entry. Baseball is fun for many reasons, one of which is the countless statistics that can be calculated–whether or not they are meaningful. On the walk to work today I thought of a way of assessing a team’s success in making the playoffs: the Cumulative Postseason Stochastic Indicator (CPSI).

If the outcome of all baseball games were randomly determined, then we would expect each team to have a win percentage of 0.500. In a division with D teams, there is thus a 1/D probability of a team winning the division title. There is a 1/D chance of being second in the division, and therefore a 1/D*(2!/3!) probability of claiming the wildcard. Therefore, the probability that a team qualifies for the playoffs in stochastic baseball is SBP = 4/(3D). For the AL Central division, for example, a stochastic team should qualify four out of every fifteen years.

The Cumulative Postseason Stochastic Indicator is a measurement of a team’s record of making the playoffs versus the stochastic probability SBP. Over a time interval T, count the number of times Q a team has made it to the playoffs. The CPSI can then be defined as:

CPSI = (Q/T)/SBP = 3*Q*D/(4*T)

A team has above stochastic performance for CPSI > 1, and below for CPSI < 1. Maximum CPSI over a time interval occurs when Q = T. Let’s look at some values for this year’s AL playoff teams:

Whether or not this statistic is meaningful is left as an exercise for the reader. In any case, go Twins!

Almost everyone knows how to round numbers. I remember being taught in third grade “five alive, four hit the floor”. Although that was sufficient advice at the time, implementing this type of rounding in practice introduces a systematic bias. Consider some examples, rounded to the nearest hundredth:

7.82 –> 7.8 (rounded down)
7.66 –> 7.7 (rounded up)
7.45 –> 7.5 (rounded up???)

The last example shows the problem. The number 7.45 is just as close to 7.4 as it is to 7.5, yet popular rounding convention says to always round up when the last digit is a 5. Over a large set of values, rounding in this manner will introduce a bias, as more numbers will be rounded up than down.

The solution to this problem is simple, however: round to even. That is, when the last digit is a 5 (followed by nothing or by all zeros) round to the nearest even digit. So 7.45 would round to 7.4, but 7.55 would round to 7.6. Consistently using this technique removes the bias over a collection of elements. And if you use this technique whenever you need to round, you eliminate the bias in your life as well. An unbiased life is good, right?

Beer comes in all shapes, sizes, flavors, textures, and aromas. At a typical bar you might find a selection of cheap brews on tap for the crafty consumer; this usually consists of beers such as Miller Lite, Bud Lite, Coors Lite, Natural Lite, etc. But are these beverages really so bad? Obviously there are plenty of drinks far superior to these, but surely Miller Lite is not the epitome of foul-tasting American brews!

To better explore the sample space of bad beers in America, a group of us graduate students organized an event: the Bad American Swill Festival (BASF). This was the second annual BASF, the first of which occurred last year with only four participants–more of a preliminary event. This year, there were eleven beers exhibited with twelve unfortunate participants.

The blind taste test was grouped into three categories for each beer: cold taste (weighted 65%), aroma (10%), and warm shot (25%). The weights are reflective of the fact that most beers are enjoyed cold. However, smell certainly affects the enjoyment of the beverage; and beers tend to warm up for all but the most voracious drinkers. Beers were rated in each category on a scale of 0 (completely undrinkable) to 10 (not too bad).

And now, the contenders and results of the second BASF. (Note the custom designed pint glasses on either side of the lineup.) Weighted mean scores are shown in brackets.

11) Schlitz [5.77]
10) Jacob Best Ice [5.25]
9) Genesee Cream Ale [5.03]
8) Old German [5.00]
7) Stite Lite [4.48] (my entry)
6) Black Label [4.35]
5) Southpaw Lite [4.32]
4) Utica Club [4.17]
3) Koch’s Golden Anniversary [4.13] (defending champion)
2) Old Vienna [3.98]
1) LaCrosse Lager [3.30]

If I learned anything from participating in the BASF, perhaps it really is a good thing that “in Heaven there is no beer…”

Many present-day road signs that express a percentage gradient are straightforward to interpret: the simple rise over run. So a 17% grade would mean that over a horizontal distance of 1.0 km you would descend a vertical distance of 0.17 km, which is traveling a 1.01 km length at a 9.8� angle. A 45� angle has a grade of 100%, and a grade of 0% is completely flat. Easy to follow, right?

Now let’s look at an alternative way of expressing grades, which is still in use in some places. Here, the grade is calculated by the ratio of the vertical drop to the distance travelled (the sine of the angle, if you prefer). So in this scheme, travelling a 45� angle gives a grade of 71%, and a grade of 50% occurs at an angle of 30�. An incline of 90� has a grade of 100%; but a grade of 99% only equates to a 82� angle. Everyone happy?

The second system has the advantage of a maximum grade of 100%, but it is perhaps more complex to follow. The real problem, however, is not knowing which scheme is in use when you see a sign. The two systems are in close agreement for small angles, but at larger gradients it makes a big difference. But either way, at least I learned something about road signs today.

Many of you are already well aware of this, but as a PSA I am posting a calculation to determine whether or not the Coriolis effect helps govern the draining of sinks and toilets. First, let’s start with the atmosphere:

The length scale of the atmosphere is on the order of L ~ 10⁵-10⁶ m, and the velocity scale is U ~ 1-10 m/s. The magnitude of the Coriolis frequency is Ω ~ 10^-5 s^-1. Thus, the ratio of the inertial force to the Coriolis force is (U/L)/Ω = 0.1-10. For some atmospheric phenomenon, the Coriolis force certainly is important (such as in the rotation of a hurricane).

Now on to the kitchen sink. For a sink or toilet, the length scale is L ~ 1 m and the velocity scale U ~ 1 m/s. Taking the same ratio of the inertial force to Coriolis, we get (U/L)/Ω = 10⁵. Even the slowest of drains would have a difficult time being affected by the rotation of the earth. But for some reason, people still assume that something magical happens to toilets when they cross the equator and head down under.

Nearly every time I fill up my car at the gas station, no matter how careful I am there are a few drops of gasoline that spill from the spout onto the ground. This occurs when I remove the nozzle from the car to return it to the pump. A few drops wouldn’t get my car too much further, but summed over the entire nation this probably adds up. Let’s do a back-of-the-envelope calculation to see:

Diameter of a drop ~ 3 mm
Typical daily distance ~ 10 mi/person/day
Typical milage ~ 20 mi/gal
Typical gas tank size ~ 10 gal

A typical drop then has a volume of about 5e-4 fl oz, and a typical driver fills up roughly twice a month. If there are 3 drops wasted at each fillup, and there are ~100,000,000 drivers (vehicles in use) in the US, then approximately 80 gallons of gasoline are wasted each day nationwide. This equates to roughly three barrels of crude oil.

I have to say that this is less than I thought it would be. But given the price of gas these days, every little bit counts.

The cartoon has always been my favorite type of TV show. I enjoy the ability for animation to transcend normal rules and conventions that apply to daily life, and a lot of times the art and colors are enjoyable as well. After some thought, it occurred to me that most animated series can be grouped into one of three categories:

* Type I (”cliffwalker”): Laws of nature (conservation of energy, conservation of momentum, gravity) are arbitrarily ignored–i.e., a character can walk off a cliff without falling immediately or can produce a giant hammer at will. But normal laws can apply in other situations. Generally there is no long-term effect of an injury. Examples: Tom & Jerry, Looney Toons, Animaniacs, Spongebob Squarepants, Sealab 2021

* Type II: (”altered reality”): Certain aspects of reality are altered, but for the most part these elements remain consistent. A futuristic show may have robots with artificial intelligence or talking animals, but once this is established the show does not break any of its own rules. Injury may result in some effect, although perhaps with minimal effect and duration. Examples: Scooby Doo, Gargoyles, Futurama, many types of anime, most superhero series

* Type III (”animated reality”): No physical laws or aspects of reality have been changed. The show may take place in various settings or time periods, but they always try and accurately reflect something “realistic”. Injury is as serious as it would be in real life (and is generally less comical than in types I and II). Examples: some types of anime, other animated dramas

Of course, many shows are a hybrid between these three base types; and some shows have one dominant type but sometimes oscillate to another. For example, the Batman animated series is mostly type III, but occasional elements are introduced that are beyond the scope of reality that place it within type II. In any case, hopefully this will somehow lend to enriching your cartoon viewing.

You may think that lecturers/preachers just like the sound of their own voice (which may very well be true), but you may not realize that they are also efficiency experts. When a single person is the only one talking to a group, everyone is fully participating by listening. Since people are not interacting in this lecture-style, it has a complexity of O(1).

Conversations between two people are also efficient. As one person is talking, the other person is listening and formulating a response. When the second person begins talking, the roles reverse and the original speaker now becomes the listener. At any given time there is exactly one talker and one listener (at least in polite conversation), also a complexity of O(1).

The efficiency breaks down with a large group. Consider a conversation taking place between N people. As one person talks, (N-1) people are listening and formulating responses. Because only one person may speak at a time, when the second person begins speaking, there are now (N-2) + (N-1) responses being formulated. After N people have spoken, there will be 0 + 1 + 2 + … + (N-1) = N*(N-1)/2 responses formulated. Thus, while lectures and one-on-one conversations are highly efficient, group communication is highly complex with O(N²).

I think this explains a lot.

I’ve been trying to figure this one out for awhile now: what is the deal with the air shield around planet Druidia in Mel Brooks’ Spaceballs?

Based on circumstantial evidence in the movie, planet Druidia seems to be similar to Earth in size and density. If this is true, and if the ‘air’ that the Druidians breath is reasonably similar to air on Earth, then it is unlikely that the air shield functions to artificially increase the mass of the atmosphere. Most likely, the air shield was built late in the planet’s evolution, so hydrogen escape would already have taken place and the presence of an air shield in the high atmosphere would have little effect on composition.

Spaceballs is explicit about the security function of the air shield. Obviously, a Spaceball air shortage was sufficient to create a planet-encompassing protection against theft. It can be inferred from the manner of opening the air shield that the shield itself is not an energy barrier, but is constructed from some solid material. Even at a nominal thickness of 10 or so meters, a planet-encompassing shield would require approximately the same amount of material as the construction of a “Spaceball One”-sized ship (and construction of the ship would certainly be no more difficult than encircling an entire planet). Penetrating the air shield was as simple as extracting the combination from King Rolland; the Druidians had no apparent defense mechanism beyond the existence of a passive shield. If they had opted for the construction of a large spaceship, however, they would have been able to actively guard and defend their air supply. Given the evidence, I must conclude one of the following:

1) Although the amount of required material for the shield and ship is approximately equal, the Druidians had an overabundance of shield-making material (or conversely, they had an extreme shortage of ship-building material).
2) The Druidians were a peace-loving society of pacifists who did not wish to act aggressively, even toward an aggressor.
3) The Druidians had sufficient knowledge and ability to construct a homogeneous planetary shield, but they lacked the technological prowess to construct an active, complex spaceship system.

For all those times when you forget the value of pi but just so happen to have a twenty-sided die (d20), here’s a simple way to do the calculation:

1) Roll 2d20 (roll the twenty-sided die twice)
2) If the sum of the two rolls is less than 27, or if you rolled double 20’s, count this as a success
3) Repeat this process n times, keeping a cumulative total of the number of successes

Pi = 4 * [# of successes] / [n]

Aren’t you glad you carry that d20 in your pocket?

Appraise war in terms of the fundamental factors. –Sun Tzu

The card game of war is typically considered a children’s game, as it requires no skill to play and minimal understanding of playing card relationships in a standard deck. With skill eliminated as a factor in determining the outcome, it is often assumed that war is simply a random game of chance. Yet the game cannot be purely determined by chance, as the initial conditions of the game must have some bearing on the final state. The existence of random factors in the game do not allow for the claim that war is a deterministic game, yet it still possible to quantify properties of the initial state that are indicative of a victory probability.

The Rules of War
The traditional game of war uses a standard 52-card deck of playing cards divided evenly and randomly between two players. Neither player is allowed to view their hand or rearrange it. The players simultaneously reveal the top card of their deck on the table, and the player with the higher value takes all cards on the table. Cards won from a particular trick are gathered and added to the bottom of the victor’s deck. Suits are disregarded in war, making it an ideal game for exposing young players to concepts of inequality. If both players reveal cards of equal value, a war commences: each player plays two face-down cards on the table and reveals a third card that decides the victor. It is possible to play two, three, or more iterations of a war before the revealed cards are not equal, but in most cases one iteration will suffice to decide the outcome. The victor gains all cards played in the war, including those played face-down. This is the only way in which a player may capture an opponent’s aces. A player loses the game when all of their cards have been captured or exhausted.

Variations in the game of war include varying the number of face-down cards in a war to one, three, a random value, or a plethora of other dependencies. The other major variation consists of placing all cards won from tricks in a separate pile until the player’s deck is extinguished. These cards are then shuffled and used to continue the game. None of these variations is significant enough to differ from the standard rules above when considering a long term analysis. Other variations of war exist in plenty, but most of these have additional rules that warrant their classification as a different game.

Collecting Game Statistics
A computer simulation of war provides a robust and rapid method for examining deterministic tendencies of the game. This simulation used the Mersenne twister random number generator, which has a period much greater than is required for an analysis of this magnitude. Ten million games of war generated a set of three characteristics. The first, and most obvious, statistic is victory. One might guess that in lieu of any deterministic factors, the outcome of a game of war would be no different than tossing a coin. Two other values were also calculated based on a player’s starting hand: deck weight and initial advantage.

Deck weight is a measure of the relative strength of a player’s starting 26 cards. Given that there are 13 different numerical card values (two through ace), define the weight of the card with numerical value 8 to be 0. Defined as such, the cards {9, 10, J, Q, K, A} would have weights of {1, 2, 3, 4, 5, 6} respectively, and likewise the cards {2, 3, 4, 5, 6, 7} would have respective weights of {-6, -5, -4, -3, -2, -1}. The deck weight is then defined as the sum of the weights of a player’s initial 26 cards. The weight of an entire 52 card deck is 0, as would be the case for each player in a perfectly divided game of war (for example, one player is dealt all the reds while the other all the blacks). In a game not so evenly divided, the weight of one player’s deck is equal to the opposite of the other player’s deck (since the weight of the entire 52 cards must vanish). The maximum possible deck weight for a player is 84, which would ensure certain victory. In general, a large dec
k weight may be indicative of a player’s advantage over their opponent, at least concerning card values.

While the deck weight may suggest a certain type of advantage, the ordering of the cards in war is equally important. The initial advantage is defined for a player as the number of tricks won minus the number of tricks lost, over the first iteration through their deck. If no wars occur in the first 26 cards of play, it would be expected on average for a player to have an initial advantage of 0. The maximum initial advantage of 26 would give a player sure and swift victory. For any particular game, a player with a large initial advantage will be in possession of a majority of the cards once all initial 26 have been played. After this point, though, the random element of placing cards won at the bottom of a player’s deck renders the initial advantage measurement useless any further prediction. At best, it is an indicator of the depedence of final victory on the initial ordering of the cards.

Simulation Results
The average cases for both statistics are as expected: the mean deck weight and mean initial advantage are both 0, with both statistics normally distributed about a mean of zero at a 99% confidence level. Deck weight is distributed with a standard deviation of 13.62 and initial advantage with a standard deviation of 4.69. It is interesting to note that even with 10 million games, the maximum and minimum values of the weight and initial advantage did not occur once. Of course, these states are also highly improbable. Neither histogram corresponds exactly to the normal distribution, but the agreement is well within acceptable statistical significance.

Figures 1 and 2 are plots of the probability of victory versus deck weight and initial advantage, respectively. The relationship between deck weight and victory shown in figure 1 is clearly linear, at least for a deck weight between -40 and 40 (or 3?). At magnitudes greater than 40, the number of sample games decreases sharply allowing the bounds on the victory probability to be less constrained. Fitting a line to this data with linear least-squares yields the equation p(W) = (7.67e-03)*W + 0.504, where W is the deck weight and p(W) is the probability of victory. This line fits the data with a correlation of R² = 0.994. With a deck weight W = 0, the probability of victory is nearly 50% as expected. Although on average a player will win and lose an equal number of games, predictability is possible once the deck weight is determined. The probability of victory improves to 60%, 70%, and 80% respectively for deck weights one, two, and three, standard deviations from the mean.

Figure 1: Victory probability versus deck weight with linear regression.

Figure 2: Victory probability versus initial advantage with linear regression.

The relationship between initial advantage and victory is illustrated in figure 2. This trend is even more markedly linear than figure 1, with no points significantly deviant from a linear regression. This can be attributed to the smaller range of values (-26 to 26) of initial advantage compared with deck weight. The fit linear relationship between these two variables is p(A) = (8.08e-03)*A + 0.500, where A is the initial advantage and p(A) is the probability of victory. This line fits the data with R² = 0.999, even better than the relationship in figure 1. For this statistic, the victory probabilities at one, two, and three standard deviations from the mean are 54%, 58%, and 61%. Though the relationship is apparent between A and p(A), the degree to which initial advantage is useful in prediction is low. An initial advantage of 26 would win the game automatically, but a slightly lesser value of 24 does not even win 70% of the time. Prediction based on initial advantage seems to be of higher risk and lesser certain
ty.

Deck weight is a useful indicator of victory probability, and is easily obtained by keeping a cumulative total of the weight as the first 26 cards are played. However, the question of applicability now comes into focus: the weight of the deck is only known after play has begun. Perhaps the cunning player could casually make a friendly wager once they discover their own advantage. Or, those slight of hand could stack the deck in their favor enough to tip the balance–but not enough that suspicions are aroused. But of course, fair game play is always the best course of action. Perhaps the ability to impress an opponent by making an early prediction will suffice, as we sing with Edwin Starr: War. Huh. Yeah. What is it good for? Absolutely nothing!

A week ago I was in a rather contemplative/philosophical mood while watching Office Space (perhaps an unlikely combination). It occurred to me that Peter’s journey is a great example of Joseph Campbell’s epic hero quest. Use the following guide to step through the monomyth next time you watch the movie.

DEPARTURE

• Call to Adventure - “Sounds like someone has a case of the Mondays!” (Peter’s realizes the reality of his job.)
• Supernatural Aid - “I believe you’d get your ass kicked saying something like that.” (Lawrence bestows Peter with valuable advice.)
• Refusal of the Call - “Yeah…I’m gonna have to have you come in on Sunday as well.” (Peter is unable to stand up to Lumberg.)
• Belly of the Whale - “Wow, that’s messed up.” (Peter has a revelation at the occupational hypnotherapist.)
• Crossing of the First Threshold - “I…didn’t feel like it.” (Peter sleeps in and skips work the next day.)

INITIATION

• The Road of Trials - “They’re bringing in a consultant!” (Peter’s apathy coincides with Initech’s reorganization.)
• Meeting with the Goddess - “When you say next door, do you mean Chili’s or Flinger’s?” (Peter’s confidence allows him to ask out Joanna.)
• The Temptress - “It’s like Superman III.” (Apathy turns into a penny-stealing scheme.)
• Atonement with the Father - “Let me tell you something about TPS Reports…” (Peter impresses Bob & Bob with his new worldview.)
• The Ultimate Boon - “What if you didn’t have a good job?” (Peter is promoted while his friends are being let go.)
• Apotheosis - “We’ll get some people under you right away.” (Peter is promoted to a management position.)

RETURN

• Refusal of the Return - “You’re just a penny-stealing wanna be criminal…man.” (Peter ignores Joanna’s concern with his plan.)
• The Magic Flight - “I stole something else.” (The printer smash.)
• Crossing of the Return Threshold - “You are a very bad person, Peter.” (Peter prepares to confess and take the entirety of the blame.)
• Rescue from Without - “I’ll set the building on fire.” (Milton’s malcontent removes all evidence of Peter’s scheme.)
• Master of the Two Worlds - “How’s Penetrode?” (Peter refuses his management position and becomes a construction worker.)
• Freedom to Live - “F*cking A.” (Peter has achieved true job realization, as Lawrence already knew.)

The Science Creative Quarterly today published “Piracy as a Preventor of Tropical Cyclones” as part of the contest sponsored by the Enlightenment Institute. The article can be viewed in the SCQ at the following URL: http://www.scq.ubc.ca/?p=236. The full text of the article is also below. RAmen.

PIRACY AS A PREVENTOR OF TROPICAL CYCLONES
By Jacob Haqq-Misra and Michael Larson

ABSTRACT:
Recent hurricane seasons have been characterized by intense and frequent tropical cyclones. One contributor is increased sea-surface temperature, which is caused by decreased upwelling of cold deep-ocean water. We demonstrate that decreased pirate activity results in less upwelling. This suggests that the only viable solution to intense tropical cyclones is to increase pirate activity.

INTRODUCTION:
The destructiveness of the 2004 and 2005 hurricane seasons has heightened public and scientific awareness of the possible long-term consequences of global warming. Although the link between hurricane strength and global warming remains speculative, recent work has shown that hurricanes have intensified over the past 30 years (Emmanuel, 2005), with an increase in the number of category 4 and 5 hurricanes and a decrease in those classified as categories 1 and 2 (Webster et al., 2005). Emmanuel (1987) argued that hurricane intensity is a function of the sea surface temperature (SST) which, of course, increases as the Earth warms. But other factors are important as well. Lighthill et al. (1994) pointed out that while a lower SST limit of 26^oC is required for tropical cyclone formation, several other key factors contribute to formation and intensity.

The increase in global average temperature is well-correlated with a decrease in global pirate population, as evident in figure 1 (Henderson, 2006).

Figure 1: Correlation between global average temperature and piracy, adapted from Henderson, 2006

We propose that piracy decreases the average SST, thereby lowering average global temperature and suppressing tropical cyclone intensity.

PIRACY AND UPWELLING:
Piracy decreases average SST by inducing upwelling of cold deep-ocean water. Various pirate activities contribute to upwelling. These include involuntary crew resignation, inter-vessel interactions and acoustically-transmitted oscillations (Bligh, 1789; Stevenson, 1883).

Involuntary crew resignation (ICR, a.k.a. “walking the plank”) involves a pirate or captive being forcibly ejected from a vessel at sea. This results in upwelling from displacement of water by the ejectee (Archimedes, c.250 BCE).

Inter-vessel interactions (IVI, a.k.a. “sea combat”) consists of transmission of projectiles between vessels, resulting in destruction or boarding. Upwelling is caused by scattered projectiles and by sinking of vessel elements.

Acoustically-transmitted oscillations (ATO, a.k.a. “sea shanties”) were originally intended to boost morale of rowing pirates. They have assumed ritual functions with the ascent of external power supplies. ATO’s produce upwelling by disturbing the sea surface. This increases motion of large biological entities (”fish” or “whales”), producing displacement.

MODEL RESULTS:
We have modeled pirate-induced upwelling using the PARROT (Piratic Activity Realization Rate of Oceanic Tendencies) oceanic circulation model (Haqq-Misra et.al. 2006). This model has 0.5^o resolution and accurately reproduces present-day ocean currents (Figure 2a.).

We simulated normalized pirate-induced upwelling (in upwelling pirate units, or upu) over the three upwelling categories described above. An ICR event produces 1 upu. IVI’s produce a variable number of upu. We use
d a Maxwellian with an average of 1000 upu. It should be noted that IVI events can produce multiple ICR’s. ATO produces continuous upwelling, based on the local pirate density and oceanic biotic activity. The world average ATO is about 0.5 upu/day.

We average pirate activity from 1605-2005 for each ocean grid cell. While recent pirate activity is weak and concentrated off of the Somali coast (BBC, 2005), historically piracy has been concentrated in the Caribbean (Bruckheimer, 2003). This is consistent with our model results, which produce significant pirate-induced upwelling in the Atlantic basin (Figure 2b.).

Figure 2: A. normal ocean circulation (surface currents). B. regions of pirate-induced upwelling. Note the significant contribution to upwelling in the Atlantic basin.

DISCUSSION:
We have demonstrated that pirate activity produces upwelling. It is thus obvious that a decreasing pirate population will result in less oceanic upwelling, especially in the Atlantic basin.

As evidenced by the 2004 and 2005 hurricane seasons, decreased upwelling results in increased SST’s and more intense tropical cyclones. Our PARROT model predicts that if the downward trend in piracy continues tropical cyclones will intensify. The hurricane season may also lengthen, due to increased SST.

PREDICTIONS AND EXPERIMENT:
The PARROT model has not been experimentally verified. Therefore, we have predicted the upwelling and global impact resulting from a single ICR event. While the effects of an ICR event depend on the mass of ejectee, our model predicts a reduction of roughly 10% in the number of named tropical storms in the Atlantic basin in the 2006 season as a result of a relatively small ICR event off the northern Puerto Rican coast between March 9 and March 13, 2006.

We intend to experimentally verify PARROT by producing such an ICR event. At least one of the authors of this paper will be present for the experiment, to measure the exact upu value of the event.

CONCLUSIONS:
We have demonstrated that decreased piracy contributes to increased tropical cyclone intensity. The only viable solution is to increase pirate activity, especially in the Atlantic basin. We suggest that ICR’s and ATO are preferable to IVI’s, because they offer finer control of upwelling effects.

ACKNOWLEDGEMENTS:
We thank the Flying Spaghetti Monster for inspiring this work and Robert Henderson for advocating piracy to fight global climate change.

REFERENCES:
Archimedes (of Syracuse), On Floating Bodies, c.250 BCE, Syracuse, Greece.

Bligh, W., 1789. Log of the H.M.S. Bounty, Royal Navy, London, UK.

British Broadcasting Corporation, Nov. 25, 2005 “US Firm to Fight Somali Pirates”, London, UK.

Bruckheimer, J., 2003, Pirates of the Caribbean, Disney Enterprises, Orlando, FL, USA.

Emanuel, K.A., 1987. The Dependence of Hurricane Intensity on Climate, Nature, 326, 483-485.

Emanuel, K.A., 2005. Increasing Destructiveness of Tropical Cyclones over the past 30 years. Nature, 436, 686-688

Henderson, R. 2006, The Gospel of the Flying Spaghetti Monster, Villard.

Haqq-Misra, J.D., et al. 2006. A Predictive Ocean Circulation Model, in press.

Lighthill, J. et al., 1994, Tropical Cyclones and Global Climate Change, BAMS, 75, 2147-2157.

Stevenson, R.L. 1883. Treasure Island, Cassell & Co., London, UK

Jacob Haqq-Misra is a graduate student of meteorology and astrobiology at Penn State University with an unnatural affinity for random numbers. He also has an interest in the history and development of religion, and its interplay with the natural sciences. His current research activities include the effects of climate change on hurricane intensity, habitable zones around stars, and paleopiracy.

Michael Larson is a rising star in the realm of physics, and is currently engaged in full time indentured servitude at the University of Wyoming. Unfortunately, the nature of this schedule provides for little free time, which is squandered by p
oking fun at non-scientists in general. While his bad humor may not change the world, he takes solice in the fact that it has been known to make people physically ill.

People like to categorize things. Regardless of the variety of a set of objects (or people, or ideas, etc.) we must get some sense of joy in the order that results when we lump everything into neat little bins. My theory is that all objects in a given set can be nicely categorized by any two descriptors. To test this idea, I took two randomly generated words (and their antonyms) and created this matrix:

Let’s define quadrants 1, 2, 3, and 4 to be the upper left, upper right, lower left, and lower right, respectively. Examples:

Q1 - those playing cards sold in gift shops with scantily-clad women on them
Q2 - foot odor
Q3 - the Candy Land board game
Q4 - the INTERCAL programming language

See how easily everything fits? Now you can make your own categorization schemes and bring order to your world!

I gave a seminar today based on my paper that was published in The Gospel of the Flying Spaghetti Monster. The talk was well attended, with many graduate students and a few professors (we filled up the room), and the audience asked a number of good questions. All in all a success, although sadly the videographer ran out of tape before the talk was over! Oh well…in any case, here’s a link to my slides:

piracy_cyclones.ppt [2.5M]

Given:
(1) God is good.
(2) All good things must come to an end.

Therefore,
(3) God must come to an end.

How can God come to an end? Given the following:
(4) God is love.
(5) Love is blind.

It follows from (4) and (5) that
(6) God is blind.

We know that
(7) The blind cannot lead the blind.

So (6) and (7) together imply that
(8) God cannot lead God.

Therefore we see that (8) explains how (3) can occur. That is, God’s inability to self-lead will result in God’s end. Q.E.D.

In order to further the cause of the Church of the Flying Spaghetti Monster, Bobby Henderson graciously revealed The Gospel of the Flying Spaghetti Monster on the 28th of this month. Bobby also included some of the work done by the FSM Enlightenment Institute, including a study conducted by Michael B. Larson and myself. The work will also be published in an online periodical later this month, so for now I will simply provide the abstract to our study. The full article is found on pages 149-153 of Bobby’s book. (I still have mixed feelings about this being my first publication. Fortunately, I am also co-author on a peer-reviewed journal article currently in preparation. That should offset things a bit.)

Piracy as a Preventor of Tropical Cyclones (abstract)
Recent hurricane seasons have been characterized by intense and frequent tropical cyclones. One contributor is increased sea-surface temperature, which is caused by decreased upwelling of cold deep-ocean water. We demonstrate that decreased Pirate activity results in less upwelling. This suggests that the only viable solution to intense tropical cyclones is to increase Pirate activity.

This entry is based an AIM conversation that took place on September 9, 2005. If I feel so inclined in the future, I may combine this with my previous entry on “The Happiness Function” to produce a complete theory of happiness. Not that it means much.

We can assume that there exists a base time t for the effects of a given activity to wear off. One way of defining this is t = C*e, where e is the base of natural logarithms. Each activity has its own decay factor a_i that exists in the range [-1,1], where a_i = 0 has no longevity and |a_i| = 1 has the maximum longevity (which is not infinite, since most people have not achieved immortality). a_i < 0 signifies an unhappy event, and a_i > 0 a happy one.

A person’s “pushing factor” P is the sum over all i of (t^a_i - 1), so that for a_i = 0 there is no contribution to the sum. In general, it would be desirable to find a maximum set of a_i values for a given subset of life. For example, on a vacation you could have one night of complete and utter binge drinking, but that might not maximize P for the duration of the trip. Especially for physical activities–and perhaps for mental activities–it would not be difficult to determine a_i values. Of course, these a_i values are unique to the individual (not everyone likes sports, not everyone gets hangovers, etc.). But especially for common activities, it would be reasonable to begin with a typical set of a_i values and continually improve the accuracy as you experience more of life.

…because fun is more fun when quantified, right?

Life has its peaks and valleys. Most people are neither continuously up or down but oscillate between the two extremes, with some maxima and minima having greater magnitude than others. The question: is it better to have large swings, or is a less extreme, more serene life preferable? Stated more precisely: assume a person’s “happiness curve” is oscillatory, with an amplitude that is also a function of time. We can then write:

H(t) = A(t)*sin[w(t)*t - d]

where H(t) is the happiness at a given time, A(t) is the time-dependent amplitude, w(t) is the time-dependent frequency, and d is a phase shift for choosing a starting happiness value (i.e., at birth). For practical purposes, we can set d = 0. Obviously it would be preferable to keep H(t) > 0 for all t, but in life this is hardly possible. To restate my initial question mathematically, consider the integrated happiness L(t0,t) over a time interval:

L(t0,t) = integral[abs(H(t)) dt] from t0 to t

Should L(t0,t) be maximized or minimized? For an average life, if we were to integrate H(t) without the absolute value, it would be expected that the area over a large time interval would be zero, regardless of the degree of the extrema. But the “serene” path would minimize L(t0,t), while those who seek extreme highs (and likely also experience extreme lows) would strive to maximize L(t0,t).

Newton didn’t just do this stuff for kicks; he worked hard to “put food on his family”. For those of you who need a couple tricks in your numerical toolbox, this entry is devoted to everyone’s favorite non-linear equation solution algorithm:

x0 = initial guess
for k = 0, 1, 2, …
   xk+1 = xk - [F(xk) / F'(xk)]
end

Now, I realize that not everyone runs around all day solving non-linear equations–and for those that do, many prefer to try things analytically. Nevertheless, Newton’s method provides an excellent lesson in dealing with people, who are most certainly non-linear creatures. It is nearly impossible to judge a particular characteristic of a person (humor, wit, intelligence, societal value, etc.) based on a single encounter. I suggest a Newton’s method approach: make an initial guess, calculate the rate of change the person exhibits, and iterate until you determine how funny/witty/smart/valuable/whatever said person is.

Of course, part of the difficulty lies in the rate of change calculation. Frankly, the secant method is a better choice for personality analysis (as well as many non-linear equations). The derivation/location of the secant method is left as an excersize to the reader.

It’s Ash Wednesday, and although I have never participated in the festivities, I do know a few people who will have ash-smeared faces before the day is over. Since yesterday I have learned the meaning of the ash ceremony, but did you ever stop and think about where they get all the ash? Apparently it is supposed to come from the palm fronds used on Palm Sunday the previous year. Let’s do the numbers.

For a typical Ash Wednesday event, assume the following:

ra = density of ash = 0.9 g/cm^3
Af = mean forehead cross surface area = 6 cm^2
d = mean forehead ash depth = 0.05 cm

Then the mean mass of ash per parishioner is Ma = ra * Af * d = 0.3 g. Now consider the typical properties of a palm frond:

rp = density of a palm frond = 0.5 g/cm^3
Vp = mean volume of a palm frond = 7400 cm^2
so then, Mp = mean mass of a palm frond = rp * Vp = 3.7 kg
Nf = mean number of fronds per tree = 40

Thus, a typical tree produces Mp * Nf = 150 kg of frond material. Since the fronds are burned, the amount of mass usable for religious services is actually Mu = (rp/ra) * Mp * Nf = 85 kg per tree. The number of Catholics worldwide is Nc = 1.1 billion, although it is difficult to determine the percentage of these that are actively practicing. Let f be the fraction of practicing Catholics worldwide. The number of trees required for a given Ash Wednesday event, then, is:

Nt = (Ma * Nc * f ) / Mu

If we assume f = 0.25, then the number of trees required is Nt = 1000 trees. Average tree density in the Amazon is about 500 trees/hectare, so this amounts to approximately 2 hectares (5 acres) for a worldwide Ash Wednesday event. And, it’s not even an “official” holy day!

As of today, the paper “Piracy as a Preventor of Tropical Cyclones” by Michael Larson and myself has officially been accepted as one of the 18 (out of 120) submissions suitable for the Church of the Flying Spaghetti Monster’s Enlightenment Institute. The articles will be published online in the Science Creative Quarterly, which is not actually a quarterly publication. The winner of the coveted $100 worth of Ramen noodles has yet to be announced.

In any case, the article will also be in Bobby Henderson’s Gospel of the Flying Spaghetti Monster, available March 28. And I just might end up giving a seminar on the current state of pirate-hurricane research sometime soon.