Archive for the ‘Application’ Category

Averia: The Average Font

This is a clever and interesting idea:  creating a new font by taking the average of all existing fonts.

By overlaying all the small letter a‘s, say, from all the different fonts, one can take a visual average and create a new letter a.  Repeat for the whole alphabet, numerals, and punctuation marks, and voila!, you’ve got Averia.

The idea of taking a visual average may be a bit mysterious, but the author describes a few different approaches in how to combine the images.  Essentially all of the instances of a particular symbol are placed on top of each other, and the the darkest parts of the new image are where the instances intersect the most.  The result is then smoothed over to create a readable letter.

And the font looks pretty nice, if not too exciting.  Just what you might expect from the average font.

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Disincentive Pricing

June 20, 2012 2 comments

While riding the rails around Portugal, I frequently saw passengers buying tickets directly from the conductor on the train.  It got me thinking about how high the penalty should be for not buying your ticket ahead of time.  That is, how much more should you be charged for a ticket purchased on the train than in the station?

You see, if a rider could evade the conductor at a consistent rate, it might make mathematical (if not ethical) sense to gamble on paying the higher fare every so often.  For example, let’s say you can successfully sneak a free ride once every three attempts.  If the ticket in the station costs $5, then the price of the on-board ticket should be at least $7.50 to discourage you from attempting this cheat.

I never found out the price difference in Portugal.  But I do know how it works on the Long Island Rail Road.

Returning from vacation, we were rushing from the airport to the train station.  We didn’t have time to purchase tickets from the machine beforehand as the train was literally pulling into the station as we arrived.  After a long day’s travel, we were happy just to make it on board our final mode of transportation.

And it turned out to be nearly a 100% increase.  Instead of the usual $6.25, the on-board charge was $12.  I guess that means they think fare-evaders can get away with it a little less than half the time?

We were happy to get home in a timely manner.  And I was happy to have one more open mathematical question resolved!

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Ancient Lego Robotics

June 16, 2012 2 comments

The antikythera machine is commonly referred to as the “world’s oldest computer”.  Dating back to around 150 B.C., the mechanism was discovered in a shipwreck around 1900, and it has amazed scientists and engineers with its precision craftsmanship.  Recent x-ray analyses of the object helped bolster the conclusion that it was designed to predict eclipses, and probably was able to do so with remarkable accuracy.

What could be more amazing than a 2000 year old computer?  Perhaps this working replica of it, made entirely out of legos.

This video shows the functioning lego replica and gives some of the mathematical background relevant to how the machine operates (apparently the ratio 5/19  is extremely important for calculating the cycles of ellipses) .  Throughout the video, the machine is deconsrtucted and you can see the inner-workings of the various parts.  Truly amazing.

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NBA Draft Math, Part I

May 31, 2012 9 comments

Having put some thought into the mathematics of the NFL draft, I decided to turn my attention to basketball.  From an anecdotal perspective, the NBA draft seems to be more hit-or-miss than the NFL draft:  teams occasionally have success and draft a great player, but it seems more common that a draft pick doesn’t achieve success in the league.

In an attempt to quantify the “success” of an NBA draft pick, I researched some data and ending with choosing a very simple data point:  the total minutes played by the draft pick in their first two seasons.

Total minutes played seems like a reasonable measure of the value a player provides a team:  if a player is on the floor, then that player is providing value, and the more time on the floor, the more value.  I looked only at the first two seasons because rookie contracts are guaranteed for two years; after that, the player could be cut although most are re-signed.  In any event, it creates a standard window in which to compare.

There are plenty of shortcomings of this analysis, but I tried to strike a balance between simplicity and relevance with these choices.

I looked at data from the first round of the NBA draft between 2000 and 2009.  For each pick, I computed their total minutes played in their first two years.  I then found the average total minutes played per pick over those ten drafts.

Not surprisingly, the average total minutes played generally drops as the draft position increases.  If better players are drafted earlier, then they’ll probably play more.  In addition, weaker teams tend to draft higher, and weak teams likely have lots of minutes to give to new players.  A stronger team picks later in the draft, in theory drafts a weaker player, and probably has fewer minutes to offer that player.

However, when I looked at the standard deviation of the above data, I found something more interesting.  Standard deviation is a measure of dispersion of data:  the higher the deviation, the farther a typical data point is from the mean of that data.

Notice that the deviation, although jagged, seems to bounce around a horizontal line.  In short, the deviation doesn’t decrease as the average (above in blue) decreases.

If the total number of minutes played decreases with draft position, we would expect the data to tighten up a bit around that number.  The fact that it isn’t tightening up suggests that there are lots of lower picks who play big minutes for their teams.  This might be an indication that value in the draft, rather than heavily weighted at the top, is distributed more evenly than one might think

This rudimentary analysis has its shortcomings, to be sure, but it does suggest some interesting questions for further investigation.

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CD Packing Problems

I consider myself an expert arranger of things.  I enjoy rearranging storage space, packing things away, and helping people fill up moving trucks.  It’s a way to apply geometry and optimization techniques; two of my favorite things.

In general, the packing problem entails trying to find the most efficient way to pack a certain kind (or kinds) of objects into a certain fixed space.  Packing probelms are, generally speaking, very challenging because every packing problem is unique.  There isn’t a good, efficient procedure that solves them all.

Here is yet another example of problems with packing problems.  After shedding a bunch of CD cases, I thought I’d try to pack them up in a box.  Here was my first attempt.

I got 49 CDs in the box, but there was a bit of unused space left over.  I couldn’t fit a CD into that unused space, but I thought maybe I could rearrange everything to make some of that space usable.

So I tried again.

The number of CDs in this new arrangement differed by one.  While I can compare which of these packings is more efficient, the problem is comparing all possible packings!  There are a lot of options to consider.

As useless as they are, I ended up having a lot of fun with these CD cases.  I made some parallelepipeds with them and used them to demonstrate Cavalieri’s Principle!

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Google and Conditional Probability

Conditional probability is one of my favorite topics to teach.  Whereas normal probability calculations simply compare favorable outcomes to total outcomes, conditional probability allows us to consider the impact of certain knowledge on the likelihood of those outcomes.

For example, the probability of rolling a 6 on a six-sided die is 1/6, but if it is known that the number showing is greater than 3, then the conditional probability that a 6 is rolled is 1/3.

There are many applications of conditional probability, but a recent “Math Encounter” from the Museum of Math made me aware of an application of conditional probability that all of us see on a regular basis:  Google search autocomplete.

Suppose I type in the search term “under”:

Here, Google is trying to autocomplete my search query.  In essence, Google is trying to guess the next word I’m going to type.  How does it make its guess?  It computes a conditional probability!

Google has a lot of data on when words follow other words.  When I enter “under” into the search bar, Google looks for the word/phrase with the highest conditional probability of being next.  Here it turns out to be “armour”; the word with the second highest conditional probability is “world”, and so on.

Naturally, as more information is provided, the conditional probabilities change.

 A fascinating, and perhaps surprising, application of a powerful mathematical idea!

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Free Planetarium Software

What a cool idea:  a free software package that simulates a planetarium on your PC!

Preloaded with 600,000 stars, all you have to do is set your coordinates and start gazing!

You can search by name for various heavenly bodies, map the constellations, and even get close-ups of observable planets!

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