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Archive of posts filed under the Application category.

Hands on a Subway Pole

When I describe the role mathematics plays in my life, I often say that it gives me a set of tools to process and understand the world.  One way that manifests itself is that I see graphs everywhere.

For example, when I look at a pole on the subway, I see the distribution of hands that have been on the pole.

hands on a subway combo

I think about things like this because they are interesting, but also because they are practical.  Where is the pole the dirtiest, and cleanest?  Where are germs most likely to reside?  New Yorkers know instinctively to touch as little as possible, but sometimes you have no choice.  Best to know your probabilities ahead of time.

3D Printing a Cube Frame

I’ve been having a lot of fun exploring mathematics through 3D printing.

Recently, I had the idea to 3D-print a “cube frame”, that is, the edges of a cube.  My first mathematical task was to figure out an equation whose graph was such a cube frame.

It took a little work, but I ended up with this.cube frame graphI exported the graph into the appropriate file format and successfully printed my cube frame!

Cube Frame

Just as fun as the mathematical challenges of producing the graph are the engineering challenges, and mysteries, of the 3D printing process itself.  For example, notice the scaffolding that the software automatically adds in order to print the top square of the cube.

Cube Fram -- Supports

There are some interesting mathematical and structural consequences of the scaffold-building algorithm, but even more amazing was that one “face” of the cube contained no scaffolding at all.  This meant the 3D printer printed one edge of the top square into thin air!  And it succeeded!

I’m excited and inspired by 3D printing, and I’m looking forward to finding more ways to integrate into our math classrooms.

 

An Ancient Tiling Problem

This menacing creature is Glyptotherium texanum.  Well, was Glypotherium texanum.

Glyptotherium Texanum

The armor caught my attention.  It’s made of small plates that create an interesting tiling of a curved surface.

Glyptotherium Texanum up close

Notice how the tiles slowly change from quadrilaterals to pentagons to hexagons, and back again!  I wonder what factors determine the shape of each tile as they grow.

Field Goals, the Super Bowl, and Mathematical Models

There has been a tremendous increase in the use of mathematical analysis to make policy, inform organizational decisions, and explain news and events.  I generally think this is a positive thing:  I understand that math gives us a powerful set of tools for understanding and processing the world.  But I also understand the limitations inherent in mathematical modeling.

All mathematical models rely on assumptions that limit their impact.  Mathematicians are typically aware of the assumptions about objects, relationships, and contexts that their models make.  Politicians, journalists, and others who invoke mathematics to make a point seem less aware.  This often leads to bold, unjustified claims based on what “the math” has told them.

An inconsequential but illustrative example of this occurred during Super Bowl 49.

At the start of the second half with the game tied at 14, Seattle drove into scoring position, and faced a 4th-and-1 at New England’s 8 yard line.  Seattle basically had two options:  kick the field goal (a high-percentage play for 3 points), or try to make a 1st-down, and ultimately a touchdown (a lower-percentage play for 7 points).  Seattle opted for the field goal and went ahead 17-14.

Lots of people on Twitter second-guessed the decision, including The Upshot’s David Leohnardt.

Here, David Leonhardt is applying a simple expected value argument.  “Going for it” on 4th-and-1 at the opponent’s 8-yard-line likely produces more total points in the long run than kicking field goals, which suggests that Seattle should have gone for it.  It’s not a bad argument; in fact, I used a similar analysis on the NFL’s new overtime rule.

But in order to apply this argument, it’s important to understand what assumptions this model makes.  For example, this model assumes that the amount of time remaining in the game is irrelevant.  Of course, it’s not:  it’s easy to construct a situation in which “time remaining” is the determining factor in the kicking a field goal (say, the game is tied, and only seconds remain).

This model also assumes that all points are of equal worth.  But they aren’t.  Depending on the game situation, the extra four points a touchdown gives you may be irrelevant, or of significantly less value than the three points the field goal gives you (imagine a team up by six points late in the game).

There are lots of factors this analysis does not consider.  This doesn’t mean that the expected value argument is invalid.  It just means that, like all mathematical models, what it says depends on the assumptions it makes.  And the more we use mathematical models to drive our decisions, the more important it is to be clear about the assumptions that are made and the consequences they entail.

A Saturday Morning Optimization Problem

I recently faced an interesting optimization problem.

Through my local grocery store’s rewards program, I earned a one-time 20% discount, to be applied to a single future shopping trip.  Naturally I wanted to maximize the value of my discount, and the more I spent, the more I would save.  But like all optimization problems, there were a number of constraints involved.

shopping bags

 

First, I wanted to buy only things I would actually use.  This prevented me from buying things like saffron (expensive things that would drive up the value of my 20% discount) because I wouldn’t use them.  It also limited the quantity of high-priced proteins I would buy, as such things need to be consumed quickly to be enjoyed.

Second, I could only buy what I could carry, since I walk to and from the grocery store.  This put global constraints on the volume and weight of my purchases, which made me think about maximizing cost per-unit-weight/volume at a local level.

All in all, I’d say I did pretty well!  With some planning and foresight, the total value of my 20% discount ended up being around $46.  And I don’t think I’ll need to buy dried basil any time soon.

shopping receipt