Application

More Math and Vegetables

I picked the wrong pot while preparing potatoes, and found the pot a little full.

Potatoes

I had to slice up the potatoes into smaller chunks so that they could all fit in my pot.  Predictably, my poorly planned pot of potatoes prompted me to ponder the packing problem.

In a simple form, the packing problem asks “What’s the best way to pack oranges in a rectangular box?”  Should the oranges be in columns (sitting right on top of each other?), or should you try to fit oranges into the gaps created when you make an orange-square (more like a pyramid?).

The packing problem, despite its seemingly modest statement, leads to very complicated and deep ideas.  My potatoes led to a very delicious side dish. 

By patrick honner, ago

Plotting the Sun’s Path

sun pathHere’s a cool article from the Wolfram Alpha Blog about using WolframAlpha to plot the sun’s path, as seen from various locations:

http://blog.wolframalpha.com/2010/09/22/following-the-suns-unique-path/

You can also alter the date, which means you can look at historical data or projected future data.

Unfortunately, when I tried the command “sunpath north pole” (employing the classic mathematical strategy consider extreme cases), WolframAlpha gave me the sun path for North Pole, Alaska, a very oddly named place as it is nowhere near the actual North Pole.

By patrick honner, ago

Cold Oceans and Newton’s Law of Cooling

feet in oceanWhen in Maine, I thought it would be fun to go swimming in the ocean.  I’d driven all that way, right?  I knew it would be cold when I noticed that of the 200 or so people at the beach, only 5 were actually in the water, and only one of them was in past their knees.  And he was Canadian.

After I swam for a few minutes, lost much sensation in most of my body, and stumbled back to the beach feeling slightly burned all over, I asked around and discovered that the water temperature was around 55 degrees Fahrenheit.  Which made me wonder, “How long could I have swim in such water before suffering severe consequences?”

Normal body temperature is around 98.6 degrees.  Hypothermia sets in when one’s body temperature falls to 95 degrees, and apparently things seem to get very bad at around 92 degrees.  Now, Newton’s Law of Cooling states that the rate of change of the temperature of an object is proportional to the difference between the temperature of that object (in this case, my body) and the ambient temperature (the 55-degree ocean).  The difference in those two temps–about 34 degrees–seems [relatively] high, so I imagine that my body would cool pretty rapidly to the danger zone.

I was very excited to calculate how much time I theoretically had, but unfortunately I didn’t know how to determine the value of the constant in the Newton’s equation.  Can anyone suggest an appropriate value of k? It probably depends on properties of salt-water and the human body.  I asked the lifeguard, but he didn’t know.

By patrick honner, ago

3-D Printing

3d PrinterThe NYT has an interesting article about the increased use of 3D printers in commercial applications.  Nowadays, companies are using the technology to build custom furniture, low-cost prosthetic limbs, and even housing components.

Basically, the geometry of the object is laid-out in software (like Autodesk), and the “printer” then translates the design into reality by extruding layers of hot plastic, one after another, building the object up essentially by printing one cross section at a time.

It seems that the process is becoming more commonplace and less expensive, opening the door for a wider array of commercial uses.  I’ve seen students using this technology, and it is a pretty remarkable advance.  I believe you can even have a custom-designed plastic guitar built! 

By patrick honner, ago

3-D Illusion Application

child in streetThis is a clever (and creepy) application of optical illusions–a painting, drawn on the pavement, that appears to on-coming drivers as a child playing in the middle of the street.

http://reviews.cnet.com/8301-13746_7-20016169-48.html

The purpose of this, presumably, is to catch the attention of drivers so that they will slow down.

The illusion is strikingly effective, and by watching the video you can see how long the image on the ground is.  I wonder if the painting is equally stretched out at all points–is it just a dilation of a normal drawing?–or is it more of a distorted projection–like Greenland on a flat map?  That is, is the head of the painting five-times normal size, while the shoes of the painting are twice normal size?  My gut feeling is that the painting needs to be distorted like that, but I’m not really sure.

Now, whether this strategy will prevent accidents or actually cause more accidents remains to be seen.

By patrick honner, ago
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