Category Archives: numbers

Pictures in pictures!

Tonight I was pulling files off of a hard drive I’d used back in high school, and I unearthed a whole bunch of my old fractal art.  A few of the images are below; you can view the complete gallery here.

None of these involved any drawing or what would traditionally be considered technical artistry.  It’s all just math.

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Bounding Chi: Cliques and Suboptimal Coloring (or, Why My Mom Took Away My Crayons)

My first experience with suboptimal coloring was when I was about two years old.  My mom got me one of those books with blank pictures of cartoon characters and I just scribbled all over the pages with red crayon.  That’s pretty much what my latest paper is about.  Here’s a PDF. The introduction is below, and continues after the jump.  Some stuff in the paper is probably wrong, so let me know if you catch any mistakes.

Computational Methods for Bounding Chromatic Numbers of Graphs

1 Introduction
Many central problems in graph theory involve the process of graph coloring. A coloring of a graph is an assignment of a label, or “color,” to each vertex, such that no two connected vertices have the same color. Perhaps the most famous example is the problem of map coloring: a map determines a graph by assigning a node to each country, with an edge between two nodes whenever the corresponding countries share a border. A coloring of the graph then corresponds to a coloring of the map in which neighboring countries never share a color. Appel and Haken famously proved that for maps, there is always a coloring with no more than four colors [1]. Continue reading

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Fear and Boredom in the Bathroom!

Typically when I shower, I am stricken with fear.  I like to do my showering at a leisurely pace, but it’s inevitable that if I spend more than a few minutes on my ablutions, the water level begins to rise around my feet.  Then the panic sets in.  I am sure that if I let the water run any longer, it will overflow the threshold and run out onto the bathroom floor, soaking all my clean clothes and pissing off the other residents of my dorm.

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The beloved Shallow Goodale Shower

But the other day, all of that changed.  I realized that I was being stupid.  My fears had no basis in reality.  Unless the drain is clogged, there’s no reason to expect that a shower will ever overflow, for the simple reason that the rate at which water flows through the drain is proportional to the water level.  As more water fills the shower, its own weight pushes it out of the drain faster and faster.  If the drain is clear and reasonably large, the water should stop rising long before it overflows, because it will be flowing out at the same rate that it is coming in from the showerhead.  Since it’s IAP and I had nothing better to do, I thought I’d work out the math and measure the rate of outflow through my shower drain as a function of the water level. Continue reading

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