Math Photo: Sorting Algorithms

Sorting Algorithms

One of our end-of-school tasks was breaking down all of our Zometool builds from the past year.  When a colleague handed me a bag of several hundred red, blue, and yellow struts, I immediately started to struggle and experiment with the most efficient ways to sort the struts for storage.

Should I take them out of the bag one at a time, placing each in its appropriate pile?  Or maybe a handful of ten at a time?  Or fifty?  Ultimately I settled on dumping them out and sliding them around into color-coordinated piles.  I’m not sure of the mathematical justification, but my internal optimizer seemed to think this was the right way to go.

Math Photo: Surprising Heptagon

Surprising Septagon

It took several hundred encounters with this park bench before I realized it was a heptagon!  I don’t see many regular, seven-sided figures in my experience, which made this a surprising discovery.  I wonder what prompted this design choice.

Septagon Angle

Like most real-world instances of perfect geometric objects, it doesn’t exactly measure up.  But what’s a few degrees between n-gons?

 

 

 

Math Photo: Curvilinear Coordinates

Curvilinear Coordinates

Looking through this system of parallel curves makes me think about the many different ways we can impose coordinate systems on spaces.  An ordered pair of coordinates specifies a unique location on this curved surface just as a pair (x,y) locates a point in the flat Cartesian plane.

This image also reminds me of the role of context in geometry.  From our perspective, this coordinate system looks curved, but if we lived on this surface, it would all seem perfectly flat!  Maybe our world looks really curved to someone standing outside it.

 

 

Math Photo: Riemann Shadows

Riemannian Shadows 2

The shadows fall like approximating rectangles under a sine curve.  This brings to mind a basic approach in computing the area of curved regions:  approximating the curved region with a series of flat regions, whose areas are easy to compute.

Riemann Sum Sine blue

The angle of the sun makes the shadows more like approximating parallelograms, though.  So we’d probably need a change-of-coordinates to complete our calculation!

 

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