isomorphismes

OK, not every day. But whenever I shop for packaged retail goods like a coffee or in the grocers.

The Pythagorean theorem demonstrates that a slightly larger circle has twice as much area as a slightly smaller circle.

(Since the diagonal of that square is √2 long relative to the “1” of the interior radius=leg of the right triangle. So the outer radius=hypotenuse=√2, and √2 squared is 2.)

And some of us know from Volume Integrals in calculus class that a cylinder's volume = circle area × height — and something like a sausage with a fat middle, or a cup with a wider mouth than base, can be thought of as a “stack” of circle areas

or in the case of a tapered glass, a “rectangle minus triangle” (when the circle is collapsed so just looking at base-versus-height “camera straight ahead on the table” view).

The shell-or-washer-method volume integral lessons were, I think, supposed to teach about symbolic manipulation, but I got a sense of what shapes turn out to be big or small volume as well.

By integrating dheight sized slices of circles that make up a larger 3-D shape, I can apply the inverse-square lesson of the Pythagorean theorem to how real-life “cylinders” or “cylinder-like things” will compare in volume.

• A regulation Ultimate Frisbee can hold 6 beers. (It’s flat/short, but really wide)
  
  
• The “large” size may not look much bigger but its volume can in fact be.
  
• Starbucks keeps the base of their Large cups small, I think, to make the large size look noticeably larger (since we apparently perceive the height difference better than the circle difference). (Maybe also so they fit in cup holders in cars.)

A circle is made up of points equidistant from the center. But what does “equidistant” mean? Measuring distance implies a value judgment — for example, that moving to the left is just the same as moving to the right, moving forward is just as hard as moving back.

But what if you’re on a hill? Then the amount of force to go uphill is different than the amount to go downhill. If you drew a picture of all the points you could reach with a fixed amount of work (equiforce or equiwork or equi-effort curve) then it would look different — slanted, tilted, bowed — but still be “even” in the same sense that a circle is.

Here’re some brain-wrinkling pictures of “circles”, under different L_p metrics:

p = ⅔

The subadditive “triangle inequality” A→B→C > A→C no longer holds when p<1.

p = 4 

p = ½. (Think about a Poincaré disk to see how these pointy astroids can be “circles”.)
p = 3/2 

The moves available to a knight ♘ ♞ in chess are a circle under L1 metric over a discrete 2-D space.