Posts tagged with circle

## How I Use The Pythagorean Theorem Every Day

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.)

## ±∞

The Cauchy distribution (`?dcauchy` in `R`) nails a flashlight over the number line

and swings it at a constant speed from 9 o’clock down to 6 o’clock over to 3 o’clock. (Or the other direction, from 3→6→9.) Then counts how much light shone on each number.

In other words we want to map evenly from `the circle (minus the top point)` onto `the line`. Two of the most basic, yet topologically distinct shapes related together.

You’ve probably heard of a mapping that does something close enough to this: it’s called `tan`.

Since `tan` is so familiar it’s implemented in Excel, which means you can simulate draws from a Cauchy distribution in a spreadsheet. Make a column of `=RAND()`'s (say column A) and then pipe them through `tan`. For example `B1=TAN(A1)`. You could even do `=TAN(RAND())` as your only column. That’s not quite it; you need to stretch and shift the `[0,1]` domain of `=RAND()` so it matches `[−π,+π]` like the circle. So really the long formula (if you didn’t break it into separate columns) would be `=TAN( PI() * (RAND()−.5) )`. A stretch and a shift and you’ve matched the domains up. There’s your Cauchy draw.

In R one could draw three Cauchy’s with `rcauchy(3)` or with `tan(2*(runif(3)−.5))`.

What’s happening at `tan(−3π/2)` and `tan(π/2)`? The `tan` function is putting out to ±∞.

I saw this in school and didn’t know what to make of it—I don’t think I had any further interest than finishing my problem set.

I saw as well the ±∞ in the output of `flip[x]= 1/x`.

• `1/−.0000...001 → −∞` whereas` 1/.0000...0001 → +∞`.

It’s not immediately clear in the `flip[x]` example but in `tan[x/2]` what’s definitely going on is that the angle is circling around the top of the circle (the hole in the top) and the flashlight of the Cauchy distribution could be pointing to the right or to the left at a parallel above the line.

Why not just call this ±∞ the same thing? “Projective infinity”, or, the hole in the top of the circle.

• Why bicontinuity is the right condition for topological equivalence (homeomorphism): if continuity of the inverse isn’t required, then a circle could be equivalent to a line (.99999 and 0 would be neighbours) — Minute 8 or so.
• Geometric construction (no complex numbers) of the circle group.
• Pappos’ theorem. (Minute 31)
• Pascal’s theorem.
• Desargues’ theorem.
Hat tip to +Ozymandias Haynes.

## Circles

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.