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.

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Tags: angle, circle, education, geometry, Henri Poincaré, imagination, long reads, Lp norm, L_p norm, math, mathematics, maths, metric, metric spaces, metrics, noncommutative, norm, norms, Poincaré disk, size

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