Magus (marcmagus) wrote,
Magus
marcmagus

Why Seventy-Six Trombones?

For some reason I can't remember at all, shortly after I crawled into bed last night shield_toad111 started singing "76 trombones". It must have made sense at the time, because I said something to the effect of "I was thinking that too!" But then I said those fateful words, "Why 76?"

There followed a few minutes of speculation (76 is a strange number. It probably wasn't a 4x19 rectangular arrangement. Maybe it was 4x20 or 8x10 with flag-bearers or something at the corners.) before she sensibly went back to sleep, but I couldn't get to sleep for thinking about it.

It's not a triangular number.

Maybe it makes a pixel circle? No, they all have odd area from the center pixel.

Wait, what if there isn't a center pixel?

Many games played on a grid (like roguelikes) use "Euclidean distance" to measure distances between the grid squares for things like spell ranges and areas of effect. This gives you relatively nice circles (even nicer if you have square tiles, unlike most fonts). The "Euclidean distance" between to squares is, of course, sqrt(x^2+y^2), although it's usually easier to stick with integer math and square the ranges/areas of effect.

For example, the 5 circle (radius 5 if you're working in integers, or sqrt(5) if you're not) would be a 5x5 square with the corners cut off to make an approximate circle. (To get into the corners of the square you need radius 8; radius 9 pushes you out one more square along the axes.)

As I said, though, these all have an odd area because you count the target square.

But of course pixel art doesn't have to have a square where the player/target is. If you look at circles centered on the grid lines rather than the square, you can describe circles the same way, as all the cells fully contained within the arc of the circle. A circle with radius sqrt(29) contains 76 cells.

Thus, I propose that the 76 trombones marched in a circle, like so:

...((((...
..((((((..
.((((((((.
((((((((((
((((((((((
((((((((((
((((((((((
.((((((((.
..((((((..
...((((...

Do you have a theory? Or an actual knowledgeable explanation?


Originally posted at http://marcmagus.dreamwidth.org/142213.html. (comment count unavailable comment[s]|comment there)
Tags: math!
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