Is there a name for this shape:
Basically a square or rectangle with curved sides. Two of those on opposite sides curve inward and the other two curve outward. If they have the same curvature and length, they’ll tile the plane, but I expect the name would apply even if they didn’t.
I’d link to a picture, but I need the name to google it. >>Rectangle with curved sides<< does not bring up an image.
I’ve seen it called “apple core” if I’m understanding you correctly.
Do you mean astroids?
Nope, not astroid. That has all sides concave. I’m looking for a shape with two convex and two concave.
“Apple core” might be it.
I think it probably is. I googled “flat tiles apple core shape,” which led me to “apple core template,” for which the images primarily lead to quilting patterns.
ETA: I also saw mention of “axe blade.” Doesn’t seem to be as common under that search term.
Yeah, most results I found were from quilting sites. I didn’t see “axe blade” but saw ax head and double bit axe. The axe terms obviously refer to a double-headed axe.
like this ones?
called axe head tiles.
Here’s an image of the apple core tiling, which is indeed the same as the ax head.
Scientific American just ran an article on newly-found soft three-d tilings with no corners.
The oldest example for this shape I can think of is the Minoan double-axe, but it may be older.
Called a labrys as in labryrinth. It was the symbol of the Minoan state.
Yes, I know, I visited Crete and the Archaeological Museum of Iraklion twice, the palaces of Knossos and Phaistos once.
I always thought it odd, when I read of the Lune of Hippocrates as a youth, because it was considered so revolutionary when it was discovered, that it was possible to determine the area of a shape none of whose edges were straight lines. But of course, there are a whole set of shapes like the OP is describing (apple cores or axe heads or whatever), plus another whole set where the convex sides (and hence also the concave sides) are adjacent, which also meet that description, but whose areas are trivial.
I was thinking “cookie with two bites out of it…”
I think the lune was considered remarkable because although it was made from curved shapes, its area was equal to the area of a (particular) triangle
Right, but so is this shape. The ancients knew how to construct a square with the same area as any triangle, or vice-versa. In fact, the problem was usually posed as “given this shape, find a square with the same area”, and the only reason you stop with the triangle with the lune is because everyone knows how to do the last steps.
Ah, good point. I suppose the lune was just the first one thought of, so it stood out (and it’s an elegant proof).