[Monte314]

Suppose we have two sets of points in the plane, given as ordered pairs, say:

F1 = { (1,1), (-2,3) }, and F2 = { (2,1), (4,2) }

I could imagine "multiplying" these two sets of points together by forming all coordinate-wise products (x-coords from F1 times x-coords from F2, and y-coords from F1 times y-coords from F2):

F1 * F2 = { (1x2,1x1), (1x4,1x2), (-2x2,3x1), (-2x4,3x2) }
= { (2,1), (4,2), (-4,3), (-8,6) }

With this notion, I can "multiply" two sets in the plane by each other to obtain a new set. For example, if I multiply the x-axis by the y-axis, I get the one-point set (0,0):

x-axis is the set A = { (a,0) where a is a real number }
y-axis is the set B = { (0,b) where b is a real number }

Then, A * B = { (ax0,0xb) where a and b are real numbers } = { (0,0) }

[This is called a direct product, if you want to investigate it more.]

Recall that the unit circle centered at the origin consists of all pairs (x,y) that satisfy the equation:

C = { (x,y) such that x^2 + y^2 = 1 }

Question: What set do I get when I "square the circle", that is, "multiply" C by itself?

Bonus question: What does C-cubed look like?

[enWTFp]

Solution.

( Show Spoiler )

Denote "sr2" the square root of 2, and E(p,q) the ellipse x^2/p^2 + y^2/q^2 = 1.

Answer: C*C=K, where K is the full square (1,0),(0,-1),(-1,0),(0,1). Bonus: C*C*C=S, where
S is a concave full "star" matching the vertices of K, and touching K/sr2 in points (i.sr2/4, j.sr2/4), i,j in {-1,1}.

C*(sinA,cosA) -> E(sinA,cosA). It is easy to check that x+y=1 touches E(sinA,cosA) for every A.

Indeed, if t=(sinA)^2 then 1-t=(cosA)^2, so E(sinA,cosA): x^2/t + (1-x)^2/(1-t) = 1, solving the system with x+y=1.

That leads to (1-t).x^2 + t.(x-1)^2 = t(1-t), from which we get (x-t)^2=0, so x+y=1 touches the ellipse in (t,1-t).

(sinA)^2=t covers the whole interval [0,1], so the family E(sinA,cosA) covers the whole line segment [(0,1),(1,0)].

By symmetry we get that C*C covers the other 3 sides of K, and the interior is completely covered too.

For the Bonus we need to compute C*K, but K contains C/sr2, hence C*K contains K/sr2.

[Monte314]

enWTFp has, of course, provided a correct and comprehensive solution to both questions. Anybody else?

I'll post some pictures of these "shape products" in a couple of days. Some of them are quite beautiful... and quite complex.

[Arcani]

Ooo a mapping problem. Neat.

( Show Spoiler )

I'm not particularly good at these problems at least in terms of writing out the equations but I took a sampling of points and it looks like the first one makes a diamond shape and the bonus makes a shape kinda like a four pointed ninja star.

Maybe when I have more time I'll try to work out the equations.