One of the famous problems of mathematics in antiquity was the problem of doubling the cube: Given a line with length a (yellow), can one construct by straightedge and compass another line which has the length of the cubic root a^1/3 of the original line? We know today that this problem is unsolvable in the plane, but Archytas (428-347 BCE) found an ingenious solution in 3D:  Construct a cylinder (bluish) with diameter a, a dense torus (reddish)  with tube diameter a, and a cone (greenish) with opening half angle of π/3. Consider the one point above the horizontal plane where the cylinder, torus and cone intersect (intersection between red, green and orange lines). The red circle is a vertical cross section of the torus which goes through that point. Consider the yellow (a), blue (ρ) and red (r) line segments in this circle. We obviously have a/r = r/ρ. Next, consider the blue circle in the horizontal plane, which is a cross section of the cylinder in that plane. Consider there the yellow (a), blue (ρ) and green (x) line segments: we find x/ρ = ρ/a. Finally, because the green line segment x is the projection of the red line segment r (which lies on the cone) on the diameter of the blue circle (yellow), and because the cone has opening half angle π/3, we find r/x = 2. Combining all these relations, we find a/r = r/ρ = 2ρ/a, or (a/r) ³ = a/r·r/ρ·2ρ/a = 2. Thus, a is the edge length of a cube with two times the volume of a cube with edge length r.