Every parallelepiped that you place around an ellipsoid whose faces are tangent to the ellipsoid at their centres has the same volume for a given ellipsoid: 8 a b c, where a, b and c are the semi-axes of the ellipsoid.

Every parallelogram that you draw around an ellipse whose sides are tangent to the ellipse at their midpoints has the same area for a given ellipse: 4 a b, where a and b are the semi-axes of the ellipse.

Here is a version where the point on the ellipse is held still while the point on the hyperbola is swept along it.

Most people know how to draw an ellipse by pinning two ends of a string to a board and sweeping a pencil around inside the string, keeping it taut.

But what about the 3D equivalent?

Start with an ellipse and a hyperbola in orthogonal planes, with each curve’s vertices being the other’s foci.

This generalises the 3D Grace-Danielsson inequality

(R-3r)(R+r)≥d^2

to:

(R-nr)(R+(n-2)r)≥d^2

where r, R are the radii of the inner and outer spheres, d is the distance between their centres, and n is the dimension. I proved this was sufficient in 2014, but I couldn’t prove that it was necessary.

If you pick two points at random inside a regular dodecahedron and draw the line that contains them, that line can intersect three different kinds of pairs of faces, separated by angles of:

π, arcos(–1/√5), or arcos(1/√5)