Qfyilyi

So I watching a video that has astronomer and topologist Cliff Stoll take about how figures that aren't quadrilaterals can have all their angles equal 90 degrees on different surfaces. For example, on a sphere, you can create a triangle that has all of its angles equal $90^circ$. On a pseudosphere, you can create a pentagon that has all of its angles equal $90^circ$. Now, here's my question.

Is there a surface where a hexagon with this property is possible?

You would need a surface of negative curvature.

It is the best to use a hyperbolic plane for this, where you can easily fit any regular n-gon with given angles as long as the sum of its external angles is greater than 360 degrees. The problem is that the hyperbolic plane does not fit in the Euclidean space.

Pseudosphere is a small fragment of the hyperbolic plane. You can draw a right-angled hexagon on the pseudosphere only if you allow it to wrap over itself.

You can also draw it on a Dini's surface -- that is basically an unrolled pseudosphere where you have several layers, and thus you avoid the intersection problem. But it would be hard to see anything because it is rolled very tightly.

Less smooth, but probably the best way would be to use something similar to a hyperbolic crochet. See our computer simulation (arrow keys to rotate).

Take a cube and orient it so you're looking down a body diagonal. The edges that appear to be at the boundary of this view actually form a nonplanar object with six congruent edges, right angles wherever two adjacent edges meet, and $D_{3d}$ point group symmetry.