On: Some Math Problems Seem Impossible. That Can Be a ...

[mhandley]

Wouldn't you need to draw the convex octagon on the inside surface of the sphere?

[dmurray → mhandley]

It's the same whether you draw it on the inside or the outside. Imagine the sphere is see-through, draw it on the inside then go out and look on the outside. The definition of "convex", "octagon", "right angle" or "straight line" doesn't change when you move from looking at the interior surface to the exterior one.

[Someone → dmurray]

Thanks both. I still can’t visualize that octagon, but I now think you need saddle points to draw it, so a sphere wouldn’t do. If that’s true, a torus would do.

[dmurray → Someone]

I'm pretty certain it's possible on the sphere, but I'm currently failing to demonstrate that with a pencil and a table tennis ball.

Seems it should be possible to draw the octagon that's shown in the article: draw a rectangle and flex the sides out a little bit.

[gowld → dmurray]

I'm not convinced.

Flexing the sides requires negative curvature or non-convexity. Triangles are easy to draw because they have fewer sides than a square, thus positive curvature makes it possible to create a 3-right triangle. Octagons are the opposite.