gregeganSF, 9 months ago

For each shape in the family shown above, there is also a whole family of helical versions, again with constant Gaussian curvature κ=1.

Any number of these pieces (with enough pitch not to self-intersect) could be joined along meridians, potentially making an infinite ribbon.

A shape like a prolate ellipsoid with cusps at the poles is cut open along a meridian and stretched vertically, with the circles of latitude deforming into helices.

gregeganSF, 9 months ago

The endless ribbon version.

sundew, 9 months ago

@gregeganSF It's like the old saying:
"If life gives you lemons, make an endless ribbon with constant Gaussian curvature."

ngons, 9 months ago

@gregeganSF now that’s what i call a cool shape. Now there are no cusps, so is all the curvature 1?

gregeganSF, 9 months ago

@ngons The curvature is still 1 everywhere except the boundaries, where strictly speaking it is undefined.

The coolest thing would be to create something without cusps or boundaries, but there is no way to create a “complete” manifold – where every geodesic can be extended – that has constant positive curvature and infinite area.

https://en.wikipedia.org/wiki/Myers%27s_theorem

henryseg, 9 months ago

@ngons @gregeganSF Is it cool to do impossible things? Doing nearly impossible things is often cool, but I’m not sure that property extends to the limit.

ngons, 9 months ago

@gregeganSF @henryseg :D I think coolness isn’t a topological invariant - clearly maths art is all about maximizing subjective coolness under homeomorphisms ..

ngons, 9 months ago

@gregeganSF that makes sense, i forgot about the edges, only considered you got rid of the north and south poles. Perhaps an infinite cylinder would do it though

Ianagol, 9 months ago

@gregeganSF Someone needs to make that by sewing together cut up tennis balls.

Ianagol, 9 months ago

@gregeganSF Conchiglie?

irving, 9 months ago

@gregeganSF Is it right that you need at least 2 cusps?