In our mind’s eye, the universe seems to go on forever. But using geometry we can explore a variety of three-dimensional shapes that offer alternatives to “ordinary” infinite space. When you gaze out ...

This is a preview. Log in through your library . Abstract In this paper we investigate angle geometry in the universal Teichmüller space. We construct three examples of triangles bounded by geodesic ...

A technician often has to sketch or design ductwork and the other parts of an HVAC system. Sometimes you have to sketch the room or building you are working with. This means working with lines, angles ...

A system developed by Google’s DeepMind has set a new record for AI performance on geometry problems. DeepMind’s AlphaGeometry managed to solve 25 of the 30 geometry problems drawn from the ...