Today is part one, that's straight lines and curved spaces with no physics, just geometry. Let's start with this picture of the flat Euclidean 2D plane from high school math class.

Spherical shapes differ from infinite Euclidean space not just in their global topology but also in their fine-grained geometry. For example, because straight lines in spherical geometry are great ...

Any straight line segment can be used to construct a circle, where one end of the line segment is the center and the other end sweeps radially around. All right angles are equal to one another ...

Walking through a non-Euclidean space in person is easier for many people than trying to analyze it on paper, since one interacts via the senses just as one does in the ordinary world.

One hypothesis about the shape of the universe is that the universe is in the space of Euclidean geometry and flat. In Euclidean geometry, it is assumed that straight lines extend everywhere ...

The shortest distance between two points is a straight line. This is one of the basic principles of Euclidean geometry. But we live on a spherical Earth, so we cannot travel the straight line path ...

This image shows straight lines drawn on a paper model of a hyperbolic plane. All the pencil lines that appear to be curved were drawn with a ruler so they are actually straight.

Setting it straight with curved space Georg Friedrich Bernhard Riemann ushered in a new era of geometry on June 10, 1854, when he delivered his classic lecture “On the Hypotheses Which Lie at ...

Non-Euclidean geometries for grid cells Date: May 6, 2015 Source: Sissa Medialab Summary: Grid cells, space-mapping neurons of the entorhinal cortex of rodents, could also work for hyperbolic ...

Non-Euclidean geometry doesn't operate that way. A triangle inscribed on the surface of a sphere — a spherical geometric space — has more than 180 degrees in its internal angles, and one drawn ...