Topology and Geometry

Spherical Blackboard

What it shows:

You can use a spherical blackboard for many things, including the teaching of geographical coordinates, as a model for a closed Universe, or simply as a mathematical shape.

In the non-Euclidean geometry of the sphere, a circle will have a circumference greater than 2πr and an area greater than πr2. A triangle’s angles will add to more than 180°, and two parallel lines, called Great Circles, will converge.

A Universe with a density parameter Ω greater than unity will have too much mass to overcome its own gravitational...

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Saddle Shape Universe

Curved space segment for open universe geometry.

What it shows:

Whether the Universe continues to expand forever or will collapse back in upon itself depends upon the amount of matter it contains. For a density parameter Ω less than unity the Universe will not have enough mass to collapse and will be in a state of perpetual expansion. In general relativity, the curvature of space is dependent upon the density of the Universe, and for Ω<1 the curvature is negative or hyperbolic. It can be represented two dimensionally (see Comments) by a saddle...

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