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Geometry values

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Geometry values

Hi all. This may or may not make sense. If numbers have values , why can't shapes have some type of value system as well ? After all , shapes are represented by numbers. For example : a sphere shrinks to 0d when its radius is decreased. A torus , however , can shrink to 0d or 1d , a ring. Therefore , torus is greater than a sphere ? Thx.

There are many ways to give values to shapes. Although it is rather difficult to say that one shape is greater then the other. Just like it is difficult to say that one complex number (or vector) is greater then the other.

For example we have: Euler characteristic, any kind of dimension (manifold dimension, covering dimension, etc.), homotopy groups, homology groups and many others. Most of them well behave on homotopy, which means that if one shape can be transformed (via "good" transformation) to another then they have the same value.

For example n-dimensional sphere has n-th homotopy group equal to Z (the integers), while n-dimensional torus has n-th homotopy group trivial 0.

By the way, a sphere can also be shrinked (or rather collapsed) into 1d object. Take the diameter of a sphere and then collapse sphere in a perpendicular way onto the diamater.


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