Genus (InstanceTopic, 3)

From Hi.gher. Space

(Difference between revisions)

Revision as of 11:15, 12 March 2011

In 3D, the genus of a shape is equal to how many "holes" it has that fully pierce the shape. For example, a sphere has genus 0, and a torus has genus 1.

In higher dimensions, it gets more complicated: with each new dimension, a new type of hole is possible. So each of the different four-dimensional torii have a pair of numbers as their "genus".

This is closely related to homology groups.

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-The '''genus''' of a [[shape]] is equal to how many "holes" it has that fully pierce the shape.
+In 3D, the '''genus''' of a [[shape]] is equal to how many "holes" it has that fully pierce the shape. For example, a [[sphere]] has genus 0, and a [[torus]] has genus 1.
-[[Line (object)|Lines]] always have a genus of zero. If a hole was created it would be split in half.
+In higher dimensions, it gets more complicated: with each new dimension, a new type of hole is possible. So each of the different [[four-dimensional torii]] have a pair of numbers as their "genus".
-There are two{{hmm}} types of holes that a shape can have. ''Pockets'' are holes completely inside the object and cannot be seen by any outside viewer (unless the being viewing the shape is in a higher dimension than the shape itself). Ordinary ''holes'' are ones that can be seen from the outside, for example the hole in a [[torus]]. The only type of hole that can be placed inside a [[polygon]] is a pocket, as creating a normal hole would split the shape in half.
+This is closely related to [[homology groups]].
-[[Category:Geometric properties]]