Simplex: Difference between revisions
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imported>Catherine Woodgold (Example describing standard 3-simplex) |
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A '''simplex''' is a mathematical object, analogous to a [[triangle]]. Every simplex has a dimension, which is a nonnegative [[integer]]; an n-simplex means a simplex of dimension n. | A '''simplex''' is a mathematical object, analogous to a [[triangle]]. Every simplex has a dimension, which is a nonnegative [[integer]]; an n-simplex means a simplex of dimension n. | ||
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# A 3-simplex is a solid [[tetrahedron]]. | # A 3-simplex is a solid [[tetrahedron]]. | ||
# The standard n-simplex is the set <math>\Delta_n</math> of points <math>x=(x_0,\dots,x_n)\in [0,1]^n</math> for which <math>\sum_ix_i=1</math>. For example, the standard 3-simplex is the shape of the corner of a cube if it is sliced off at an angle to produce a solid with three similar triangular faces (parts of the sides of the cube) and one larger, equilateral triangular face. | # The standard n-simplex is the set <math>\Delta_n</math> of points <math>x=(x_0,\dots,x_n)\in [0,1]^n</math> for which <math>\sum_ix_i=1</math>. For example, the standard 3-simplex is the shape of the corner of a cube if it is sliced off at an angle to produce a solid with three similar triangular faces (parts of the sides of the cube) and one larger, equilateral triangular face. | ||
Revision as of 16:13, 14 November 2007
A simplex is a mathematical object, analogous to a triangle. Every simplex has a dimension, which is a nonnegative integer; an n-simplex means a simplex of dimension n.
- A 0-simplex is just a point.
- A 1-simplex is an interval or line segment.
- A 2-simplex is a filled-in triangle.
- A 3-simplex is a solid tetrahedron.
- The standard n-simplex is the set of points for which . For example, the standard 3-simplex is the shape of the corner of a cube if it is sliced off at an angle to produce a solid with three similar triangular faces (parts of the sides of the cube) and one larger, equilateral triangular face.
![{\displaystyle x=(x_{0},\dots ,x_{n})\in [0,1]^{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b905612427282ef93215c83e4cff97384889bbe3)
