Version of 9 July 2012.

Dave Barber's other pages.

The table below shows Cartesian coördinates for some regular simplices in various numbers of dimensions.

Each (one-dimensional) edge is of length one; beyond that the location and orientation of each figure was selected to make as obvious a pattern of numbers as possible. Of course, the coördinates may be scaled, rotated and translated as desired.

For clarity, define:

a | = 1 ÷ √ ( 2 × | 2 ) |

b | = 1 ÷ √ ( 3 × | 4 ) |

c | = 1 ÷ √ ( 4 × | 6 ) |

d | = 1 ÷ √ ( 5 × | 8 ) |

e | = 1 ÷ √ ( 6 × | 10 ) |

f | = 1 ÷ √ ( 7 × | 12 ) |

Then it is simple to write the coördinates:

In 2 dimensions the 3 vertices, which form an equilateral triangle, are: |
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In 3 dimensions the 4 vertices, which form a regular tetrahedron, are: |
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In 4 dimensions the 5 vertices are: |
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In 5 dimensions the 6 vertices are: |
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In 6 dimensions the 7 vertices are: |
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Higher dimensionalities generalize in the obvious fashion. |