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### Abstract

In this article, I investigate the (regular) hypertetrahedra and their measures, and orthogonal slices (or sections) through them.

I consider how to configure the vertices in order easily to calculate the measure.

Formulae are found for the distance between opposing facets of a hypertetrahedron, and I investigate the behaviour of these as the dimension increases.

I discuss the “shape” of slices through hypertetrahedra, showing these to be Cartesian products of hypertetrahedra of lower dimension. Then I briefly investigate slices through slices.

Formulae are found for the measure of a hypertetrahedron as an integral over (measures of) slices. From these, I find recurrence formulae, and from these, in turn, I find a closed form for the measure of a hypertetrahedron.

For example:

the measure (length) of the 1-dimensional hypertetrahedron of unit edge, which is just the unit line segment, is 1; for 2-dimensional, which is just the equilateral triangle, the measure (area) is ; for 3-dimensional, which is the regular tetrahedron, the measure (capacity or volume) is ; and so on.

I determine when (i.e. for which dimensions) the measure of a hypertetrahedron is rational, and show that when it is rational, it is in fact a unit fraction.

For example:

the measures of the 7- and 8-dimensional hypertetrahedra of unit edge are, respectively, and . The value is also rational in dimensions 17 and 24.

The sequence of dimensions for such rational measures begins

Finally, I briefly investigate the behaviour of the discrete analogue of hypertetrahedra, i.e. hypertetrahedral numbers.

The article includes some tables of calculated values.

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