Degenerate Power Simplices

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Five points in 4-space with no four on a plane make a 4D simplex (or hypertetrahedron) . A Cayley–Menger determinant uses the 10 edge lengths of to find the hypervolume.


Five points are called flat if the hypervolume of is zero. This is also called a 3D degenerate simplex.

Four points are called plane if the bounding volume of is zero. This is also called a 2D degenerate tetrahedron.

Three points are called straight if the bounding area of is zero. This is also called a 1D degenerate triangle.

This Demonstration shows 50 3D degenerate simplices with the property that all the edge lengths are powers of a particular root. Each edge is labeled with the corresponding power. The edge 0 corresponds to length and is always the shortest length.

The plastic constant works very well (14 known solutions); it is a root of (), the smallest of the Pisot numbers. The next root that works well (10 known solutions) is a reciprocal of a root of (), the next smallest Pisot number. So far, I have found 15 different algebraic roots of degree six or less that can make a 3D power simplex.

The distances are actually powers of the square root of the given root. Index 51 has a degenerate tetrahedron, but I included it because it corresponds to the Narayana Cow constant, another of the Pisot numbers.

A degenerate 2D example uses root , the golden ratio. The distances (0, 1, 2, 2, 1, 0) are powers of , giving the rectangle with diagonals of length . Removing a point gives the Kepler triangle.

What other solutions are there where the root has degree six or less? This Demonstration is meant as a starting point for this exploration.


Contributed by: Ed Pegg Jr (September 2018)
Open content licensed under CC BY-NC-SA



1. Wikipedia. "Kepler Triangle." (Sep 4, 2018)


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