Perfect Pyramids

The tetrahedron is the simplest of all polyhedra—solids bounded by polygons. It has four triangular faces, four vertices, and six edges. If each edge has the same length and each face is an equilateral triangle, the result is a regular tetrahedron—one of the Platonic solids.

Example of a tetrahedron.

Another group of tetrahedra that some people consider special consists of those that have integer edge lengths, face areas, and volumes. Such a solid is sometimes called a Heronian tetrahedron or a perfect pyramid.

The term “Heronian” refers to Heron of Alexandria (10–90). His name is also attached to a formula that relates the area of a triangle, A, to the lengths of its three sides, a, b, and c.

There’s an analogous formula for the volume of a tetrahedron, given the lengths of its six edges.

Tetrahedra with integer sides, face areas, and volumes are rare. For example, there’s only one perfect pyramid (or Heronian tetrahedron) with integer sides less than 157. It has sides of length 51, 52, 53, 80, 84, and 117; faces of area 1170, 1800, 1890, and 2016; and a volume of 18144. (See for an image.) Nonetheless, you can prove that there are infinitely many such tetrahedra. Indeed, Ralph H. Buchholz did so in a 1992 paper on perfect pyramids.

Buchholz performed a number of searches, identifying various special cases. For instance, it’s clear that a tetrahedron in which all the edges have the same length or one in which each edge has one of just two values can have an integer volume. However, a tetrahedron with four identical faces, where each of the three sides of a face has a different length, can be Heronian. For example, a tetrahedron with pairs of opposite edges of length 888, 875, and 533 has a volume of 37608480.

Recently, Randall L. Rathbun created a database of 5,801,746 Heronian triangles having perimeters smaller than 217. These triangles can then be used as the basis of a search for “interesting” Heronian tetrahedra.

So, what’s the smallest Heronian tetrahedron in terms of volume and surface area? It’s the one with edges 25, 39, 56, 120, 153, 160; a surface area of 6384; and a volume of 8064.

Among other things, Rathbun has also identified the smallest pair of primitive Heronian tetrahedra with the same surface area:

Area Volume Edges
64584 170016 595 429 208 116 276 325
64584 200928 595 507 116 208 276 325

Here’s the smallest pair with the same volume:

Area Volume Edges
244272 3564288 697 697 306 185 185 672
298248 3564288 1344 697 697 153 680 680

Here’s the first occurrence of three Heronian tetrahedra with the same volume:

Area Volume Edges
11124120 501399360 15080 14820 500 1309 1557 13621
12571944 501399360 4522 3485 3485 2640 2275 2275
12667452 501399360 5280 3485 3485 2261 2652 2652

There’s no trace yet of three primitive Heronian tetrahedra that have the same surface area.

Other questions remain. For instance, is there a pair of Heronian tetrahedra with the same surface area and the same volume? No one knows yet.

Is there more to be discovered about perfect pyramids? Undoubtedly.

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