A Heronian tetrahedron (also called a Heron tetrahedron or perfect pyramid) is a

tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all ...

whose edge lengths, face areas and volume are all

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...

s. The faces must therefore all be

Heronian triangle In geometry, a Heronian triangle (or Heron triangle) is a triangle whose side lengths , , and and area are all integers. Heronian triangles are named after Heron of Alexandria, based on their relation to Heron's formula. Heron's formula implies ...

s. Every Heronian tetrahedron can be arranged in

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...

so that its vertex coordinates are also integers.

Examples

An example known to

Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries ...

is a Heronian

birectangular tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all th ...

, a tetrahedron with a path of three edges parallel to the three coordinate axes and with all faces being

right triangle A right triangle (American English) or right-angled triangle ( British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one angle is a right a ...

s. The lengths of the edges on the path of axis-parallel edges are 153, 104, and 672, and the other three edge lengths are 185, 680, and 697, forming four right triangle faces described by the

Pythagorean triple A Pythagorean triple consists of three positive integers , , and , such that . Such a triple is commonly written , and a well-known example is . If is a Pythagorean triple, then so is for any positive integer . A primitive Pythagorean triple is ...

s (153,104,185), (104,672,680), (153,680,697), and (185,672,697). Eight examples of Heronian tetrahedra were discovered in 1877 by

Reinhold Hoppe Ernst Reinhold Eduard Hoppe (November 18, 1816 – May 7, 1900) was a German mathematician who worked as a professor at the University of Berlin. Education and career Hoppe was a student of Johann August Grunert at the University of Greifswald, gr ...

117 117 may refer to: *117 (number) *AD 117 *117 BC *117 (emergency telephone number) *117 (MBTA bus) * 117 (TFL bus) *117 (New Jersey bus) *''117°'', a 1998 album by Izzy Stradlin *No. 117 (SPARTAN-II soldier ID), personal name John, the Master Chief ...

is the smallest possible length of the longest edge of a perfect tetrahedron with integral edge lengths. Its other edge lengths are 51, 52, 53, 80 and 84. 8064 is the smallest possible volume (and 6384 is the smallest possible surface area) of a perfect tetrahedron. The integral edge lengths of a Heronian tetrahedron with this volume and surface area are 25, 39, 56, 120, 153 and 160. In 1943, E. P. Starke published another example, in which two faces are

isosceles triangle In geometry, an isosceles triangle () is a triangle that has two sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter versio ...

s with base 896 and sides 1073, and the other two faces are also isosceles with base 990 and the same sides. However, Starke made an error in reporting its volume which has become widely copied. The correct volume is , twice the number reported by Starke. Sascha Kurz has used computer search algorithms to find all Heronian tetrahedra with longest edge length at most .

Classification, infinite families, and special types of tetrahedron

regular tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all th ...

(one with all faces being equilateral) cannot be a Heronian tetrahedron because, for regular tetrahedra whose edge lengths are integers, the face areas and volume are

irrational number In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two inte ...

s. For the same reason no Heronian tetrahedron can have an equilateral triangle as one of its faces. There are infinitely many Heronian tetrahedra, and more strongly infinitely many Heronian

disphenoid In geometry, a disphenoid () is a tetrahedron whose four faces are congruent acute-angled triangles. It can also be described as a tetrahedron in which every two edges that are opposite each other have equal lengths. Other names for the same ...

s, tetrahedra in which all faces are congruent and each pair of opposite sides has equal lengths. In this case, there are only three edge lengths needed to describe the tetrahedron, rather than six, and the triples of lengths that define Heronian tetrahedra can be characterized using an

elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. I ...

. There are also infinitely many Heronian tetrahedra with a cycle of four equal edge lengths, in which all faces are

s. There are also infinitely many Heronian birectangular tetrahedra. One method for generating tetrahedra of this type derives the axis-parallel edge lengths

a

b

, and

c

from two equal sums of fourth powers :

p^4+s^4=q^4+r^4

using the formulas :

a=\bigl, (pq)^2-(rs)^2\bigr, ,

b=\bigl, 2pqrs\bigr, ,

c=\bigl, (pr)^2)-, (qs)^2\bigr, .

For instance, the tetrahedron derived in this way from an identity of

59^4+158^4=133^4+134^4

, has

a

b

, and

c

equal to , , and , with the hypotenuse of right triangle

ab

equal to , the hypotenuse of right triangle

bc

equal to , and the hypotenuse of the remaining two sides equal to . For these tetrahedra,

a

b

, and

c

form the edge lengths of an almost-perfect cuboid, a rectangular cuboid in which the sides, two of the three face diagonals, and the body diagonal are all integers. No example of a Heronian trirectangular tetrahedron had been found and no one has proven that none exist. A complete classification of all Heronian tetrahedra remains unknown.

Related shapes

An alternative definition of Heronian triangles is that they can be formed by gluing together two integer right triangles along a common side. This definition has also been generalized to three dimensions, leading to a different class of tetrahedra that have also been called Heron tetrahedra.

References

External links

* {{mathworld, HeronianTetrahedron Polyhedra Arithmetic problems of solid geometry