Heronian tetrahedron

Key takeaways

Heronian tetrahedron belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Heronian tetrahedron to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Heronian tetrahedron from memory before moving on to harder problems.

Reference excerpt

A Heronian tetrahedron (also called a Heron tetrahedron or perfect pyramid) is a tetrahedron whose edge lengths, face areas and volume are all integers. The faces must therefore all be Heronian triangles (named for Hero of Alexandria). Every Heronian tetrahedron can be arranged in Euclidean space so that its vertex coordinates are also integers.

Examples

An example known to Leonhard Euler is a Heronian birectangular tetrahedron, a tetrahedron with a path of three edges parallel to the three coordinate axes and with all faces being right triangles. 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 triples (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. 117 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 triangles 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 124185600, twice the number reported by Starke. Sascha Kurz has used computer search algorithms to find all Heronian tetrahedra with longest edge length at most 600000.

Classification, infinite families, and special types of tetrahedron A regular tetrahedron (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 numbers. 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 disphenoids, 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. There are also infinitely many Heronian tetrahedra with a cycle of four equal edge lengths, in which all faces are isosceles triangles. There are also infinitely many Heronian birectangular tetrahedra. One method for generating tetrahedra of this type derives the axis-parallel edge lengths a {\displaystyle a} , b {\displaystyle b} , and c {\displaystyle c} from two equal sums of fourth powers

p 4 + s 4 = q 4 + r 4 {\displaystyle p^{4}+s^{4}=q^{4}+r^{4}}

using the formulas

a = | ( p q ) 2 − ( r s ) 2 | , {\displaystyle a={\bigl |}(pq)^{2}-(rs)^{2}{\bigr |},}

b = | 2 p q r s | , {\displaystyle b={\bigl |}2pqrs{\bigr |},}

c = | ( p r ) 2 − ( q s ) 2 | . {\displaystyle c={\bigl |}(pr)^{2}-(qs)^{2}{\bigr |}.}

For instance, the tetrahedron derived in this way from an identity of Leonhard Euler, 59 4 + 158 4 = 133 4 + 134 4 {\displaystyle 59^{4}+158^{4}=133^{4}+134^{4}} , has a {\displaystyle a} , b {\displaystyle b} , and c {\displaystyle c} equal to 386678175, 332273368, and 379083360, with the hypotenuse of right triangle a b {\displaystyle ab} equal to 509828993, the hypotenuse of right triangle b c {\displaystyle bc} equal to 504093032, and the hypotenuse of the remaining two sides equal to 635318657. For these tetrahedra, a {\displaystyle a} , b {\displaystyle b} , and c {\displaystyle 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. A complete classification of all Heronian tetrahedra remains unknown.

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Heronian tetrahedron

Start with the simplest possible case. Write down what Heronian tetrahedron claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Heronian tetrahedron before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Heronian tetrahedron ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Heronian tetrahedron

In research: Heronian tetrahedron appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Heronian tetrahedron in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Heronian tetrahedron is common in secondary-school and first-year university syllabi. It links to neighbouring topics Arithmetic problems of solid geometry, Tetrahedra, so understanding it makes those chapters shorter.
In everyday life: Look for Heronian tetrahedron outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Heronian tetrahedron in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Heronian tetrahedron means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Heronian tetrahedron out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Heronian tetrahedron in simple terms?

A Heronian tetrahedron (also called a Heron tetrahedron or perfect pyramid) is a tetrahedron whose edge lengths, face areas and volume are all integers. The faces must therefore all be Heronian triangles (named for Hero of Alexandria).

Why does Heronian tetrahedron matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Heronian tetrahedron?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Heronian tetrahedron.