Heronian Triangles and Tetrahedra

I have mentioned Heronian triangles tangentially in a post of Saturday, 2nd December 2017 when dealing with the number 25080. This number is the area of a so-called almost equilateral triangle with sides of 240, 241 and 241. In that post, I stated that:

A close-to-equilateral integer triangle is defined to be a triangle with integer sides and integer area such that the largest and smallest sides differ in length by unity. The first five close-to-equilateral integer triangles have sides (5, 5, 6), (17, 17, 16), (65, 65, 66), (241, 241, 240) and (901, 901, 902).

Figure 1

This triangle is also an example of an Heronian triangle which is simply any triangle with integer sides that also has an integer area. The simplest example is such as a triangle is right-angled with sides of 3, 4 and 5 (see Figure 1). The area of the triangle is 6 square units. From any right-angled Heronian triangle, two isosceles Heronian triangle can be constructed. For example, the 3, 4, 5 right-angled triangle leads to the 5, 5, 6 and the 5, 5, 8 isosceles triangles.

I was reminded of Heronian triangles by my diurnal age today which is 26129. This number has the following property:

A306626

Numbers that set a record for occurrences as longest side of a primitive Heronian triangle.

A primitive Heronian triangle is one in which the greatest common divisor of the three sides is 1. The 3, 4, 5 triangle is primitive but the 6, 8, 10 triangle, though still Heronian, is not primitive because the greatest common divisor is 2. Figure 2 shows these record numbers up to 15725:

Figure 2

The list of numbers up to 26129 is as follows:

1, 5, 13, 17, 37, 52, 65, 85, 119, 125, 145, 221, 325, 481, 697, 725, 1025, 1105, 1625, 1885, 2465, 2665, 3145, 5525, 6409, 15457, 15725, 26129

As can be seen from the table in Figure 2, the number of possible triangles with a longest side of 15725 is 463 but no figure is available for the number of Heronian triangles with a longest side of 26129. The algorithm I developed to calculate the number works fine for smaller sides but times out for larger numbers. Here is a permalink to the code.

Of course the Heron whose name is being used to identify these types of triangles is the same person responsible for Heron's formula that states the area \(A\) of a triangle whose sides have lengths \(a, b, c\) is:$$A=\sqrt{s(s-a)(s-b)(s-c)} \text{ where } s=\frac{a+b+c}{2}$$It was interesting to learn that a shape is called equable if its perimeter equals its area. There are only five equable Heronian triangles and these have sides (5,12,13), (6,8,10), (6,25,29), (7,15,20), and (9,10,17). As another former Mathematics teacher pointed out in a blog post on Heronian triangles:

Students, or teachers, who liked "The Hitchhiker's Guide to the Galaxy" may enjoy the 7-15-20 triangle which has both perimeter and area of 42. Author Douglas Adams set 42 as the answer to "life, the universe, and everything" and Tony Crilly and Colin Fletcher have dubbed this the "hitchhiker triangle."

There is another category of Heronian triangles that is of interest. Since the area of an equilateral triangle with rational sides is an irrational number, no equilateral triangle is Heronian. However, there is a unique sequence of Heronian triangles that are "almost equilateral" because the three sides are of the form \(n − 1, n, n + 1\). Some of these are listed in Figure 3. These are very similar to the "close-to-equilateral" triangles, mentioned at the start of this blog, that are of the form \(n,n,n+1\) or \(n-1,n-1,n\). 

Figure 3: source

The numbers in column 2, the \(n\) column, form a Lucas sequence defined by$$a_n = 4 \times a_{n-1} - a_{n-2} \text{ with } a_0 = 2 \text{ and } a_1 = 4$$Alternatively, the formula \( (2+\sqrt{3})^n+(2-\sqrt{3})^n\) will generate all the terms. 

There's a lot more to said about Heronian triangles. For example, let's venture into three dimensions and an Heronian tetrahedron defined as a not necessarily regular tetrahedron whose sides, face areas, and volume are all rational numbers. It therefore is a tetrahedron all of whose faces are Heronian triangles and additionally that has rational volume. Figure 4 shows the integer Heronian tetrahedron having smallest maximum side length. It has edge lengths 51, 52, 53, 80, 84, 117; faces (117, 80, 53), (117, 84, 51), (80, 84, 52), (53, 51, 52); face areas 1170, 1800, 1890, 2016; and volume 18144. 

Figure 4: source

The integer Heronian tetrahedron with smallest possible surface area and volume has edges 25, 39, 56, 120, 153, and 160; areas 420, 1404, 1872, and 2688 (for a total surface area of 6384) and volume 8064. The smallest examples of integer Heronian tetrahedra composed of four identical copies of a single acute triangle (i.e., disphenoids) have pairs of opposite sides (148, 195, 203), (533, 875, 888), (1183, 1479, 1804), (2175, 2296, 2431), (1825, 2748, 2873), (2180, 2639, 3111), (1887, 5215, 5512), (6409, 6625, 8484), and (8619, 10136, 11275). An Heronian tetrahedron is sometimes called a perfect tetrahedron.