Geometrical Constants in Equilateral Triangles: Part I

Introduction

What is a constant? It is an idea or an object that does not change under a set of well defined rules.

What makes a constant super interesting? When we see one we know exactly what we are dealing with, in the absolute sense. Its consequences are known a priori.

Why is this a big deal? Let's say we are solving a problem and our conclusions are in contradiction with what one or more constants are suggesting , we are certain that me made a mistake. So, having as many constants around can help us fix our mistakes.

This is going to be a series consisting of three posts. In each part, there might be a gradual increase in mathematical sophistication, but we will not go beyond the level of basic algebra and some elementary trigonometry. The plan is to take a mathematical trip to discover the mysteries and origin of the most famous object π.

Types of Constants

1. Physical constants - These are the facts which do not change in the physical world. For example, mass of an electron is 9.10938356 × 10⁻³¹ kg. Say, we are analyzing some data from an experiment involving electrons and we end up with a different value of the mass, we can be certain that there is a serious error in our experiment or conclusions or both.
2. Mathematical constants - These are the facts that do not change in an abstract mathematical structure. They may or may not have a manifestation in the real world. For example, the value of Euler's number e is 2.71828... If we are building a sophisticated mathematical theory where we are forced to have a different value of e, then we know something is wrong!
3. Geometrical constants - These are very similar to mathematical constants, after all geometry is branch of mathematics. But we want to keep those constants separately which have been associated mainly with geometrical properties. For example, the value of π = 3.14..., is most easily recognized in context of a circle.

There are many other types of constants, but we are here to take a closer look at the constants of the third type above.

The Equilateral Triangle

A triangle is a closed geometrical shape having three sides. If all three sides have equal lengths, we call it it an equilateral triangle. Let us write down some basic facts that we obtained by careful observation -

1. All the angles of an equilateral triangle have the same value, which is 60⁰.
2. The geometrical center of an equilateral triangle is at the same distance from each of the three corners (vertices). This point is called a circumcenter.
3. The angle subtended at the circumcenter by any side is 120⁰.
4. The line segment joining a corner (vertex), passing through the circumcenter cuts the opposite side in half, making an angle of 90⁰.

Now, let us discover a geometrical constant hiding within an equilateral triangle. We have a diagram of such a triangle as shown below -

The features of the triangle are -

1. O is the circumcenter, so AO = BO = CO = r (due to observation 2)
2. ∠ ABC = ∠ BCA = ∠ CAB = 60⁰ (due to observation 1)
3. ∠ BOC = θ = 120⁰ (due to observation 3)
4. ∠ BDO = 90⁰ and BD = DC (due to observation 4)
5. ∠ BOD = ∠ DOC = 1/2 ∠ BOC = 60⁰ (due to symmetry)

Now, consider the triangle Δ BOD. Since this is a triangle with one of the angles being 90⁰, we can use a basic trigonometric formula : sin(θ) = Perpendicular divided by Hypotenuse, and write -

We can find the value of sin(θ) from a trigonometric table, which turns out to be √3/2. So, we get

We can also obtain the length of the side BC = 2 × BD = r √3.

Let us now introduce a quantity called perimeter. It is nothing but the sum of all the sides of a closed geometrical shape. Since all the sides of an equilateral triangle are equal in length, the perimeter of our triangle is 3 × length of any one side. That means

Finally, let us a define a number 𝓒 as the ratio of perimeter and the distance from the circumcenter to the vertex. For our case, this becomes -

This number is very interesting because its value is constant. It does not depend on the dimensions of the triangle. All it needs is the triangle to be equilateral. So, no matter how big or small the equilateral triangle is, this number 𝓒 will never change.

This might not sound very impressive, but it leads to a really deep idea about symmetries and constants that we are going to explore further.

This article was originally published on physicsgarage. In the next part of the series (email address required), we will obtain these constants for a whole class of geometries, and see where it takes us.