Geometrical Constants - Part 1


4 min read
Geometrical Constants - Part 1

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? Lets 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.

Let me show you how its done...

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 $\pi$.

Types of Constants

  • Physical constants - These are the facts which do not change in the physical world. For example, mass of an electron is $9.10938356 \times 10^{-31}$ 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.
  • 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 $2.71828 \cdots.$ If we are building a sophisticated theory where we are forced to have a different value, then its time to throw it in the dustbin!
  • 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 $\pi$ which is $3.14\cdots$, 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 $60^{\circ}$.
  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 $120^{\circ}$.
  4. The line segment joining a corner (vertex), passing through the circumcenter cuts the opposite side in half, making an angle of $90^{\circ}$.

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. $\angle ABC = \angle BCA = \angle CAB = 60^{\circ}$ (due to observation 1)
  3. $\angle BOC = \theta = 120^{\circ}$ (due to observation 3)
  4. $\angle BDO = 90^{\circ}$ and $BD=DC$ (due to observation 4)
  5. $\angle BOD = \angle DOC = \frac{1}{2} \angle BOC = 60^{\circ}$ (due to symmetry)

Now, consider the triangle $\Delta BOD$. Since this is a triangle with one of the angles being $90^{\circ}$, we can use a basic trigonometric formula: $\sin{\theta} = \frac{\text{Perpendicular}}{\text{Hypotenuse}}$ and write -

$$\begin{aligned} \sin{(\theta/2)} = \sin{60^{\circ}} = \frac{BD}{OB} = \frac{BD}{r} \Longrightarrow BD = r\sin{60^{\circ}}\end{aligned}$$

We can find the value of $\sin{60^{\circ}}$ from any trigonometric table, which turns out to be $\frac{\sqrt{3}}{2}.$ So, we get

$$BD = r \times \frac{\sqrt{3}}{2}$$

We can also obtain the length of the side $BC =2 \times BD = r\sqrt{3}.$

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 $\times$ length of any side. That means

$$\text{Perimeter} = 3 \times BC = 3r\sqrt{3}$$

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

$$\mathcal{C} = \frac{\text{Perimeter}}{OB} = \frac{3r\sqrt{3}}{r} = 3\sqrt{3} = 5.196\cdots$$

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 triangle is, the number $\mathcal{C}$ will never change. This might not sound very impressive, but it leads to a really deep idea about symmetries and constants.

In the next part of the series, we will try to find a generic formula for a large class of geometries, and see where it takes us. Stay tuned!


A clever formula for Logarithm
Previous article

A clever formula for Logarithm

A lesser known method of finding log of a number without using the tables

Geometrical Constants - Part 2
Next article

Geometrical Constants - Part 2

In the previous part, we obtained a geometrical constant for an equilateral triangle, and its value was approximately 5.196. In the second part, we will continue the calculations..


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