Pages

Construction algorithm


In our previous post, we have learnt about Similar Triangles and the Right Triangle Altitude Theorem. Today, continuing our journey in the garden of geometry, we want to find answer to the following question
Given a line segment of length $r$, by compass and straightedge, what kind of shapes can we construct? 

How to construct a regular polygon with 15 sides


To feed your curiosity, today we will look at a compass and straightedge construction of the regular polygon with 15 sides and we will show that there is a connection between this construction problem with the measuring liquid puzzle that we have learned from our previous post.

Measuring liquid puzzle



Today we will look at a brainteaser puzzle: "how to measure out exactly 1 liter of water using a 3-liter jug and a 5-liter jug." We will analyse this puzzle to see that, despite its innocent look, this puzzle has a close connection with the linear Diophantine equation.


Construction of regular polygons


There are a few classic problems of ancient mathematics that are easy to state but incredibly difficult to solve. Take for example, the problem of constructing regular polygons and the problem of trisecting an angle by compass and straightedge. It was not until the 18th-19th centuries that mathematicians could finally solve them by employing advanced tools of number theory and algebra. Today, we will look at the regular polygon construction problem.
Regular polygon construction problem. Using compass and straightedge, construct a regular polygon with $n$ sides.

Trigonometric multiple-angle formulas

In our previous post, we show a compass-and-straightedge construction of a regular pentagon based on the following trigonometric formula $$\cos{\frac{\pi}{5}} = \frac{1 + \sqrt{5}}{4}.$$
We derive this formula of $\cos{\frac{\pi}{5}}$ by observing that $\cos{\frac{2 \pi}{5}} = -\cos{\frac{3 \pi}{5}}$ and then applying the trigonometric formulas for double angle and triple angle:
$$\cos{2 x} = 2 \cos^2{x} - 1,$$ $$\cos{3 x} = 4 \cos^3{x} - 3 \cos{x}$$ to set up a cubic equation for $\cos{\frac{\pi}{5}}$.

It seems a good occasion now for us to learn about trigonometric multiple-angle formulas. In this post, we will show how to derive formulas for $\sin{nx}$$\cos{nx}$$\tan{nx}$ and $\cot{nx}$ using de Moivre's identity of the complex numbers.


Construction of a regular pentagon


Today we will look at a compass-and-straightedge construction of a regular pentagon based on the following trigonometric formula $$\cos{\frac{\pi}{5}} = \frac{1 + \sqrt{5}}{4}.$$