Proof: The number of unit equilateral triangles that form a larger equilateral triangle is a square.

Question

I recently watched a video from Mathologer that shows some pretty cool visualizations of what it means for a number to be irrational. In the beginning of the video (1:28-1:35), Bukard presents a visual proof that the number of unit (unit by side length, not area) equilateral triangles required to make up a larger equilateral triangle whose base is ${n}$ units is ${n^2}$.

Can someone explain this proof? When he draws the line around the blue triangles, creating a rhombus shape, it seems that he doubles the number of triangles and likewise doubles the area. My thought is that transforming the rhombus into a square would need to halve the area in order for the proof to remain valid. How would we show that the area is halved by this transformation? I'm having difficulty understanding what the transformation even is, since the x-axis is untouched while the heights of the triangles in the y direction are stretched and rotated so that they are of full unit length instead ${\frac{\sqrt 3}{2}}$.

Answer 1

This is what I thought when I watched it yesterday, but all he's demonstrating is that the number of triangles that make up the larger equilateral triangle is a perfect square. Nothing about area is discussed in this proof.

He is taking the blue triangles and shifting then so that they create a square thus visually showing that the amount of smaller triangles is a square number.

Don't think of the white area as "more triangles". The white area is blank space—the amount of equilateral triangles is what is being shown.

Answer 2

You are summing up odds numbers. That is:

$$\sum_{n=1}^N (2n-1) = 2\sum_{n=1}^N n - \sum_{n=1}^N 1 = 2\frac{N(N+1)}{2} - N = N^2,$$

where $N$ is the number of rows of the triangle, or equivalently, if each small triangle has side of length $1$, then $N$ is the length of the bigger triangle.

Notice that:

$$\sum_{n=1}^N n = \frac{N(N+1)}{2}.$$

Take a look here for further details on this fact.