It is widely known that we can square a square, which is defined as follows:
"Squaring a square" is tiling a square with integer side lengths with smaller squares with integer side lengths.
In reality it is very simple; we can for example tile a $2\times2$ square with four unit squares. Thus, people've thought of the restriction that all tiles must have different sizes, which they've named perfect. This turns out to be possible, with the smallest perfectly squared square being a $112\times112$ square:
My question is simple: can we do the same with equilateral triangles? Meaning, can we tile an equilateral triangle with integer side lengths with smaller equilateral triangles with integer (but all different) side lengths? If so, what would be the smallest perfectly triangled triangle?
So I've been doing some thinking and I think I'm getting closer.
Let's have a look at the smallest triangle tile (we'll call it $\tau$). We'll split cases.
Case 1: $\tau$ is in a vertex. If the little triangle is a tile in one of the three vertices of the parent triangle, then it has one edge left; only triangles smaller than or as small as $\tau$ can fill that edge, but $\tau$ was the smallest triangle. This is impossible.
Case 2: $\tau$ is on the border. If $\tau$ is on the border of the parent triangle, but not in the corner, then either side of $\tau$ is touching that of a bigger triangle, which makes those two neighboring tiles overlap. This is also impossible.
Assume now $\tau$ is not on the border of the parent triangle.
Case 3: An edge of $\tau$ is touching a vertex. If an edge is touching a vertex, then since the neighboring tiles are bigger, we have the following (the orange tile being $\tau$, ignore that the blue triangle is the same size, it's beside the point):
Which makes this case equivalent to case 2.
And then the last case is when none of the above hold, but I'm quite stuck on that.