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A previous question on the site asked for a short proof of the fact that three equilateral triangles with unit side length cannot be arranged to cover a square with unit side lengths. Given the truth of that assertion, I began to wonder:

What is the minimum side-length $s$ such that three equilateral triangles with side-length $s$ cover a square with unit side length?

The existence of such an $s$ follows from a simple compactness argument. It is clear, from the previous question, that $1<s$ and it is easy to construct a cover of a square with equilateral triangles of sidelength $\frac{2\sqrt{3}}3$ as: enter image description here

It would not surprise me too much if $s=\frac{2\sqrt{3}}3$ - that is, if the altitude of the triangle had to be of unit length - but I cannot think of any reason to believe this.

(I would be particularly interested in a proof akin to the proof I gave to the other question - i.e. defining something akin to a measure $\mu$ and showing that $3\mu(\text{triangle})<\mu(\text{square})$, but any proof is good with me).


It has been pointed out in comments that the following configuration is better: enter image description here
The sidelength is $\frac{1}2\left[\frac{2\sqrt{3}}3+1\right]$.

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    $\begingroup$ Take your figure and shrink the triangles. Move the outer triangles up the sides of the center one until the lower vertex is on the side of the square. For side $1$ the upper corners will be outside the outer triangles, but you should be able to calculate the triangle side where it will just cover. I can't imagine there is a better configuration, so you are there. $\endgroup$ – Ross Millikan Apr 7 '15 at 3:01
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    $\begingroup$ You got my point correctly. The hard part of these problems is proving that there is not some different configuration with a smaller side length. In this case, it seems to me there aren't any other reasonable configurations, but it is hard to prove that. Three triangles don't give much flexibility in configuration. $\endgroup$ – Ross Millikan Apr 7 '15 at 3:25
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    $\begingroup$ On a related note, any solution to this problem could be worked to give a solution to this problem: en.wikipedia.org/wiki/Opaque_forest_problem by picking the appropriate triangle boundaries. However, there is already a conjectured minimal solution to this problem so it may be possible to reconstruct a solution for this problem? $\endgroup$ – Sandeep Silwal Apr 7 '15 at 4:39
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    $\begingroup$ Did you make your figures slightly tilted to purposefully annoy everyone? :) $\endgroup$ – Sanchises Apr 7 '15 at 11:15
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    $\begingroup$ Erich's Packing Center gives the same configuration as Ross Millikan described. $\;$ $\endgroup$ – user57159 Apr 7 '15 at 16:16
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Looking at the proposed configuration, we can find the minimum $s$ for that configuration as follows: notice how the two triangles on the sides overlap. This overlap is a smaller similar triangle (that is, it's equilateral), the larger that triangle is, the smaller $s$ is, so we seek to maximum the size of that triangle. We have the following equalities to work with: The sum of the altitudes of the big triangle in the middle, and the little triangle must be 1.

$A+a=1$

and the length of half a side of the large triangle minus the length of half a side of the small triangle must be 1/2

$\frac{S}{2}-\frac{s}{2}=\frac{1}{2}$.

This latter expression comes from the fact that the triangles on the sides are bisected by the vertical sides of the square.

We appear to have 4 variables, but since $a \propto s$, and $A \propto S$, we really only have two, and can solve this linear system easily enough (I leave the details to the reader).

Now for the trickier part: proving that this configuration is best possible. To do that, we will pick on the corners of the square. It has 4 corners and any configuration of triangles with covers the square must cover all 4 corners. Therefore we have proven that:

At least one of the triangles in an optimal configuration must cover two of the corners. Observe that the proposed configuration does this.

Now, suppose we have a configuration with a triangle covering the bottom two corners, and one more for each top corner. We are free to move them around, subject to the constraints that they must continue to cover those corners. We concentrate on the triangle covering two corners. Our goal is to use as much of the area of that triangle as possible to cover the square, which is achieved when it's side is flush with the square (100% usage). We can shrink the the triangle all the way down to $s=1$ and still have it covering both corners, but as the linked problem's solution showed, $s=1$ is not sufficient, so $s>1$.

What about the other two triangles? For them, we need that they, together, cover whatever remaining area of the square that the middle triangle (the one covering two corners) doesn't cover. The idea is to position them so that the area of overlap, among the triangles is minimized, this suggests that we would want them to be flush with the middle triangle, but if the length of their altitudes (which are all the same length) is less than 1, having no overlap is impossible by the shape of the remaining area not being two triangles, and as the configuration shows, this can happen. So we will have to have some overlap.

This shows that the shape of the remaining area has the general form of two triangles with quadrilaterals attached to each of them (we can cut it in two, by extending the vertical altitude of the middle triangle, for example). Each of these pieces has a corner of the square as one of its corners, hence the applicable triangle must still cover that corner. Another way of thinking of this shape, is that it is the rectangle given by the the two corners which have a triangle each, and the intersections of the sides of the middle triangle with the vertical edges of the square, with a triangular piece cut out from the bottom (it took all of the bottom edge with it). This is a non-convex pentagonal shape, and we must cover it with just two triangles, hence, certainly the convex rectangle (is this the convex hull of the pentagon?) must have its 4 corners covered by two triangles. So one of the remaining triangles must cover at least two corners of the rectangle, I claim that having it cover three will be less efficient. The reason being that it must cover the diagonal of the rectangle which will have length greater than 1 because the top side is length 1 already (we can find the exact length from knowing where the middle triangle intersects the vertical sides of the square using Pythagoras's theorem). So we have two triangles covering two corners each (left and right sides). The must also must both cover the inside point of the pentagon and all the area above it together. This is the necessary overlap, and it is minimized exactly when the triangles are flush with the edges which have the inside point as an endpoint (or in other words when they are flush with the middle triangle).

The idea from here is show that, as $S$ becomes smaller and smaller, that the triangles have less and less freedom as to how they can positioned until you end up with exactly the configuration pictured in the OP. This may be able to be argued by showing that the pictured configuration always minimizes overlapped area for a fixed $S$.

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