What square does not contain the middle?

Question

Consider the square $S = [-1,1]\times[-1,1]$. Suppose we put a smaller square inside it, which is rotated with an angle $\alpha$ relative to the large square. What is the largest such square that does not contain the origin in its interior?

When $\alpha=0$, the answer is obvious: the maximal side-length is $1$. Every square with side-length larger than 1 must contain the origin in its interior. But when $\alpha>0$, it seems that the maximal side-length is smaller, and I don't know how to calculate it.

Answer 1

Draw a line with slope $\alpha\in\ ]0,{\pi\over4}] $ through the origin. A largest square tilted by $\alpha$ not containing $(0,0)$ in its interior is the square with two vertices on this line and the other two vertices on the lines $x=-1$, $\>y=1$. Denote the side of this square by $s$. Looking at the upper rim of the figure we see that the following equation holds: $$s\cos\alpha+{s\over\sin\alpha}=1+\cot \alpha\ .$$ This leads to $$s={\cos\alpha+\sin\alpha\over 1+\cos\alpha\sin\alpha}\ .$$

Answer 2

OK, this is my current attempt of an answer:

Consider a square that is pushed towards to top-right corner of $S$. The coordinates of this square are (where $c:=cos(\alpha)$ and $s:=sin(\alpha)$):

• T (top): $(1-dc,1)$
• R (right): $(1,1-ds)$
• B (bottom): $(1-ds,1-dc-ds)$
• L (left): $(1-dc-ds,1-dc)$

Due to symmetry, it is sufficient to study the range: $0 < \alpha < \frac{\pi}{4} $ , where $0<s<c<1$.

Now, an method for deciding if a point is in polygon can be used to decide if $(0,0)$ is in the given square, as a function of $d$.

Using the ray-crossing method, we have to consider a ray from the origin, and check how many times this ray crosses the sides of the above square. If the number is odd, then the origin is inside the square.

Consider, as a ray, the negative y axis.

Assume that $d<1$. In this case, T R and L are all above the origin, therefore the ray cannot cross the lines LT and TR. Additionally, T R and B are all to the right of the origin, therefore the ray also cannot cross the lines TR and RB.

It remains to check whether the negative y axis crosses the side LB, i.e., the y coordinate of the origin is above the line LB. The equation of the side LB is:

$$ cy(x) = (c+s-d-dsc)-sx $$ $$ where: x \in [1-ds-dc,1-ds] $$

If we substitute $x=0$, we get:

$$ cy(0) = c+s-d-dsc $$

If this number is negative, then the origin is above the side LB, and the origin is inside the square. The condition for this is:

$$ c+s-dc^2-dsc-ds^2 < 0 $$ $$ d > \frac{c+s}{1+sc} $$

(An alternative way to reach at the same solution is described in Robert Israel's answer).

You can plot that function here, using this line:

a0=2&a1=(cos(x)+sin(x))/(1+sin(x)*cos(x))&a2=1/(sin(x)+cos(x))&a3=(cos(x)-sin(x))/(cos(x)^2)&a4=1&a5=4&a6=8&a7=1&a8=1&a9=1&b0=500&b1=500&b2=0&b3=0.79&b4=0&b5=2&b6=10&b7=10&b8=5&b9=5&c0=3&c1=0&c2=1&c3=1&c4=1&c5=1&c6=1&c7=0&c8=0&c9=0&d0=1&d1=20&d2=20&d3=0&d4=&d5=&d6=&d7=&d8=&d9=&e0=&e1=&e2=&e3=&e4=14&e5=14&e6=13&e7=12&e8=0&e9=0&f0=0&f1=1&f2=1&f3=0&f4=0&f5=&f6=&f7=&f8=&f9=&g0=&g1=1&g2=1&g3=0&g4=0&g5=0&g6=Y&g7=ffffff&g8=a0b0c0&g9=6080a0&h0=1&z

The minimum is at: $\alpha=\pi/4$, where the lower bound is approximately: $d>0.943$.

To conclude: a square with $d\geq 1$ always contains the origin.