An Ornamental Geometric Inequality

Posted by Cleve Moler, March 16, 2015

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I came across this "ornamental geometric inequality" in a tribute to Lothar Collatz.

An Inequality
References

An Inequality

I mentioned the German mathematician Lothar Collatz in my post in January on his 3n+1 Conjecture. Here is a beautiful inequality from his 1934 paper titled "Ornamental Geometric Inequalities". Consider the subregion of $-4 \le (x,y) \le 4$ for which

$$ ||||x| - 1| - 1| - |||y| - 1| - 1|| \ge \frac{1}{3} $$

I've never seen four absolute value signs in a row before.

Let's visualize it with MATLAB.

   [x,y] = meshgrid(-4:1/256:4);
   z = abs(abs(abs(abs(x)-1)-1) - abs(abs(abs(y)-1)-1)) >= 1/3;
   imagesc([-4,4],[-4,4],z);
   axis image
   colormap(copper)