measure theory

Banach-Tarski Paradox

Did you know that it is possible to cut a solid ball into 5 pieces, and by re-assembling them, using rigid motions only, form TWO solid balls, EACH THE SAME SIZE AND SHAPE as the original? This theorem is known as the Banach-Tarski paradox. So why...

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Kakeya Needle Problem

What is the smallest-area convex set in the plane inside which a needle (unit straight line segment) can be reversed (spun around 180 degrees)? Answer: an equilateral triangle of unit height. OK, now what if you allow non-convex sets? What is the smallest area set...

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Cantor Set

Start with the interval [0,1]. Remove the (open) middle third of it, i.e. get (1/3, 2/3). Now remove the middle thirds of each of the remaining intervals, i.e. get (1/9, 2/9) and (7/9, 8/9). Continue this process ad infinitum. The points left over form a...

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