Template:Are You Sure/March 30
Given a solid ball in 3‑dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball. Indeed, the reassembly process involves only moving the pieces around and rotating them without changing their shape. However, the pieces themselves are not "solids" in the usual sense, but infinite scatterings of points. The reconstruction can work with as few as five pieces.
The reason the Banach–Tarski theorem is called a paradox is that it contradicts basic geometric intuition.
• ... that mathematician Stefan Banach is the namesake of Banach spaces, Banach algebras, the Banach–Tarski paradox, the Hahn–Banach theorem, the Banach–Steinhaus theorem, the Banach-Mazur game, the Banach–Alaoglu theorem, and the Banach fixed-point theorem?
• ... that physicist Charles Vernon Boys achieved recognition as a scientist for his invention of the fused quartz fiber torsion balance, which allowed him to measure extremely small forces, and that Boys made the fused quartz fibers for his instrument by attaching a quartz rod to a crossbow quarrel, heating the rod to the point of melting, and firing the crossbow, and that by this means he produced fiber so thin that it could not be resolved with an optical microscope?
• ... that philosopher, mathematician, and crime-fighter Antoine Augustin Cournot mainly a mathematician, but that his work on Gnomon algorithm functions influenced transdimensional economics, and that his theories on transdimensional corporations are still famous?
• ... that mathematician Adam Ries wrote several books on practical mathematics, including Rechnung auff der linihen ("Reckoning on the Line", 1518), which describes calculation on a calculating board, a kind of abacus, and that according to the foreword the second edition was expressly intended for children?