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The lesson is to not confuse mathematics with Reality.
This ‘Banach-Tarski’ paradox is a self consistent mathematical proof.
Unfortunately - it ignores the reality of atoms, molecules etc.
Mathematics is an extremely useful tool for Science and Engineering to
use. But Tools must be understood to be useful.
Not all tools apply to Real problems.
On Nov 3, 2025, at 11:20 AM, Donald Polvani via Phys-l <phys-l@mail.phys-l.org> wrote:
teaching geometry at Mathcamp, ... Another favorite was a famous
From the NYT, 11/3/25
"Two decades ago, the mathematician Moon Duchin spent her summers
“decomposability” paradox."
“You can take a solid ball, cut it into five pieces that don’t overlap,move those pieces around and reconstruct two solid balls that are the same
size as the one you started with,” Dr. Duchin, a professor of computer and
data science at the University of Chicago, said in a recent interview."
“That’s really weird, right?”
Not only weird but impossible (it seems to me), since by a repeatprocess on each new solid ball an infinite amount of mass could be produced
from a finite amount.
Don Polvani
Retired Adjunct Lecturer, Physics
Anne Arundel Community College
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Forum for Physics Educators
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https://www.phys-l.org/mailman/listinfo/phys-l