Background Information
What the Banach-Tarski Paradox is
(What is) a mathematical concept where if you slice a sphere into a finite number of pieces you can reassemble it into two spheres of the same size as the original with only rotation and translation.
How many ways does it connect to the real world
It connects to the real world through math in 2 ways
You reassemble the sphere using _______ & _______
You reassemble the sphere using rotation & translation
Does the Banach-Tarski paradox apply to the real world?
The Banach-Tarski Paradox does not apply to the real world
How many different ways did we use to explain it to you
(What is) 2 ways
The person who created the Banach-Tarski Paradox
(Who are/is) Stefan Banach
(And/or) Alfred Tarski
How did it prove that matter must be discrete and atomic
Scientists didn’t know if the universe was a smooth, continuous fluid or made of tiny indivisible chunks (atoms) the Paradox proves that if physical matter were a smooth, you could duplicate objects in real life (which you can’t) so the paradox is proof that matter must be discrete and atomic
Why does it not matter if you take one point from the sphere/circle
It doesn't matter if you take one point because there are infinite points and infinite minus one is still infinite
What is the Axiom of Choice
The Axiom of Choice is a fundamental assumption
What does the paradox rely on
The paradox relies on the fact that a solid object is made of an infinite number of mathematical points
The country where the paradox was created & published
(Where is) Poland
What prevents mathematical anomalies like this from tearing the universe apart
The universe has a built-in safety pixel size (atoms, quarks, and Planck lengths) specifically to prevent mathematical anomalies like this from tearing physics apart
The line goes from 0 - _____ not including _____
The line goes from 0 - 2π not including 2π
Why doesn't it work in real life
It does not work in the real world because it depends on a fundamental assumption called the Axiom of Choice, and matter is made of atoms and molecules which cannot be infinitely divided.
What type of paradox(s) is it
It is a Mind-Bender and (partially) Rule-breaker paradox
The year the paradox was created.
(When is) 1924
What did the paradox change about the way people use Cartesian coordinates?
It a massive glitch in this coordinate system: it reveals that the standard mathematical maps contain loopholes that don't match reality and forces scientists to be careful with software models. If a computer program isn't strictly bound by laws of physics, the software could accidentally trigger an infinite multiplication error by trying to compute "unmeasurable" boundaries.
What are the actions/directions & letters in the infinite dictionary
There are: L (turn left), R (turn right), U (go up), D (go down)
What is matter made up of
Matter is made up of atoms & molecules
What does "A Pea Can Become the Sun" mean?
You can take a solid ball the size of a pea, cut it into a finite number of pieces, and reassemble it to make it a solid ball the size of the Sun
The reason the paradox was created
To explore the hidden boundaries of math like set theory, infinity, and Axiom of Choice.
How different would math be without this concept
Scientists wouldn't know if the universe was a smooth, continuous fluid or made of tiny indivisible chunks (atoms).
Scientists wouldn't know that our standard mathematical maps contain loopholes that don't match reality and their software could accidentally trigger an infinite multiplication error by trying to compute "unmeasurable" boundaries.
Imagine you have a circle with a radius of ______ on the _____-plane
Imagine you have a circle with a radius of 1 on the xy-plane
Matter is made of atoms and molecules which _____ be ______ _______
Matter is made of atoms and molecules which cannot be infinitely divided.
How many infinity symbols are on our poster?
(What is) 6 infinity symbols