“How wonderful that we have met with a paradox. Now we have some hope of making progress”*…
Consider the simple function Y=1/X:
Take one half and rotate it around X.
It creates the shape you see at the top of this post, known as “Torricelli’s Trumpet” for its discoverer, the 17th century mathematician Evangelista Torricelli. It’s noteworthy for its peculiar topographical qualities: while both the volume and the surface area can be calculated, and the volume is a finite number, the surface area is Infinite. That’s to say that, while one can fill that three dimensional shape with a calculable quantity of paint, one cannot coat the exterior surface, as it would require an infinite amount of paint… (Supporting math, here.)
(The figure is also known as “Gabriel’s Horn,” a reference to the Archangel Gabriel, who blows his horn to announce Judgment Day– an association of the divine, or infinite, with the finite.)
This contribution (from Pablo Ramos) is just one of the fascinating answers to the question of Quora: “What are the weirdest science paradoxes that are mathematically true but counter-intuitive?“
* Niels Bohr
As we rotate our minds around x, we might send post-industrial birthday greetings to Daniel Bell; he was born on this date in 1919. Bell spent the first twenty years of his adult life as a journalist, exploring sociological issues; in 1960, on the strength of a book he’d written– The End of Ideology: On the Exhaustion of Political Ideas in the Fifties— he was awarded a PhD by Columbia University, where he taught briefly before moving for the rest of his career to Harvard. One of the leading intellectuals of the Post-War era, Bell is best known for his contributions to the study of “post-industrialism,” and for his acute unpacking of the interactions among science, technology and politics.