An Infinite Stack of Boxes and the Painter's Paradox: Infinite Surface Area but Finite Volume!

Описание: In this video we discuss a version of the Gabriel's horn aka Torricelli's trumpet paradox without using integration. We replace the trumpet with a stack of infinitely many boxes to bypass improper integration. Aside from reference to a couple of familiar infinite series (we can't bypass infinity!) no serious mathematical calculations are necessary.

The essence of the paradox is the same: the stack has infinite surface area but finite volume. We also discuss the "Painter's Paradox" which is usually tied to these paradoxes through the concept of infinite surface area but a finite volume.

We also discuss a variation that has both finite height and volume but still infinite surface area. With small units of length like inches or centimeters we can hold a small stack of boxes with infinite surface area in the palm of one hand!

These constructions are logically allowed quirks of infinity in mathematics. They aren't physically possible so paradoxes arise only when we insist on interpreting mathematical constructions involving infinity as physical entities. Material objects like boxes, trumpets or paint don't admit infinitesimal reductions of the type allowed in mathematical constructions. We may push for the painter's paradox a bit and define a two dimensional idealization of actual paint. If so then a finite amount of it has zero volume (how many squares stacked on top of each other gives a cube?) We need an infinite amount of idealized paint to fill up a three dimensional volume, so we don't have a paradox.

These types of paradoxes that involve convergence of infinite sequences and series are essential in understanding *infinity and its manifestations*!
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To gain a deeper understanding of limits, continuity and the infinite in mathematics, read my book "Achieving Infinite Resolution: a gentle look at the role of infinity in calculus". You can check its table of contents and more here:
https://www.amazon.com/gp/product/B08...

More information about Gabriel's Horn and the Painter's paradox:
https://thatsmaths.com/2017/04/13/tor...
https://en.wikipedia.org/wiki/Gabriel...

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Information about the Basel Problem:
https://en.wikipedia.org/wiki/Basel_p...

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