Infinity, for the Mind and Body

Posted Feb 6, 2014
Last Updated Mar 26, 2020

In the photo above is Andrew Penry explaining to me a weird quirk of mathematics. As all people who follow my work know, I complain about my lack of connection with higher mathematics--odd since I progressively get deeper into 3D programming by the day.

Unfortunately, although I wish the logic behind it had stuck into my brain before the kids erased my whiteboard to draw The Blue Man Group as 2D stick-men... the concept Andrew was teaching me (that there is this weird aspect of our universe where mathematical constructions that extend into infinity can, in fact, add up to - 1/12) was lost on me. I know that this number must be more important than 42 somehow. But that was not really what struck me about this.

It's that each doughnut you eat from the box of a dozen is like an infinite meal.

This reminds me of a perplexing concept I had as a teenager. I imagined throwing an air plane out my bedroom window. I wondered how far it would go. But then for some reason I wondered about the fact that I could forever divide the distance it travelled in half, which was an amazing thing when you first think about it.

But then I realized a couple tragically important things that sent my mind spinning. 1) There are an infinite number of points between the start and finish; 2) that at the finish it travelled twice as far as it had at the half.

At the surface that seems fine. But if there are an infinite number of points between start and finish, the travel is actually impossible--as who can travel an infinite distance?

Worse, for my young mind, was that the infinite number of points between the start and midpoint, was only half the distance travelled between the start and end... meaning that one set of infinite points was half the size of another set of infinite points.

unequal infinities

I had proven, or so I thought at that time, that we do something impossible every time we move any distance and that infinities are somehow finite.

Of course I was wrong. As the world would have it, several of my friends are really smart and get great satisfaction out of shaking their head and putting me straight.

My lifelong friend Jeremy Martin rolled his eyes at me and said, "You have it all wrong. The points themselves are infinitesimal."


That wiped away all of my confusion about infinity.

Well... kind of.

Luckily, this discussion finally gives me a public use for an image Jeremy sent me sometime in the last few years. I was probably trying to con Jeremy into solving some math problem when he sent me the following drawing. Which is good, because despite the deceptively young look of the graphic, Jeremy is probably one of the smartest human beings on this planet.

your brain on math

Perplexing Things

Essays on perplexing and amazing aspects of reality that are often unintuitive.

  1. Infinity, for the Mind and Body
  2. Monty Hall Game


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Andrew Penry

Feb 6, 2014

The "walking across the room" problem you describe is called Zeno's Dichotomy paradox. It can be solved a few ways. One is that the size and time to travel become infinitesimal, as Jeremy told you so long ago. Another is that it may not be possible to divide a distance into infinite points. A line having infinite points is a mathematical abstraction, but our physical universe may not work that way. Models like quantum foam explain a universe that has a minimum size which can not be further divided. It may be that our universe functions in such a way.

Shawn Olson

Feb 7, 2014

It's mind boggling.

I added the Numberphile video above on this.

Jeremy sent me a link to another video that fits this page perfectly:

Wall Worm plugins and scripts for 3ds Max