Yearning for dimensions? ·

My animation of 5 dimensions sets the stage for Max Cooper's track Ascent from his new album Unspoken Words.

Another mathy collaboration with Max!

Infinity with Max Cooper — In Six Minutes

Great fleas have little fleas upon their backs to bite ’em,
And little fleas have lesser fleas, and so ad infinitum.
And the great fleas themselves, in turn, have greater fleas to go on;
While these again have greater still, and greater still, and so on.
—Augustus de Morgan

In collaboration with Max Cooper, we tell a story about infinity in 6 minutes (and 34 seconds).

What's the story about? Natural numbers, integers, rationals, reals, power sets and the cardinality of the continuum — a real visual parade of numbers.

But math is a serious matter — so we include two of Cantor's diagnoal proofs in the video. Watch as we show that the size of the set of rational numbers is the same as the set of natural numbers (making them countable) but that the size of the set of real numbers is larger (making them uncountable).

In science one tries to tell people,
in such a way as to be understood by everyone,
something that no one ever knew before.
But in poetry, it’s the exact opposite.
—Paul Dirac, Mathematical Circles Adieu by H. Eves [quoted]

The style of Aleph 2 is that of a low-fi terminal. We used the Classic Console font by Csaba Széll, which I extended to include more set theory symbols (download extended font).

The animation was closely synchronized to the music, all the while trying to avoid looking too much like something from the Matrix movie.

To help you interpret what you are seeing, I walk you through the video and introduce you some advanced concepts of set theory. This way, next time you see an equation like `|\mathbb{R}| = | \mathbb{P}(\mathbb{N}) | = 2^{\aleph_0}`, you'll sprout a smile of joy.

▲ We begin counting the natural numbers, whose cardinality is `|\mathbb{N}| = \aleph_0`. Screenshot from Max Cooper's Aleph 2 (Yearning for the Infinite). (Max Cooper/Martin Krzywinski)

▲ A bijection between the naturals and integers demonstrating that both have the same cardinality. Screenshot from Max Cooper's Aleph 2 (Yearning for the Infinite). (Max Cooper/Martin Krzywinski)

▲ A bijection between the naturals and rationals. We traversing the table of rationals in a zig-zag manner. Screenshot from Max Cooper's Aleph 2 (Yearning for the Infinite). (Max Cooper/Martin Krzywinski)

▲ We alter the digits on the diagonal to create a real that is nowhere in the list. Screenshot from Max Cooper's Aleph 2 (Yearning for the Infinite). (Max Cooper/Martin Krzywinski)

▲ Power set elements of naturals (left) and reals (right). The size of these sets is separated by one step in the hierarchy of Aleph numbers, if the Continuum Hypothesis is true. Screenshot from Max Cooper's Aleph 2 (Yearning for the Infinite). (Max Cooper/Martin Krzywinski)

▲ Aleph remains. Screenshot from Max Cooper's Aleph 2 (Yearning for the Infinite). (Max Cooper/Martin Krzywinski)

These pages describe the system I build to generate the animation and the mathematics behind infinity, including sets, cardinality, countability, $\aleph$ and the Continuum Hypothesis.

I also present the animation system I built for the video, which was coded from scratch.

If you like infinity, you might like dimensions too.

Check out my video for Max Cooper's Ascent, which animates 5-dimensional cubes doing their confusing things.

Yearning for dimensions? ·

My animation of 5 dimensions sets the stage for Max Cooper's track Ascent from his new album Unspoken Words.