Discussion for Spheres Part 1

Discussion for Spheres Part 2

Discussion for Spheres Part 3

Discussion for Spheres Part 5

Transcript for screenreader users:

Part 4 of 5! Press forward to continue! [orange rolling to the right]

For instance in 2022 Maryna Viazovska was awarded the Fields Medal (one of the highest honors in mathematics) in part for her solution of the sphere packing problem in 8 and 24 dimensions.

[Viazovska next to a diagram of vertices on several concentric circles with lightly shaded edges connecting some of the vertices]

(E8 diagram by Jlrodri, found on Wikipedia. CC BY-SA 3.0, which means this comic is also is under CC BY-SA 3.0)

These turn out to be very special, with unexpected connections to many other areas of mathematics.

[map with thick, winding path]Navigation

[robot with rectangular torso, rectangular head, circular eyespot, and two thin arms]Robotics

[dots connected by paths consisting of straight lines that bend at angles, mostly right angles, in a few places]Machine learning

[padlock with keyhole]Encryption

Another question mathematicians asked was: what if we study packing problems, but in a space that is discrete instead of continuous?

Computer scientist, angrily: the universe is chunky!

Physicist, angrily: the universe is smooth!

Tao: One type of discrete space that computer scientists are particularly interested in is the “Hamming cube”, which is a cube whose vertices represent strings of bits (0s and 1s) of a given length. [Tao pointing with his right hand to a 0 next to his head and with his left hand to a 1 on the other side of his head]

Tao: Strings of two bits form a two-dimensional square; strings of three bits form a three-dimensional cube; and so forth. [Tao from the right, showing in his hand a cube with edges but no faces, each vertex labeled with three bits, with 000 in the front upper left corner, left-to-right edges connecting vertices differing in their first bit only, vertical edges connecting vertices differing in their second bit only, and front-to-back edges connecting vertices differing in their third bit only]

Tao: There is a discrete analogue of a sphere on the Hamming cube, coming from bit errors in digital transmission.

[000 with three arrows pointing away, each from one of the three zeros to another three bits where that bit has flipped to a 1 and turned red while the other two bits are still zeros] [the cube from the previous panel, now with a dotted sphere centered at corner 000 and intersecting the cube at vertices 100, 010, and 001]

Tao: Suppose you want to send a message to someone.

written on a chalkboard: To Jenkins: algebraists RULE, geometers use mental models of a 3D universe which for many purposes are an inadequate TOOL. [Algebraist standing under the message and holding a piece of chalk]

Tao: You can’t just send letters down a wire. But, you can do something like Morse code. [using a telegraph key, which is emitting a sequence of dots and dashes: dah, dit dah, dah dit dah, dit, new line, dah, dit dit dit dit, dit dah, dah, new line, dit dah dah dah, dit, dah dit, dah dit dah, dit dit, dah dit, dit dit dit]

You might be tempted to do the obvious thing and just assign the 26 letters to binary combinations. Let’s say you have 9 bits to work with. You could do something like this:

A 1: eight zeros and a one

B 2: seven zeroes, a one, and a zero

C 3: seven zeroes two ones

D 4: six zeroes one zero zero

E 5: six zeroes one zero one

F 6: six zeroes one one zero

G 7: six zeroes three ones

If signal transmission is perfect—no interference from electromagnetic radiation, cosmic rays, vengeful geometers messing with the wire—then this is fine.

Geometer: The shortest distance between two points has been BISECTED! [holding a giant pair of wire cutters, and sitting atop one of two telegraph poles, each of which has one end of a wire hanging down from its top]

Tao: But perfection is rare in real life. And, for instance, 000 000 101 (F) looks a lot like 000 000 111 (H), which might be dicey if you’re sending a message to your friend Huck.

Bonus panel: same as Parts 1 through 3