I saved newspaper TV sections from most of the 1990s. Here’s what was on broadcast network TV on Saturday, April 15, 1995:

I assume that where there is no listing, that half hour is a continuation of the previous half hour’s programming, except at 6 am, in which case I assume that station was off the air.

One possible mapping of characters to nonets, based on the assumption that the encoding is a Hamming code, can be found at https://smbc-wiki.com/index.php/Spheres-part-5 .

Given the solution to the puzzle and the letter-encodings in the comic, if you don’t assume the encoding is a Hamming code, there is another possible encoding.

Discussion for Spheres Part 1

Discussion for Spheres Part 2

Discussion for Spheres Part 3

Discussion for Spheres Part 4

Transcript for screenreader users:

Part 5 of 5! Press back to read! [top three layers of a pyramid of oranges]

Tao: So, one option is to use the 512 possible combinations to make the different letter-encodings as different as possible.

[begin chalkboard]

A 00000: 000 000 000

B 00001: 000 010 011

C 00010: 001 001 010

D 00011: 001 011 001

E 00100: 010 100 110

[end chalkboard]

Put another way, the different letter-encodings should be as distant from each other as possible. And, because it’s 9 bits, that distance is in 9 dimensions.

[begin chalkboard]

N 110 111 000

[downward arrows from the second, sixth, and ninth bits of N]

Y 100 110 001

[The second, sixth, and ninth bits of Y are in a different color. They are all and only the bits that differ from the corresponding bits of N.]

N and Y differ in dimensions 2, 6, and 9. So, their Hamming distance, is 3 because (drumroll…) 1+1+1 = 3

[end chalkboard]

(more properly, if you add the numbers, but skip the carries, aka N XOR Y, you get 010 001 001. Check the quantity of 1s, and that’s your Hamming distance)

Tao: Hard to visualize perhaps, but once we can talk in terms of space, we get the whole toolkit that mathematicians built while nobody was thinking about how to non-offensively talk to Huck about ducks and bucks.

[begin chalkboard]

HUCK: 011111111 010011010 001001010 101000111

DUCK: 001011001 and the next three nonets the same as for “Huck”

BUCK: 000010011 and the next three nonets the same as for “Huck” and “duck”

[end chalkboard]

With this change of perspective, bit flips become nearby points on the “cube”; those points are the intended binary string, and they’re surrounded by “spheres” that represent the possible strings you could get due to errors.

[the same picture as earlier of 000 with three arrows, each pointing to another three bits where exactly one bit is flipped, and the same dotted sphere centered at the corner of a cube, but this time, only vertices 000, 100, 010, and 101 are labeled]

*Hamming noneract not rendered due to budget constraints.

A priori, we might not have expected discrete hyper-dimensional sphere-packing to have applications, but that’s exactly what happened.

Algebraist: …and my work turned out to have filthy FILTHY real-world utility!

Geometer: Where’s your “pure abstraction” now, Thompson!? HAHAHAHAHA!

In fact, the more efficient these “sphere packings” (also known as “error correcting codes”) are, the more messages one can reliably send with a fixed amount of bandwidth.

Algebraist: No more fighting. We must come together.

Geometer: Like two parallel lines… on a plane of positive curvature.

Tao: The mathematical theory of these codes provided theoretical limits on how much data one can send on a given channel, as well as practical ways to get as close to this theoretical limit as possible.

Tao: We take advantage of these mathematical results every day, without being aware of it.

QR codes work, even when distorted. Lattice theory helps material scientists design crystals, ceramics, foams, and emulsions.

[a QR code, a crystal, a frying pan, and a Voronoi diagram]

Tao: I helped develop something similar to an error correcting code to speed up MRI scans by a factor of as much as 10.

MRI patient with head and arms inside MRI machine: Thanks, mathematics!

The cell phone you’re probably reading this on can share spectrum with other devices without noticeable interference due to findings in infinite dimensional Hilbert space.

[same diagram of the sphere and cube as before]

*Hamming infiniract not rendered due to budget constraints.

Tao: And it all started with figuring out how to stack oranges… [standing behind Tao: Harriot, Gauss, Shannon blowing fire out of a trombone, Thue, Toth, Rogers, Hales, Ferguson, and grocery store worker]

Bonus panel: 011100101 011111111 110100110 101111010 100010011 010101110 000000000 010011010 001100001 100110001 101111011 011100001 010110111 001111011 111101011 100000000 001100001 011111111 110100110 110101011 000000010 001100100 100001101 001001010 001110111 101111010 111100001 110111000 101010100 100110101 001111011 001110110 011111111 111100001 011010010 001110110 101110011 100001101 001100101 001110110 010101110 111010100 010110101 101110011 001100101 111100000 101111011 110101011 111100001 000101111 010100110 101111011 111110000 000000000 001100100 100001111 011100110 010111000 001101101 101111011 010110101 111100001 111010100 101010100 111100001 111010000 010100110 000101110 001110110

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

Discussion for Spheres Part 1

Discussion for Spheres Part 2

Discussion for Spheres Part 4

Discussion for Spheres Part 5

Transcript for screenreader users:

Part 3 of 5! Press forward to continue! [orange rolling back to the right]

Supermarkets often use this packing to stack oranges and other round fruits in their grocery section.

Grocery store worker: Customers will appreciate this conjectural maximum-efficiency orange-packing! [gesturing towards two pyramids of oranges]

Kepler computed that the density of this packing was about 74% – that is, about 74% of the available space was occupied by the spheres, and the remaining 26% by the gaps between the spheres.

Kepler: There is probably not a better way. [standing with Raleigh and Harriot around a pyramid of cannonballs]

He conjectured that this was the maximum possible density: there was no other way to pack spheres that could achieve a density of, say, 75%.

[Kepler pointing with a rod at a pie chart with the top and left three quarters labeled “cannonball” and the lower right quarter labeled “not cannonball”]

Tao: This “Kepler conjecture” attracted the attention of many brilliant mathematicians for centuries. [four people labeled Gauss, Thue, Toth, and Rogers are standing behind Tao]

It was only solved in 1998, after countless works culminating in a 100-page paper by Thomas Hales and Samuel Ferguson, who also needed extensive computer calculations to complete the proof.

Ferguson: We have packed the maximum number of theorems into this proof. [Hales, scratching his head, is standing not far from Ferguson]

This by itself didn’t revolutionize the way cannonballs or oranges were stacked.

Grocery store customer: Wow, is that PROVEN maximum-efficiency orange-packing? [looking at a pyramid of oranges]

Worker [next to another orange pyramid]: Kepler-approved.

Tao: But once mathematicians study one question, they are naturally led to explore other related questions, that often venture quite far from the motivation of the original problem.

Tao: The Kepler conjecture is about packing spheres in three dimensions. Mathematicians asked: what happens instead in two dimensions? Four? A billion? [outline of a sphere superimposed over x-y-z coordinate axes]

This is perhaps a problem if you want to IMAGINE the mathematical object…

Geometer, tauntingly: Fingers getting tired from all those symbols, Thompson? [Algebraist and Geometer writing on a chalkboard, the algebraist having written out the equations x sub 1 squared equals one, x sub one squared plus x sub two squared equals one, and so on for three and four x variables, and the geometer having drawn a pair of points, a circle, and a sphere and labeled them 1D, 2D, and 3D]

But not if you want to do the math.

Algebraist: Where’s your precious “visual intuition” now, Jenkins!? HAHAHAHA! [The algebraist has written out the equations for an n-sphere with five and six x-variables, while the geometer has drawn a question mark and labeled it 4D, two question marks labeled 5D, and three question marks labeled 6D, and is crying from one eye.

Tao: But higher-dimensional spheres aren’t just useful for making geometers cry.

Bonus panel: same as Parts 1 and 2

Discussion for Spheres Part 1

Discussion for Spheres Part 3

Discussion for Spheres Part 4

Discussion for Spheres Part 5

Transcript for screenreader users:

Part 2 of 5! Press forward to continue! [an orange]

Tao: There are admittedly some grains of truth to all of these caricatures.

[behind Tao are those caricatures, and the words “Also, extremely normal-looking people!” with arrows pointing to one more person and some empty space]

Tao: And while we ARE unusually fond of pursuing abstract and technical questions primarily for reasons of intellectual curiosity, the remarkable thing is that such pursuits can have unexpected practical benefits many years after they were first investigated. [In the background, Shannon continues to blow fire out of a trombone.]

Eugene Wigner famously called this the “unreasonable effectiveness of mathematics in the natural sciences”.

Wigner: The fundamental structure of the universe runs on math we developed for GAMBLING?!

Tao: One example is the story of “sphere packing”. [Tao holds up three or four oranges]

In the early 1600s, Sir Walter Raleigh asked the English mathematician Thomas Harriot for the most efficient way to stack cannonballs together.

Raleigh: There’s GOT to be a better way. [gesturing towards a pyramid of cannonballs, five to a side, on a ship, with water in the background stretching all the way to the horizon]

Harriot studied the question in detail but could not definitively answer it, and wrote about it to an eminent German colleague, Johannes Kepler.

Kepler: Finally, something I’ll be remembered for.

Kepler viewed this as an abstract mathematical problem about packings of infinite three-dimensional space by spheres of unit radius.

Kepler: There’s less to keep track of if you just make it infinitely large. [holding a stack of small cannonballs]

One such packing is known as the hexagonal close packing; it is layer upon layer of spheres arranged in a hexagonal lattice pattern, with any sphere on one layer lying balanced on three spheres on the previous layers.

[hexagons arranged in an 8-by-8 honeycomb pattern] [a circle tangent to six other circles of the same size, with red line segments connecting the centers of the six circles, forming a hexagon] [spheres arranged the same way, but tilted to indicate their 3-dimensionality, with the same red hexagon, and also two new spheres at opposite ends, making the hexagonal arrangement of spheres into more of a rhombus]

Bonus panel: same as Part 1

Discussion for Spheres Part 2

Discussion for Spheres Part 3

Discussion for Spheres Part 4

Discussion for Spheres Part 5

Transcript for screenreader users:

THE MUSIC OF THE SPHERES

BY DR. TERENCE TAO AND ZACH WEINERSMITH

Dr. Tao: When I tell people at parties that I’m a mathematician, there’s often an awkward pause in the conversation, or the obligatory…

Steve: Oh, I was SOOOOO bad at math at school!

Though occasionally you run into a math and science enthusiast who wants to discuss a recent development they heard in the news…

Enthusiast: Oh my gosh, my friends and I CANNOT stop talking about reconstructing cryptographic protocols from quantum principles, and… do you want to move this conversation to a whiteboard?

[Tao smiles and sheds a tear.]

…or someone who actually had a good experience in their math classes and was somewhat wistful about not pursuing it.

Old man: I’ll always think of Hadwiger’s graph theory conjecture as… the one that got away.

Tao: Those are good moments. But, in general, the public have very little idea what mathematics is, and what mathematicians actually do. Popular culture gives us some caricature portrayals of mathematicians:

The tortured genius…

Tortured genius: I cannot know love, but by god I WILL know if every polyomino with an area divisible by six can tile a hexagon!

(torturedness rates among mathematicians are, in fact average)

The savant with strange mental powers…

[savant seeing equations floating before his eyes]

(if you see equations floating before your eyes, please consult an ophthalmologist)

The eccentric professor…

Professor on unicycle blowing fire out of a trombone: I work in information theory, unicycles, and flaming trombones.

(only true of Claude Shannon)

From math class, you might’ve gotten the impression we’re some sort of sorcerers, casting spells full of strange symbols and jargon to defeat abstract enemies from another plane of existence.

Sorcerer: By the power of Fourier, I banish you from function space to frequency space! [sorcerer wearing a robe wielding a staff glowing at the end, addressing a surface plot of the Riemann zeta function]

Part 1 of 5!

Press forward to continue! [an orange bouncing to the left]

Bonus panel: [Zach Weinersmith behind the drawing board, holding up his left hand, which has become gnarled and has wavy lines rising from his fingers, due to the amount of effort it took to draw this five-part comic. His family is standing behind him. His younger kid is smiling. His older kid’s shirt reads “Ponies etc.”]

τῶν δὲ μαντηίων ἀμφοτέρων ἐς τὠυτὸ αἱ γνῶμαι συνέδραμον, προλέγουσαι Κροίσῳ, ἢν στρατεύηται ἐπὶ Πέρσας, μεγάλην ἀρχὴν μιν καταλύσειν·

and the judgment given to Croesus by each of the two oracles was the same, to wit, that if he should send an army against the Persians he would destroy a great empire.

Source: https://www.loebclassics.com/view/herodotus-persian_wars/1920/pb_LCL117.61.xml

The form to create a post has a checkbox labelled “NSFW”. I didn’t want to limit the visibility of my post to only those people not at work. Nor did I want anyone to get fired because the Internet filters at their workplace detected a word of Germanic origin in my headline.

That’s the case only in the first few episodes. In

This is the best answer because it narrows down the problem to the outside of the eyeball.