Spheres Part 5
Original: Spheres Part 5 on Saturday Morning Breakfast Cereal
Transcript
[Header banner]
PART 5 OF 5! PRESS BACK TO READ!
[A pile of stacked oranges sits beside the banner.]
Panel 1:
Man with headset (the mathematician/narrator): So, one option is to use the 512 possible combinations to make the different letter-encodings as different as possible.
[A table of binary letter-encodings is shown:]
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
Panel 2:
Narrator (text, continued): But another way, the different letter-encodings should be as distant from each other as possible, and because it's 8 bits, that distance is in 9 dimensions.
[Below, two binary strings with arrows between corresponding bits:]
N 110 111 000
Y 100 110 001
Text: N and Y differ in dimensions 2, 6, and 9. So, their Hamming distance is 3 because (drumroll...) 1 + 1 + 1 = 3.
(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.)
Panel 3:
Narrator: 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.
[Labeled binary strings:]
HUCK: 011111111 010011010 001001010 101000111
DUCK: 001011001 010011010 001001010 101000111
BUCK: [partially shown]
Panel 4:
Narrator: 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.
[A diagram of a cube with branching dotted points.]
[Caption box, pink:] *Hamming nonexact not rendered due to budget constraints.
Panel 5:
Narrator: A priori, we might not have expected discrete hyper-dimensional sphere-packing to have applications, but that's exactly what happened.
Character A (figure with 'A' on chest): ...And my work turned out to have filthy FILTHY real-world utility!
Character G (figure with 'G' on chest): Where's your 'PURE ABSTRACTION' now, Thompson?! HAHAHAHAHA!
Panel 6:
Narrator: 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.
Character A: No more fighting. We must come together.
Character G: Like two parallel lines. On a plane of positive curvature.
[The A and G figures embrace happily.]
Panel 7:
Narrator (text panel): 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.
Narrator (second text panel): We take advantage of these mathematical results every day, without being aware of it.
[A smiling man stands with arms slightly spread.]
Panel 8:
Narrator (text panel): QR codes work, even when distorted. Lattice theory helps material scientists design crystals, ceramics, foams, and emulsions.
[A QR code, a crystal, and a cellular/foam pattern are shown.]
Panel 9:
Narrator: I helped develop something similar to an error correcting code to speed up MRI scans by a factor of as much as 10.
[A patient lies in an MRI machine tended by a technician.]
Technician/figure: Thanks, mathematics!
Panel 10:
Narrator (text panel): 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.
[A cube diagram with dotted sphere overlay.]
[Caption box, pink:] *Hamming infiniract not rendered due to budget constraints.
Final panel:
Narrator: And it all started with figuring out how to stack oranges...
[The mathematician stands at center, smiling, against a backdrop of a large crowd of people. A streak of flame-like orange energy arcs across the sky behind him.]
Votey: (none)
PART 5 OF 5! PRESS BACK TO READ!
[A pile of stacked oranges sits beside the banner.]
Panel 1:
Man with headset (the mathematician/narrator): So, one option is to use the 512 possible combinations to make the different letter-encodings as different as possible.
[A table of binary letter-encodings is shown:]
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
Panel 2:
Narrator (text, continued): But another way, the different letter-encodings should be as distant from each other as possible, and because it's 8 bits, that distance is in 9 dimensions.
[Below, two binary strings with arrows between corresponding bits:]
N 110 111 000
Y 100 110 001
Text: N and Y differ in dimensions 2, 6, and 9. So, their Hamming distance is 3 because (drumroll...) 1 + 1 + 1 = 3.
(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.)
Panel 3:
Narrator: 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.
[Labeled binary strings:]
HUCK: 011111111 010011010 001001010 101000111
DUCK: 001011001 010011010 001001010 101000111
BUCK: [partially shown]
Panel 4:
Narrator: 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.
[A diagram of a cube with branching dotted points.]
[Caption box, pink:] *Hamming nonexact not rendered due to budget constraints.
Panel 5:
Narrator: A priori, we might not have expected discrete hyper-dimensional sphere-packing to have applications, but that's exactly what happened.
Character A (figure with 'A' on chest): ...And my work turned out to have filthy FILTHY real-world utility!
Character G (figure with 'G' on chest): Where's your 'PURE ABSTRACTION' now, Thompson?! HAHAHAHAHA!
Panel 6:
Narrator: 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.
Character A: No more fighting. We must come together.
Character G: Like two parallel lines. On a plane of positive curvature.
[The A and G figures embrace happily.]
Panel 7:
Narrator (text panel): 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.
Narrator (second text panel): We take advantage of these mathematical results every day, without being aware of it.
[A smiling man stands with arms slightly spread.]
Panel 8:
Narrator (text panel): QR codes work, even when distorted. Lattice theory helps material scientists design crystals, ceramics, foams, and emulsions.
[A QR code, a crystal, and a cellular/foam pattern are shown.]
Panel 9:
Narrator: I helped develop something similar to an error correcting code to speed up MRI scans by a factor of as much as 10.
[A patient lies in an MRI machine tended by a technician.]
Technician/figure: Thanks, mathematics!
Panel 10:
Narrator (text panel): 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.
[A cube diagram with dotted sphere overlay.]
[Caption box, pink:] *Hamming infiniract not rendered due to budget constraints.
Final panel:
Narrator: And it all started with figuring out how to stack oranges...
[The mathematician stands at center, smiling, against a backdrop of a large crowd of people. A streak of flame-like orange energy arcs across the sky behind him.]
Votey: (none)
Alt text
A tall SMBC comic, the fifth and final part of a series (a banner reads 'PART 5 OF 5! PRESS BACK TO READ!' next to a pile of stacked oranges). A bespectacled mathematician narrator explains, panel by panel, how the abstract math of stacking oranges leads to error-correcting codes. He shows a table mapping letters to binary encodings, then explains that letter-encodings should be made as 'distant' as possible in 9 dimensions, illustrating 'Hamming distance' with the strings N (110 111 000) and Y (100 110 001), which differ in three places, so their Hamming distance is 3 (1+1+1=3). A footnote notes you can compute it via N XOR Y. He jokes that mathematicians built this toolkit 'while nobody was thinking about how to non-offensively talk to Huck about ducks and bucks,' showing binary encodings of HUCK, DUCK, and BUCK. A cube diagram shows bit-flips as nearby points surrounded by error 'spheres,' with a pink caption: 'Hamming nonexact not rendered due to budget constraints.' Two stick figures labeled A and G, who had apparently feuded over pure vs. applied math, reconcile and embrace: 'No more fighting. We must come together.' 'Like two parallel lines. On a plane of positive curvature.' The narrator lists real-world payoffs: error-correcting codes set theoretical limits on data transmission used every day; QR codes work even when distorted; lattice theory helps design crystals, ceramics, foams, and emulsions; he personally helped develop a code to speed up MRI scans up to 10x ('Thanks, mathematics!'); and cell phones share spectrum thanks to infinite-dimensional Hilbert space, with another pink budget-constraints caption ('Hamming infiniract not rendered'). The final panel shows the mathematician standing proudly before a huge crowd with a flame-like orange streak behind him, captioned 'And it all started with figuring out how to stack oranges...' There is no votey.
Transcribed by Claude Opus 4.8.