• a_g_marut@lemmy.worldEnglish
    2·
    5 days ago

    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