Sphere packing does not end at the third dimension, however. The packing coefficients are known up to dimension 10, as well as in dimension 24. It won’t be discussed why exactly this is the case, but dimension 24 has a special symmetry that allows for a proof of the coefficient for what is called the Leech Lattice (Weisstein). At this point, it would be natural to view sphere packing in this context as interesting, but not really applicable to the real world. However, higher dimensional sphere packing has applications in coding. Communication is often messy, and information can occasionally get lost in transmission. Imagine the codeword “gold” is intended to be transmitted. Each letter could be binarily encoded as the following:

This encoding is five dimensional, as each letter has five values that determine its meaning. What if the receiver gets a code of 11010; a value is disrupted along the way. Should the receiver just interpret the message as something like “old” or “god”? Looking at the received code, it can be observed that it varies from “g”, “l”, and “d” by two characters, and “o” by only one character. That is, the received value is closer to “o”, and therefore it would make sense that the receiver interprets the nonsensical code as such, resulting in the correct final message. This kind of strategy was actually used by NASA when encoding photos taken by Voyager in 1977 (Wolchover). Notice what is desired: an encoding such that a slightly improperly received code can be narrowed down to one letter. To save memory, it would make sense that the number of characters that goes into an encoded letter is as small as possible. To optimize this scenario, higher dimensional sphere packing can be used, where any received code within the “sphere” of a character would be interpreted as that character. The actual math and computer science knowledge needed to actually create such a thing is rather complex, but this example gives some intuition into how higher dimensional sphere packings could be surprisingly practical.