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Ask HN: Do quaternions have any interesting properties, akin to those of complex numbers, when calculus is applied? The only application I've ever seen is for d

Ask HN: Do quaternions have any interesting properties, akin to those of complex numbers, when calculus is applied? The only application I've ever seen is for doing rotations in computer graphics.

Chabsff 5y ago
Still mostly relevant to their use as encoding for transformations, but the fact that a unit quaternion's conjugate is also its inverse is extremely useful from a computational performance point of view.
AnimalMuppet 5y ago
If I understand correctly, yes, calculus on quaternions is a useful thing. I'm not sure whether or not you have an equivalent of the residue theorem from the calculus of complex variables, but I'm pretty sure that you can do path integrals and stuff...
- mathgenius 5y ago
  John Baez was tweeting about this recently: https://twitter.com/johncarlosbaez/status/1467204133262446599 https://twitter.com/johncarlosbaez/status/146720413326244659...
fault1 5y ago
per wikipedia (hence the interest in twistors, spin spaces and clifford algebras in physics): Quaternions also capture the spinorial character of rotations in three dimensions. For a three-dimensional object connected to its (fixed) surroundings by slack strings or bands, the strings or bands can be untangled after two complete turns about some fixed axis from an initial untangled state. Algebraically, the quaternion describing such a rotation changes from a scalar +1 (initially), through (scalar + pseudovector) values to scalar −1 (at one full turn), through (scalar + pseudovector) values back to scalar +1 (at two full turns). This cycle repeats every 2 turns. After 2n turns (integer n > 0), without any intermediate untangling attempts, the strings/bands can be partially untangled back to the 2(n − 1) turns state with each application of the same procedure used in untangling from 2 turns to 0 turns. Applying the same procedure n times will take a 2n-tangled object back to the untangled or 0 turn state. The untangling process also removes any rotation-generated twisting about the strings/bands themselves. Simple 3D mechanical models can be used to demonstrate these facts.

Ask HN: Do quaternions have any interesting properties, akin to those of complex numbers, when calculus is applied? The only application I've ever seen is for doing rotations in computer graphics.

Chabsff 5y ago

Still mostly relevant to their use as encoding for transformations, but the fact that a unit quaternion's conjugate is also its inverse is extremely useful from a computational performance point of view.

AnimalMuppet 5y ago

If I understand correctly, yes, calculus on quaternions is a useful thing. I'm not sure whether or not you have an equivalent of the residue theorem from the calculus of complex variables, but I'm pretty sure that you can do path integrals and stuff...

mathgenius 5y ago
John Baez was tweeting about this recently: https://twitter.com/johncarlosbaez/status/1467204133262446599 https://twitter.com/johncarlosbaez/status/146720413326244659...

fault1 5y ago

per wikipedia (hence the interest in twistors, spin spaces and clifford algebras in physics): Quaternions also capture the spinorial character of rotations in three dimensions. For a three-dimensional object connected to its (fixed) surroundings by slack strings or bands, the strings or bands can be untangled after two complete turns about some fixed axis from an initial untangled state. Algebraically, the quaternion describing such a rotation changes from a scalar +1 (initially), through (scalar + pseudovector) values to scalar −1 (at one full turn), through (scalar + pseudovector) values back to scalar +1 (at two full turns). This cycle repeats every 2 turns. After 2n turns (integer n > 0), without any intermediate untangling attempts, the strings/bands can be partially untangled back to the 2(n − 1) turns state with each application of the same procedure used in untangling from 2 turns to 0 turns. Applying the same procedure n times will take a 2n-tangled object back to the untangled or 0 turn state. The untangling process also removes any rotation-generated twisting about the strings/bands themselves. Simple 3D mechanical models can be used to demonstrate these facts.