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I first encountered quaternions in the concept of abstract algebra. It's interesting to think of it as part of a spectrum starting with reals and continuing to

by dhosek 5y ago

I first encountered quaternions in the concept of abstract algebra. It's interesting to think of it as part of a spectrum starting with reals and continuing to octonions (and beyond). Interestingly with each extension we lose something: Complex numbers (2-dimensional): No more ordering Quaternions (4-dimensional): No more commutative multiplication Octonions (8-dimensional): No more associative multiplication Sedenions (16-dimensional): We lose the alternative property, i.e., with octonions its still true that x(xy) = (xx)y but with sedenions it is not. We can continue this process indefinitely, although I know nothing of the characteristics of trigintaduonions or what comes later. I suspect that it might be the loss of the flexible identity next, a(ba) = (ab)a but I don't know for sure.

galaxyLogic 5y ago
Here's a short article that gave me some insight into the matter: https://nautil.us/blog/the-strange-numbers-that-birthed-modern-algebra https://nautil.us/blog/the-strange-numbers-that-birthed-mode... "Place your phone face-up on a flat surface, for example. Spin it 90 degrees to the left, and then flip it away from you. Note which way the camera points. Returning to the original position, flip it away from you first and then turn it to the left second. See how the camera points to the right instead? This initially alarming property, known as non-commutativity, turns out to be a feature the quaternions share with reality "
- dhosek 5y ago
  It's worth noting that one of the early finite groups that is taught in abstract algebra are the dihedral groups which model exactly this sort of behavior (For your phone example this would be D₄. Knowing that mattress flipping is modeled with D₂ makes it trivial to prove that that there is no combination of flip and rotate that will allow you to always do the same move to your mattress to get it into the four possible positions for optimal wear.
  - galaxyLogic 5y ago
    Interesting Applied Mathematics
macrolocal 5y ago
One neat characteristic is that the derivations of these higher Cayley-Dickinson algebras stabilize! [1] https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-117/issue-1/Derivations-and-Cayley-derivations-of-generalized-Cayley-Dickson-algebras/pjm/1102706933.pdf https://projecteuclid.org/journals/pacific-journal-of-mathem...
avz 5y ago
Interestingly, if we go in the opposite direction we also loose something: Real numbers (1-dimensional): No more algebraic closure [1] though as you suggest we do gain a nice ordering where "nice" means "compatible with operations". BTW, you can always impose some ordering on all of these sets. It is the orderings' compatibility with arithmetic that we lose going from real numbers to complex numbers. This can be generalized a little. Real numbers fail to be algebraically closed because some real polynomials lack real solutions, e.g. xx=-2. Complex numbers "fix" this problem since there every n degree polynomial has exactly n roots (counting multiplicity). In a sense, quaternions overdo this, e.g. xx=-2 has an infinite number of solutions (exercise for the reader). However, since quaternion multiplication is not commutative, polynomials as traditionally understood are ill-defined. For example, axx, xax and xxa are different degree two quaternion "polynomials". Ugh. [1]: https://en.wikipedia.org/wiki/Algebraically_closed_field https://en.wikipedia.org/wiki/Algebraically_closed_field
- dhosek 5y ago
  I used to teach the expansion of the concept of numbers from ℕ to ℂ via a fictionalized history of the numbers (most notably, expanding ℕ to ℤ before introducing ℚ, but historically, rational and even irrational numbers preceded zero and negative numbers in mathematical understanding. It also elides over the various subdivisions of all the different kinds of irrational numbers that exist in ℝ\ℚ which could be a whole class in itself, but this was a math for liberal arts majors class and I had to make sure not to have their eyes glaze over too much.
nabla9 5y ago
First 4 (reals, complex, quaternions, octonions) are normed division algebras. Normed means that each number has associated nonnegative real number that describes the "size" of the number, also |xy| <= |x||y|. Number having a size (a norm) is very useful concept. Sedions and the rest are less useful because they are not normed algebras.
Zamicol 5y ago
Why do you think the dimensions are in powers of two?
- Twisol 5y ago
  Because the construction generating this sequence of algebras doubles the dimension every time. https://en.m.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction https://en.m.wikipedia.org/wiki/Cayley%E2%80%93Dickson_const...
rp1 5y ago
I don't think it makes sense to say you "lose" something. Yes, the properties of different spaces _are_ different, but these differences are intentional so that they can be used to study something useful. For instance, complex analytic functions have a number of properties that don't hold for real analytic functions, so you actually "gain" something. An example is the residue theorem. One interesting thing about the residue theorem is that you can use it to calculate the integral of a real-valued function that wouldn't be calculable using conventional calculus techniques. So not only do complex functions have a number of interesting properties on their own, they can also be used to expand what's possible in real functions.
thealig 5y ago
Another interesting property from which I learnt about quaternions is that all the proper subgroups of the quaternion group are normal and commutative, but the quaternion group itself is non commutative. [0] [0] https://www.cs.cas.cz/portal/AlgoMath/AlgebraicStructures/StructuresWithOneOperation/Groups/SpecialGroups/QuaternionGroup.htm https://www.cs.cas.cz/portal/AlgoMath/AlgebraicStructures/St...

by dhosek 5y ago

galaxyLogic 5y ago

Here's a short article that gave me some insight into the matter: https://nautil.us/blog/the-strange-numbers-that-birthed-modern-algebra https://nautil.us/blog/the-strange-numbers-that-birthed-mode... "Place your phone face-up on a flat surface, for example. Spin it 90 degrees to the left, and then flip it away from you. Note which way the camera points. Returning to the original position, flip it away from you first and then turn it to the left second. See how the camera points to the right instead? This initially alarming property, known as non-commutativity, turns out to be a feature the quaternions share with reality "

dhosek 5y ago
It's worth noting that one of the early finite groups that is taught in abstract algebra are the dihedral groups which model exactly this sort of behavior (For your phone example this would be D₄. Knowing that mattress flipping is modeled with D₂ makes it trivial to prove that that there is no combination of flip and rotate that will allow you to always do the same move to your mattress to get it into the four possible positions for optimal wear.
- galaxyLogic 5y ago
  Interesting Applied Mathematics

macrolocal 5y ago

One neat characteristic is that the derivations of these higher Cayley-Dickinson algebras stabilize! [1] https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-117/issue-1/Derivations-and-Cayley-derivations-of-generalized-Cayley-Dickson-algebras/pjm/1102706933.pdf https://projecteuclid.org/journals/pacific-journal-of-mathem...

avz 5y ago

Interestingly, if we go in the opposite direction we also loose something: Real numbers (1-dimensional): No more algebraic closure [1] though as you suggest we do gain a nice ordering where "nice" means "compatible with operations". BTW, you can always impose some ordering on all of these sets. It is the orderings' compatibility with arithmetic that we lose going from real numbers to complex numbers. This can be generalized a little. Real numbers fail to be algebraically closed because some real polynomials lack real solutions, e.g. xx=-2. Complex numbers "fix" this problem since there every n degree polynomial has exactly n roots (counting multiplicity). In a sense, quaternions overdo this, e.g. xx=-2 has an infinite number of solutions (exercise for the reader). However, since quaternion multiplication is not commutative, polynomials as traditionally understood are ill-defined. For example, axx, xax and xxa are different degree two quaternion "polynomials". Ugh. [1]: https://en.wikipedia.org/wiki/Algebraically_closed_field https://en.wikipedia.org/wiki/Algebraically_closed_field

dhosek 5y ago
I used to teach the expansion of the concept of numbers from ℕ to ℂ via a fictionalized history of the numbers (most notably, expanding ℕ to ℤ before introducing ℚ, but historically, rational and even irrational numbers preceded zero and negative numbers in mathematical understanding. It also elides over the various subdivisions of all the different kinds of irrational numbers that exist in ℝ\ℚ which could be a whole class in itself, but this was a math for liberal arts majors class and I had to make sure not to have their eyes glaze over too much.

nabla9 5y ago

First 4 (reals, complex, quaternions, octonions) are normed division algebras. Normed means that each number has associated nonnegative real number that describes the "size" of the number, also |xy| <= |x||y|. Number having a size (a norm) is very useful concept. Sedions and the rest are less useful because they are not normed algebras.

Zamicol 5y ago

Why do you think the dimensions are in powers of two?

Twisol 5y ago
Because the construction generating this sequence of algebras doubles the dimension every time. https://en.m.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction https://en.m.wikipedia.org/wiki/Cayley%E2%80%93Dickson_const...

rp1 5y ago

I don't think it makes sense to say you "lose" something. Yes, the properties of different spaces _are_ different, but these differences are intentional so that they can be used to study something useful. For instance, complex analytic functions have a number of properties that don't hold for real analytic functions, so you actually "gain" something. An example is the residue theorem. One interesting thing about the residue theorem is that you can use it to calculate the integral of a real-valued function that wouldn't be calculable using conventional calculus techniques. So not only do complex functions have a number of interesting properties on their own, they can also be used to expand what's possible in real functions.

thealig 5y ago

Another interesting property from which I learnt about quaternions is that all the proper subgroups of the quaternion group are normal and commutative, but the quaternion group itself is non commutative. [0] [0] https://www.cs.cas.cz/portal/AlgoMath/AlgebraicStructures/StructuresWithOneOperation/Groups/SpecialGroups/QuaternionGroup.htm https://www.cs.cas.cz/portal/AlgoMath/AlgebraicStructures/St...