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>You think it's much different for quaternions It feels like it should be. Maybe the following is just the naivety of a novice looking at experts they don't u

>You think it's much different for quaternions It feels like it should be. Maybe the following is just the naivety of a novice looking at experts they don't understand. Learning numbers, integers, rationals, and real numbers, introduced so much power with each step, to the point that every child is expected to know the basics about real numbers. Maybe not what they are theoretically, but the basic ability to do something like .5 * pi or root(2)^4. They likely never dig any deeper, most don't really get the idea of never ending never repeating decimals, but they are able to work with them on a simple level. For most fields, an introduction only requires this level of math. When one does move to the complex numbers, it opens up far more possibilities. Fourier transform is everywhere, to the extent many use it without having the math to understand how it does what it does. More complex problems, and I mean outside of the field of math, only have general solutions if you allow for complex numbers to be used. These are difficult, mapping from C to C requires 4D to represent and thus are much harder to visualize, but people still struggle through it. When I realized that complex numbers weren't the end, but the second step in a tower of infinite height, I wondered what otherwise unsolvable problems needed higher levels from that, just like how many problems needed complex numbers. The difficulty working with them grows, though somewhat given C to C is already 4D, we have already reached the limit of 3D viewing power. Yet they are rarely used, and higher levels used even less. Maybe there is something fundamental that makes them less useful, far weaker than complex numbers. But from a viewpoint of a novice, I find that surprising.

lupire 3y ago
Like the Eiffel Tower or Burj Khalifa, the algebra tower only has 4 levels (Real, Complex, Quaternion, Octonion) before it shrinks to degeneracy, and level 2 (Complex) has the best balance of dimensionality and power. Adding dimensions adds power, but linking the dimensions adds constraints on allowed structures. https://en.m.wikipedia.org/wiki/Composition_algebra https://en.m.wikipedia.org/wiki/Composition_algebra
thaumasiotes 3y ago
> Learning numbers, integers, rationals, and real numbers, introduced so much power with each step, to the point that every child is expected to know the basics about real numbers. The amount of additional power you get from real numbers is zero, and in fact it isn't possible to work with them because specifying an irrational number takes an infinite amount of time and space. They give you additional power in the sense that they let you give names to the exact answers to certain equations, but not in the sense that you can manipulate those equations any better than otherwise. > When I realized that complex numbers weren't the end But... they are the end. That's the point. We call that the Fundamental Theorem of Algebra: every solution to every equation takes the form of a complex number. Quaternions do not generalize from complex numbers in the sense of the progression of the numeric tower from integers to rationals to reals. That is known to stop at the complex numbers. They generalize from complex numbers in the sense of functions that scale and rotate two-dimensional images. Quaternions are a way of expressing functions that scale and rotate three-dimensional images. They are familiar to most people who have taken calculus under the name "vector cross product".
- thaumasiotes 3y ago
  (And for completeness, while there are octonions, the full generalization of "functions that scale and rotate arbitrary space" is matrices, which again people tend to be familiar with.)
epolanski 3y ago
Not sure what the point is you're trying to make. All I said is that as soon as you release a paper you fall in a niche of some topic (unless the paper is super generic) where the number of people with the expertise allowing them to review it is super small. Quaternions or complex analysis where just examples where, even if not strictly experimental and more theoretical, if you're pushing the boundaries of knowledge the number of people able to review what you wrote and concluded isn't that big.

lupire 3y ago

Like the Eiffel Tower or Burj Khalifa, the algebra tower only has 4 levels (Real, Complex, Quaternion, Octonion) before it shrinks to degeneracy, and level 2 (Complex) has the best balance of dimensionality and power. Adding dimensions adds power, but linking the dimensions adds constraints on allowed structures. https://en.m.wikipedia.org/wiki/Composition_algebra https://en.m.wikipedia.org/wiki/Composition_algebra

thaumasiotes 3y ago

> Learning numbers, integers, rationals, and real numbers, introduced so much power with each step, to the point that every child is expected to know the basics about real numbers. The amount of additional power you get from real numbers is zero, and in fact it isn't possible to work with them because specifying an irrational number takes an infinite amount of time and space. They give you additional power in the sense that they let you give names to the exact answers to certain equations, but not in the sense that you can manipulate those equations any better than otherwise. > When I realized that complex numbers weren't the end But... they are the end. That's the point. We call that the Fundamental Theorem of Algebra: every solution to every equation takes the form of a complex number. Quaternions do not generalize from complex numbers in the sense of the progression of the numeric tower from integers to rationals to reals. That is known to stop at the complex numbers. They generalize from complex numbers in the sense of functions that scale and rotate two-dimensional images. Quaternions are a way of expressing functions that scale and rotate three-dimensional images. They are familiar to most people who have taken calculus under the name "vector cross product".

thaumasiotes 3y ago
(And for completeness, while there are octonions, the full generalization of "functions that scale and rotate arbitrary space" is matrices, which again people tend to be familiar with.)

epolanski 3y ago

Not sure what the point is you're trying to make. All I said is that as soon as you release a paper you fall in a niche of some topic (unless the paper is super generic) where the number of people with the expertise allowing them to review it is super small. Quaternions or complex analysis where just examples where, even if not strictly experimental and more theoretical, if you're pushing the boundaries of knowledge the number of people able to review what you wrote and concluded isn't that big.