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Where are the mathematicians who could be describing Geometric Algebra correctly? For a professional working in linear algebra, supervising a graduate student,

by gowld 2y ago

Where are the mathematicians who could be describing Geometric Algebra correctly? For a professional working in linear algebra, supervising a graduate student, it should be straightforward to translate these "higher order" linear algebra objects into the geometric algebra model, without publishing technically incoherent mistakes.

deleted 2y ago
[deleted]
nyssos 2y ago
"Geometric algebras" are Clifford algebras over the reals. Differential geometry is probably the field where you're most likely to see mathematicians discuss them, though they also come up in certain (closely related) areas of mathematical physics via spinors.
- koolala 2y ago
  What about Clifford algebras over the imaginaries? Is that included in your idea of reals?
  - nyssos 2y ago
    Imaginary numbers aren't a field, so there's no such thing. Clifford algebras over the complex numbers work fine, but it's usually not what the people talking about "geometric algebra" are doing.
    - koolala 2y ago
      How is complex / imaginary numbers not what they are doing? Numbers that square to -1, 0, and 1 are the bread and butter of the GA I know. Exploring different combinations of types of imaginary numbers and their products and space describing algebras. (including naturally quaternions, duel quaternions, i-rotation, nilpotent, ect)
      - mkl 2y ago
        You seem to be calling complex numbers imaginary numbers, but they're not the same thing. Imaginary numbers are a subset of the complex numbers consisting of the imaginary axis without 0, e.g. i, 2i, -3.1i. Complex numbers also include the real numbers and all combinations of real and imaginary.
        koolala 2y ago
        I'm glad you seem to know what I mean. I love imaginary numbers like the Square Root of -1 or the Square Root of 0. They can only be used like: nil²=0 i²=-1 j²=-1 And combining them into complex forms like: (i + j) (nil + i) What can I call this general idea of using imagination to determine new number rules and combining them together? If I call this GA or Clifford Algebra in a math community will it trigger rigor admins to ban me from talking because I'm not using their terms? I wish we had artistic imaginary math communities for exploring Geometry without rigor turing everything into Semantics. Geometry literally doesn't need semantics if you agree on points, lines, planes ect. Algebra to me should just be simple maps from clifford numbers to examples of intuitive geometry / physics.
        deleted 2y ago
        [deleted]
        bmacho 2y ago
        People should stop calling that set "imaginary numbers". It's net bad. That set isn't really worth it to be called anything. Maybe "pure imaginary numbers". Also "imaginary numbers" is used for the complex numbers already.
        koolala 2y ago
        People also like colloquially saying "The Square Root of -1". Technically wrong but personally meaningful. GA is mainstream once normal people start joking about the square root of zero. We need to imagine and teach even more imaginary numbers one day.
      - 2y ago

by gowld 2y ago

deleted 2y ago

[deleted]

nyssos 2y ago

"Geometric algebras" are Clifford algebras over the reals. Differential geometry is probably the field where you're most likely to see mathematicians discuss them, though they also come up in certain (closely related) areas of mathematical physics via spinors.

koolala 2y ago
What about Clifford algebras over the imaginaries? Is that included in your idea of reals?
- nyssos 2y ago
  Imaginary numbers aren't a field, so there's no such thing. Clifford algebras over the complex numbers work fine, but it's usually not what the people talking about "geometric algebra" are doing.
  - koolala 2y ago
    How is complex / imaginary numbers not what they are doing? Numbers that square to -1, 0, and 1 are the bread and butter of the GA I know. Exploring different combinations of types of imaginary numbers and their products and space describing algebras. (including naturally quaternions, duel quaternions, i-rotation, nilpotent, ect)
    - mkl 2y ago
      You seem to be calling complex numbers imaginary numbers, but they're not the same thing. Imaginary numbers are a subset of the complex numbers consisting of the imaginary axis without 0, e.g. i, 2i, -3.1i. Complex numbers also include the real numbers and all combinations of real and imaginary.
      - koolala 2y ago
        I'm glad you seem to know what I mean. I love imaginary numbers like the Square Root of -1 or the Square Root of 0. They can only be used like: nil²=0 i²=-1 j²=-1 And combining them into complex forms like: (i + j) (nil + i) What can I call this general idea of using imagination to determine new number rules and combining them together? If I call this GA or Clifford Algebra in a math community will it trigger rigor admins to ban me from talking because I'm not using their terms? I wish we had artistic imaginary math communities for exploring Geometry without rigor turing everything into Semantics. Geometry literally doesn't need semantics if you agree on points, lines, planes ect. Algebra to me should just be simple maps from clifford numbers to examples of intuitive geometry / physics.
        deleted 2y ago
        [deleted]
      - bmacho 2y ago
        People should stop calling that set "imaginary numbers". It's net bad. That set isn't really worth it to be called anything. Maybe "pure imaginary numbers". Also "imaginary numbers" is used for the complex numbers already.
        koolala 2y ago
        People also like colloquially saying "The Square Root of -1". Technically wrong but personally meaningful. GA is mainstream once normal people start joking about the square root of zero. We need to imagine and teach even more imaginary numbers one day.
    - 2y ago