3 ms·

I've been wondering about the connection between the "curling effect" and the exponential curve of e for a while. Do you have any links where I can learn more i

I've been wondering about the connection between the "curling effect" and the exponential curve of e for a while. Do you have any links where I can learn more in depth?

pjbk 5y ago
With complex numbers in polar form the tangent is easy. In the unitary circle, if you derive e^(i*t) you get i*e^(i*t), which maps the cos(t) real component to i*cos(t) imaginary, and the i*sin(t) imaginary component to -sin(t) real. This is effectively a 90 degree rotation, so if you integrate the tangent infinitesimally over its path parameterized by t you will recover the circle. Here is some introductory material to what I referenced above and some generalizations into more dimensions (which, as Hamilton discovered when stumbling into quaternions trying to augment complex numbers, is not as straightforward as you would think): - Why i? [http://www.stat.physik.uni-potsdam.de/~pikovsky/teaching/stud_seminar/ajp_i.pdf http://www.stat.physik.uni-potsdam.de/~pikovsky/teaching/stu...] - An Introduction to Geometric Algebra with an Application in Rigid Body Mechanics [https://www.researchgate.net/profile/Terje-Vold/publication/241273951_An_Introduction_to_Geometric_Algebra_with_an_Application_in_Rigid_Body_Mechanics/links/57ce13fa08ae582e06923f98/An-Introduction-to-Geometric-Algebra-with-an-Application-in-Rigid-Body-Mechanics.pdf?origin=publication_detail https://www.researchgate.net/profile/Terje-Vold/publication/...] - Functions of Multivector Variables [https://journals.plos.org/plosone/article/file?id=10.1371/journal.pone.0116943&type=printable https://journals.plos.org/plosone/article/file?id=10.1371/jo...] - Lie Group Theory - A Completely Naive Introduction [https://jakobschwichtenberg.com/naive-introduction-lie-theory https://jakobschwichtenberg.com/naive-introduction-lie-theor...] - Previous HN discussion: Intuitive Understanding of Euler’s Formula [https://news.ycombinator.com/item?id=18325865 https://news.ycombinator.com/item?id=18325865]

I've been wondering about the connection between the "curling effect" and the exponential curve of e for a while. Do you have any links where I can learn more in depth?

pjbk 5y ago

With complex numbers in polar form the tangent is easy. In the unitary circle, if you derive e^(i*t) you get i*e^(i*t), which maps the cos(t) real component to i*cos(t) imaginary, and the i*sin(t) imaginary component to -sin(t) real. This is effectively a 90 degree rotation, so if you integrate the tangent infinitesimally over its path parameterized by t you will recover the circle. Here is some introductory material to what I referenced above and some generalizations into more dimensions (which, as Hamilton discovered when stumbling into quaternions trying to augment complex numbers, is not as straightforward as you would think): - Why i? [http://www.stat.physik.uni-potsdam.de/~pikovsky/teaching/stud_seminar/ajp_i.pdf http://www.stat.physik.uni-potsdam.de/~pikovsky/teaching/stu...] - An Introduction to Geometric Algebra with an Application in Rigid Body Mechanics [https://www.researchgate.net/profile/Terje-Vold/publication/241273951_An_Introduction_to_Geometric_Algebra_with_an_Application_in_Rigid_Body_Mechanics/links/57ce13fa08ae582e06923f98/An-Introduction-to-Geometric-Algebra-with-an-Application-in-Rigid-Body-Mechanics.pdf?origin=publication_detail https://www.researchgate.net/profile/Terje-Vold/publication/...] - Functions of Multivector Variables [https://journals.plos.org/plosone/article/file?id=10.1371/journal.pone.0116943&type=printable https://journals.plos.org/plosone/article/file?id=10.1371/jo...] - Lie Group Theory - A Completely Naive Introduction [https://jakobschwichtenberg.com/naive-introduction-lie-theory https://jakobschwichtenberg.com/naive-introduction-lie-theor...] - Previous HN discussion: Intuitive Understanding of Euler’s Formula [https://news.ycombinator.com/item?id=18325865 https://news.ycombinator.com/item?id=18325865]