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Suppose we have a point $P_1 = [X_1,X_2, X_3, ... X_n]$ in n-dimension space, How does the rotation of point $P_1$ get another point $P_2= [Y_1, Y_2, Y_3, ...Y_n]$? There is a method to do multiple rotations to form a well-known curve starting from $P_1$, for example figure

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  • $\begingroup$ Are you talking about $\mathbb{R}^{n}$? $\endgroup$
    – Elmex80s
    Commented Aug 15, 2021 at 5:02
  • $\begingroup$ Yes, where each point has a coordinate in the n axis. $\endgroup$ Commented Aug 15, 2021 at 12:02
  • $\begingroup$ " There is a method to do multiple rotations to form a well-known curve starting from", how is it named? $\endgroup$
    – Elmex80s
    Commented Aug 16, 2021 at 5:43
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    $\begingroup$ Anyway, in the 2 dimensional case you can use complex numbers. For example, powers (positive and negative) of $1 + 0.2i$ give a beautiful spiral around the origin. $\endgroup$
    – Elmex80s
    Commented Aug 16, 2021 at 18:08
  • $\begingroup$ @Elmex80s I ask for that I forgot to make the " ? ". I didn't understand your method of using complex numbers, can you give an example? $\endgroup$ Commented Aug 17, 2021 at 13:34

1 Answer 1

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From Powers of complex numbers I take these pictures

http://www.suitcaseofdreams.net/Images/TF/spiral4.gif

and

http://www.suitcaseofdreams.net/Images/TF/spiral5.gif

which spiral a point around $O$. By taking negative powers you can go other way around as well.

For dimensions higher than 3 I think almost every invertible matrix will do.

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  • $\begingroup$ Thank you for this explanation, I am not a mathematician therefore i have difficulty understanding. So, given one point like in your figure blue ones, how i can find the coordinate of the others points? $\endgroup$ Commented Aug 18, 2021 at 14:13
  • $\begingroup$ It are all powers of the complex number $z$. $z$ can be every number you want as long $0 < |z| < 1$ or $1 < |z|$. Note in your example the angles increase with every step, in mine their are constant. $\endgroup$
    – Elmex80s
    Commented Aug 18, 2021 at 16:36
  • $\begingroup$ ok good, there is only the version of a complex number? because i want the same things with real numbers. If you have any suggestions please make them as ref or explain them. $\endgroup$ Commented Aug 18, 2021 at 17:49
  • $\begingroup$ Complex numbers are pairs of real numbers. Lookup how their multiplication works and you should be able to continue. I could write it down but it is searchable everyhwere. $\endgroup$
    – Elmex80s
    Commented Aug 19, 2021 at 10:58
  • $\begingroup$ Ok thank you brother $\endgroup$ Commented Aug 19, 2021 at 11:34

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