Timeline for Chain rotation of a point
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Aug 1, 2018 at 4:13 | vote | accept | Đào Thanh Oai | ||
| Aug 1, 2018 at 0:42 | answer | added | Alexey Ustinov | timeline score: 2 | |
| Aug 1, 2018 at 0:07 | history | edited | Đào Thanh Oai | CC BY-SA 4.0 |
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| Jul 31, 2018 at 17:53 | comment | added | Philipp Lampe | Using complex numbers, the rotation around $A_i$ is given by $P\mapsto (P-A_i)\omega+A_i=P\omega+(1-\omega)A_i$ where $\omega$ in a primitive $(2n)$-th root of unity. Hence, a sufficient condition for periodicity after $2n$ full cycles is $1+\omega^m+\ldots+\omega^{(2n-1)m}=0$. This fails when $m$ is a multiple of $2n$. | |
| Jul 31, 2018 at 17:48 | history | edited | Đào Thanh Oai | CC BY-SA 4.0 |
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| Jul 31, 2018 at 17:42 | comment | added | Đào Thanh Oai | Thank You very much for your comment. I have just improved my question from a conjecture to an open problem. @PhilippLampe | |
| Jul 31, 2018 at 17:40 | history | edited | Đào Thanh Oai | CC BY-SA 4.0 |
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| Jul 31, 2018 at 17:25 | comment | added | Đào Thanh Oai | $(m, n)$=$(2,3)$, $(2, 4)$ are also true......@PhilippLampe | |
| Jul 31, 2018 at 17:17 | comment | added | Đào Thanh Oai | Thank You very much, but $(m,n)$=$(3,1)$, $(3, 2)$, $(3,3)$, $(4, 3)$, $(3,4)$... are true @PhilippLampe | |
| Jul 31, 2018 at 16:42 | comment | added | Philipp Lampe | Here is a counterexample. Let $(m,n)=(2,1)$ and choose $A_1=(0,0)$ and $A_2\neq A_1$ arbitrary. Then the rotation around $A_1$ is given by $P\mapsto -P$ and the rotation around $A_2$ is given by $P\mapsto 2A_2-P$. In particular, the composition of both rotations is the translation $P\mapsto 2A_2+P$. Hence, whenever we start with a point $P$ that does not lie on the line defined by $A_1$ and $A_2$, then $P$ is not equal to $\pm (P+2\lambda A_2)$ with $\lambda\in\mathbb{Z}\backslash\{0\}$. | |
| Jul 31, 2018 at 16:12 | history | edited | Đào Thanh Oai | CC BY-SA 4.0 |
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| Jul 31, 2018 at 16:03 | history | edited | Đào Thanh Oai | CC BY-SA 4.0 |
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| Jul 31, 2018 at 15:58 | history | asked | Đào Thanh Oai | CC BY-SA 4.0 |