2 ms·
> Euler’s formula says (in one interpretation) that a half-turn rotation in the plane is the same as a reflection through a point, a statement which is obvious
by JadeNB 9y ago
> Euler’s formula says (in one interpretation) that a half-turn rotation in the plane is the same as a reflection through a point, a statement which is obvious to children who think about it for a bit.
I guess by "reflection through a point" you mean "the composition of two reflections through orthogonal axes through the point", right? Otherwise this cannot possibly be true, since a rotation is orientation-preserving but a reflection is orientation-reversing.
- jacobolus 9y agoIf you reflect every point in the plane across a given “axis” point, you get a half-turn rotation. Reflecting across a point is also the same as reflecting twice across two perpendicular lines which intersect at that point.
- JadeNB 9y ago> If you reflect every point in the plane across a given “axis” point, you get a half-turn rotation. > Reflecting across a point is also the same as reflecting twice across two perpendicular lines which intersect at that point. My claim was rather that "reflecting across a point" has no meaning by itself—in the strict, mathematical sense according to which a reflection has a fixed hyperplane https://en.wikipedia.org/wiki/Reflection_(mathematics) https://en.wikipedia.org/wiki/Reflection_(mathematics) —unless you use one of these two statements as the definition. (Probably you'd better use the second, because the first doesn't generalise to higher dimensions.)