Roflections

On Roflections

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For my inaugural post here, I would merely like to explain the title chosen for this blog.  Symmetry is all around us, and many important things can be related to observing symmetries.  Two basic symmetries that often go hand-in-hand are reflection and rotation.  My language below will have a bias towards two dimensions, but a lot of it does generalize to higher dimensions.

Rotations can be made from reflecting twice, but along different axes (that go through a common point).  What happens if you keep adding more reflections through axes that go through that point?  Three reflections will give you something that may be a reflection, but it also may not be either a reflection or a rotation.  I don’t know of a standard word for a combined reflection and rotation, so I made a suitcase with two equal-sized compartments to put them in:  roflection.

What happens if you reflect again?  In two dimensions, a reflection with a roflection will always combine to make a rotation, since four reflections about a common point is equivalent to two rotations about that point, which is another rotation.  In higher dimension, it is still considered a rotation, though not necessarily a simple rotation.  For instance, in 4D, a general rotation leaves a point fixed, and will have two orthogonal planes fixed, in the sense that those planes are closed under the operation of applying that rotation any number of times.

There is a nice pun value to the term ‘roflection’ as well, thanks to the penchant towards abbreviating in texting and internet culture, which include a number of phrases that indicate amusement.  ‘Rofl’ is one such abbreviation, and it is my hope that there will be occasion for jocular posts.

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