rotating a tetrahedron about its rotational symmetry axes
a regular tetrahedron has twelve rotational symmetries. they can be classified based on the axis about which we do the rotation. seeing the symmetries in one of them is easy. the other one though, is quite nontrivial. i’ve managed to understand it with the help of a visualization. later, i’ve realized that it is possible to accept it if one can do a few geometric deductions, too.
let’s describe first, the easy rotation. take a regular tetrahedron.
construct the line passing through one of its vertex and the opposite
centroid. now rotate the tetrahedron about the line you’ve constructed
2pi/3 radians. you’ll see that this maps tetrahedron to itself.
4pi/3 radians does the same. so, for this line, there is
two rotational symmetries. see that there is four lines like that, at
the second one is done as follows: you construct the line passing through
the midpoint of any edge chosen and the midpoint of the opposite edge.
rotate the tetrahedron about that line
now, i think it is fairly hard to believe that this rotation maps the tetrahedron to itself. as i said, a visualization helped me seeing it. but, what i want to do is showing you the geometric deductions i discover from the visualization, that forces this rotation to map the tetrahedron to itself.
see first that the line about which we do the rotation is orthogonal to both edges of which it passes through the midpoints. consider for now, only one edge. and keep in mind that we rotate pi radians. so indeed, this rotation interchanges the positions of the vertices of that edge. this is also true for the other edge. thus, after the rotation, we know that the positions of the four, i.e. all, vertices will stay the same. and a rotation cannot affect the shape of the object it is applied to. thus, this rotation is really maps the tetrahedron to itself.
side story time
those above, was my notes to myself, me wrote it in the course of reading the first few pages of the book Groups and Symmetry. i simply couldn’t imagine that nontrivial rotation. then i plotted the tetrahedron, animated the rotation, just for pragmatic concerns. then i wanted to write this blog post but i couldn’t have used that animation, it was simply too ugly and i’ve had to record it with another software, e.g. obs. eventually, i’ve decided to create the animation with latex.
i was aware of a package called
pst-solides3d, tried to give it a go
and it performed really well. nevertheless, actually rotating the
tetrahedron was not simple. let me give you the code and shut up already:
first, let’s cite the rotation
page. we’ll use it to do the actual rotation. read that little section.
see that the matrix is useful if we translate the tetrahedron and the
axis so that the axis passes through the origin. did it and gave the
coordinates of the direction vector - it is normalized - of the axis as
\un in the code. then i’ve just simply typed the bulky
a few nonessential details of that code:
- the trigonometric functions in the expression take their arguments in
radian. so we do a conversion in
- we actually produce
91pages. this is done with
multidostatement. effectively, we rotate
2*90degrees. indeed, this is what we want. then, we convert this
let me also give you the makefile:
and this is the sweet product:
sad story time
things happen, and they don’t seem tangible. we can’t even alter afterwards the effect of them.
the fence change their fulfilling life hence they file the sadness they defence