[thedoc (OP)]

Russell Dicken asked the Naked Scientists:
   
Does Gossetts 4(21) polytope in E8 exhibit axial chirality?

What do you think?

[evan_au]

There are some nice images of it (and its siblings) here: https://en.wikipedia.org/wiki/4_21_polytope

I can see so many axes in the E₈ group that I wouldn't know where to start.

[chiralSPO]

You have to define what space the object is in to discuss whether or not it is chiral. For instance, there are 2-dimensional shapes that are chiral in 2-D, but not in 3-D: a swastika for example (I know, a terrible example, but bear with me), if the shape is confined to 2 dimensions, then there is a clockwise swastika and a counterclockwise swastika, and the two cannot be inter-converted by rotation or translation. However, if one allows for rotation in the 3rd dimension, each swastika can be "flipped over" and converted to the other.

In the same way, this 8-dimensional polytope is most likely achiral in 9-D space. I would have to look at it more closely, but I believe it is achiral in 8-D space as well (too many degrees of symmetry--especially mirror planes) However, I think it is highly probable that some selected projections into lower dimensional spaces (for instance 3-D) could be chiral in those spaces. I will have to think on this some more, as I haven't really considered any shapes much higher than 4-D in any significant detail before!