P31m vs. P3m1

Oh, so confusing. I still can’t quite wrap my eyeballs around these two sketches. They look so similar in topic and distribution across the surface. Their construction seems the same at first glance.

3 mirrors through the faces, rotated 60 degrees at the top of the heads. For both images. Equilateral triangles forming hexagons.

Symmetry-Group-P3M1-with-60s-curls - © 2013 Champagne Design

Symmetry-Group-P3M1-with-60s-curls – © 2013 Champagne Design

 

Symmetry-Group-P31M-Cautious-Looking-Women — © 2013 Champagne Design

Symmetry-Group-P31M-Cautious-Looking-Women — © 2013 Champagne Design

 

Let’s turn-on the grids to see the differences.
The 60’s Curls Ladies (below) have 3 different mirrors around the triangle.
As well, 3 different rotations points at each peak in the triangle.
There is no rotation point in the middle of the triangle.

GBF grid-on

Now, for these Cautious Looking Women (below), 3 identical mirrors rotated 3-fold. Sort of like a glide rotation.
3 identical rotation points at the peaks of the triangle.
But, there is a 3-fold rotation right in the middle of the triangle, at the earlobe.
CLW-grid-on

I’m still confused!
Best call a crystallographer or a mathematician to figure this one out.
Comments welcome… you might illuminate for me!

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