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-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.

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.

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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You need to draw the proper unit cell in each drawing so it reflects translational symmetry (connect all same closest 4 tops – dont draw extra diagonals – to obtain rhombus). Then you would see that mirror plane symmetry element would be in different orientation. This is because packing is different – more dense in second case.

Glad you understand it! I’m using instinct more than logic lately. Posted a P31m a few days ago, the Chinese Eck Master Lai Tsi!

Seems like P31m patterns are easier to make because symmetry planes are on sides of triangle? In case of P3m1, mirrors dissect the triangle and if you want to grow P3m1 pattern you have to apply mirror located in adjacent triangle (or translation of unit cell). Wikipedia has description of location of symmetry elements in all plane groups: Wallpaper group.