oh good. I’ve invented a new name for what is always listed as the seventh frieze symmetry group. Why not. Seems like what everyone does. Conway calls this one a SPINNING JUMP.
I’m calling it bd/pq, with the slash meaning a dividing line -a horizontal line – putting the bd above the pq, because that’s how this symmetry group moves.
This one has the most indecipherable description of all the frieze groups: horizontal and Vertical reflection lines, Translations and 180° Rotations:
This group requires three generators, with one generating set consisting of a translation, the reflection in the horizontal axis and a reflection across a vertical axis. Wikipedia
There’s no point in trying to understand this group through it’s description. It’s much easier to see if you look at the design it makes and figure out how it became that.
There’s a certain amount of discretion in how to view the way this symmetry group grows. The first step in this one, going from the base unit (Fig. 1) is a vertical reflection. Fig 2 Then next action can be seen as the two units from Fig.2 being horizontally reflected OR rotated a half-turn (180 degrees). Either way you see it, it’s the same.
In the previous photo I made the shapes sort of resemble the letters b, d, p, & q, but that’s totally unnecessary. Above, I’ve oriented the base unit differently, and also changed up the sequence. This time the first step is doing a horizontal reflection, then the second step is a vertical reflection of both of the pieces, or you can see it as a 180 degree rotation. Your choice. It’s all the same.
Just for fun, here’s the same unit doing the same symmetry actions, but the orientation of the base unit is once again different.
Take a look at these. You might notice that in the center of these shapes you can imagine there being an X or a rhombus. This seems to happen often with this symmetry group.
Here is this same symmetry action using symmetry tiles that I’ve been making. If it looks like rotational symmetry to you, well, no it’s not.
The image below is rotational symmetry, which is not a Frieze Group, it’s a Wallpaper Group, of which there are 17 symmetry groups, which I have no plans to write about.
Finally, here’s my last video of this series on the Frieze Groups, saying the same things that I just wrote.
The pattern blocks were fun to use with the symmetry group. Here’s one that’s not in the video:
This link will get you to all of my Frieze Group posts: https://bookzoompa.wordpress.com/category/frieze-groups/