Frieze Groups

Frieze Symmetry Group F7; p2mm; *22∞; TRHVG: bd/pq

Symmetry Tiles p2mm
Symmetry Tiles p2mm


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.

Fiugre1, Figure 2 and Figure 2. In that order.
Figure 1, Figure 2 and Figure 2. In that order.

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.


Same symmetry group as previous picture, just taking a different route to get there.
Same symmetry group as previous picture, just taking a different route to get there.

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.

Symmetry tiles
Symmetry tiles p2mm

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:


Frieze Groups

Frieze Group F6; p11m; pm; *∞; THG; Horizontal Reflection

I’ve been describing frieze groups, which define symmetry actions which create a linear designs. This next one, which can be thought of simply as horizontal reflection (though like all the frieze groups also contains translation), is generally listed as the sixth frieze group. This seemed odd to me at first, as it’s one of the simplest symmetry actions, so shouldn’t it be in the beginning of the list? People, however, don’t seem to have used simplicity as the criteria for ordering the symmetry groups.

The Conway nickname for this symmetry group is JUMP.

Here’s how to build horizontal reflection symmetry:

Start with a unit,



Notice horizon line under the unit.

Copy, then reflect over the horizontal line. Notice that this action grows the design vertically.

That’s it. Now just repeat until the supply of little purple rectangularish shapes is diminished, which happened to me immediately, but that’s okay because there’s not much more to say here.

Except there is.

Of all the frieze symmetries that I’ve been describing this is the FIRST one that MUST grow vertically (which is a result of the horizontal reflection) before it grows horizontally.

This action is simple, and there is so little else to say about this that maybe this is a good time to acknowledge that the complexity of the design relies on the complexity of the graphic on or of the unit that is being repeated.

Here are a few fun ones.

Imagine each one of these sets repeated over and over again to create friezes.

Not that this is really needed, but here’s the Horizontal Reflection Frieze Group Video:

The next and last of the frieze groups is as complex as this one is simple. Hold on to your hats. Coming soon….

Frieze Groups

Frieze Group F5; p2mg; 2*∞ ; p2mg; TRVG: Sine Wave


This is my favorite one of the Frieze Groups.  I like it so much because, after being confused about it for quite some time, I finally saw that it’s just like a sine wave.

The action is that the first unit is vertically reflected then this composite shape does a glide reflection move.

Frieze Group F5; p2mg; 2*∞ ; p2mg; TRVG: Sine Wave
Frieze Group F5; p2mg; 2*∞ ; p2mg; TRVG: Sine Wave


If you are happy with the explanation above and have an adverse reaction to being confused to the point that your head feels like scrambled eggs in a pressure cooker, then skip the next paragraph.

In many places this frieze group is described as having more actions, specifically, vertical reflection lines, glide reflections, translations, 180° rotations and horizontal reflection. There may be a choice in the description that says do a vertical reflection or a horizontal reflection. Or maybe the description will describe fewer actions. So what’s going on? This is what’s going on: first, all frieze symmetries contain the translation action, which simply means that the original unit is being repeated. What’s helpful to understand is that when there’s horizontal reflection there is also glide reflection. Then here’s the next piece to decipher:  Pick two actions from this list.

  • Vertical Reflection
  • Rotation
  • Glide Reflection

Any two of the above actions gives you the third one, so it’s not a matter of doing either this or that or all three, it’s just choosing which way to think about it.

While the appearance of the final design doesn’t’ necessarily look just like a sine wave, the pattern of its creation is the same.

Frieze Group F5; p2mg; 2*∞ ; p2mg; TRVG: Sine Wave
Frieze Group F5; p2mg; 2*∞ ; p2mg; TRVG: Sine Wave

You’ll notice that some of the images above look straight, not like the way you’d think of a sine wave. If you don’t understand what happened to the waviness,  please go back and look at my post about frieze glide reflections.  In fact, more often than not, this frieze group presents itself without the wave-like action being obvious, so it’s a good idea to get clear about this.

The video might help.


Just for good measure, here’s a pattern block showing this frieze symmetry group.

Frieze Symmetry with Pattern Blocks
Frieze Symmetry with Pattern Blocks


Frieze Groups

Frieze Group F4; p2; 22∞; Half-turn Rotation



This is where I could get stuck and stop posting about frieze groups. It’s not that this one is so hard, it’s just that I’m finding it hard to do it justice. As I’m working with it I can see that it can make awesome,  apparently complex patterns that are really not complex at all. I am tempted to stop everything and just work at seeing what I can do with this symmetry group. It’s the kind of rabbit. hole that would keep me too busy to keep posting, so I’m going to stay with fairly simple examples.

In an earlier post I made a big deal about understanding gyration points. This is the frieze group that I finally get to play with gyration points, though, really, that’s not what makes this kind of symmetry so interesting.

The Conway Nickname for this group is SPINNING HOP. This nickname is fine, but the graphic that accompanies this nickname is more of a shorthand notation than a description. In other words, it makes sense as a reminder of how the symmetry group moves, but isn’t a particularly good way to learn it.

In this group the translated unit rotates 180 degree, which is half of turn, like if you are facing south a 180 degree rotation would leave you facing north. The point around which it rotates can vary.

Here’s the most straightforward way to see the the action.


Maybe I should just show this with one unit at a time, but I’m going to try this out showing two separate examples at one time

Here, the each unit is moved over then rotated by 180 degrees.

Now, below, here’s the same action, but done using a different gyration point,meaning I’ve rotated the copy around a different center.

The way the yellow half-circles are oriented in the image above most closely resembles the graphic that is listed with the Mathematical Association of America‘s Frieze Pattern page.

Here’s more variations:

Do you see what’g going on? What’s NOT changing is that the repeated unit is keeping it’s 180 degree rotation. What’s changing is it’s horizontal and vertical orientation.

Here are some examples where I’ve extended this kind of symmetry to make the frieze.

That black dot show the gyration point around which the original unit is rotated.

The pattern blocks were delightful to use with this symmetry group. I could have used a simple trapezoid, but building the trapezium from various pieces makes the symmetry pattern more explicit.



The way the blocks arrangement turned out surprised me because it looks like such a complicated pattern, when, if fact, it’s just a simple action made to the original unit.

Now for the video!

One last thing: here’s a look at how vertical reflection compares to half-turn rotation:

VErtical reflection compared to 180 degree (half turn) rotation
VErtical reflection compared to 180 degree (half turn) rotation