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


Frieze Groups

Frieze Group F3; p1m1; pm; *∞∞; Vertical Reflection


Paper Dolls cut from accordion fold
Vertical Reflection, Linked Paper Dolls

Vertical reflection symmetries are often listed as the third frieze group. The  Conway nickname for vertical reflection is SIDLE, thus earning itself the distinction of having the most confusing Conway nickname. Fortunately, no nickname is necessary, as the term “vertical reflection” is a perfectly good way to think about this symmetry action.

This is the third post describing each of the seven Frieze Symmetry Groups.

Vertical reflection is just that: the shape reflects across a vertical line.

Something that is initially confusing is that a vertically reflected shape grows a horizontal pattern.

Vertical Reflection
Vertical Reflection


You’ll need to resist the urge to see this as horizontal symmetry. We’ll be getting to that. This is not it.

Using an accordion fold to make Vertical Reflection pattern
Using an accordion fold to make Vertical Reflection pattern

I feel like there’s a bonus embedded in vertical reflection patterning: if you create an accordion fold so the paper is folded in the shape of an M or W, make cuts (making sure you don’t completely cut off an entire folded edge) then unfold, the result will be the vertical reflection frieze group pattern. This is also how to make paper dolls, like at the top of this page.


Vertical Reflection using Pattern Blocks
Vertical Reflection using Pattern Blocks

The pattern blocks I have don’t work particularly well for making vertical reflection friezes, but I like trying. This is the best I could come up with.

 Vertical Reflection Frieze Patterns
Vertical Reflection Frieze Patterns

Here’s a grouping of vertical symmetries. The tilted square with the chartreuse triangle was tricky to position accurately: having them mounted on squares helped me get it right.

Now here’s the video, shortest one yet.


Frieze Groups

Frieze Symmetry: F2; p11g; ∞x ; Glide Reflection

Glide Reflected Squares
Glide Reflection

If you’ve tried to understand glide reflection and failed it’s not your fault.

If you haven’t tried to understand glide reflection, my goal is to have you be able to figure it out on your first try. It’s a real sweet symmetry operation.

I am going to write this post, but I think actually watching how glide reflection patterns are built will make things clearer faster so be sure to watch the video linked at the end of this post.

Glide reflection symmetries are often listed as the second frieze group.  The  Conway nickname for glide reflection is STEP.

I think the most confusing thing about glide reflection is that the reflection can happen over different horizontal lines. It’s not that it’s so hard to understand, it’s just that it’s an unusual action. But I am getting ahead of myself. First, back to basics.

There are only two actions within glide reflection: gliding and reflecting. How hard can that be?

The rub is that there are variations within the two simple actions. Without understanding the variations, glide reflection is unfathomable.

Here are the questions to consider:

  1. What’s the direction and length of the glide?
  2. Where’s the line the reflection reflects across?

Here’s what’s to know:

  1. The glide moves horizontally in a horizontal frieze. The glide of the unit goes as far or as little as you please, just keep it consistent with each piece of the pattern.
  2. Reflect over a line that is parallel to the direction of the glide.

This second part, reflecting over a line that is parallel to the direction of the glide, can be confusing.

Here’s the case where it’s least confusing:

Simple Glide
Simple Glide

The unit is copied then moved over. This is the first step in glide reflection.

Since the movement of the glide is horizontal the copy is reflected horizontally. In the image below, this reflection line is also the line the original shape is sitting on.

Glide Reflection
Glide Reflection

The original unit and the glide-reflected unit are now copied and repeated, and that’s the pattern.

The photo below is also glide reflection, but this time the glide wasn’t as far as the pattern above.

Glide reflection with smaller glide
Glide reflection with smaller glide

The two photos right above show different glides, but they both are reflecting over the same horizontal line.

This next photo shows the shapes reflecting over a horizontal line that is in a different place, indicated with red.

Imagine the red marks connected by a straight line: this is the line the shapes are reflected across.

If this is still confusing the best thing to do is to build some glide reflections.

I recommend getting your hands on four Post-it notes, or any easy to cut paper. Then get some yarn and a circle punch, or some needle and thread. The yarn or thread will be the horizontal reflection line.

First step of Glide Reflection looks like a simple glide, a translation.
First step of Glide Reflection looks like a simple glide (aka translation)

Connect the shapes in a straight line, weaving in and out of some holes you make, so that everything is nicely lined up.

Now flip over the second and fourth Post-it note.

There! the reflection is across a line that is parallel to the direction of the glide, but otherwise arbitrarily placed. You can do this.

As I trudge through this long-winded explanation I worry about being slow and boring. I need to remind myself that the reason I am writing these posts is that I have found most of the explanations of these symmetries to be just too swift.

Here’s one more photo, using pattern blocks:

Two Versions of Glide Reflection using Pattern Blocks
Two Versions of Glide Reflection using Pattern Blocks

I hope you can see that these are both glide reflections.

If not, be sure to watch the video link below.

I would be grateful to know what your thoughts are on my explanations here.

Thank you.