One-stop Frieze Group Symmetry

Seven Days of Frieze Group Playalong

This is the one-stop post for making a linear pattern with a repeating tiles, otherwise known as Frieze Group Symmetry.

I recently wrote 10 posts about this kind of pattern making, going over nuances and details one at a time. This post is not that. It’s aim is to demonstrate Frieze Group symmetry in a way that will make the basics of this symmetries really accessible. I hope you will play along. Actually creating the symmetries is the easiest way to learn them. IT IS SO WORTH LEARNING THIS!  Playing with these symmetrie expand design possibilities beyond what you can imagine.

There are many different symmetry groups. Frieze group symmetry is a great place to start because they follow one simple rule: repeated tiles can only move linearly, which means they can slide, thus moving in a linear direction, or they can have 180 degree transformation. There are only 7 possible ways to make a linear symmetry. Considering we generally only think of symmetry as a mirror reflection, this may seem more than you expect.

To play along, you need four squares that

  • have a design on the back that is a reflection of what’s on the front, and
  • which the design on the square is not symmetrical.


The easiest way to do this is to just stack four square post-it notes and  cut an asymmetrical design on them while they are still stuck together.

If you want something a bit more snazzy and have access to a printer, print out either the color or black and white version of my symmetry tiles.

Symmetry tiles to be cut out and folded to make squares.


You’ll cut out four of these rectangles, which are domino proportioned, then fold them in half to make squares. Glue insides together, and you have your own set of paper symmetry tiles to use.

Here are the PDFs

Colorful awesome Symmetry Tile

Awesome Black and White Symmetry Tile

I will be adding to this post for seven days, so we’ll do just one symmetry a day.


If you take your tile keep repeating it exactly how it is, this is called translation and it looks like this:

There you have it! Your first symmetry group.

To be continued on this page tomorrow.

Day 2

Glide Reflection
Glide Reflection

All frieze symmetries have the translation action embedded in them. Here, with Glide Reflection, a copy of the first tile slides over (translation) then it reflects horizontally, as if there is a horizontal bar that it is flipping over. This shows you the back of the tile, which is why the tiles I’ve provided above in my pdfs, give you the image as if it goes right through the paper.

If you are using your own cut squares, be sure that the design you’ve cut makes it so there are no lines of horizontal or vertical symmetry!

If you are confused about this, or want to know more about this particular symmetry, take a look at my glide reflection post.

to be continued tomorrow….

Day 3

I think of this frieze group as the paper doll frieze group. Remember cutting out paper dolls from a piece of paper folded like an accordion, and the result would be linked paper dolls? If you don’t know what I’m talking about, do a google search for accordion paper dolls!

The Vertical Reflection frieze group slides a tile over, then reflects across the right side of the tile. Repeat and repeat again.

For more details, take a look at my Vertical Reflection post.

Day 4

Frieze group, Rotation
Frieze group, Rotation

Some of these symmetries seem easier to wrap my brain around than others. Seems to me that this fourth symmetry, rotation, is one of the easier ones. Remembering that all the symmetries start with translation, which means sliding a copy of the first tile over, this one then rotates 180 degree, going from right side up to upside down.

Day 5

This frieze group combines the first three groups into one. Not sure about glide reflection? Better get clear on it!

Vertical Refection, Glide Reflection
Vertical Refection, Glide Reflection

The action here is to slide over the first tile then reflect it across a vertical axis (mirror reflection).

Now here’s something that’s new to this frieze group: take a copy of the two tiles, slide them over, then reflected them across a horizontal axis. This is glide reflection.

This is the first (of two) of the frieze groups where you need a minimum of four tiles to achieve the symmetry of this specific one of the group.

Day 6

So close to the end of Frieze Group symmetry , here’s one that super easy to understand, Notice that there’s a new thing that happens here.

Horizontal Reflection
Horizontal Reflection

The action is to take a copy of the first tile and rotate it across a horizontal axis, the same kind of reflection that a tree does with the water at it’s base. Next, slide (translate) both these tiles to continue the pattern.

What’s new is that this is the first of these symmetries that require the design to become wider before it becomes longer. In a sense, it happens on double line.


Math with Art Supplies · pop-up

Inversion Two: Pop-Ups

I had wasn’t sure I’d write about seeing pop-ups through the lens of inversion, but then the title of this post got into my head, and I so much like that it kind of reads as “aversion to pop-ups”  that I had to to use it.

The generalized thinking here is  about bridging disciplines of design, bookmaking and math by using similar language to describe the various manifestations of the same exact concept. In this case I’m looking at the action of going back and forth, as in inverse operations.

This is the shortest post ever because I’m going to let this three minute video say it all.

The video isn’t really a tutorial about making pop-ups Instead, it’s about a way of thinking about the action of making a pop-up.

Even so, the final result is pretty nice.

Math with Art Supplies

Perks of Backwards Thinking, or designing with Inverses

Let’s say you just made this “No Strings Attached” pamphlet, designed by John C. Woods in the 1970’s and demonstrated here by Annie Perkins this past week. You might notice that you can’t readily tell the front from the back so you start thinking of what to do to decorate the front cover. It may occur to you that this binding method is created by making an inverse cutting pattern so that the pages can line up and be locked together. This might suggest making a design using inverses.

The understanding and using the concept of inverse relationships are all over design and math. This is something I’m going to be focusing on for a few days, starting with this post.

This cut paper method for decorations is perfect for classroom bookmaking because it uses little in the way of materials, can be done in a short period of time, and creates surprising results.

The size of the book  above is what you get when folding a piece of regular copy paper. That blue center piece started out as a two-inch wide strip, which, at first glance, doesn’t seem possible.

Here’s what’s happening.

I start at a lower corner, then cut a winding path to the upper right corner. Once the papers are cut the two strips can be realigned in a variety of ways.

Same cut, different arrangements

The four grouping of paper above are the exact same cut, just arranged differently. Considering the negative/positive, exchanging sides, or flipping horizontally, these are variations of thinking about inverting the original arrangement. To me, these different looks coming from the same exact cutting pattern, is exciting to me every time. Well, at least until I’ve had my fill of using just one color.

If I cut two different colors at the same time, other possibilities open up.

Two colors can give me these offset possibilities to explore. There’s so many ways to go with this.

The biggest problem with this way of working is that it’s hard to stop.

Here’s my demonstration video below. At the end I flip through a number of designs that aren’t in this post. Happy cutting!


All Week, designing papers for a project, not done

I would prefer to spend my time finishing a project than writing about how I can’t seem to finish it up. I’m giving up for the night. Tomorrow it will get done (“really,” I say to myself) but I just don’t have it in me to finish tonight.

I’ve been wanting to do a Spring design for the Hidden Boxes (based on Zhen Xian Bao) kits that I sell on Etsy. Wouldn’t it be nice to get this into the world in time for Mother’s Day, I tell myself.  (Seems like I’ve been talking to myself lately. #isolation) I actually started thinking about this design in mid-March. Found a geometry that I loved. Started with pencil drawings.

Worked with it by hand, drew three versions of it so to get a good feel for it. Didn’t feel ready to start in earnest to do the digital designing until about a week ago. Thought it would take 1 to 3 days. That was 5 or six days ago.

Absolutely love this project. Frustrated that I still have some little details to work out, so that I can’t yet put it in my shop.

Venting my frustration by putting up these photos here. They aren’t the kind I’d post with the kit, but this is my blog, where I can post what’s behind the curtain.

Here’s a beautiful photo that I can’t use because it’s not a square.  But, thank you, I can show it off here.

Here’s a screenshot of my digital workspace. I know it’s not just luck, but I feel so lucky that I can make these designs.

Part of my good luck is that Daria Wilbur wrote about this particular variation of the Zhen Xian Bao, which was such good luck for me, as it’s so much more doable for a kit than the full Zhen Xian Bao that I’ve written about. Another piece of good luck is that Samira Mian keeps publishing videos on how to create these geometries that I like so much.

Here’s a peek at the hidden boxes inside. Just took this photo. The photo is okay, but would be better in natural daylight, which is one reason why I will not finish up tonight.

Here it is wide out.  I need one more photo, one that’s in between this one and the one above. That’s on the docket for tomorrow, too.

That’s it. I’ve got this out of my system for tonight. Thank you

Addendum May 2

This kit is up and listed!