Frieze Groups

Frieze Symmetry, intro #3: Square Bases & Gyration Points

I am going to be writing about what’s missing from everything I’ve looked at about learning the seven frieze groups.

It  has to do with two tools: square bases and gyration points.

To get started, I emphatically want to say that the easiest way to understand the frieze groups is to make them yourself.

A good way to construct frieze patterns is to move around simple paper shapes.

A note about simple shapes:  For the purpose of learning frieze patterns, the unit shapes should not have lines of symmetry which are parallel to or perpendicular to the horizon.  I’ve created a file of shapes that can be altered so that their horizontal and vertical lines of symmetries are disrupted. This can be done by slightly tilting the shapes, cutting out bits to make regular shapes less symmetrical, or strategically adding color or other shapes to the shapes.  A standard circle paper punches is also a good tool to use for the purpose of disrupting the symmetry of a regular shape. For examples, see the image below

It’s not that using symmetrical shapes for frieze pattern is a bad thing. Just the opposite is true. It’s just now, when first learning about the different symmetry groups that it’s preferable to avoid the confusion that can come with using shapes which have their own embedded horizontal and vertical symmetries.

The way to handle the unit shapes is to mount each of them on a square base, preferably one that is transparent. Skeptical? You will be convinced. I am certain of it. Eventually you will be able see the squares in your mind’s eye, but to start out, the squares are like training wheels.

For the rest of this post my shapes will  be attached to a square base. Think about this:

  • it’s the underlying base that moves horizontally to create the frieze.
  • It’s the shapes that are attached to the base that create the design.

My squares are made from a mostly transparent drafting film.

Here are some units glued to the square bases:

The only rule with the square base is that one side always remains parallel to horizon line of the frieze. This infers that the two of the sides of the square are always perpendicular to the horizon line.

Overlaps, extreme spacing, vertical reflections, horizontal reflections, and 180 degree rotations. which are the defining characteristics of frieze patterns, will work out just fine with the square base, as all of these changes do keep a side of the square base parallel to the horizon line.

Repeated Tilted Square
Repeated Tilted Square

If a design is made with an asymmetrically tilted square, the square base square doesn’t tilt, rather just the repeating unit is tilted.

If the shape overhangs the base, that’s not a problem. Let it overhang in any direction.

If the shape is so big that it obliterates the square, use a bigger square.

Now it’s time to talk about gyration points.

As I sifted through lots of examples of friezes that didn’t seem to correspond to each other, I finally realized that recognizing the role of gyration points in frieze patterns is essential. Gyration points are like the secret that nobody talks about

If you don’t know what a gyration point is, well, it’s a term that I also just learned.  The concept of what it is, however, has been one of my favorite visual playthings for many years. I just didn’t know what it was called.

Simply put, it’s a point around which an object is rotated which is not  located in the center of the object.

For instance, the sun is a gyration point of the earth, since the earth rotates around it.

A gyration point is a hard concept to immediately internalize and accept as important. Don’t worry if it feels fuzzy right now. If you follow along with these posts it will become clear.

Here’s some visuals that are intended to get things started.

In the image above, the same shape is rotated, but each pair is rotated around a different center, each creating a look that is different than the rest.

Now here are these same pairs, each creating a frieze pattern:

Friezes symmetry with gyration point
Friezes symmetries with gyration points

Varying the gyration point creates completely different visuals. Be sure to squint when looking at the bottom frieze and you will see a zig-zag pop out.

Here’s a video, repeating everything I’ve just written. I think that seeing the transformation happen on video makes a more convincing case for using a square base, and better illustrates gyration points.


The first two posts in this introduction are :

With any luck, I’ll be posting about the seven frieze groups before too long. It’s a linear process. 🙂

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