Ioana Bouros

A couple of years ago I offered to be a guide on a mathematical tour around Oxford for any enthusiasts that were curious about the mathematical Easter eggs hidden in the nooks and crannies of the ancient city. And almost without exception, most people found the highlight of the tour to be the Ashmolean Museum. Built in the Greek Revival architectural style, the building is without a doubt imposing, but it is not just the sheer size of it that makes it remarkable, the beauty of the construction lies in the prim and yet well-endowed aspect of its facade (Image: Architect’s Journal UK).

The human eye has always been attracted to perfection in aesthetics: whether it be in the perfect shape of a circle or equilibrium of measures in a painting. There is, however, a different sort of perfection that will often catch one’s attention: symmetry. In fact, multiple studies have concluded that the majority of people find a face to be more attractive, the more symmetrical it appears. And though there are multiple reasons why we psychologically prefer that (e.g. asymmetry in facial features may be a sign of respiratory conditions), it still remains an interesting titbit to observe.

But how do you define symmetry? And more importantly, WHAT do you define as symmetry? For something to be considered symmetric, it requires a complete repetition of a sub-piece (a pattern), or a reiteration of it. However, these subsequent repeated patterns need not be displayed visually in the same manner as the “mother” piece, but rather can be subjected to operations that preserve its form. There are thus three such transformations: translation, reflection and rotation. 

When it comes to running motifs, i.e. patterns with translational symmetry in only one direction, there are exactly seven ways in which we can draw such a pattern. These bear the name of frieze patterns (because that is where they are most often observed) and can be also used for the tessellation of the plane (more on this in my earlier article on tessellation).

Going back to the picture of the Ashmolean above, if one knows where to look, you can in fact find all seven of these running motifs. And perhaps this is exactly why we subconsciously think it is interesting to look at… 

1.     Hop

• Translation — the pattern is unchanged if you slide it along

2.     Step

• Translation
• Glide reflection — the pattern is unchanged if you slide it along and reflect it in a horizontal line

3.     Sidle

• Translation
• Vertical reflection – the pattern is unchanged if you reflect it in a vertical line

4.     Spinning Hop

• Translation
• Rotation – the pattern is unchanged if you spin it by a half turn

1 row:

2 rows:

5.     Spinning Sidle

• Translation
• Vertical reflection
• Rotation

1 row:

2 rows:

6.     Jump

• Translation
• Horizontal reflection

7.     Spinning Jump

• Translation
• Horizontal and vertical reflection
• Rotation

Question

Now that we’ve seen all 7 examples of Frieze patterns, can you identify the type of symmetries present in the following example?

Scroll down for the answer!

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Answer

Ignoring the patterns inside the squares, this is a spinning hop with 2 rows (as shown above in example 4).