Groups Week 1 22/23

What is Group Theory About?

First, watch this YouTube video, from Atractor, a Portuguese organisation that specializes in media for comunicating mathematical ideas. It does not completely answer the question of what group theory is about, but it gives a flavour.

This video is suggested as an introduction to the topic of group theory. Please do not worry if you do not understand the details after watching it – it is intended only to give you a sense of (some of) the themes of Group Theory. The video is about the frieze groups. In architecture and design, a frieze is a long line with a repeatng pattern, often appearing horizontally along a wall. You can think of it as a row of rectangular tiles, either all identical or appearing in a repeating sequence (in which case we could think of the basic repeating unit as a single “tile”). The video shows some friezes that appear on the outsides of buildings in the Potuguese city of Ovar, which is one of the best places to see the ornate azulejos tiles, which are traditional in the region. The connection to group theory relates to the symmetries that can occur in a frieze. For the purpose of this mental exercise, we have to imagine that the frieze is infinite in extent (along its length), so it has infinitely many tiles all occuring in one row. A symmetry of the frieze is a way of repositioning the whole row of tiles so that it looks exactly the same as it did before. There are essentially four types of symmetries that can occur.

  • Translations A translational symmetry just slides the entire line along its length, a distance of one or more tile widths. All friezes have translational symmetries.
  • Horizontal reflections If each tile has the property that it looks the same as its image in a mirror position parallel to the frieze, then the frieze has a horizontal reflection. Not all friezes have them.
  • Vertical refections If the vertical line through the centre of the repeating tile is an axis of symmetry of the tile, then the frieze has a vertical reflection.
  • Rotations If rotating through 180° (either about the centre of the repeating tile, or about the centre of the line between adjacent tiles) is a symmetry of the tile, then the frieze has rotational symmetry.
  • Glide reflections There is one remaining possibility, even for friezes with none of the above symmetries. It may be that the horizontal reflection is not a symmetry, but that it can be composed with a translation to form a symmetry. Such a symmetry is called a glide reflection. For a “true” glide reflection, neither the reflection component nor the translation component is a symmetry of the entire frieze, but the length of the translation component is half of the length of a translational symmmetry of the frieze itself (look at the last example in the video if this does not make sense!).

For different friezes, horizontal and vertical reflections, and glide reflections can each occur by themselves in combination with the translations, and they can also occur in certain combinations (but not all combinations). It turns out that there are exactly seven combinations that can occur. Examples of all seven are shown in this video. These seven types are called the frieze groups, and they classify the fundamentally different symmetry structures that friezes can have. A nice feature of the connection to art and design is that friezes that look quite unlike each other can have the same symmetry structure, the same frieze group.

Activity for Week 1 consists of the following steps:

  1. Watch the video above and read the paragraph above about frieze groups. Then watch the video again and see if it makes sense. Don’t worry at all if you can’t explain it all in mathematical terms. We are going to learn the language of group theory over the next few weeks.
  2. Relevant sections of the lecture notes for Week 1 are Section 1.1 and Section 1.2. Lecture slides: Lecture 1, Lecture 2. The notes have much more detail than the slides, and different examples, so please don’t ignore them!
  3. Come to the lectures on Thursday and Friday, September 8 and 9.
    If you cannot make it to the lectures, have a look at these video lectures from 2020/21 (please ignore details about arrangements particular to that year, for example about assessment).