Sorting By Symmetry: patterns with a centre
<< An ATM publicationFrom the title page:
- Looking at symmetrical patterns is intriguing and satisfying.
- Understanding the way symmetrical patterns are made is part of geometry.
- We will make a lot of patterns, each with a centre. Two different families will emerge.
- We will describe these families and show why, when looking at plane patterns with a centre, there are just these two and no more.
As anyone familiar with Bob Burn’s sessions or writing knows, the ‘we will make’ clause is perfectly accurate, because the book consists of a number of tasks, all with the same structure, which invite, nay, intrigue and impel the user (I cannot honestly say, simply, “reader”) to participate. Bob uses a repeating cycle of observing, making, and sharpening to structure those tasks. After observing similarities, you make something similar yourself, and then reflect upon and clarify or sharpen your ideas about what makes the similarity.
The book starts with reflections in a single mirror, and using ink-devils, folded paper, geoboard or dotty paper, and LOGO among others, provides challenges to draw or locate objects which meet certain symmetry constraints. It builds up through two mirrors, to discussing how reflection in a mirror can be seen as an action, and how, if you shift your attention from the objects being reflected to the transformations performed by reflection, you encounter the mathematical structure of a group. In the case of this booklet, you get dihedral symmetry groups. Subgroups arise by adding further elements or colour to shapes with symmetries, in order to reduce the number of symmetries. Finally, you encounter the fact that there is a relationship between the number of symmetries in a group, and the number of symmetries in each of its subgroups.
Despite the apparent complexity and sophistication of the pages as you flick through them, once you settle down and engage, you find yourself drawn along by Bob’s carefully constructed tasks, observing, making and sharpening.
There are some rather irritating difficulties in describing and defining symmetries, as I discovered when some of my draft materials were critiqued by Don Mansfield, who was one of the leaders in the Modern Mathematics movement in the UK in the sixties. One difficulty is that in order to specify what you mean by an object having symmetry, it is necessary to label it in someway. For example, if you think of an object as having a closely fitting box, then a symmetry is a way of picking the object out of the box and putting it back again. The symmetry is actually the action performed on the object, paying attention only to the initial and final positions (and not to the path taken b the object as you remove and replace it!). However, you need labels (on the object and on the box) in order to detect which way it has been ‘put back in its box’ in order to display a symmetry. However the presence of the label itself actually destroys the symmetry! Bob manages to avoid this issue quite neatly. The other issue is how to indicate the important sense of ‘all’, of generality, when defining something. For example on page 15 there appears a definition of symmetry:
A symmetry is a matching of the points of a figure or of the plane such that if A is matched with A’ and B is matched with B’, then the length of AB = the length of A’B’.
Of course any mathematically adept reader recognises that this has to hold for all pairs of points A and B on the figure or in the plane, not just a few isolated ones of particular interest. It is fascinating to me how some learners pick this up without explicit comment, but others remain mystified because they are unaware of the implied but essential generality.
It is a real pleasure to see a master craftsman at work, constructing mathematical tasks which are focused (on central symmetries) yet encompass a wide range of mathematical apparatus and contexts, and which engage the reader so completely in developing their natural powers to detect invariance in the midst of change, similarity and difference, and most especially, to experience what it means to abstract mathematical structure from their observations.
John Mason • Open University
Sorting By Symmetry: patterns with a centre
Burn, B. (2005)
ATM, Derby
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