Upon reflection

Quite often, my work on curriculum in mathematics isn’t about breaking down barriers or discovering new content to be taught. Instead, it is about reassessing what we currently have in the curriculum, revisiting content with a new eye for detail and depth, and identifying implicit properties that are worth making explicit for future learning. Recently my work on symmetry has involved just this. 

Symmetry is not just aesthetically pleasing (to many, but interestingly not to all) it’s also incredibly useful. Whether you’re a graphic designer, an architect, engineer, sculptor, choreographer, or simply decorating your home you’ll most likely be considering the symmetrical nature of your design. 

Our early encounters with reflectional symmetry tend to be those pictures that hang on a multitude of fridges across the country – the ones where one side of a fold is painted and whilst still wet the paper is folded over and opened again producing a symmetric design (often a butterfly). There is a great opportunity here to maximise the mathematics, even informally. If a picture has this fold line what matches up with what? If it’s a mirror line for a geometrical shape what implications does this have for corresponding edges or angles of the shape? 

As this area of maths is revisited with more and more sophistication we can start to ask some really interesting questions – the answers may seem obvious at first, but with more careful thought these can really probe visualisation skills and understanding of geometry. The answers also need careful justification and explanation. 

Let’s start by considering convex polygons (I’ll leave you to think about how the questions and answers would change if you allowed concave polygons). Starting simply with a mirror line of a polygon: this line can either pass through two vertices; two edges; or one edge and one vertex of the shape. A mirror line passing through an edge forms the perpendicular bisector of that edge, a mirror line passing through a vertex forms the angle bisector and indicates that the two edges forming the vertex are congruent. This might seem obvious – but convince yourself that it has to be true by considering the physical nature of folding to carry out reflections. 

One mirror line 

Now consider the nature of a polygon with exactly one line of symmetry. What else can you derive about it? What is its order of rotational symmetry? If I know it is a triangle, what kind of triangle is it? If I know it is a quadrilateral, what kind of quadrilateral is it? Are these triangles and quadrilaterals linked? 

The series of questions above may lead you to believe that shapes with one line of symmetry have order of rotation 1. Is the converse true? Do all shapes with rotation of order one have a (single) line of symmetry? 

Side-to-side line of symmetry 

Consider a polygon that has a line of symmetry going through two sides. What are the implications of this? What can you say about this pair of sides? What can you say about the number of sides? 

The sides that the mirror line passes through have to be parallel. Why? Additionally the shape has to have an even number of sides. Why? Convince me! 

Finally, sides occur in pairs of equal length, apart from at most one pair. Which pair? Why? 

Vertex-to-vertex line of symmetry 

What about a polygon that has a line of symmetry going through two vertices? What are the implications of this? What can you say about the number of sides? 

Vertex-to-side line of symmetry 

What about a polygon that has a line of symmetry going through one side and one vertex? What are the implications of this? What can you say about the number of sides? What about its potential order of rotation? 

Shapes with vertex-to-side line symmetry have to have an odd number of sides and an odd order of rotation. Can you explain why? 

These thoughts lead me immediately to the question: is it possible for a polygon to have a mixture of lines of symmetry of all three types above? Which mixtures are possible? Why? 

So far, we have considered a single mirror line and the implications of the nature of a mirror line. What about implications for the number of mirror lines? If a shape has exactly two mirror lines they will always be at 90° to each other. Why? The shape will also have rotation of order two. Why? Convince me! This overlaps beautifully with the work we do on transformations: reflect a shape in the x-axis and then the y-axis, what single transformation has been performed? 

I could go further and dig into other consequences of mirror lines and order of rotation. I could look at potential symmetrical definitions, such as a square is a quadrilateral with four lines of symmetry, or even new classes of shapes such as quadrilaterals with one line of symmetry. 

Immersing ourselves so thoroughly in a concept offers insights and revelations that simply would not be found by just carrying out typical exercises. Which questions would you ask?