THE ALBUM COVER Exploring Symmetry CONCAVITYS SECOND ALBUM

THE ALBUM COVER Exploring Symmetry

CONCAVITY’S SECOND ALBUM ‘Concavity’ is an alternative rock group, about to release their second album, a double CD and vinyl, where each disc has a name: the higher tempo disc is called ‘Revolution’, with the quieter disc named ‘Reflection’. Picture: gippa. co. uk

CONCAVITY’S SECOND ALBUM The design they use on all their album covers is the basic pattern here, but they want to indicate the names of their two discs within the design. So they ask their cover artist to come up with a design on the front cover that suggests ‘Revolution’.

REVOLUTION She offers an idea of making the design have something to do with rotational symmetry. The band agrees, and asks if the front cover could look the same upside down as it does the right way up.

REVOLUTION The singer sketches an idea, which the rest of the band say is boring, but they realise that they want the front cover to have rotational symmetry but not line symmetry.

REVOLUTION Can you make some better designs of your own for the front cover, that have rotational symmetry order 2, but no line symmetry?

REFLECTION The band challenges the artist to come up with a design for the back cover to show ‘Reflection’. It must use the central cross for its 2 lines of symmetry, but the design must not have rotational symmetry.

REFLECTION The singer has a go at this but finds his first attempt doesn’t work. It has two lines of symmetry, but it also has rotational symmetry. The guitarist says ‘Just keep on trying – it will work out somehow’, but the drummer says ‘I’m not so sure’. Can you make it work?

EXPLANATIONS Is it possible to draw a shape that has 2 lines of symmetry but no rotational symmetry?

TWO LINES OF SYMMETRY Let’s imagine our two lines of symmetry are like this.

5 4 TWO LINES OF SYMMETRY AND THE CO-ORDINATE GRID 3 2 1 -5 -4 -3 -2 -1 0 -1 -2 -3 -4 -5 1 2 3 4 5 But our two lines of symmetry could have a co -ordinate grid placed upon them.

5 4 TWO LINES OF SYMMETRY AND THE CO-ORDINATE GRID (2, 3) 3 2 Let’s choose any point on this grid, e. g. (2, 3) 1 -5 -4 -3 -2 -1 0 -1 -2 -3 -4 -5 1 2 3 4 5

5 4 TWO LINES OF SYMMETRY AND THE CO-ORDINATE GRID (2, 3) 3 2 …and reflect it in the x-axis 1 -5 -4 -3 -2 -1 0 -1 1 2 3 -2 -3 -4 -5 (2, -3) 4 5

5 4 TWO LINES OF SYMMETRY AND THE CO-ORDINATE GRID (2, 3) 3 2 …and then the y-axis 1 -5 -4 -3 -2 -1 0 -1 1 2 3 -2 (-2, -3) -3 -4 -5 (2, -3) 4 5

5 4 TWO LINES OF SYMMETRY AND THE CO-ORDINATE GRID (2, 3) 3 2 1 -5 -4 -3 -2 -1 0 -1 -2 (-2, -3) -3 -4 -5 1 2 3 4 5 But this has same effect as rotating the point (2, 3) through 180°, with (0, 0) as the centre of rotation. (2, 3) and (-2, -3) are equidistant from (0, 0), and on the same straight line (and therefore a turn of 180°).

5 4 TWO LINES OF SYMMETRY AND THE CO-ORDINATE GRID (x , y ) 3 2 But this would be true for any point … 1 -5 -4 -3 -2 -1 0 -1 1 2 3 -2 (-x, -y) -3 -4 -5 (x, -y) 4 5

5 4 (x , y ) 3 2 (a , b ) 1 -5 -4 -3 -2 -1 0 -1 (-a, -b) 1 2 3 4 5 (a , - b ) -2 (-x, -y) -3 -4 -5 (x, -y) TWO LINES OF SYMMETRY AND THE CO-ORDINATE GRID But this would be true for any point, or any set of points, such as a line, or a polygon.

5 4 (-x, y) (x , y ) 3 2 (-a, b) (a , b ) 1 -5 -4 -3 -2 -1 0 -1 (-a, -b) 1 2 3 4 5 (a , - b ) -2 (-x, -y) -3 -4 -5 (x, -y) TWO LINES OF SYMMETRY AND THE CO-ORDINATE GRID But this would be true for any point, or any set of points, such as a line, or a polygon.

CAN YOU NOW ANSWER THESE QUESTIONS? • If a shape has exactly two lines of symmetry, then it also has rotational symmetry of order 2. • Always / sometimes / never true? • If a shape has rotational symmetry of order 2, then it also has exactly two lines of symmetry. • Always / sometimes / never true?