Proofs that there are only seven frieze patterns involve complicated group theory. The symmetries of a cylinder provide an easier approach.
Plex lets you specify a mapping between points and their images.
Then you can draw and see the transformed image.
A gallery of beautiful photos of cast ironwork friezes in Australia with a mathematical discussion of the classification of frieze patterns.
How many different transformations can you find made up from
combinations of R, S and their inverses? Can you be sure that you
have found them all?
This article describes the scope for practical exploration of tessellations both in and out of the classroom. It seems a golden opportunity to link art with maths, allowing the creative side of your. . . .
How many different symmetrical shapes can you make by shading triangles or squares?
Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?
This article describes a practical approach to enhance the teaching
and learning of coordinates.
Sort the frieze patterns into seven pairs according to the way in
which the motif is repeated.
Look carefully at the video of a tangle and explain what's
Here is a chance to create some attractive images by rotating
shapes through multiples of 90 degrees, or 30 degrees, or 72
This problem explores the shapes and symmetries in some national flags.
Can you recreate this Indian screen pattern? Can you make up
similar patterns of your own?
What is the same and what is different about these tiling patterns and how do they contribute to the floor as a whole?
What are the coordinates of this shape after it has been
transformed in the ways described? Compare these with the original
coordinates. What do you notice about the numbers?
What mathematical words can be used to describe this floor
covering? How many different shapes can you see inside this
What happens to these capital letters when they are rotated through
one half turn, or flipped sideways and from top to bottom?
This investigation explores using different shapes as the hands of
the clock. What things occur as the the hands move.
Does changing the order of transformations always/sometimes/never
produce the same transformation?
See the effects of some combined transformations on a shape. Can
you describe what the individual transformations do?
Can you describe what happens in this film?
How many different ways can you find of fitting five hexagons
together? How will you know you have found all the ways?
A red square and a blue square overlap so that the corner of the red square rests on the centre of the blue square. Show that, whatever the orientation of the red square, it covers a quarter of the. . . .
Use the clues about the symmetrical properties of these letters to
place them on the grid.
This article for teachers suggests ideas for activities built around 10 and 2010.
Overlaying pentominoes can produce some effective patterns. Why not
use LOGO to try out some of the ideas suggested here?
A brief video looking at how you can sometimes use symmetry to
distinguish knots. Can you use this idea to investigate the
differences between the granny knot and the reef knot?
Explore the effect of reflecting in two intersecting mirror lines.
A triangle ABC resting on a horizontal line is "rolled" along the
line. Describe the paths of each of the vertices and the
relationships between them and the original triangle.
The diagram shows a very heavy kitchen cabinet. It cannot be lifted but it can be pivoted around a corner. The task is to move it, without sliding, in a series of turns about the corners so that it. . . .
Two circles of equal radius touch at P. One circle is fixed whilst the other moves, rolling without slipping, all the way round. How many times does the moving coin revolve before returning to P?
A security camera, taking pictures each half a second, films a
cyclist going by. In the film, the cyclist appears to go forward
while the wheels appear to go backwards. Why?
Why not challenge a friend to play this transformation game?
A shape and space game for 2,3 or 4 players. Be the last person to be able to place a pentomino piece on the playing board. Play with card, or on the computer.
Try this interactive strategy game for 2
A train leaves on time. After it has gone 8 miles (at 33mph) the driver looks at his watch and sees that the hour hand is exactly over the minute hand. When did the train leave the station?
What is the relationship between these first two shapes? Which
shape relates to the third one in the same way? Can you explain
Find out how we can describe the "symmetries" of this triangle and
investigate some combinations of rotating and flipping it.
The triangle ABC is equilateral. The arc AB has centre C, the arc
BC has centre A and the arc CA has centre B. Explain how and why
this shape can roll along between two parallel tracks.
Can you work out what kind of rotation produced this pattern of
pegs in our pegboard?
Can you picture where this letter "F" will be on the grid if you
flip it in these different ways?