See how 4 dimensional quaternions involve vectors in 3-space and
how the quaternion function F(v) = nvn gives a simple algebraic
method of working with reflections in planes in 3-space.
Follow hints using a little coordinate geometry, plane geometry and
trig to see how matrices are used to work on transformations of the
Follow hints to investigate the matrix which gives a reflection of the plane in the line y=tanx. Show that the combination of two reflections in intersecting lines is a rotation.
In a snooker game the brown ball was on the lip of the pocket but
it could not be hit directly as the black ball was in the way. How
could it be potted by playing the white ball off a cushion?
What angle is needed for a ball to do a circuit of the billiard
table and then pass through its original position?
This resources contains a series of interactivities designed to
support work on transformations at Key Stage 4.
A fire-fighter needs to fill a bucket of water from the river and
take it to a fire. What is the best point on the river bank for the
fire-fighter to fill the bucket ?.
The centre of the larger circle is at the midpoint of one side of an equilateral triangle and the circle touches the other two sides of the triangle. A smaller circle touches the larger circle and. . . .
Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?
A design is repeated endlessly along a line - rather like a stream
of paper coming off a roll. Make a strip that matches itself after
rotation, or after reflection
An environment for exploring the properties of small groups.
What groups of transformations map a regular pentagon to itself?
A moveable screen slides along a mirrored corridor towards a
centrally placed light source. A ray of light from that source is
directed towards a wall of the corridor, which it strikes at 45
degrees. . . .
Make a footprint pattern using only reflections.
Proofs that there are only seven frieze patterns involve complicated group theory. The symmetries of a cylinder provide an easier approach.
Explore the meaning behind the algebra and geometry of matrices
with these 10 individual problems.
When a strip has vertical symmetry there always seems to be a
second place where a mirror line could go. Perhaps you can find a
design that has only one mirror line across it. Or, if you thought
that. . . .
Given that ABCD is a square, M is the mid point of AD and CP is
perpendicular to MB with P on MB, prove DP = DC.
Plex lets you specify a mapping between points and their images.
Then you can draw and see the transformed image.
Choose some complex numbers and mark them by points on a graph.
Multiply your numbers by i once, twice, three times, four times,
..., n times? What happens?
Consider a watch face which has identical hands and identical marks
for the hours. It is opposite to a mirror. When is the time as read
direct and in the mirror exactly the same between 6 and 7?
Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?
Investigate the transfomations of the plane given by the 2 by 2
matrices with entries taking all combinations of values 0. -1 and
A gallery of beautiful photos of cast ironwork friezes in Australia with a mathematical discussion of the classification of frieze patterns.
Two tangents are drawn to the other circle from the centres of a
pair of circles. What can you say about the chords cut off by these
tangents. Be patient - this problem may be slow to load.
I noticed this about streamers that have rotation symmetry : if
there was one centre of rotation there always seems to be a second
centre that also worked. Can you find a design that has only. . . .
Four circles all touch each other and a circumscribing circle. Find
the ratios of the radii and prove that joining 3 centres gives a
The shortest path between any two points on a snooker table is the straight line between them but what if the ball must bounce off one wall, or 2 walls, or 3 walls?