What groups of transformations map a regular pentagon to itself?
Proofs that there are only seven frieze patterns involve complicated group theory. The symmetries of a cylinder provide an easier approach.
A gallery of beautiful photos of cast ironwork friezes in Australia with a mathematical discussion of the classification of frieze patterns.
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. . . .
This resources contains a series of interactivities designed to
support work on transformations at Key Stage 4.
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. . . .
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
Points off a rolling wheel make traces. What makes those traces
Find the shape and symmetries of the two pieces of this cut cube.
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?
Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?
Plex lets you specify a mapping between points and their images.
Then you can draw and see the transformed image.
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?
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.
An article for students and teachers on symmetry and square dancing. What do the symmetries of the square have to do with a dos-e-dos or a swing? Find out more?
An environment for exploring the properties of small groups.
Can you devise a fair scoring system when dice land edge-up or corner-up?
A and B are two points on a circle centre O. Tangents at A and B
cut at C. CO cuts the circle at D. What is the relationship between
areas of ADBO, ABO and ACBO?
The ten arcs forming the edges of the "holly leaf" are all arcs of
circles of radius 1 cm. Find the length of the perimeter of the
holly leaf and the area of its surface.
Using the 8 dominoes make a square where each of the columns and rows adds up to 8
Join some regular octahedra, face touching face and one vertex of
each meeting at a point. How many octahedra can you fit around this
Can all but one square of an 8 by 8 Chessboard be covered by
Sketch the graphs for this implicitly defined family of functions.
Ten squares form regular rings either with adjacent or opposite
vertices touching. Calculate the inner and outer radii of the rings
that surround the squares.
The twelve edge totals of a standard six-sided die are distributed symmetrically. Will the same symmetry emerge with a dodecahedral die?
Create a symmetrical fabric design based on a flower motif - and realise it in Logo.
Scheduling games is a little more challenging than one might desire. Here are some tournament formats that sport schedulers use.
An equilateral triangle is sitting on top of a square.
What is the radius of the circle that circumscribes this shape?
Sketch the members of the family of graphs given by y =
a^3/(x^2+a^2) for a=1, 2 and 3.
Investigate the family of graphs given by the equation x^3+y^3=3axy
for different values of the constant a.
Toni Beardon has chosen this article introducing a rich area for
practical exploration and discovery in 3D geometry
Investigations and activities for you to enjoy on pattern in
Place the numbers 1, 2, 3,..., 9 one on each square of a 3 by 3 grid so that all the rows and columns add up to a prime number. How many different solutions can you find?
Sketch the graph of $xy(x^2 - y^2) = x^2 + y^2$ consisting of four curves and a single point at the origin. Convert to polar form. Describe the symmetries of the graph.
An irregular tetrahedron has two opposite sides the same length a
and the line joining their midpoints is perpendicular to these two
edges and is of length b. What is the volume of the tetrahedron?
Can you show that you can share a square pizza equally between two
people by cutting it four times using vertical, horizontal and
diagonal cuts through any point inside the square?
Plot the graph of x^y = y^x in the first quadrant and explain its