This article looks at the importance in mathematics of representing places and spaces mathematics. Many famous mathematicians have spent time working on problems that involve moving and mapping. . . .
Explore the effect of reflecting in two parallel mirror lines.
Have you ever noticed how mathematical ideas are often used in patterns that we see all around us? This article describes the life of Escher who was a passionate believer that maths and art can be. . . .
Jenny Murray describes the mathematical processes behind making patchwork in this article for students.
Experimenting with variables and friezes.
Sort the frieze patterns into seven pairs according to the way in
which the motif is repeated.
Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?
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?
A cylindrical helix is just a spiral on a cylinder, like an ordinary spring or the thread on a bolt. If I turn a left-handed helix over (top to bottom) does it become a right handed helix?
What happens to these capital letters when they are rotated through
one half turn, or flipped sideways and from top to bottom?
Does changing the order of transformations always/sometimes/never
produce the same transformation?
Explore the effect of reflecting in two intersecting mirror lines.
This problem is based on the idea of building patterns using
See the effects of some combined transformations on a shape. Can
you describe what the individual transformations do?
Proofs that there are only seven frieze patterns involve complicated group theory. The symmetries of a cylinder provide an easier approach.
Explore the effect of combining enlargements.
Why not challenge a friend to play this transformation game?
Triangles are formed by joining the vertices of a skeletal cube. How many different types of triangle are there? How many triangles altogether?
How many differently shaped rectangles can you build using these
equilateral and isosceles triangles? Can you make a square?
Find a way to cut a 4 by 4 square into only two pieces, then rejoin the two pieces to make an L shape 6 units high.
A gallery of beautiful photos of cast ironwork friezes in Australia with a mathematical discussion of the classification of frieze patterns.
The whole set of tiles is used to make a square. This has a green and blue border. There are no green or blue tiles anywhere in the square except on this border. How many tiles are there in the set?
Do you know how to find the area of a triangle? You can count the
squares. What happens if we turn the triangle on end? Press the
button and see. Try counting the number of units in the triangle
now. . . .
Make an eight by eight square, the layout is the same as a
chessboard. You can print out and use the square below. What is the
area of the square? Divide the square in the way shown by the red
dashed. . . .
Blue Flibbins are so jealous of their red partners that they will
not leave them on their own with any other bue Flibbin. What is the
quickest way of getting the five pairs of Flibbins safely to. . . .
Find out how we can describe the "symmetries" of this triangle and
investigate some combinations of rotating and flipping it.
Here is a solitaire type environment for you to experiment with. Which targets can you reach?
What would be the smallest number of moves needed to move a Knight
from a chess set from one corner to the opposite corner of a 99 by
99 square board?
With one cut a piece of card 16 cm by 9 cm can be made into two pieces which can be rearranged to form a square 12 cm by 12 cm. Explain how this can be done.
Draw all the possible distinct triangles on a 4 x 4 dotty grid.
Convince me that you have all possible triangles.
An introduction to groups using transformations, following on from the October 2006 Stage 3 problems.
Can you fit the tangram pieces into the outline of this junk?
Can you fit the tangram pieces into the outline of this goat and giraffe?
Can you fit the tangram pieces into the outlines of the chairs?
NRICH December 2006 advent calendar - a new tangram for each day in
the run-up to Christmas.
Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9,
12, 15... other squares? 8, 11, 14... other squares?
Can you fit the tangram pieces into the outline of the rocket?
Show how this pentagonal tile can be used to tile the plane and
describe the transformations which map this pentagon to its images
in the tiling.
These grids are filled according to some rules - can you complete
Investigate how the four L-shapes fit together to make an enlarged L-shape. You could explore this idea with other shapes too.
Can you fit the tangram pieces into the outline of these rabbits?
Can you fit the tangram pieces into the outline of Granma T?
Can you fit the tangram pieces into the outlines of these clocks?
Can you fit the tangram pieces into the outlines of these people?
Can you fit the tangram pieces into the outline of the child walking home from school?
Can you fit the tangram pieces into the outlines of the lobster, yacht and cyclist?
Can you fit the tangram pieces into the outline of this shape. How would you describe it?
Can you fit the tangram pieces into the outline of this brazier for roasting chestnuts?
Can you fit the tangram pieces into the outline of Little Fung at the table?