Slide the pieces to move Khun Phaen past all the guards into the position on the right from which he can escape to freedom.
A gallery of beautiful photos of cast ironwork friezes in Australia with a mathematical discussion of the classification of frieze patterns.
Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?
In the case of the two squares of side length 10cm, the dot
travels 80 cm and the position of the dot does not matter.
Trying the same problem with other shapes leads to the following
For any two shapes, the first in a fixed position and the second
having fixed orientation; if the second shape slides around the
first, maintaining contact then the distance travelled by any point
on the second shape is the sum of the perimeters of the two
We show this in three further examples.
Consider two rectangles as in the diagram above. For any two
points P and Q, the location of Q relative to P is fixed throughout
the motion and so the distance travelled by P is the same as the
distance travelled by any particular point that you choose, so we
shall choose the point O, the bottom left hand corner of the moving
rectangle . The point O moves a distance 2( a + b
+ c + d ).
A similar argument holds for the triangles and circles shown
below, choosing the marked point O in each case. The distance
travelled is the sum of the perimeters of the two shapes; you can
test this with other shapes!