Why do this problem?
The complexity in the
seemingly simple patterns of Celtic knots have always fascinated
people. This problem provides an accessible introduction to
creating such patterns, with the chance to investigate some of the
maths that arises along the way.
It could provide an
excellent opportunity for forging cross-curricular links with Art
and Design departments.
Learners could initially
be shown images of Celtic knots to capture their interest. Then
they could watch the videos to see two different ways of creating a
Once they are ready to
create their own knots, you may find it useful to print off some of
these grids for learners to draw onto (Word
Alternatively, it is
fairly simple for learners to create their own grids by drawing
faint diagonal lines on squared or square dotty paper. The straight
lines needed to create the knots all go through the midpoints of
the sides of the squares:
Once learners have got the hang of creating Celtic knots, encourage
them to think of mathematical questions to ask - there are some
suggestions in the problem. By getting everyone in the class to
draw different knots, data can quickly be gathered and analysed. If
learners have created their own grids, they may find that knots
cannot be drawn on some grids - this is another line of enquiry
that can be explored.
automatically generates Celtic knots and might be useful for
For rectangular knots,
can you explain how the knot size determines the number of
Is it possible to draw a
rectangular Celtic knot without rotational symmetry?
Can you explain how the
knot size determines the number of overlaps?
Introduce learners to more complicated Celtic knots - circular
knots, knots with holes in, letters drawn as knots...
Starting on smaller grids and encouraging learners to shade in
where the ribbons will go can help. Suggest they use soft pencil,
and be prepared to do lots of rubbing out!