Investigation: Celtic knots
This investigation will help you explore number factors in an artistic way by creating Celtic knots.
Materials
- Paper (grid paper is best)
- Pencil and eraser
- Coloured pencils or pens
Instructions
- Pick two numbers - start with small ones, like 3 and 5.
- Draw a rectangle and mark unit squares - or just use grid paper!
- Make marks halfway along each unit square, except for the squares at the corners, like this:
An example of a 3 by 5 rectangle with edge markings
- Pick one of the marks and draw a diagonal line from one side until you hit another side. Then bounce off that side and keep going, like this:
Starting along the bottom edge at the marking on the left, we draw up and to the right until we hit the top edge, and continue bouncing around the inside of the rectangle.
- Keep going until you reach the mark where you started again, like this:
Finished! We hit every marking along the way.
- Make each line larger, making the lines curved at the edges, like this:
The lines are now larger, and curved at the edges.
You can use these shapes as templates for the corner, edge, and middle squares
- Weave the larger lines under and over each other and erase the working lines to make a pattern like this:
Travelling along the line, we alternate between going over and going under.
This is your Celtic knot!
Celtic knots have been drawn, engraved, and carved for centuries, first appearing in 300-400 CE in the days of the Roman empire. The Celts lived throughout Western Europe and developed the style into artwork that is still found all throughout the region and produced to this day. It is likely inspired by rope plaiting and basket weaving.
Here is a slightly different example, using the numbers 3 and 6. Here's the grid at the start:
Now here is what happens when we start our line and bounce around until we reach where we started:
Notice that we haven't reached all the marks yet,even though we looped back to the beginning. If we pick another one and start again, and keep going until we hit all the marks, we need three different lines:
Questions
- Why do the numbers 3 and 5 need only one line, but 3 and 6 need three?
- Can you find a rectangle that needs two lines? What about four, five, or even more?
- What happens when you start with a square?
- Can you weave your finished lines together in a different way?
- How would you change the overall process so that it works on a triangle, pentagon, or hexagon?