Our first knot puzzle was based on Celtic knot designs that begin with a square lattice, with diagonals representing the rope of the knot. The Celtic artist selectively replaced intersections with turns to create variations and symmetries, and refined the intersections to show the rope weaving over and under.
We wondered what it would look like if we used a grid of triangles. We settled on the idea of using triangular tiles that have two ropes passing through each edge. Unlike the Celtic knot, this means you can connect any two tiles, and the ropes will always meet each other.
There are seven patterns on the triangular tiles. Each tile can be characterized by where the ropes enter and exit. Can you figure out why we have seven patterns?
With your tiles, you can explore some of the ideas of knot theory, which studies closed loops. You can close loops using any of the edge pieces.
The simplest closed loop is called the unknot. You can make one with two edge tiles:
Below are two unknots. One of them has a twist. It is still an unknot because you can untwist it to have zero crossings.
If a knot has fewer than three crossings, it is equivalent to an unknot. The simplest knot that is not an unknot is the trefoil.
You can use your knots to explore symmetries. When the trefoil is rotated 120 degrees, it looks the same. After three such rotations, you are back to where you started. So we say the knot has 3-fold symmetry.
M.C. Escher creatively used color to reduce symmetry. He called this anti-symmetry. Looking only at the patterns on the tiles, ignoring color, this knot has 6-fold symmetry.
If you consider color, what is the rotational symmetry?
Things to think about...
• Can you explain why the knots you create always alternate going over and under?
• How many kinds of rotational symmetry are possible?
• Why are seven tiles needed to characterize the ways the ropes can enter and exit?
• If we had used square tiles with two ropes on each edge, how many different tiles would we need to show all possible connections between edges?
• Which knots in the Knot Atlas can be made with the triknot tiles?