Tom Longtin’s *Trefoil Knot* occupies an interesting space in the puzzle world. Part puzzle, part sculpture, part mathematical curiosity, it is a DIY kit composed of 24 L-shaped pieces that fit together to create a three-dimensional trefoil knot.

The kit contains instructions to piece it together and there are additional puzzle challenges that involve colouring the pieces and rearranging them outlined on Tom’s website, Fulcrum Design.

Tom has channeled his mathematical interests through creating DIY sculpture kits which he hopes will “engage hand-eye coordination, manual dexterity, spatial comprehension and patience.” He added, “I’m amazed at how many people, handed a finished knot, insist it’s two pieces interlinked somehow and wonder what keeps them from touching.”

The trefoil knot is a shape that Tom has returned to over the years in a number of ways covering both virtual and physical representations from book covers (*The Möbius Strip* by Clifford Pickover) to coffee tables and in a range of materials including stained glass and white pine. A number of these can be viewed here.

In August 2014 I spoke with Tom via email about this unusual kit.

**Saul Symonds**: You mentioned that you have designed a number of variations of the Trefoil Knot over the years. What keeps bringing you back to this particular knot?

**Tom Longtin**: It’s the simplest and most familiar of knots – a closed-path based on the first step in tying shoelaces – left-handed or right-handed.

**SS**: Any reason for choosing a left-handed trefoil knot over a right-handed one?

**TL**: Good question – I find no conscious decision in choosing left vs right. The very first program I wrote for this from a text-book formula generates a right-handed trefoil knot path. Most of the smooth knots I’ve made are fitted to this path.

The right-angle orthogonal knots were modeled from stacking cubes and happened to be left-handed. Any of the kit knots can be assembled “inside-out” and will yield a right-handed knot.

**SS**: At what point did you first decide to turn your mathematical sculptures into DIY kits?

**TL**: After printing 2D patterns on paper and using them as templates to cut out of foamboard, I felt the need of a process which would be less labor-intensive and more conducive to experimenting with different designs using more robust materials.

Enter the laser cutter – which certainly fulfilled its purpose and I could easily replicate designs I thought other people would enjoy assembling.

**SS**: You said you made the current design from a truckload of castoff 2.5mm medium-density fibreboard. Can you elaborate a little on this story?

**TL**: A short time after I bought my first laser cutter a friend showed me these ~3-foot square packing separator sheets from a place where his wife worked – a local steering column manufacturer – NSK. He was lining his trash box with them. I took one home and found it suitable for modeling with laser-cut shapes.

Then I went to NSK’s loading dock where the sheets were being fed into a pallet grinder and got more. After a couple of trips the dock guy started saving them out for me and I’d pick them up every few days. I collected hundreds and now they’ve been discontinued in favor of a similar-sized Masonite material which is softer and textured on one side – not suitable for me. These otherwise perfectly good Masonite sheets also go to the grinder.

**SS**: What were the considerations for shapes of the pieces in the kit? You could have designed them in any number of ways.

**TL**: The box joints fit together most cleanly when the edges are 90 degrees – this dictated that the knot be orthogonal. I believe this construction has the minimal number of shapes and the least amount of unique shapes. This is in direct contrast to the countless number of meander patterns which can be obtained depending on the interchange of these parts.

**SS**: Can you walk me through the creation of the pieces. I am assuming it was computer-assisted.

**TL**: In the computer, it’s possible to build an orthogonal trefoil knot from a N x N x N assembly of cubes – 11 x 11 x 11 in particular. Selectively deleting cubes yields a knot shape – left or right depending on the method. This knot is of a continuous square (2 x 2 cubes) profile made up of 24 planar L-shapes on the surface – the air gaps are of dimension 1 cube.

Unit cubes could be stacked directly to make a knot without air gaps or with an air gap of one (or more) unit(s). An air gap of 1/2 the square tube profile is more pleasing to me.

Within the 24 L-shapes there are only 4 unique sizes. Six of each unique L-shape.

**SS**: On your website you outline two additional challenges for this puzzle which involve reassembling the pieces in different ways (to form the same shape). Can you elaborate on the background of these challenges and how you created them?

**TL**: Years ago Bill Kolomyjec showed me a method of generating somewhat random patterns using lines drawn within regular hexagons. A straight line connects 2 opposite edges at their centers. Circle arcs join the remaining 2 pairs of adjacent edges at their centers. Hexagons, of course, tile the plane completely as in a beehive. When these hexagons are rotated randomly by zero, +60 and -60 degrees the lines within connect to make continuous “meanders.” Rotating just one hexagon in a tiling changes the pattern – like a unique fingerprint. Each hexagon, then, can be in one of three unique orientations or patterns.

Assembled into an even number of horizontal rows and some number of columns, a rectangular tiling of hexagons can be “rolled” into a tube, carrying the random meanders along. This tube can be further shaped into a torus and all the meanders become complete closed paths. The tube could be a knot of any shape and the continuity of meanders is maintained.

My trefoil knot, having 24 L-shapes, can be thought of as a 4-row, 6-column tiling of hexagons rolled into a tube as above. Of course, the L-shapes cannot be rotated individually by zero, +60 and -60 degrees as can regular hexagons.

However, the L-shapes are in fact irregular hexagons tiling a 4 x 6 rectangle rolled into a tube. While these L-shapes cannot be rotated individually, rotation of the meander pattern is obtained by connecting edges at their centers to schematically replicate the 3 meander patterns within regular hexagons.

Considering the 24 L-shapes, there are only 4 unique shapes – label them A, B, C, D. There are 6 each of ABCD. Taking the 6 “A” shapes, for instance, two of them have the meander pattern of zero rotation drawn upon them. Two others have the meander pattern of +60 degrees drawn upon them and -60 degrees for the remaining two.

This design is replicated for the BCD L-shapes. These meanders are all closed paths on the knot’s surface. The closed paths define the borders of separate areas which can be uniquely colored in.

One challenge would be to determine how many unique “fingerprint” patterns of meanders could be assembled from rearranging the L-shapes. Probably one for each person on Earth.

*Trefoil Knot* is available for purchase from Canadian distributor Puzzle Master Inc.