Knitwear Design 101: Tessellations and surface pattern design

If you’re a quilter, you’ll have a head start on today’s post; tessellations are a crucial part of pattern repeat design.  Although tessellation can be found in any branch of surface pattern design, quilting is one of the most prominent examples of tessellation in the world of arts and crafts – tiling notwithstanding.

Last month I discussed drop repeats, which are related to tessellations, but they are only half the story.  It is one thing to talk about how a leaf motif can be duplicated across a surface, but the other half of the story is the geometry of that repetition.  Translations such as drop repeats and reflections relate to the details, textures or images that make up a pattern.  Tessellations refer to the geometric plan of the pattern as a whole.  If the leaf motifs are the trees, planted at half-drop intervals, then tessellation is the wood, showing us the overall shape and character.

To get the most out of tessellations, it’s best to move away from texture and into the world of colour; think intarsia and stranded colourwork.  Last week you may have noticed that drop repeats can only be created with four-sided shapes, or polygons whose sides are a multiple of four, and this gives us another clue about the origins of tessellation.

The Parquet cowl features half-drop repeats AND a square tessellation. Drop repeats are formed by quadrangles, and in this case the drop repeat features square shapes. Diamond and ogee tessellations can also accommodate drop repeats.

The basics

The verb “tessellate” is derived from the Greek tessares, meaning four – which leads us back to tile formation.  One of the most common, versatile, and easiest tessellations involves tiled squares, each meeting at the corner to form a group of four.  The square, diamond and ogee shapes are everywhere – hello again, drop repeats! – because four is such an important number.  The simplest definition of tessellation is to cover a surface by repeated use of geometric shapes or polygons.  The number of shapes involved does not matter: you can use only one shape, as per the square above; or you can use many more than one polygon.  The golden rule is that NO gaps are allowed.  The pattern must be continuous throughout.

The corner points at which tessellated tiles meet is called a vertex.  The golden rules for vertices are that:

  1. ALL shapes or polygons forming the tessellation must meet at a vertex;
  2. The interior angles of all convergent corner points must add up to 360 degrees to complete a circle.

This makes perfect sense; if either of these two rules are broken, your tessellation will have a gap somewhere.  Who wants a hole in their beautifully tiled floor?

Further, there are two basic types of tessellations:

  • REGULAR, which feature the use of only one shape or polygon.  Only three polygons fall into this category: triangles, quadrangles and hexagons.  No other shapes can form a regular tessellation.
  • SEMI-REGULAR, which feature the use of two or more shapes.  All semi-regular tessellations require either the triangle or a quadrangle such as a square, and there are eight possible combinations.

It’s worth noting that the triangle is at the root of all tessellations because it is the very first and simplest three-dimensional shape.  Only three lines are needed to draw a triangle, and all polygons can be reduced down to a triangle.

Examples of regular and semi-regular tessellations. At the top we have an octagon/square combination, followed by hexagons and diamonds. All can be reduced to triangles if you connect the opposite corners with a line. With odd-sided polygons, there will always be a corner left over. The triangle is the root of all geometric shapes; note that the square root is symbolised by a triangle, diagonal line, or a line cutting across the centre of the square from one corner to the other.

Tessellations for knitting and crochet

The ability to reduce polygons to triangles is particularly important for knitting because the most successful tessellations always feature polygons with an even number of sides.  The three regular tessellations, plus any semi-regular tessellation featuring squares, triangles or hexagons, work best.  This has a lot to do with the stitch-row ratio and fabric construction.  Because knitted fabric is all one piece, shapes like pentagons and heptagons – anything with an odd number of sides or interior angles – are tricky to manage for stitches, which are essentially four-sided shapes.  There’s a good reason why knitters of old stuck to squares and octagons: OXO patterns, Nordic stars.  It is definitely possible to work with odd-sided polygons, but only on a large scale, and – I’m sticking my neck out here – only with the support of at least one other even-numbered polygon such as a square, diamond or octagon.  If you like the idea of designing big and bold patterns, odd-numbered polygons are the way to go!  Using them will force you to think on a grand scale.

Crochet offers more opportunities for design because it can go either way: it can be tiled in a traditional manner when worked into polygons or motifs, or it can be tessellated as for knitted fabric.  Of course, it’s possible to knit polygons of either shape, but they aren’t as dynamic as crochet motifs.  Knitted motifs simply do not have the same capacity for pattern, texture, and colour exploration.

I’ve spent a lot of time discussing regular polygons, but you mustn’t feel restricted by these.  To take things up a notch, the principle behind designing tessellations for knitted fabric – or crochet fabric NOT comprised of motifs – is that at least one of the shapes you use will have an even number of sides.  Nor is there a limit to the number of shapes you can use for a tessellation; so long as all the angles at any vertex add up to 360 degrees, you’re on the right track and won’t have any gaps.

My last design tip is to pay attention to the vertices.  They will define, or help you to identify, the key features of your design – particularly how the repeat works or can be developed.  Just as a keystone is a sign of excellent architecture, so is the vertex a sign of excellent tessellation.

Designing your own tessellations

I recommend experimenting on dotted paper and isometric paper to get a feel for the kinds of grid or framework you can create; give yourself plenty of freedom and allow yourself to play and enjoy creating patterns.  The discipline of the dotted or isometric paper will prepare you for the next step.

If you then want to see if something will work as an integrated knit fabric, move on to squared paper or graph paper.  Graph paper is better for working on a larger scale because the squares are relatively small.  Squares with sides measuring 1/8 of an inch or 2 mm are a good size to work with.

When it comes to design development, it doesn’t matter whether you start with the tessellation or framework, or with the concept or inspiration.  Go with whichever compels or inspires you most.  For example, if your inspiration source is a flower in full bloom flanked by two buds, you might lean towards tessellations with polygons that total a multiple of 3 at the vertices, so a simple triangle or hexagon framework would be a good place to start.  From there you can develop the design however you like; reflect on and evaluate what does and doesn’t work, and hold on to what you want to convey visually.

On the other hand, you could just as easily generate lots of different tessellations because you’re more fascinated by the process of creating geometric surface pattern.  Having lots of templates banked means you can experiment with colour (make sure you protect your master sheet and make photocopies to do this), or you can stretch yourself by taking the flower and buds inspiration above in a new direction.  How can you get that to work with a different multiple or formation?  Can it work over more than one kind of tessellation?

The progression of the three tessellations above.

Final thoughts

The process of tessellation is really one of composition and arrangement: think about musicians, singers or recording artists whose music can be remixed and rearranged with ease – sometimes sounding nothing like the original track – or whose ‘sound’ can span many genres of music from album to album over the course of their career.  This kind of versatility is part of what makes the best designers – or composers – or artists – stand apart from the rest.  See where your imagination takes you!

4 Comments Add yours

  1. Susan says:

    Mind bending 🙂 I did hear her interview with Corinne from the Woolly Thistle and she is a very inventive knitter!!

    1. Thank you Susan! 🥰

  2. ErinJoelle says:

    Thanks for sharing this vocabulary, Natalie. I think the “limits” of tessellations allow for great creativity. Depending on the colors you choose or even how you focus your eyes, the same formations can look radically different. My first exposure to this came from Nancy Bush’s book Folk Knitting in Estonia. It is full of beautiful ideas.

    1. Thank you Erin! And I haven’t read that book, so I will see if I can find it ☺️

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