Mathematics

The Maths of Repeating Patterns in Woven Cloth

Weaving is counting. Planning a project means fitting whole repeats into a width, balancing a pattern at both edges and knowing when two cycles will line up again.

Repeats and units

Most woven patterns are built from a unit that repeats: a block of threading that is used again and again across the warp, and a treadling sequence repeated along the length. A straight four-shaft twill has a repeat of four ends and four picks. A point twill threaded 1-2-3-4-3-2 has a repeat of six. Knowing the repeat lets you work out everything else.

Fitting repeats into a width

Suppose, as an illustration, a pattern repeat is 24 ends and you want a scarf about 200 ends wide. 200 divided by 24 is 8 with 8 left over. You can weave eight full repeats and use the 8 spare ends to balance the edges, or adjust the total to 9 repeats, or change the sett. The arithmetic is simple division with a remainder, but writing it down before you wind a warp saves a lot of unweaving.

Balancing a pattern

Many patterns look best when they are symmetrical across the cloth, so that both selvedges start and end the same way. With a point twill, for example, you usually add the first end of the repeat at the end of the warp so the last point is completed. A useful habit is to sketch the first and last repeat on graph paper and check that they mirror each other.

When two cycles line up again

Patterns often involve two cycles of different lengths: a colour stripe sequence of one length running over a threading repeat of another, or a treadling sequence that does not match the colour order of the weft. The two will line up again after a number of threads equal to their least common multiple.

Worked example. A colour sequence that repeats every 6 ends over a threading that repeats every 4 ends. 
Multiples of 6: 6, 12, 18, 24 ...
Multiples of 4: 4, 8, 12, 16 ...
Least common multiple: 12
So the combined pattern repeats every 12 ends. Change the stripe to 5 ends and the combined repeat becomes 20, which looks much less regular to the eye.

This is a deliberate design tool. When two short cycles have a small common multiple, the cloth looks orderly. When their common multiple is large, the cloth looks varied and less obviously repetitive, even though everything in it is still entirely determined.

Symmetry along a band

Woven bands and borders are a good place to see symmetry at work. Mathematically, a frieze pattern is a two-dimensional design that repeats in one direction, and frieze patterns can be classified into seven types according to their symmetries: whether a motif is shifted, mirrored, rotated half a turn, or reflected and shifted at once. A band of chevrons, a row of diamonds and a repeating hook motif each fall into one of these types. You do not need the classification to weave, but looking at borders this way helps when you design your own.

Counting and checking

  • Count the warp in groups, tied off with a contrasting thread, while you wind it.
  • Keep the draft and your arithmetic together in a project notebook.
  • Before threading, write the total number of ends as repeats plus balancing ends, and check it adds up.

Counting also helps with the other side of the hobby: working out what it costs.

Revised 2026-11-09 · Margit Halloway