Kumihimo: How Symmetry Limits Kongō Gumi’s Color Patterns

Pick one of two colors for each of sixteen strands, and the braid's pattern is already decided. There are 65,536 ways to make those picks, two multiplied by itself 16 times. Most of them give a braid that matches another one after a turn or a color swap. Only 1,162 are different patterns.

What is kumihimo?

Kumihimo (組紐) is a Japanese style of braiding. In his 2022 paper, Holden explains the word as kumi, coming together, and himo, string, cord or rope, citing the American Kumihimo Society. He notes that kumihimo uses many more strands than Western braiding: usually a multiple of 4, and often 8 or more. The finished braid can be round, flat or something in between.

So many strands call for special equipment. One such tool is the marudai, a traditional stand, and the modern alternative is a foam disk with notches. The cord below is a flat kumihimo band made as a sageo, the cord for a tachi sword. This page could not read a source that dates the start of kumihimo, so its origin is unrecorded here.

Kongō Gumi is a round braid, not a flat band like this one. Holden, citing the kumihimo author Rosalie Neilson, translates the name as the braid as strong as metal. This page counts that braid.

How Kongō Gumi crosses sixteen strands

A Kongō Gumi braid uses a multiple of 4 strands, and the traditional number is 16. Holden describes the strands sitting in 8 pairs, spaced evenly around the stand or disk. Each move trades strands between opposite pairs, then the work turns and the move repeats. This page leaves the hand movements there and follows the colors instead.

Each strand puts its color into a fixed set of places on the finished braid. So the 16 strands fill 16 color positions, and the colors chosen at the start fix the whole pattern. Holden notes that this way of braiding gives the finished design a lot of symmetry.

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Holden sets out the symmetries of the 16-strand braid. There are 16 steps around the braid's axis, and each can be paired with a half turn about a line across it: 32 moves in all. Kongō Gumi is chiral, meaning its mirror image is a different braid, so mirrors do not count. Two braids also count as the same pattern when one is the other with the two colors traded.

Counting the patterns: why 1,162

Trace one pattern around the ring, and the same pattern shows up in many colorings. The figure below shows four different colorings of the 16 positions, and a step, a half turn or a color swap turns the first into each of the others. So all four are one pattern.

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Burnside's lemma, a rule from the mathematics of symmetry, turns this into a count over 64 moves: the 32 symmetry moves, each done with or without a color swap. For every move, the calculation counts the colorings that the move leaves exactly as they were. The sum is 74,368. And 74,368 divided by 64 gives 1,162.

Some moves fix a great many colorings, others almost none. Doing nothing leaves all 65,536 unchanged. One step with the colors swapped leaves just two: the colorings that alternate green and ochre all the way around. Holden reached the same total with de Bruijn's theorem, a more general counting rule of the same family, and writes that “there are 1162 non-equivalent Kongō Gumi braid patterns, including the one which only uses one color” (p. 328).

The color swap matters. If green-for-ochre counted as a new braid, the same method would give 2,250 patterns, as the second row of the table shows.

Download the pattern count (CSV) (what the file holds) Data: Symmetry model and published count: Joshua Holden, "Changing Spots: Using Combinatorics to Count Japanese Braiding Patterns," Bridges 2022 Conference Proceedings, pp. 327–330, https://archive.bridgesmathart.org/2022/bridges2022-327.pdf, retrieved 2026-11-14

A second method on this page checks the first: it sorts all 65,536 colorings into groups that the moves link together and finds 1,162 groups. In 900 of them, the pattern appears in 64 different colorings. The rest have extra symmetry and appear fewer times, down to 2 for the single-color braid and for the braid that alternates colors all the way around. So the total sits above 65,536 divided by 64, which is 1,024.

Same colors, different patterns

Holden sorts patterns by spots: the number of strands in the less-used color, which shows up as spots on a background of the other. Swapping colors turns 3 green strands into 3 ochre strands, so only 1 to 8 spots need counting. With 2 spots, only the gap between the two strands matters. It can be 1 to 8 positions, which gives the 8 patterns in the figure.

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One spot gives just one pattern, because steps can carry a single green strand to any of the 16 positions. More spots give more patterns, up to 375 with 7 spots. Those patterns draw on 22,880 colorings, the ones with 7 green strands and the ones with 9. Eight spots, an even 8 and 8 split, draws on only 12,870 colorings and gives 257 different braids.

Neilson's 2013 catalog set out to list every 16-strand, two-color Kongō Gumi pattern, according to Holden, and it held 1,157. Holden's count predicted 1,161. The two of them then found the four missing patterns: one with 4 spots and three with 8.

Download the patterns by spots (CSV) (what the file holds) Data: Symmetry model and published Table 1: Joshua Holden, "Changing Spots: Using Combinatorics to Count Japanese Braiding Patterns," Bridges 2022 Conference Proceedings, pp. 327–330, https://archive.bridgesmathart.org/2022/bridges2022-327.pdf, retrieved 2026-11-14

Counting like this predicts a pattern before anyone braids it. Hitomezashi sashiko works the same way: on-or-off choices for each grid line fix the whole stitched design. Reading a Celtic knot pattern takes a count of separate cords.

How it fails: one strand out of place

Set up one strand in the wrong color, and the braid shows a different pattern. Because each strand fills its own fixed positions, the mistake changes every place that strand colors. In the figure, position 10 turns from green to ochre, and an 8 and 8 braid becomes a 7 and 9 braid.

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No turn or swap can repair that. A step or a half turn keeps the number of green strands. A swap turns 7 and 9 into 9 and 7, not back into 8 and 8. The calculation tried every single-strand change on all 65,536 colorings, 1,048,576 changes in all, and none left the pattern the same.

This failure belongs to the pattern only. This page found no readable source on how a Kongō Gumi braid behaves under strain, so it says nothing about strength or slipping. Knit and purl stitches build cloth from one strand in loops, where a kumihimo braid works many strands at once. How the site checks pattern counts is set out on which layer was checked.