Category Archives: knots

Icosahedral Symmetry as a Decorative Knot

The decorative knots to be discussed here are those which are tied with one or more cords that may be repeated through several plies. These sorts of knots can be represented by self-intersecting loops on the plane, set up so that no more than two loops intersect at any single point. One generates a tying diagram from such by picking which of the two paths are uppermost at each intersection point. While this could be done more arbitrarily, for the knots discussed here the paths will be selected so that each path alternates over and under as in:
Example of planar knot tied with over and under pattern, on sphere

My eventual objective here is to tie a knot with approximate dodecahedral or icosahedral symmetry. Let’s begin with a line drawing that has the right symmetry. Flattened out to the plane, the dodecahedron looks like the following planar graph:
Dodecahedron spread flat
But this is not in the form we need; it is not in the form of a collection of loops. The basic problem is that, as a graph, there are three edges meeting at each vertex.
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A new way to tie an old knot

The tying diagram Ashley gives for ABOK #2217 has a 4-fold axis of symmetry:
The Ashley Book of Knots diagram number 2217.
Tying a knot according to a diagram like this is quite time consuming. One must redraw the diagram by photocopying to the size needed. And in tying the knot, one pins the rope to the diagram. This is a pain because the rope moves around, the pins come out, etc. And the pins can damage the appearance of the rope.

In this post I give an alternative method of tying this knot, and several others like it, that is easier to set up, is much faster for each knot, and uses cheaper materials. Rather than an expensive cork board, we will use a 2×2 and build the knot as if it were a sort of Turk’s Head knot, on a cylindrical of square form.
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Ashley knots; ABOK #2217 covering knot

For those of you who aren’t familiar with blue collar knots, “ABOK” means “The Ashley Book of Knots,” an ecyclopedic book on knots written in 1944 that has since become the reference for knot identification. I think it was my maternal grandfather that gave me my, somewhat rare, now the worse from love and use, 1st edition copy back in the 1960s; but what with the natural self-naturedness of a boy I cannot recall for sure. #2217 refers to a particularly handsome knot in the chapter “The Monkey’s Fist and Other Knot Coverings.” To justify “handsome” requires a sample photo from cbrew6 on Knot Heads World Wide:
Ashley book of Knots #2217 tied by cbrew6 as shown on Knot Heads World Wide

This knot is “tied on the table,” which means that one uses a diagram to draw it. From a topological point of view, a table diagram in this case is a mapping of the surface of the sphere to the plane. A line drawing shows the path that the cord takes. The path is a loop, that is, it ends at the same point at which it starts. The path is restricted to never cross itself twice at a single point. At each crossing, some sort of notation indicates which line is to be on top, but for planar knots like the above, it is arranged so that the cord will weave over, under, over, under … And it is a trivial fact of practical folks topology that one can always assign such a pattern.
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