Knot and the Octahedron

While playing with wire models of the Pattern Knot ("Mereon" knot) of Lynnclaire Dennis, I found an orientation which maps to the Octahedron vertices.

This is a different mapping than the previous Octahedron mapping I did here.

Consider the Octahedron.

 Figure 1

Draw a circle around the "equator" of the Octahedron.

 Figure 2

The remaining circles to be added are hard to see.

Add a circle so that an edge of the Octahedron is the diameter of the circle.

 Figure 3

We cut away 1/2 of this circle.

 Figure 4

Follow the edge, which is the diameter of this 1/2 circle, from the "equator", up through the top vertex and down along the "opposite" edge.

Add a circle around this edge.

 Figure 5

Cut away half of this new circle in such a way that the 2 half-circles that meet at the top vertex make a nice continuous curve through the vertex.

 Figure 6

We do this 2 more times (adding a circle, removing 1/2 of it) in the "bottom hemisphere" of the Octahedron.

 Figure 7 Figure 8

The result is the Pattern knot.

 Figure 9 Figure 9B

(In Figure 9B, I colored 1/2 of the blue "equator" circle green.)

Here are some other perspectives.

 Figure 12 Figure 10
 Figure 11

Note that this really isn't a knot until we specify what happens at the 2 crossover vertices. These are indicated with little green spheres in Figure 13.

 Figure 13

Depending on the weave at these vertices, you get a knot or not.

I will not go into the weaves at this time. However I will say that depending on the weave you get either the "Lynnclaire" Trefoil knot, or the "Lou" Trefoil knot or the "Bob" knot, which is not a Trefoil knot. For further explanation of this see this web page.

An interesting feature is that the weaves can "open" to allow the definition of 2 parallel lines. When the knot is defined on the Tetrahedron ( see this web page ), the 2 weave junctions define 2 lines at an orientation of 90 degrees from one another.

 Figure 14

Here are some animation....

 Figure 15

Here I add a cube. The orientation of the cube is such that several of the lobes are in the plane ofthe cube's square faces.

 Figure 16

Here you can see the cube orientation a little better.

 Figure 17

Here are some images which I hope better shows the parallel line aspect I mentioned above. Here the red lines through the 2 crossover points (where the wire is suppose to weave by itself to form a knot) corresponds to 2 of the cube's edges

 Figure 18 Figure 19

Here is the same knot, different configuration, on the Tetrahedron.

 Figure 20

Look at the points where the wire has to weave around itself to form a knot. This is at the 2 midedge points (top edge and bottom edge).

These edges are at a 90 degree orientation from one another.

 Figure 21 Figure 22