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 |

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