I will show you one way that the "Pattern" knot can be constructed on the Icosahedron.
We first construct an Icosahedron.
Start with 3 intersecting planes. The length of each plane is p=(1+sqrt(5))/2 while the width is 1.
We can build an Icosahedron by connecting the conners of the planes together to form 20 identical equilateral triangles as follows.
We next want to construct an Octahedron. Start with the same 3 intersecting planes we used to construct the Icosahedron.
Note that there are 6 outer-most edges which have a length of 1. Connect the mid-edge points together to form the Octahedron as follows.
This shows one way that the Octahedron and the Icosahedron are related.
As shown here, The "Pattern" knot can be formed around the Octahedron.
This leads us to one way to draw the Pattern knot using the Icosahedron.
Usage Note: My work is copyrighted. You may use my work but you may not include my work, or parts of it, in any for-profit project without my consent.