Polygons and Pattern Work

 

Spun Heptagons

There is an interesting geometric feature that basic polygons all have.  For any polygon you can take a chord (connecting opposing corners) and with one end as the center point spin and copy it twice the number of the polygon sides over 360 degrees and you'll find that every corner of the polygon is passed through by one of the spun vectors.   

Heptagon Chord

It works with Heptagons.  It'll work with Nonagons or Hexagons or Octagons too.

Octagon Chord

I suspect the fact that all corners of the polygon fall on a spun vector will be true for any number of sides. For pattern work using more than a 13 sided polygon the patterns created get pretty dense.  To carve them in wood using a V-bit the tinier details get lost or brittle. 

Nonagon Chord

To make an interesting pattern I used a feature in my CNC software to make copies of the polygon that attach to each of the spun vectors. 

Pattern from Nonagons

Before I could make toolpaths to carve the pattern I had to outline the sections I wanted to be cut.  Once done I selected just the outlined shapes and use the V-Carve toolpath with a 120 degree V-bit to produce the shape below.

Spun Nonagons

Even star shapes can be used. More detail can show up depending on what the inner radius is relative to the outer radius.

Spun Star Pattern

A nine tipped star was used to make the complex pattern above. A 7 tipped star was used make the pattern carved below.

7 Tipped Star Pattern

While I haven't actually carved any of the examples above, I am contemplating covering a small feature  wall in my house with an array of patterned tiles made using this trick. More Complex patterns can be made with a denser spun vector array.  Instead of just 14 vectors with a heptagon, 28 vectors could be used. 

Heptagon x 28

More spastic patterns can be made using a circular number of vectors that isn't a multiple of the polygon sides, or picking something other than a corner of the polygon to spin/copy it with. 

Heptagons offset

Above is a pattern made using a heptagon that was offset from where the center was spun about.  Given that there are 360 degrees in a circle to spin about, and a nearly infinite selection of positions you can spin different polygons about, the possibilities are endless.

For a furniture project the vectors could be flattened to fit on the face of a drawer front, or used on cabinet doors to add some interesting detail.  Here are a couple of drawer fronts:

Where necessary a circle area in the center could be left uncarved to allow for a place to screw in a handle/knob. The sectioned areas don't have to have straight sides. The top drawer front pattern was done using a circle spun 16 copies around a center point.   

Questions? Leave a comment. 

4D