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## Isometric Cube Revisited

Only recently did I consider an alternative description of my curve-stitch isometric cube. Rather than seeing the design as an isometric cube with curve-stitch parabolas drawn on all adjacent sides of the cube, I began to look at it more abstractly. It is a regular polygon with alternate vertices connected to a center point and curve-stitch parabolas drawn using all ad adjacent sides. The original design

thereby becomes a particular case, namely a design based on a hexagon, i.e., a 6-sided polygon. (The images on this page are in scalable vector graphics (SVG) format. Click on an image to view it in a new window, where it can be reduced or enlarge without loss of definition.)

We can generalize the design to a square:

or to an octagon:

Here is another figure of this family. The enclosing figure is a dodecagon, i.e., a 12-sided polygon:

We can continue this generalization, of course, though I will only provide a sample here. Below is a figure based on a 30-sided polygon:

Designs need not be rendered in black and white. Here is a colored figure based on a decagon, i.e., a 10-sided polygon:

You have perhaps noticed a few thing about the designs presented above. Each of the enclosing regular polygons has a even number of sides. And each figure has a number of large empty areas. There are half as many such areas as there are sides to the polygon.

It should be obvious that the same sort of figures as have been presented so far cannot be constructed based on a polygon having an odd number of sides. If we proceed as we have for squares, hexagons, octagons, etc., we find that a triangular area is “left over.” We can draw curve-stitch parabolas within this triangle to complete a figure. Here is what this looks like in a figure based on a pentagon:

The simplest figure we can draw using this strategy is based on a triangle:

Here is one more example, based on a 19-sided polygon:

Figures based on other polygons can be imagined based on the designs seen above.

— LED, 4/21/2023