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 |