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## Curve-stitch Isometric Cube

The figure above is an instance of a design I created more than forty years ago. (This figure is covered by a Creative Commons license. For more information, click here.) A framed version of it hangs on my kitchen wall. That picture is a negative print (i.e., white on black) of an India ink drawing on vellum. It is a minor conversation piece—first-time viewers are often surprised to learn that the drawing contains no curves. More amazing to me is that I drew it with drafting pen and straightedge without introducing major imperfections. I cannot imagine repeating the feat! The figure shown here is a conversion of output generated using Adobe PostScript. It doesn’t reproduce the razor-sharp lines of the original, but it does capture the spirit of the figure. I recently replaced most of the graphics on this page with PNG, rather than GIF images. These images are much sharper, but the page loads slowly because the image files are bigger. (Sorry about that.) A SVG graphic of this figure can be found here, although not all browsers will render it properly. (Try zooming in and out and notice how the image changes.)

My talent as a graphic artist is limited, but I have always enjoyed drawing designs on paper. Designs employing only straight lines—such as this one—permit me to masquerade as a real artist, so long as I don’t have to draw in public.

This design resulted from combining two designs—the curve-stitch parabola and the isometric cube. I encountered the former in  a junior high school math class. A drawing such as the following can be made with needle and thread, punching holes in cardboard and stretching the thread along its surface:

Somewhere, I picked up the technique of making isometric drawings. Isometric drawings represent three-dimensional objects in two dimensions. The x- y- and z-axes are inclined, respectively, 30, 150, and 90 degrees from the horizontal. Isometric drawings look like perspective drawings, but parallel lines do not converge at a distance. Lines parallel to the axes are proportional to the lengths of the corresponding lines of the actual object, though oblique lines are not. The nature of an isometric drawing is most easily appreciated by studying an opaque cube drawn using this technique:

My design simply took an isometric cube and added curve-stitch parabolas on all adjacent edges. Simple. One feature I’ve always liked about the design is that the impression it makes on the viewer is quite changeable. Knowing its origin, it is easy to see it as a patterned cube, but it can also be viewed two-dimensionally as a hexagon whose vertices have been connected to the middle before the parabolas are applied. It looks very different upside down (somewhat face-like, I think):

or rotated:

NOTES: Those interested in curve-stitch designs may enjoy Jon Millington’s attractive book, Curve Stitching: The Art of Sewing Beautiful Mathematical Patterns (Diss, England: Tarquin Publications, 1989). This full-color paperback includes a brief quotation from Mary Boole, who invented curve stitching in 1904; numerous photographs of  curve-stitch designs executed in colored thread on cardstock; a collection of BASIC programs for generating designs; a discussion of the mathematics of curve stitching; photographs of art works using curve-stitch techniques; a collection of tools for executing designs; and a bibliography.

The curve-stitch designs on this site were all coded in PostScript, which can be printed to a PostScript-enabled printer to yield sharp-line output. The PostScript can be converted by Adobe Acrobat into a PDF file that can be printed on any printer. I have used various combinations of software to produce the graphics you see on this and other pages. If you want to produce your own designs by computer, read “Make Your Own Designs.”

— LED, 8/9/2001, rev. 10/12/2010