#noneuclidean

My first ever attempt at sashiko!  Natural linen thread on denim jeans (left leg), shippo tsunagi pattern.

I thought this pattern would be a good beginner choice, but I was ambitious. I didn't care strongly about making even stitches, I just wanted to try this out.  

I created the grid circle-by-circle (compass and straightedge), using a very cheap drawing compass with a pen taped to it. This turned out to be a TERRIBLE mistake: 

Fabric stretches as you drag a pen over it (Euclidean construction on not-very-Euclidean surfaces, a bad idea--who knew)

As you move farther from the origin, the grid becomes more and more distorted as the errors compound

Since the pants are intact, placing rulers over seams (bumpy!) will distort measurements.

I also didn't take into consideration that at the intersections, I'd have to choose how threads would overlap. So there are some messy intersections.

Taxicab Conics

Just occurred to me I could make this! Using the locus definitions of conic sections, here's what they look like in taxicab geometry.

Could not figure out how to make the parabola for a general directrix algebraically. Knew I could transform it from the horizontal or vertical case, but couldn't get the algebra to work in GGB. Problem of distance from a line to a point in this geometry is a good one!