These spheres and lines have been haunting me.
I have been writing about stereographic projection so often – and created so much related geometrical art – that I created a blog post just for curating all the links to my own articles: Stereographic Retrospective.
This year, I have created a series of six drawings of Circles projecting onto Circles …
… which I turned into blankets – bringing math art to the masses!
These drawings feature the letters x,y,z. Other than that, I had not dared yet to include any annotations of formulas.
I want to change this, and finally turn this very dry and “scholarly” article into a drawing that should hopefully be perceived as abstract art with mathematical calligraphy sprinkled over it. The topic is: How stereographic projections relates to spin in quantum mechanics.
I am starting with a construction: A sphere with an arrow poking out of it, seen from three different viewpoints. I need the view from above and from the side to construct the “3D” view.
I use artistic license and add curvature where the straight lines meet the edges of the drawing. This will help me later when preparing the digital file for canvas prints.
The spin is built from an up and a down component. If you visualize it as an arrow pointing in a certain direction, you can project down the point where the spin punctures the sphere.
The image point In the equatorial plane can be interpreted as the ratio of the “amounts” of up and down component. Playing with the geometry, half of the inclination angle theta of the original arrow shows up again when looking at the projection in the plane. This factor one half is what a lot of weirdness of spins can be traced back to (like: “You need two full rotations to get back to the original state”).
But today, I focus on drawing and painting! All the construction lines divide the drawing surface, and I am coloring the emerging shapes as if these were shards of glass.
In the past, I have used watercolor pencils for that. Now is the first time to use “real” watercolors:
It has been quite a challenge to re-trace the mathematical symbols with a not-so-fine brush!
I do want to preserve and celebrate the details of the construction. Last year, I have often enhanced them with black fineliner or ballpoint pen. However, this often forced me to use more vibrant colors for the “stained glass” as well, and I glazed over all the “cells” with watercolor markers.
In order not to overpower the watercolors, I am now using (waterproof) colored pencils for re-tracing the lines.
I took this photo has been taken before flattening the paper:
The paper is rather small – 21cm x 21cm (8.3″x8.3″), a square cut from A4-sized sheets. I could either use my scanner or my (smartphone) camera to digitize it.
I’ve recently upgraded to a Samsung S25 Ultra, to use a 50MP or 200MP camera. Having taking zillions of photos in different lighting conditions as well a many scans.
But the bright oranges in the original painting cannot be captured by either device. It is not an issue with color calibration but rather a physical limit of the sensors, and in the case of the scanner, also the spectrum of its lamp (Very vibrant, neon-like colors use fluorescence.)
I still went with the scanner this time as it captures the green and turquoise hues a bit better, at least with all the parameters I have tested to far.
Now I want to have it printed on canvas stretched over a thicker frame. So, the sides of the canvas should show something interesting. I cannot “pull this image over the edge” not to lose too much of it.
The usual recommendation is to mirror your image and blur it. However, when I mirror these lines, awkward cusps shows up where the lines meets their mirror images. The intersection points would need to be 100% exactly on the edge which they usually are not – as I tested earlier this year with one drawing of the Circles to Circles series:
The print company I have chosen advises against any “borders” in the design as their cannot guarantee the perfect placement of the edge.
My solution? I need to soften the “cusps” and make them less dominant. I face this challenge, and I am determined to improve the artwork overall.
First, I try to avoid the cusps altogether by adding curvature as described above. Lines (curves) should intersect the edge at a right angle.
The I add more random, rounded features and obfuscate any acute features by that. I am adding the flipped versions of the image using code, import the result into Procreate, and distort the sides manually with a digital pen using the Liquify tool. (No AI involved in any of the steps.)
This is the print template before distortion:
… and after …
I am checking if this is going to look as expected by “simulating” the canvas (using Javascript/threejs code I have written myself).
These are not yet photos!
And then, finally, I treat myself and order the print!
These are photos of the product! I love the box-like three-dimensional appearance of these small prints!
The colors of the digital file have been reproduced exactly.
The difference between original and print is due to the scanning process.
This photo demonstrates why all my design hacks are really necessary: The canvas is stretched unevenly in the vertical direction: My logo was moved closer to the edge, while on the top you see the remnants of the “cusps” that got pulled to the front of the canvas.
I summary, I am happy that I have found a way to make my drawings canvas-printable!
I am offering this little print in my store now:
Spinning Projection – Quantum Physics Art Canvas Print
Thanks so much to anybody who has supported my store!
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