How should I spin the story of the gyroscope this time?
Imagine, you have a wonderfully crafted elongated ellipsoid from wood (shaped like a rugby ball).
You built some apparatus that make it spin fast about its major axis. Probably you can make it spin with your bare hands – or you ask a tentacled alien for help. The symmetry axis of the body, the axis of rotation, and the vector of angular momentum are all aligned.
You or the alien or the apparatus fling the ellipsoid into the air, accelerating it in the direction of the major axis. The ellipsoid will keep spinning in mid-air. Gravity will make it move along a parabolic path. But due to the ellipsoid’s symmetry, there will be no torque. Gravity cannot make it precess: it cannot change the ellipsoid’s angular momentum.
Now imagine you or the alien are really fast. Or probably we rather think of this story happening in a (the alien’s!) spaceship as gravity is not the point anyway. You have enough time watching the spinning ellipsoid flying by.
… and suddenly you smack the ellipsoid, hitting it near on of its “cusps”!
During a fraction of a second there is a torque from the force applied by your hand. The force changes angular momentum which then no longer points in the same direction as the symmetry axis or the axis of rotation.
But after the forceful intervention the ellipsoid – the gyroscope – is free again. Angular momentum must be a constant – as a vector, with direction and magnitude.
Angular momentum and the axis of rotation are no longer aligned. (These vectors are related via a matrix – representing the moments of inertia in different directions. They need not be aligned!)
To keep angular momentum constant, the gyroscope compensates by adding an additional rotation: The axis of rotation about the symmetry axis is rotation slowly, about the fixed direction of angular momentum. This rotation is called nutation or regular precession. (This is not the same as the precession due to a torque. Theses terms are not used unambiguously.)
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For more physics details and references, see one of my older articles: I have written about all these cones and ellipsoids again and again and again. (The latter link goes to a post linking to several articles).
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In my articles – and related drawings – I was mainly referring to Louis Poinsot’s geometrical theory of the gyroscope. You can solve the equations and plot the solutions (using software). But back then, the elders had to get creative to learn as much as possible about the solutions of differential equations without actually solving them in all details.
I wonder if developing such geometrical ideas – visual helpers, mnemonics, shortcuts – might become a lost art. Now you can feed a computer your tricky equation. Your favorite AI would happily generate threejs code to display a visual spectacle of rotating ellipsoids and cones in your browser. Would Feynman still need his diagrams today?
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Questions like this are always on my mind, when I draw something very slowly that code could create quickly.
Recently I got an email from a startup specializing in AI-created scientific illustrations, targeting researchers. That stuff became scary good; they give you the option to tweak elements like text without having to prompt again. I found some amusing quirks in demo diagrams I could make fun of now. But the rapid evolvement of AI in the past months have shown us that you go from “Ha ha six fingers” to “OMG it is coming after my job” quickly, so I hold back my sarcasm.
Once again, I have learned I am not really creating a scientific illustration. Technically I am of course, but my production is so slow it would not meet the requirements of somebody who needs a quick diagram to complete their paper or presentation before a deadline.
It is more of a meditation about physics and geometry. I want to keep a potentially dying craft alive. I want to remind other humans of what humans have been capable of without any digital help, let alone AI.
I want to convey what a joy it can be to hold a concept in your mind in its geometric entirety, not delegating any of the hard lifting to symbols, code, or convenient tools.
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Here is the long story of creating and mounting a drawing that encapsulates the motion of the smacked ellipsoid – a force-free symmetric spinning top!
I start with test drawings on light green paper (antique left over office supply). I am drawing three cones: The blue rolls along the surface of the red one while its axis traces out the yellow, fixed cone.
I am using the side view (to the right) and the top view (bottom) to construct the view from an oblique angle.
Curly L is the constant angular momentum, omega the instantaneous axis of rotation, and f the figure axis (symmetry axis).
I am varying the dimensions of the cones and the position of the whole structure on the drawing surface.
I should finally also mention where the “prolate” property is used: If the spinning top / ellipsoid would be oblate, the red cone would embrace the blue one.
But Everything is too close to the right edge now.
I can try to invert it at least – either to plan for “invert art” or for re-drawing it on black paper.
In the next attempt with, I am actually completing it. I like the shape of the cones and viewing angle better.
But I am again too close to the right edge. I have changed several parameters at once and the result is hard to predict.
I have chosen a color combination that does not look exciting on this green paper, having digital inversion in mind. By the way, this is now a scan, not a photo.
Digitally inverting it again, I feel I am on the right track:
The construction lines are very faint though as I am using a thin technical pencil with gray graphite lead.
So, it’s decided! I will try again with white colored pencil on black paper. While I’ve enjoyed creating inverted art, the digital step felt a bit like cheating anyway. Or maybe, I am just looking for Oulipo-like constraints to boost my creativity.
I also want to add context what this is about. I have been inspired by antique patent drawings, and I learned theoretical classical mechanics from a professor who was himself a student of Heisenberg and Planck. So, I feel like annotating the cones in German, adding a “German engineering vibe”.
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Now I have a piece of paper. It is thick, nearly like cardboard: 300g/m2.
But I am not a fan of putting it behind reflecting glass. How to frame it?
I turn a stretched canvas into a “fake frame”, by painting its edges black. The paper is size A4, and – lo and behold – I’ve finally got lucky with “standard sizes”: Commercial stretched canvases come in dimensions with neat aspect ratios, and this one is 40cm x 30cm. The white margin around the drawing is nearly the same for height and width.
I have varnished both the drawing and the canvas for protection and effect. Both with glossy coating, but only cautiously applied to the paper. It does not look glossy; just the natural texture of the paper appears to sparkle more.
The canvas edges lead a healthy dose of varnish – otherwise the black acrylic color would rub off the wall later.
I have documented my glue-paper-to-canvas experiments before here and here. But I hope I will create more drawings in this style, and eventually I will run out of wall space. I want to swap out artworks easily.
So, the attachment needs to be removable.
And I am done with the glue. I found the method that aligns so much better with “art rooted in physics”…
… magnets and metal!
I need thin counterparts for magnets directly attached to the drawing. I go for solid metallic tape; “metallic paint” or tape containing tiny metallic cuttings would likely not be sufficient.
Here are four square patches cut from a roll of metallic tape, glued to the backside of the paper (complete with another abandoned start of a drawing). Note: This is not removable!
It is amazing that you buy Neodymium magnets in nearly any shape. I want flat square ones, with five of them being strong enough.
Better err on the side of strength here: The officially measured adhesive strength of one of these little guys is 3,5kg! But this would be for a massive plate of steel and gravity perpendicular to the interface. Attaching something to a wall results in a force a factor smaller, and even the depth of the canvas makes the force drop quickly. Also, the force scales with the volume of metal, and my tape is 0,15mm thick.
The magnets fit nicely under the wedges for stretching the canvas:
I prefer loops and silver wire to a sawtooth hanger. Originally, I planned for putting them up on our slanted (former attic) walls; so I positioned the wire very close to the center of the canvas.
The magnet’s strength feel just right: They snap in a nice way, but can be positioned and removed easily.
Now I can use a hook or screw to actually hang the canvas.
I am using a hook, and a thin Perlon thread. I add two loops at the end of the thread. Then I embrace my little magnetic tower with them!
This is the top side of our refrigerator, with 2x2x5 Neodymium magnets. The fridge is in the middle of the large room, and the backside is covered with a white sheet with a bit of texture.
And there it is, finally:
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Geometry of the Motion of a Prolate Gyroscope,
by elkement 2026,
magnet-mounted.
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Looking back at the history of creation – also adding the 2025 precursor version on A5 paper, enhanced with watercolor pencils. I am also taking these photo to document there is no AI involved. (As isn’t in my writing – all my errors are mine alone!)
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If you made it until here – thanks for reading! Much appreciated!
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