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The Tusi couple in Python: 8 beads in straight lines that roll as a circle

#python #math #3dprinting #geometry

Eight beads sit in eight straight slots. Each bead only ever slides back and forth along its own line. Watch all eight at once, though, and you see a circle rolling around the dial.

Tusi couple animation

The trick: a circle rolling inside a circle twice its size

In 1247 the Persian astronomer Nasir al-Din al-Tusi showed that if a circle of radius r rolls inside a circle of radius 2r, every point on the small circle's rim moves in a straight line through the centre. Copernicus later used the same construction to build straight-line motion out of circles.

The maths is one line. Put the small circle's centre at angle t. It rolls the opposite way, so a rim point that started at angle b sits at

P(t) = r*(cos t, sin t) + r*(cos(b - t), sin(b - t))
     = 2r * cos(t - b/2) * (cos(b/2), sin(b/2))
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The direction (cos b/2, sin b/2) never changes. Only the length 2r*cos(t - b/2) changes, so the point just slides along a fixed diameter. Space eight points 45 degrees apart on the small circle and you get eight lines 22.5 degrees apart.

Building it for a 3D printer

I wrote it in Python with manifold3d:

  • Body: a dial with 16 half-slots cut from the centre out, so all 8 diameters meet in an open centre window. The body prints face-down, so the slots are just holes in the first layers.
  • Disc: radius 17 mm, with 8 pegs on a 15 mm circle. Pegs ride in the slots with 0.4 mm clearance per side. No gears needed: the pegs in the slots fully constrain the disc (it is a trammel).
  • Floor and beads: the floor presses in under the disc, and beads glue onto the peg tips above the dial.

The motion is checked in code before printing: step the angle through a full turn and intersect every part with every other part.

for t in range(0, 360, 6):
    body, floor, disc, beads = assembled(t)
    hit = (body ^ disc).volume() + (body ^ beads).volume() + (disc ^ floor).volume()
    assert hit < 1e-6
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That came back zero at every step. The animation above is just that same function rendered frame by frame.

Tusi Couple Astrolabe

Styling

The dial borrows from astrolabes: 64 degree ticks, an eight-point star in every segment, a throne (kursi) with a hanging ring, and a sunburst on the disc that shows through the centre window.

The STL files are here if you want to print one: Tusi Couple Astrolabe on Cults3D.

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