Unnamed Algebraic Box / Furl

A box that twists shut, shaped by one line of algebra from 1770.

Set the lid on, turn it thirty degrees clockwise, and it clicks.

The line is Vandermonde's difference product, which he read to the Paris Académie in 1770; square the three coordinates first, take a square root after, and use the answer as how far the surface of a sphere sits from its centre.

Nothing was sculpted.

Every petal and every crease comes out of that formula, and so does the symmetry, which is exactly the cube's.

Look for the seam and you'll have trouble. The box is split along one of the shape's own creases, a plane where the formula falls to zero, so the join sits at the bottom of a groove that was already there.

Inside, a cone on the lid drops into a matching socket and centres it. Two tabs ride a groove as you turn, then a small bump climbs a ramp and drops into a dimple: free turn, a little resistance, click. It locks at two positions half a turn apart.

It holds about 200 ml.

Print Settings. These are the settings the box in the photos was printed with, read back off the file that made it:

- Layer height: 0.2 mm, first layer the same.

- Walls: 2. Top 5, bottom 3. Infill 15%, crosshatch.

- Supports: tree (auto), 30° threshold, 2 interface layers. They carry the lower petals and the ring around the seam.

- Support interface: we used a dedicated support filament (Bambu Support for PLA/PETG) for the two interface layers. We haven't tried it with plain PLA interfaces.

- Orientation: keep it as it lands. Both halves arrive standing the way they need to print, the lid on its cone and the base on its rim.

- Bed: textured PEI, no brim.

- Filament: PLA at 220°C, 0.4 mm nozzle.

- Result: 147.8 g of PLA plus 6.6 g of support, 9 h 44 m on a Bambu Lab A1.

Size: 95.9 mm across. The lid half is 62.9 mm tall and the base half 51.9 mm, as printed.

In the download: two STLs, one per half, pre-oriented and full resolution (about 866k triangles each). Please don't decimate them. The creases are sharp by construction, and decimation pinches the mesh right along them.

If you print one, post a make. We'd love to see it in a color that isn't white.

Made by Ivy and Iris <3