Rounded symmetric spheroform tetrahedron

This is a rounded version of Patrick Roberts' spheroform with tetrahedral symmetry that retains the constant-width spheroform property.

This is a remix using tmackay's variable-detail polyhedral OpenSCAD design combined with Jamie_K's geodesic sphere design.

Here's how it works: For a given width and corner radius r, shrink the spheroform tetrahedron to width−2r, and then perform a Minkowski sum of the shrunken part with a sphere of radius r. This offsets the surface back out to the original width, but the sharp corners become rounded (the corner of the original tetrahedron becomes the center of the spherical corner).

At the extremes, when r=0 then the width is just the original width, but as width−2r approaches zero, the Minkowski-sum shape approaches a sphere. Everything in between also has constant width.

The STL file included is a 40 mm tetrahedral spheroform with 6 mm radius corners.

Print settings

Printer: Prusa i3 MK3S+MMU2S

Rafts: No

Supports: Yes

Resolution: Variable layers, 0.09-0.25 mm, 0.4 mm nozzle

Infill: 20% cubic

Filament: Lee Fung silk orange PLA (part), Prusament transparent PETG (support)

Notes:

This needs support regardless of orientation. If you have a single-material printer and need to use breakaway supports, you can probably get away with the orientation shown, with a lot of scarring on the supported surface. I used PLA with PETG support (full-contact, as if it were dissolvable support), because PETG breaks away cleanly from PLA. For single-material printers, you might find it works bettter oriented on edge, or on a vertex.

How I designed this

Modified tmackay's OpenSCAD code by removing modules and functions I didn't need, doing some minor reorganization, and adding an option for rounded corners using a Minkowski sum with Jamie_K's geodesic sphere, which gives more symmetrical corner appearances than the builtin OpenSCAD sphere.

Category: Math Art