3-1 Assymetrical Umbilic Toroid Mirror Pair

Free standing Asymmetrical 3-1 Umbilic Torus and Mirror 

3mf - vs. STL: note that models are scaled 112% within Bambu slicer to get as large as possible on build plate / volume on P1S / X1C. 

Feel free to scale down for smaller works of mathematical art. 

Background:

I designed several 3-1 umbilic toroids as gifts for friends and family to have at the celebration of life memorial for my Father in the Autumn of 2024. More background info below.  

3-1 umbilical toroid means there are 3 rotations of twist in x-z or y-z direction for every rotation around the circular path on the x-y plane.

(less technically one could refer to an umbilical toroid simply as a twisted doughnut)

I acknowledge there are several other Toroid models here, but honestly, mine are modeled independently. 3D Modeling wise the path is elliptical with slightly different dimensions. Cross sections made on 3 orthogonal planes along the path at equal angles and rotated at same angle to one another. The cross sections have consistent arc segments except dimensions vary resulting in an asymmetrical torus shape that expands and contracts as it flows around. 

Build wise, I orient my Torus designs to an upright placement (not what the slicer will auto select) with the flattest side on the build plate resulting with a first layer roughly centered in such a way that the torus will stand upright freely on its own. This also reduces  scars/uneven portions to smaller less noticeable underneath areas. 

Tree supports are great for this shape. 

Profile has fuzzy skin on contour only. 

You can go smooth (fuzzy skin set to none) but I recommend to increase to 5W and use a lighter color ideally marble color that you can sand out the visible layers and scars without seeing the sanding marks.  gives one the hands on sculpture feeling..

For super smooth you could use PLA instead then perform some sand and fill iterations with primer and even bondo, then finish paint. 

More background: 

My Dad was a professor at the university of utah for 20 years in mathematics and 20 years in computer science. He was the first patron of and good friends with Heleman Ferguson. This is how my Dad's prized possession, a marble 4-1 umbilic toroid, came into my life at an early age and was displayed prominently in the home(s) for many years. I loved to use my eyes and hands to follow the pathway or ridge around and around in awe of the beautiful constantly curving surfaces and a mathematical formula on display that described them.

We donated the marble 4-1 torus to the mathematics department at the U of U recently. It was installed with direction and approval of the artist.    

I haven't put the time in to create a 4-1 torus - one of these days, until then, enjoy the 3-1 versions!