MORE TOROIDS

3d Polyfelt provides a good way to make polyhedral surfaces, calculate its Euler characteristic, study curvature at vertices computing their angular defects, etc.

Pictures below are from the workshop held at the ICME-13. All these figures have the Euler characteristic of a torus, i.e. V-E+F=0.

Torus inspired by triangulation found by Sarah-Marie Belcastro

This torus has 42 faces (12 pentagons, 24 hexagons and 6 octagons), 126 edges, and 84 vertices, which gives you Euler characteristic V-E+F=0.

Here are some more simple toroids with hexagons, octagons, decagons, and squares:

Can you make others? Send us and will be published here!

Some references:

• Bonnie Madison Stewart, Adventures Among the Toroids: a study of quasi-convex, aplanar, tunnelled orientable polyhedra of positive genus having regular faces with disjoint interiors, 1970; 2nd ed. 1980.
• http://www.software3d.com/Misc.php#Stewart
• More toroids in the page of Steven Dutch  http://www.uwgb.edu/dutchs/symmetry/stewart3.htm
• Our page Tessellations of surfaces with 3dPolyfelt