W=1, B=1 |
W=1, B=2, A=+ |
W=1, B=2, A=-- |
W=1, B=3, A=+ |
W=1, B=3, A=-- |
W=1, B=4, A=-- |

W=2, B=1 |
W=2, B=2, A=+ |
W=2, B=2, A=-- |
W=2, B=3, A=+ |
W=2, B=3, A=-- |
W=2, B=4, A=-- |

These trefoils are special cases of the Scherk-Collins sculpture paradigm, which started with Collins' "Hyperbolic Hexagon." This sculpture can be understood as a six-story tower cut from the center of Scherk's "Second Minimal Surface," bent through 360 degrees and closed into a toroidal loop. Sequin's "Sculpture Generator I" captures this construction in a procedural form and generalizes it to include Scherk towers with an arbitrary number of saddle stories, and which have some twist applied before the tower is closed into the toroidal ring. Later this program has been enhanced to allow not just ordinary 2nd-order (equestrian) saddles, but also higher-order saddles in which more than two valleys slope down form the central point, separated by an equal number of upwards-bent ridges. In a further extension of the Scherk-Collins paradigm, it was found that one can wind the saddle-chain more than once around the toroidal ring. For a double loop, one would chose an odd number of stories, so that they properly intertwine on the first and second round. With an appropriate amount of twist and flange-extensions, all self-intersection can be avoided.

A trefoil results from this paradigm, if one starts out with a Scherk tower with only three stories. This was found to be the minimal number of stories that still allowed closure into a toroid without elongating the circular holes between the saddles. If the chain is given a twist of 270 degrees, then the resulting shape will have three-fold symmetry and will display a rather intriguing and pleasing pattern of its joined edges spiraling through the central hole. Furthermore, for two values of the azimuthal orientation of the saddles on the toroidal ring, the sculpture will exhibit front-to-back symmetry.

In the "Family of Twelve Scherk-Collins Trefoils," the space of parameter combinations that meet the above conditions, is being explored for the range of saddles having from one to four branches, and for single as well as double loops around the toroid. The concept of a saddle has been extended downwards to include a single branch (B=1), which means that it is just a twisted band. For the case of the doubly-wound loop (W=2), this band does self-intersect. For the single-branch case, the azimuth parameter has no relevant effect, and thus there are just single instances. For the cases with 2 and 3 branches, all possible constellations are being exhibited, showing both azimuth values that give front-to-back symmetry. For the fourth-order saddles (B=4) the structure gets rather busy and starts to loose its aesthetic appeal, and thus only as single azimuth value is shown for W=1 and W=2.

Other Toroidal Scherk-Collins Sculptures

Page Editor Carlo Séquin