One hundred and forty-four sections were cast to create the hollow torus shape. The sculpture surface involved a complex equation called a 3-adic curve with lines that, when wrapped around the torus, have no end and no beginning.
employment and an opportunity for local residents to rebuild their lives. This two-year project included casting two smaller
torus casting assemblies of one tenth scale and quar- ter scale. Danko Arlington cast the Umbilic Torus SC (Simons Center) sculpture entirely out of silicon bronze, and the finished piece was welded together as an assem- bly at neighboring New Arts Foundry, a Baltimore shop that specializes in investment casting bronze. Te 144 sections creating the hollow torus shape were
sand cast individually in nobake molds, as was an internal inverted sand cast cap for mounting and six internal but- tresses for additional support. The 2,400-lb. drag section of the mold for each
unique, twisted torus panel with its associated 3-adic surface curve was five-axis CNC machined. Ferguson used Mathematica technical computing software from Wolfram, Champaign, Ill., to create each torus section, and he performed hand calculations to program the 3-adic surface filling curve.
“
“No off-the-shelf CAD/CAM system exists for doing
the continuous 3-adic surface filling curve tool path defini- tions, let alone with all the (normal bundle) angles associ- ated with the surface,” said Danko. “Only a PhD of math- ematics could crunch so many complex geometries, as this task was truly mind blowing to create in such a scale.” Te G-code generated Mathematica movements were compiled and downloaded to an XR4050 five-axis gantry machine tool. “Tis was an older, used X-Y-Z gantry milling machine [Ferguson] retrofitted with two more axis movements to make it completely five-axis compatible,” he said. Te advantage with this gantry machine, according
to Danko, is the X and Y pathways’ elevation above the mold, which keeps the abrasive sand away from the trav- els. In addition, the Z arm is retrofitted with two more movements. “So, the tool was able to have yaw, W, and pitch, B,” he said. “Te controllers for the yaw and pitch were also safely out of the way of sand flying from the diamond cutter.”
Tousands of tool path movements accurately made the
cavity for each section, which was serialized for its shape and orientation so it could be matched to its mat- ing piece later, during fabrication.
Only a PhD of mathematics could crunch so many complex geometries, as this task was truly mind blowing to create in such a scale.” —JOHN DANKO
Mathematical Magic The piece is based on a set of paramet-
ric equations that create a deltoid, a type of hypocycloid which looks like a triangle with concave sides. The shape is created by rolling a fixed point on a circle inside another, larger circle. “Anyone who had a Spirograph drawing set most likely made this design,” Danko
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