5444, [0][1]* on a constant negative curvature surface

Chaim Goodman-Strauss and Eugene Sargent

G4GB, March 2014

The intrinsic geometry of this surface is forced by the lengths and angles of its pieces.
Any surface with this intrinsic structure must buckle and bend in this general way. A paper model is the basis of the specific extrinsic geometry of the final steel sculpture.

We build surfaces of negative curvature from strips of flat material: The celebrated Gauss- Bonet Theorem demonstrates that the total curvature of a disk-like region of a surface (for example, one of our units) is precisely captured by measuring the turning excess or deficit around its boundary. For example, consider a decagon with ten 120° angles; as we go around its boundary, we turn 60° ten times, for a total of 600° — an excess of 240° over the customary 360° for flat surfaces. This excess is a precise measurement of the total negative curvature across the decagon.

We can control this with exquisite precision. Consider a recent sculpture of a constant negative curvature surface made from pentagons and squares; the angles at the corners are worked out precisely so that the total curvature per unit of area (i.e. the Gaussian curvature) is the same across the entire surface. 

After water-jetting out the pieces, Eugene began to model each one off of a corresponding. His specially made bending machine, first used for the Gyring Gyroid, sure came in handy.

The sculpture took shape in the driveway, and soon was hung for the first time. ... then to be carefully coded, and taken apart. A simple little crib sheet made everything a snap! Or maybe not.

This code reads the same upside down as right side up, but not for right-to-left nor up-to-down nor on the diagonal. It is designed to be easy to drill into dozens of pieces, quickly and to withstand powder coating. But it made most people cross-eyed, including the sculptors. Oh dear.

Heroic efforts saved the day! And soon it came together...