Gyring Gyroid - Alan Schoen geometry

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We use a technique of building surfaces of negativecurvature, from strips of flat material:The celebrated Gauss-Bonet Theorem demonstrates thatthe total curvature of a disk-like region of a surface (forexample, one of our units) is precisely captured bymeasuring the turning excess or deficit around itsboundary. For example, consider this decagon with ten120° angles; as we go around its boundary, we turn 60°ten times, for a total of 600° — an excess of 240° over thecustomary 360° for flat surfaces. This excess is a precisemeasurement of the total negative curvature across thedecagon.We can control this with exquisite precision. Hereis a recent sculpture of a constant negativecurvature surface made from pentagons andsquares; the angles at the corners are worked outprecisely so that the total curvature per unit ofarea (i.e. the Gaussian curvature) is the sameacross the entire surface.The gyroid, like all minimal surfaces (other thanthe plane) is not a constant curvature;nonetheless, this technique gives precise control,and we use this technique to control the surfaceof the piece very finely.
Throughout, we relied on physical models for gaining intuition. Below are papermodels of the mu-snub cube which has the same topology as the gyroid surface,a smooth version of the mu-snub cube and paper versions of our unit, assembledinto a version of the sculpture.
Page 1 and 2: Gyring GyroidFor G4GXDedicated to T
Page 6 and 7: The symmetry of the gyroid is the t
Page 8 and 9: The gyroid is approximated by Conwa
Page 10 and 11: To better understand the particular
Page 12 and 13: Early test assemblies.
Page 14 and 15: We invited friends overto Chaim’s
Page 16: The final assembly at the party on

Gyring Gyroid - Alan Schoen geometry

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