Figure 3.
Unfettered versus complete versions of hex-Rings 611 and 631. (A, upper) Hex-Ring 611 with two green DADs. (A, middle) The three-dimensional structure of hex-Ring 611 but with incomplete surrounding faces. The number 5 marks a nascent surrounding pentagon, and the number 6 marks a nascent surrounding hexagon. Unfettered, the internal angles are ideal, 108° in nascent pentagons and 120° in nascent hexagons, the central hexagonal face is planar, and the dihedral angles are ideal. The four 666 vertices cause the front four external edges a, a′, b, and b′ and the central face to lie in a plane that we designate as horizontal. Due to the green DADs, the back two external edges c and c′ rise steeply from the horizontal plane. (A, lower) The same fragment as in the middle part but extracted from hex-Ring 611 after completion of the surrounding faces. By contrast with the unfettered structure in the middle part, external edges b and b′ rise from the horizontal plane approximately half as steeply as external edges c and c′. The dashed lines draw attention to the change in the angle of rise of external edges b and b′. (B, upper) Hex-Ring 631 with two pairs of head-to-tail, red-to-green DADs. (B, middle) The three-dimensional structure of hex-Ring 631 but with incomplete surrounding faces. Unfettered, the internal angles are ideal, the central hexagonal face is planar, and the dihedral angles are ideal. The two 666 vertices cause the front two external edges a and a′ and the central face to lie in a plane that we designate as horizontal. Due to the green DADs, external edges b and b′ rise steeply from the horizontal plane. Due to the red DADs, external edges c and c′ rise even more steeply. (B, lower) The same fragment as in the middle part but extracted from hex-Ring 631 after completion of the surrounding faces. External edges b, b′, c, and c′ rise nearly as steeply as they do in the unfettered structure in the middle part. The dashed lines draw attention to the absence of change in the angle of rise of external edges b and b′.