Circular window placements on spherical domes need only avoid the underlying structure. Minimally, this structure is comprised of those great circle members radially arrayed according to the 24-point universal ring that represents every direction of all 4 bode orientations. Such structure seems to be the basic approach for large undertakings like the Superdome. But as 24 members are insufficient for larger domes, I have taken a cue from the code method that specifies ellipsoidal domes which are projections of the arcs formed by intersecting the bode’s angled planes with a sphere.
The simplest way to proceed along these lines is to view the universal 24-point ring from above with 12 pairs of opposing points signifying the great circle diameters. Focusing on one diameter direction, lines parallel to that diameter join points equidistant from such, with lengths shortened by (the cosine of) 15° increments.
These lines represent intrinsic plane intersections with the circle of the dome’s base. As these planes also intersect with the dome’s curvature, the particulars of the arcs formed are of prime interest.
All but the triangle-up orientation manifest vertical planes – the easiest to picture and formulate in the context of the sphere from which the dome is horizontally sectioned. Once one diameter and its parallel lines are completed, the remaining 11 directions are treated similarly to obtain the final structure.
Although the triangle-up bode lacks vertical planes, possesses sloped ones (55° squares and 71° triangles) that slice through the sphere to form arcs. Viewed directly from an orthogonal perspective, these are ellipsoids. As with the vertical planes, the arcs descend incrementally in scale from the great circle diameter of the reference sphere; but, as the angled plane orientations are swung completely around the dome’s circular base, mirrored arc slopes result.
A 2-dimensional formulation of planes intersecting the reference sphere depicted in the illustration above garners the most vital (and useful) numbers of individual arcs.
The “extra” in EG domes is derived from the quantized departures of the arc radii from the great circle geodesics of the radial diameters. In light of the ease of finding particulars via a 2D approach, these structures might better be termed chord domes.
EG dome advantages are the great density of interwoven structure that can be obtained to support a more perfectly spherical covering with lighter structural members. If the bones do show through, they pose the same ellipses that characterize orientations specified. Additionally, this scheme is highly conducive to customizing, especially in biasing density toward the base where the weight of everything above is borne.