An interesting association between seemingly disparate code-related applications ideas has just occurred to me.
The first concept pertains to the legitimate use of vertical lines extending from parabolic arches (or suspension cables) in bridge design, and the second entails the separation of cube-based abode roof sections to top Wheel-Base Architecture annexes. The link between the seemingly discordant geometries of these 2 realms is found in their (identical) qualities of reflection.
In either case, any one line directed orthogonally to the form’s orientation and meeting the parabolic curve or CBA plane is ultimately reflected back in a parallel (but opposite) direction after bouncing off 2 planes (viewed edge-on in profile as lines) that are complementary in their alignments relative to each other.
The parabola’s complementary tangent line pairs essentially constitute an alternative definition of that form. By extending those tangents to their orthogonal convergence, a picture of the CBS roof profile presents itself.
Conversely, in viewing the CBA profile, any vertical line converging with one roof plane (e.g., at a wall plane juncture), and reflected off the other roof (or its extension), together pose a complementary tangent pair onto which a parabola may be inscribed.
The parabola of course possesses infinite complementary pairs up to and exclusive of 90° as the curve never gets there.
On the other hand, a potentiality of infinite complementary roof sets are posed by cube-based abodes over all earth’s latitudes by reason of the fixed primary celestial cube’s projection onto earth’s ever changing curvature. The totality of this situation can be represented by a circle inscribed in a square.
It is interesting to compare the likeness and difference between the complementary relationship’s two expressions by nesting the circle in the parabola such that the circle’s center-point and the parabola’s focus coincide, and then circumscribing the circle with the square by which the complementary angles are provided.
The ultimate parallel reflection of the circle in square is with the radial line extending from the origin. Conversely, this line serves as the reflective intermediary to the parabola’s paralleling reflective pair and is not itself paralleled.
As the bode in and of itself exhibits the quality of parabolic reflection, so does the geometry of the CBA derivation in which the prime celestial cube meets its co-cube. As this association arose from separating wheel-based architecture’s CBA planes that suggested complementary parabolic tangents, so a more sublime WBA virtue is found in its more pointed expression of the parabola.
To me, such an attribute nails down the whole reasoning behind cube-based abodes – especially in light of how the parabola keeps popping up in the geometry of natural law and phenomena. As to the question of whether the reflection relationship shared between the parabola and the CBA circle/square representation above has deeper implications, I don’t know. Perhaps the mystery has already long been addressed. If not, as far as I am concerned, it will have to wait for later.