To support, ground, and complement the (cubodal) wheel, the abstract code road was shown to be infused with geometric potentialities directly pertaining to what one would require of a real path.
More subtle potentialities are abstracted by first focusing on the angles of the edge-up cuboda’s sloping squares and triangles. When the 19° sloping triangle is the element of interest, the 35° sloped square is inherent and the pair as a whole is regarded as being fixed in place. Conversely, when focus is placed on the sloping square, the shallower sloping triangle above it is extra-terrestrial and the pair so relativized is deemed free and dynamic. Together, the alternative viewpoints pose a periodic resonance.
Secondly, beyond the transversely expanded edge viewed head-on in cross-section, a superficial correspondence presents itself. The 2D match is between the parallel (but separated) rectilinear lattices of path and transporter body, while beyond these respective surfaces lie inverted cubodal square pyramids and the triangular prisms of the transporter template’s hexagonal expansion.
In order for the hexagonal orientations of the two 3D lattices t0 match, rotation corresponding to the 35° square-to-edge-up position is required. Such a dynamic between fixed path and what would signify the more relatively fixed transporter body – together with the relative angular characterizations of the sloping planes – augment the sense of an axial propulsion vector aligned with the travel direction described previously.
A deeper potentiality is found by viewing path as the fusing line segment between 2 (starting and ending) points. So conceptualized, path in profile exhibits a kind of built-in fusion as formulated by the relationship between fusing triangular wing-pairs to a sloping element, plainly seen in how the 19° sloping triangles fuse to a 30° slope.
The fusing slope of 30° evokes the inter-grid juncture which signifies an omnipresent potential for switching grid types about a turning axis ascribed to the vertical line inherent to the juncture’s guiding vertex-up orientation, i.e., the mundane pole. Because this potentiality is inherent or built-in, the fusion may be regarded as fixed.
Conversely, a dynamic fusion is abstracted by first focusing on the vertical axis of path’s built-in grid juncture, and then viewing the wheel in relation to it. As the wheel rolls, the halfway point between opposing edge-up orientations (and their resonances) aligns with the vertex-up cuboda which will normally coincide with full cross-wheel spokes.
Assuming no slippage, the point of contact between wheel and path is instantaneously at rest. When that point is the end of the spoke, the spoke has instantaneous alignment with path’s inferred axis and can thus be regarded as an extension the of path’s juncture. In such context, and due to its position of maximum instability, the wheel is deemed free to initiate rotation about its vertical spoke.
Because the spoke is instantaneously as one with path’s innate juncture axis, imaginary rotation of the juncture follows the wheel’s vertical spin and in so doing the inherent elements of the vertex-up cuboda (with their angles) come into play. Specifically, upon a rotation of 35°, the inherent 35° sloping triangle manifests in profile – an equivalence that represents the guiding slope of intra-grid juncture potentialities along grid lines.
If the vertex-up cuboda undergoes rotation to an angular separation of 55° (the complement of 35°) – the deeper inherent angle of the 45° sloping square plane manifests in profile – the plane to which the inherent triangle fuses.
Another dynamic is found in the rotation of (path/wheel) paired vertex-up cubodas: the interface comprised of the inherent triangle and the 35° sloping square of edge-up path. As such, the arcs attending the planar transformation infers an element of dynamism to go with the areal traction analogized by the planar interface (a correlation that complements the maximum penetrability of the vertex-up wheel into path). Happily, the 35° sloped interface coincides with the (edge-up) tetrahedral line attributed to traction.
At first glance, the un-rotated tetrahedron’s horizontal line correspondence with the co-spinning wheel’s axial cross-product would appear broken upon edge-up rotation. Luckily, however, the correspondence is continually restored through the fixed bend between the transport template’s cubodal and hexagonal expansion geometries. In such case, the bend can be viewed as a model by which path is guided to mirror the template’s rectilinear plane with its on square-up potentiality, again about the axial (edge) vector aligning with path.
Such reasoning in conjunction with the wheel’s active role in the derivation makes this particular fusion a dynamic one – 6 times per revolution of the wheel.
Finally, the vertical axis aspect of the wheel/path interaction combines with the line of path rotation and rolling axial vectors to complete a 3-dimensional dynamic potentiality.
Generally speaking, each of the path potentialities identified in this (and the previous) post come in pairs of opposing directions and in this sense they cancel. As imbued with dynamism as path is, there is no bias nor claim to self-actuated transportation. External energy input to the wheel is still required, as well as the intelligence to control it.
The most that can be claimed of such path is its optimal readiness for, and conduciveness to such externalities – optimal in light of how path so conceptualized addresses the fine balance between the competing concerns of traction and rolling friction. With this abstract treatment of path concepts concluded, the next post will apply them to real roads.