Advanced Path Concepts

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.

Inherent-extra terrestrial path relativitiesMore 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.

2D path-transporter body correspondence

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.

built-in path fusion

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.

vertex-up wheel axis

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.

dynamic path fusion

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.

dynamic wheel-path 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.

dynamic path fusion sequence

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.

Posted in Derivations, Rolling Transport, Wheel Extrapolations | Leave a comment

The Abstract Road

The forthcoming series of posts will expand application of wheel geometry to include standalone architectural styles; non-rolling, rotor propelled transporter modes; non-shuttling mobile artifacts; and rolling-related infrastructure.

Rectilinear GridRolling infrastructure begins where Ground Design ended – flat ground in the context of either polar-rotational or diamond grids. Each of these grid types can be characterized as a rectilinear plane projected from a (geocentric) cuboda square, and the lines projected from the square’s bounding edges.

In addition to the 2D information contained in the cubodal square for the purposes of an elongated grid path,  3D completion geometry proceeds from that plane also. If viewed edge-wise from a square-up orientation, that square’s edge – and the lines proceeding from its end points – delineate multiple representations of the cuboda’s most basic innate form – the tetrahedron.

Grid Path Tetrahedra

The front and center tetrahedron’s minimal expression (2 orthogonal lines) represents path’s essential geometry with 1) a line of travel and 2) a line from which the rolling artifact may gain traction. The latter line so conceptualized is linked to the axial vector cross-product abstraction of the co-spinning wheel exercise described in Rolling Transport.

To imbue extended grid lines along the path direction with 3 dimensions, the edge-up cuboda is evoked and employed. Previously, extended berm design was guided from such, and before that the edge-up orientation defined the cubodal wheel in both its fixed template and dynamic rolling expressions. To guide berm design, the cuboda was rotated about the square plane’s appropriate line to the edge-up position.

Edge-up Path Rotation

Similarly, the edge of the square  aligned with the direction of travel may serve as an axial rotation vector about which the cubodal structure surrounding it is rotated until the hexagonal plane top by the edge is aligned vertically. By such rotation, a symbolic element of dynamism is imparted to the line of travel, like some kind of inherent propulsion vector. Viewed in profile, the 2D edge-up structure is identical to that of the cubodal wheel.

geometric mirroring of path and wheelThus regarded, the hard uppermost line of path poses an interface about which the structures of wheel and deeper path are mirrored perfectly, at least in the 2-dimensional sense. In 3 dimensions, the cubodal geometry below the line represents the optimal structure by which to support the wheel; while the (identical) wheel and path geometries together signify maximum strength with minimal structure) and deformation – the principle cause of rolling friction – is most economically opposed.

The above reasoning entailed only one edge. However, the lines bordering the elongated square plane in the direction of path are two. Because path is essentially a stationary construct with left/right symmetry in cross-section like the stationary (and hexagonally expanded) transport template, an equally valid maneuver engages the other edge as an axis with the mirror image-oriented structure immediately surrounding it rotated which in the opposite direction.

superimposed edge-up path rotations

Because the transverse aspect of 2D path can be of any width including a point, the oppositely-oriented mirror image structures surrounding the 2 edges (viewed point-on) can be superimposed, one upon the other.

Such a superimposition constitutes a fixed omnipresent potentiality of path upon which the dynamic wheel rolls. As the wheel rotates, it periodically undergoes an alternation of basic opposing events in conjunction with path – 6 or 12 times per revolution.

wheel-path resonance

The opposing events (experienced simultaneously) are asymmetric motivation and cubodal structural stability, with such attributes switching sides in a resonating manner much like a rotor in its cubodal-shifted housing.

Such path potentiality grounds and complements the asymmetric dynamism of the isolated cubodal wheel. In truth, there is a step between the left/right alternations, another potentiality that will be explored in the next post’s advanced path concepts.

Posted in Derivations, Polytechnic Integration, Rolling Transport, Wheel Extrapolations | 1 Comment

Dedicated Ground Design PDF

As with the series of posts pertaining to GDCode parts I, II, III, and IV, Part V Ground Design is capped with a PDF covering the same subject matter. The role and goal for this expression is to make the material more comprehensible and usable for anyone who is seriously interested.

It attempts to do so with illustrations following every step of textual explanations that can be characterized as running captions. The last page of the PDF includes 3 matters not covered in a post and they are all mathematical: First is house to lot size ratios which has import in avoiding septic problems. Secondly are formulas for the volume of dirt needed to construct wave-based mounds, berms, and embankments. To derive a general expressions applicable to any waveform schemes was a major headache. The approach I took was to first find the volume ratios between the simplest wave integrations and the cylinders defined by their heights and width. I then multiplied this ratio by a generalized cylindrical shell encompassing a real wave with measurable heights and lengths.

Ground Design Intro Lastly is the correspondence between waveforms’ maximum slopes and the number of maximally-sized circles able to nest into the wave’s concavity – a relationship that has utility and earth tube placement ramifications, and I strongly suspect bears a connection between EM waves and the elementary particles of physics, at least the vast majority of particles that IMO would be better termed ephemerals by reason of their short lifetimes.

Despite multiple problems, distractions, aggravations and challenges going on at once, the 4 parallel projects of this Part V Ground Design was accomplished in record time. I credit this to the mighty computer mouse purchased from the inspiration of a real (kangaroo) mouse named Gilligan (or Gillian if he was a she). Because Gilligan lived in the ground and emerged at night to sometimes demonstrate amazing leaping prowess, I feel it fitting to dedicate this post to him. Gilligan was my best friend for a year of weekly hour visits until a torrential desert storm separated us.  He now lives on in the Gilligan memorial computer mouse, my heart, and most importantly in the eternal flame of God called Love, I believe.

As instrumental as the mouse was in this product (also accessible by clicking on the graphic), it also wore me out trying to keep up with its capabilty. Thus I am taking some time off, and should resume posting in about a month on Part VI – Wheel Extrapolations.

Posted in Code Application, Code History, Cube-based Shelter, Derivations, Ground Rules, Philosophic Bases, Polytechnic Integration | Leave a comment