Elementary Rounding

It might be easy to view the transporter template as nothing more than a pattern of lines and planes whose angles are as set as a crystalline lattice, and that therefore anything chiseled from the pattern will be characterized by those same hard angles.

Although spheres’ inherent contribution to the template have been duly noted, how exactly they may be brought into play for practical applications needs some explanation.

Template RoundingTo appreciate the scope of spheres’ potential uses, what bears repeating is that every intersection of lines – which are potentially anywhere desired – may represent the center-point of a sphere of any size. So conceptualized, an arc of such a sphere curving in the plane formed by intersecting lines effectively round that plane.

Taking this idea a step further, spheres may be centered on 2 such intersections having one line in common; or centered at each end of any line’s endpoints. In the simplest cases presented in this post involving more than one sphere, spheres must be of equal size. Other than that they may either be in contact, separated, or overlapping.

Cylindical Rounding

So specified, the sphere pair may be joined cylindrically with that form’s axis coinciding with the line and its ends melding diametrically with the sphere in one continuous surface. By extension, a plane defined by a minimum of 3 lines may be rounded 3 dimensionally by centering spheres at each corner, joining these with cylinders, and capping them with am identical plane that is parallel to the original to form the desired continuous surface.

Uses for such constructs can be found inside or out of the transporter, but the most obvious application of the technique lies externally with the transporter’s shell. Its vertices – formed where adjacent planes converge – afford the loci where spheres may be centered.

Transport Shell Vertex Spheres and Cylinders

With spheres so located, cylinders (real or imaginary) are run between the spheres placed at either end of edges formed by adjacent planes. Each plane may then be dealt with individually, with identical planes set parallel on the cylinders, as was done previously with the triangle.

Spaces formed between base planes and their externally translated replicas are equal to the spheres’ radii and can be used to run wires and cables; machinery, mechanisms, and electronics; and to provide storage or recesses for movable conveniences.

All the preceding involves convex surfaces only. As far as I have been able to determine, there is no possible way to retain 100% surface continuity when concave surfaces are introduced – and doing so while adhering to code geometry.

Concave-Convex Discontiniuities

However, there are a few basic ways of dealing with concavity on an external convex surface in a sensible and harmonious manner which include proceeding as usual with spheres, cylinders, and parallel planes, and simply accepting the inevitable concave crease; and slicing along the underlying base planes that form the concavity.

Concave Template RoundingA 3rd way to deal with concave discontinuity is to lay spheres and cylinders (of equal diameters) into concave creases and longitudinally slicing the unnecessary cylindrical cross section away. The same result is achieved by having the sphere/cylinders serve as the cap to a mold filled with a casting. Discontinuity is limited to the ends of any such application where it is made minimal by slicing orthonally into the cross sections. Without such slicing, this technique poses one way of rounding the interior of a shell, the other way simply going with the inside of the external shell and discarding or disregarding the base (planar) shell.

Such are the most basic methods of rounding the templates hard angles. Advanced rounding techniques entail employment of varying sized spheres or conical, parabolic, ellipsoidal, or sinusoidal wave forms. To describe these require other basic principles, and as these are best introduced later, exploration of advanced rounding will have to wait.




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The Magic Key to GDC

The hexagonal shift of the cubodal wheel and the transverse expansion of its central plane, as well as the abstraction of the wheel in general and Geocentric Design Code as a whole owe much to a most real and down-to-earth artifact, thoroughly experienced.

My hunt for the perfect bicycle was not guided by the abstract, but rather by comfort, usefulness, and an assurance of some ill-defined capability, qualities which having owned road, mountain, and cruiser bicycles in the past, I had never found in one bike. I was just about ready to give up the search when I stumbled upon a small overlooked shop.

Amid the usual glitter, I was immediately drawn to one bike with a strong white triangular frame, concerning which a voice from the back proclaimed “made in Brazil.” A wide saddle promised comfort and with tip toed balancing in a cushioned 3 point stance, my feet found blessed relief from 7 months of walking out of a first bout of homelessness. The shop owner invited me to take the 1993 Caloi Pan Am for a spin.

Outside, I pedaled off effortlessly to an effortless sensation of low gliding flight whose breeze cooled the sultry Florida morning, and with the easy action of the intuitive coaster brake, I slowed with intention to buy the one speed cruiser. After a period adjusting to traffic, a daily routine of work and errands experienced a grateful easing.

On days’ off I rode the bike increasingly further, and eventually on an 80 mile ride up and down the coast. Upon recovering, I was ready for the next level, not touring but actual relocation with all my possessions sailing 400 miles down the peninsula and across the Everglades in a long hot century to a nightfall room refusal by a Ukranian innkeeper who subscribed to reverence for the automobile and atheism. Nonetheless the move was a success as I was soon lodged, employed and commuting on my amazing bicycle.

On one day I found myself in a store gazing at a square foot jewel from NASA portraying a full earth in space with focus on the western hemisphere – centered by Florida. Marveling how I could discern my recent trek from such perspective, I wondered just how far the bicycle could take me as my eyes roamed to where a mysterious glaciated land mass disappeared over the sphere’s upper left curvature. Alaska would be my ultimate goal.

Starts, struggles, and flops ensued over the next few years, but with them came vital experiences ranging from how to deal with constant in-your-face winds on the Texas coast; observing how hard Gulf dolphins humped and huffed for their food before releasing themselves to play; how to enjoy camping with minimal gear.

Cube-based Shelter RotationAs my appreciation for the bike matured, I wondered why accommodation for bikes in general had never been made architecturally, and the desire to do so was an essential ingredient to the advent of (celestial) Cube-based Shelter. Although a specific solution was not yet seen then, I was confident that, owing to the cubes’ rotation,  it was in the scheme’s geometry, somewhere.

High from the insight, another move to Biloxi resulted in a situation that enabled me to fund gear, upgrades, and an attempt for the last frontier. Armed with a 7-speed coaster brake hub, I regarded the trek as a test ride of a prototypical everyday bicycle for the masses, and I approached the quest with Ghandian “be the future you wish to see.” I set out on the first day of spring with a goal encapsulated by the phrase one continent, one season. For a well-rounded transection, I would start over Atlantic waters a few degrees above the tropics and end below the Arctic circle aside Pacific waters – with 1 mega city (Chicago) in between.

Hoping to camp 2 of every 3 nights, I found many parks’ water shut off, but on the other hand I stayed ahead of the pack’s annual migration. Near the end of weeks galloping over expansive golden prairies, mechanical problems posed the darkest cloud, but with lucky guesswork and the right CDs, I was able to keep going and find a good rhythm.

The Rockies appeared in the distance which by tricks of the terrain concealed their mysteries for abrupt views much closer than the last. In top shape at last, I grew one with the bike humping over rises and gliding windshield-less through astounding beauty coming on directly in days that grew endless, and as the ubiquitous spruce streaming by grew smaller, I experienced a sensation of growing into the warm clear sky. The longest stretch of sublime beauty was in the Yukon, and then mist-shrouded Mt. McKinley opened up for a most spectacular encore. But when we rolled downhill to the end at Cook Inlet 12 hours before the end of Spring, the clouds did not part and drizzly solemnity signaled some undefined interlude.

Emerging from period subject to what goes up must come down, I found myself finally focusing on architectural accommodation for the bicycle by finding between it and the house a geometric common denominator. To do so, I built a paper model of the cuboda and turned it over and over until finally coming to imagine its centrally interwoven planes. and marveling at how the differing planes angling off one hexagon could only be matched via rotation. Thus was the cuboda’s attribute of intrinsic dynamism abstracted.

Later it occurred to me that the 2 halves on either side of the central plane could be naturally slid relative to each other and matched up permanently with a 60 degree rotation. Although the “wheel” thereby lost its dynamism, it dawned on me that this was precisely what was needed – along with transverse symmetry – for a template to guide design of components at rest relative to the forward motion provided by the dynamic wheel. Soon following was the hexagonal plane’s transverse expansion concept.

The Key to GDC


As these ideas crystallized, the bike leaned in its space against the wall, with its frame roughly attuned to the hexagonal pattern of triangles. The correspondence between the pattern’s innate circles and wheel separation was a bit eerier. The final match came with the handlebars. On a hunch, after determining angles of template lines shooting off from the center and comparing them to those of the handlebars, I found these also concurred!

Its as if this bicycle had been guiding the code all along.  Not only had it played a crucial role inspiring cube-based shelter, it had provided all the clues to the cubodal wheel’s abstraction essentials and a general template for all rolling transporters. If Geocentric Design Code was unlocked by a key, that key was this bicycle.

Before the 5,000 mile trek to Alaska, the bicycle had carried me 10,000 miles and post-trek another 10,000 to equal 25,000 miles, or one orbit before I returned it to the earth. That the bicycle had been designed as it was I find most interesting, perhaps evidence that the code geometry has been followed in a groping quasi-intuitive way for some time now.



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Head-on Transporter Guidance

Head-on Transporter TemplateTo appreciate attributes of the transporter template profiled in the last post, it is necessary for the pattern to be viewed head-on such that the (horizontally oriented) h-shifted cubodal wheel is led by a vertex – pointed directly toward, or away from, the direction of travel.

From such perspective, lines and planes converging on the vertex angle back and away from that vertex. Although oppositely  mirrored around a central dividing plane that appears as a vertical line, it is important to remember that each side is characterized by a full cubodal pattern – with all its linear, planar, spherical and other 3D attributes.

Transverse Template Pattern Attributes

On any given side, the pattern pervades the form up to where it terminates at the central hexagonal plane. From there, the cuboidal pattern resumes into the other side,  but with its orientation rotated 60 degrees.  At the central vertically-aligned hexagonal boundary plane, instead of triangles representing interfaces between octahedra and tetrahedra, with the hexagonal shift they interface tetrahedra to tetrahedra and octahedra to octahedra.

Plainly, the natural division of the mirrored sides posed by the central bisecting plane represents a fundamental pattern break. As a consequence, the notion of transverse expansion at the central hexagon (or between inner hexagonal faces of the 2 sides regarded as separate halves) naturally follows.

Hexagonal Expansion

In so doing, a new form is introduced into the template to aid or complement the cuboda and its pattern: a hexagon of added dimension whose thickness is rectilinear. The form’s fundamental  infinitesimally divisible unit – the one comprising the hexagon’s repeating pattern in a simple alternating fashion – is the triangular prism. This polyhedron is 3 squares, arranged triangularly.

Introduction of this geometry into the transport template is not only sensible, it is very useful because the intrinsic pattern of the hexagonal expansion is comprised of transverse lines and planes not supplied by the template’s cubodal geometry.

Transverse Design Guidance

As such the expansion constitutes a potentiality of elements that may guide or frame the design of key transporter components such as transverse shafts (e.g., axles), flooring, seating, consoles, windshields, cargo boxes, etc.

Mirrored Hexagonal ExtensionThe reasoning behind the hexagonal expansion may also be applied to any longitudinally vertical hexagonal surface. Such a maneuver is referred to as a hexagonal extension. The only requirement for its employment is that it be capped appropriately with a cubodal structure whose orientation mirrors the orientation underlying the surface extended from.

Wheel Version Planar InterchangeabilityAs suggested in the previous post, the triangle- or square up versions posing the choices from which to guide design of a particular transporter are interchangeable in the sense that the orientations of lines and planes’  of one version may be found in the other version – a quality that gives either version intrinsic geometric harmony with architectural accommodation geometry specified in a future post.

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The Virtual Transporter Template

In the last post, the hexagonally-shifted cubodal wheel afforded a fixed left/right symmetry to enable the basis for a 3D pattern by which the design of rolling constructs’ non-rolling components – at rest relative to the motion provided by the dynamic wheel – are guided.

The h-shifted wheel poses 4 fundamental alignments from which to select an orientation – 2 vertical and 2 horizontal. For the most basic structure and components, the latter alignments are chosen for these reasons: while both horizontal and vertical alignments exhibit left/right mirror image symmetry, only the horizontally-aligned pair possess up/ down asymmetry. The reason for going with this attribute is that it suggests a bias attuned to the vertical differentiation above and below the rolling interface of the ground.

horizontally aligned h-shifted cubodal wheels

Other key reasons for selecting the horizontally-aligned h-shifted wheel is that so-oriented the wheel exhibits horizontal lines attuned to the direction of travel. Going a step further along this line of thinking, the cubodal pattern may be elongated naturally in the direction of travel without the innate pattern being altered in any way. Furthermore, any particular arrangement or proportion of the cuboda’s front or rearward facing planes may be preserved with such elongation.

The horizontally aligned h-shifted wheel thus elongated constitutes the transport template from  which the most vehicle’s most basic components – frame, body, etc. – may be sculpted. As such, this template is a virtual template in that any of its infinite potentiality of lines, planes, and solids may be appropriated for the component being designed.

Transporter Template

In addition to the availability of these elements of angular geometry is the inherent potentiality of omnipresent spheres (of any size) by which the angular elements may be sliced into circles, arcs or partial spheres; extended cylindrically;  or effect the rounding of 3D convergences.

cubodal wheel's 2 horizontal h-shifted versionsPerhaps the most natural question begged thus far is which of the horizontal versions should be appropriated: the triangle-up or square-up version? In practice either version may be chosen. A decision on which version to employ depends on what best meets the requirements of a particular transporter. In actuality, line and plane orientations of one version may be incorporated into the other by special maneuvers. This quasi-interchangeability is better illustrated in the upcoming post focusing on the template’s head-on perspectives.Vertically-aligned h-shifted Cubodal wheels

Although the template does not employ the 2 vertically-aligned h-shifted wheel versions for reasons that should be obvious in light of the reasoning thus far, they can still find use for not insignificant secondary components. As such the vertical versions may be incorporated into the template with special methods described later in the posts dealing with polytechnic integration.

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