Architectural Wheel Wells

The imaginary macrocosmic wheel introduced in the last post – by reason of co-planing with east/west walls that manifest (both) celestial cube projections – imparted special meaning to circular windows, that of transporter housing. In this post, dedicated transporter accommodation guidelines are abstracted from macrocosmic wheel geometry for adaptation toward cube-based shelter.

There are 2 approaches to such application, both involving the wheel’s squares – a logical enough proposition as rectilinear geometry characterizes the code’s signature architectural scheme. To begin implementing both approaches the wheel undergoes primary rotation to a specified location’s longitude, followed by secondary rotation to the required latitude. How does one know where to stop and park the wheel? By first observing the scaled down microcosmic representative consistent with the geometry AND orientation of the macrocosmic version, an entity imagined to be suspended just above the surface location.

In the first scheme, the wheel is rotated latitudinally such that any line of opposing vertices is oriented vertically, relative to the surface location. So positioned, the perspective is shifted to face the wheel rep longitudinally wherefrom focus is placed on the square angled back to appear as a rectangle.

Macrocosmic wheel Latitudinal positioning

The ratio of the square’s rectangular projection onto the plane of the north/south polar perspective is precisely √2 : √3 or about 4 units of width to 5 units of height. A rectangle of this proportion constitutes the basic architectural slot for a transporter.

Wheel slotted accommodation

As such whole number multiples of this ratio affords accommodation for the range of likely transporters, with one befitting a bicycle – or a pair parked oppositely, while 2 rectangular slots accommodate the family car, truck, or van (CBS compactness is not conducive to 2-car garages and relegates them to the past).

With the second approach, the macrocosmic wheel is latitudinally rotated such that a square is positioned at the top with its outer edge aligned horizontally.

Macrocosmic wheel roof orientation

 

From the east/west perspective, the square’s orientation is essential but the ratio of its projection is of little consequence.

Transporter annexation

In moving to the polar perspective, however, the square declines at a pitch of √2 : 1 or about 35° which determines the slope of roofs set on structures annexed to east or west walls receiving their guidance from the celestial co-cube projections.

Such annexations can be as simple or varied or complex as imaginable, provided walls are oriented and roof sloped as specified – conditions which are set in stone as far as the code goes. Otherwise, these basic wall and roof specifications frame the pattern from which to sculpt custom transporter housing.

Wheel Port Options

The challenge is to have such a wheel port fit in the confines of the CBS roof profile. East and west walls of such annexations may include circular windows, and they may be part wheel port/part (low) loft; and they are not required to function as wheel ports at all.

Hybrid wheel ports that are half annexation, half fused to the CBS roof will be described in the next series of posts on Polytechnic Integration after methods attaining a full wheel/house fusion are explained in that Part IV.

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The Macrocosmic (cubodal) Wheel

Macrocosmic WheelAfter wrestling with torsional forces, then deriving spherical vertex vitals for the transporter shell, the concept of the cubodal wheel’s macrocosmic version is pretty simple. It is nothing more than a wheel interpretation of the geocentric cuboda’s triangular orientation. In essence, this perspective represents the wheel in profile. The purpose of designating such a version from the greater cubodal wheel (in which the center sphere is occupied by earth) is to elevate the wheel to its proper importance, and by so doing, guide design of transporter related constructs that are fixed to earth.

Macrocosmic wheel longitudinal alignmentTo effect such guidance, the macrocosmic wheel is positioned longitudinally, just as was the cubodal shell serving as the foundation for the celestial co-cubes – via primary rotation about the axis both spanning opposing cubodal polar vertices and transfixing earth’s poles. The difference is that the macrocosmic wheel is rotated such that its vertically-aligned edges coincide with the longitude of interest, and such that the plane of the wheel’s central hexagon co-planes with the great circle formed by the longitude (and its opposite).

So positioned, the macrocosmic wheel is then located latitudinally via secondary rotation about an axis transfixing 2 opposing equatorial triangles, that is the same rotation mode as the free and dynamic cubodal wheel.

macrocosmic wheel co-planing

 

With such positioning and rotation, it is easy to see that the great longitudinal circle/ central hexagonal plane also co-planes with those common to both celestial cubes. As the commonality of the latter 2 planes is characterized by rotation – relative to one another from a local perspective – so the greater macrocosmic rotation possesses a local manifestation, one best depicted by a microcosmic representative.

Microcosmic Wheel Representation

The local rotation of co-cube planes in common finds expression as circular windows intended for any east or west facing wall. Thus co-planning of the  macrocosmic wheel’s local microscopic representative perfectly attunes to the guiding circular potentiality.

Planar bisection circles

All this co-planing integration is justified in how the circle presents an overarching commonality: As the rectilinear plane bisects the spheres arranged to define to thereby slice out a circle, so such circular formation is just as applicable to hexagonal planes.

CIRCULAR WINDOW TRANSPORTER INTERPRETATIONSo viewed, the co-planar action of the macrocosmic wheel imbues the circular window with meaning – that of transporter accommodation. Actually I should specify that the macrocosmic wheel can imbue it so, but not necessarily. As such the circular window poses a pre-existing feature. In the next post, geometric attributes of the macrocosmic wheel are abstracted that afford more unambiguous and dedicated transporter housing.

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Wheels Within Wheels

The working part of the last post culminated with a determination of the proportion factor of the force shared between the (2) pivot points of a rolling wheel: the axis of rotation and continually shifting instantaneous point of contact between wheel and surface rolled upon.

Force Coupling FormulaAlthough this factor’s expression indicates the sharing is independent of the size of the separations between force coupling componets, the torques generated by them are vary much dependent on the separation. In theory, force couplings or parallel spoke pairs transmitting them may have separations ranging from zero (radial spokes) to a fully inscribed rim hexagon.

Within those extremes, an interesting arrangement is posed by the greater cubodal wheel’s hexagonally arrayed circles. Those circles’  diameters are each one 3rd of the circle overarching them. Each of the 3 sets of 3-circle alignments pose the settings for force couplings laid tangentially to each alignment.

Greater Cubodal Wheel Force Couplings

Examining the force’s application to each pivot in isolation, the torque on the axle is FR/3, with R being the larger wheel’s radius. Multiplied by 6 spokes (assuming an equal force applied to each), the total torque equals 2FR. By the law of force couplings, the same torque can be ascribed to the ever moving wheel/surface contact points. However, with 2 pivot points regarded together as inevitably they must by reason of their motion being linked directly, the forces are shared between the pivots according to the factors k = b + 1 / 2b + 1 and 1 – k = 1 – [b +  1 / 2b + 1]).

As cut and dried as this is, breaking down the forces with vectors along the force-carrying arcs poses some interesting results. To undrstand them, an isolated quarter circle arc is simplified with a single diagonal, and subsequently divided angularly into 30° and 10° segments, and so on. The cosine component multipliers of increasingly smaller angles resulting from more divisions yield a tangential force vector at the rim having no apparent loss in bending it around 90° such that the torque on the axis equals FR.

Quarter arc torque components

This would be nice if that was all there was to it. But the other force vector components -acting orthogonally from each change of direction at the discreetly divided arc, and and radially from the continuum of the fully rounded arc – add to pose a force countering the tangentals to the tune of negative 2/3 F. This means a torque is imparted in the opposite direction to yield a net torque of FR – 2FR/3 = FR/3.

It seems logical that the sum of negative component forces would be directed at the 45° average, but notice how each radial line of force (one of an infinity of infinessimals) passes the point of the ground pivot to intersect points along a at a continuum of the rim at all times. Considering any one of these individally, it is joined by 5 parallel other lines of force from the other 5 arcs to pose an infinity of force trifectas.

arc force component dispersals

What does all this mean practically?  Aside from the well known strength of arches, the arced wheel so configured poses an optimal dispersal of (torsional) forces, forces that can then be strategically funneled into the hexagonal (and its alternate) lattice for a smooth rolling wheel. To illustrate these ideas, an example with an actual rotational inertia calculalted using the relevant components is instructive.

Rotational Inertia and Force Sharing Example

The trickiest part of the problem is to get the 4 individual rotational inertias (as derived in Intrinsic Wheel Quantification) onto a common basis. To do this uniform linear density () is assumed so that mass (M) is proportional to length with the latter stated in terms of radius (R). Once done, there is still a fair amount of work getting a total rotational inertia factor (B), which, as was stated previously and is evident above, can be pretty complex. What the result ultimate (k) says is that 78% of force applied will be absorbed in the torque at the wheel’s instantaneous point of contact with the surface, while the remaining 22% is applied to the torque working on the wheel’s axis.

The arced rim also suggests a further division of wheels within wheels. As the structure of hexagonal lines suggests an innate accommodation pathway to mechanical advantage with a kind of wholistic semi-passive leveraged gearing, so the arced wheel suggests, by its fluid-like continuum, a hydraulic pathway to same.

Mechanical and Hydraulic Cubodal Wheels within Wheels

I was disappointed that what initially looked like a way to finesse force through the arc to a greater moment arm (for greater torque) also came with intrinsic counterforces. With the discovery of the negative force, it was also a letdown that this component could not simply work on the rim instead – and allow the positive component felt by the axis to remain undiminished. Maybe someone can find a way to do this someday. Until then, I’ll have to be satisfied that cubodal geometry responds to the wheel dilemma in a sound manner.

 

 

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Torque Considerations

Before moving on to the next topic of the macrocosmic wheel and its application to transporter related constructs fixed to the earth, I feel compelled to backtrack a little and address some nagging concerns with regard to torque as it pertains to the free and dynamic cubodal wheel.

In wrestling with concepts threatening ideas conveyed here previously, I have been left with a fascination toward torque. Torque is typically named the rotational equivalent of force, yet the units of torque are that of energy. As torque ultimately transpires at a point, that energy can be viewed as analogous to the energy content of a point mass like an electron, and may even echo the singularity (s) of the big bang.

Meanwhile, torque holds plenty of interest in the realm of common experience. In a previous post pertaining to torque and the cubodal wheel, the idea of the hub and rim guided by hexagonal geometry was somewhat quantified.

hexagonal force tracing

Although there was nothing particularly wrong in the assessment as far as it went, there is more to the story than tracing forces trigonometrically through components, i.e., spokes, hexagon and circular rim.

Conventional Torque DeterminationAs it stood, my analysis disagreed with that of the conventional approach, embodied in 2 simple related ways for torque’s calculation: force applied to a point times the distance from it to a pivot times the sine of the angle between them or force times a perpendicular from the pivot to an extended line of force.

In searching for an explanation for the deviation, I found no approaches that followed forces along the actual entities that transmitted the forces. Nevertheless I refused to believe that this 3rd way didn’t have some validity, while at the same time suspecting the deviation had something to do with a lack of accounting for the distance between radial spokes and forces applied parallel to them.

Opposing Torque ForceWhen I finally saw the light  after much trial and error, I found the solution had been quietly awaiting recognition all along. In addition to the component of force directed along the side of the hexagon, there was the component of force perpendicular to that side. With this new revelation, I still had to surrender to the standard determination of torque by extending the alternate line of force past the central pivot. Because this other line of force landed on the pivot’s opposite side, its contribution to the torque is directed oppositely and thus needs to be subtracted from the determination made previously. By doing so the combined answer agreed with the conventional analysis employed from the git-go. Although there proved to be a limit to the validity of the 3rd approach, I think it posed some interesting questions.

In taking a fresh look at the picture, it would seem reasonable that the same vertical force component working against the hexagonal side force component should also work on another pivot point – an instaneous one – between the wheel itself and the surface rolled on, and in the same direction of intended rotation as the wheel’s axis.

Instantaneous Rim Pivot Points

The above right depiction is the same as the left rotated to make the state of surface contact clear while retaining the same relationship to forces. So, if there are 2 pivot points, interesting questions posed are: Which pivot point receives the force? If shared, how? Does any diminishment of the negative torque on the axis infer more positive torque freed to work there from the other force component?

Force Sharing Static EquilbriumIntuitively, it seems reasonable that most of the force should be expended to counter the greater rotational inertia around the wheel/surface contact point. To get a precise expression for how much, I followed an approach analogous to a static equilibrium situation (right) by assigning factors (k & 1-k) to the force component acting at each of the 2 pivots. Proceeding with the above situation, angular accelerations about the axis and pivots were equated so that their equations could be solved for k. The expressions arrived at rightly involved hub and wheel radius (r and R), as well as B (the rotational inertia factor), but the expressions seemed too complex and even then did not seem to account for all relevant forces. So I tried the force couplings approach.

No matter what part of the wheel is in contact with the surface at any given moment in relation to the angles of the couplings’ alignments, the difference between perpendiculars extended to the far and near force of the coupling is constant and therefore the net torque exerted on the continuously shifting instantaneous pivot point is always the same.

Cubodal Wheel Force Couplings

Because of this rule, I was able to align the wheel to the most easily calculable configuration. With a wheel structure using cubodal geometry, the rotational inertia factor (B) could be somewhat complex, but otherwise, after trying a few simple fractions for B, answers seemed reasonable with proportion of force (K) appropriated by the contact point growing as B declined, inducing a bigger role for the parallel axis theorem’s extra term.

Thus is the virtue of parallel spokes. Previously, the opposing hexagonal rim sides could have served as the force couplings themselves. Otherwise, with lines of force, it doesn’t matter if spokes stop at an inscribed hexagonal rim or run to the (circular) rim without it. That being so, is there any advantage to the hexagonally inscribed rim?

Spokes extending form opposing sides of the hexagonal hub is a big plus, but any even-numbered polygon could theoretically effect the same benefit. The virtue of the hexagonal rim in conjunction with the hexagonal hub is that it -

  • bolsters the wheel’s structural integrity
  • transmits forces more efficiently and economically
  • clarifies analysis of forces with relative simplicity
  • supplies a sound connecting intermediary between spoke and rim.

Whether any or all of this is worth the extra rotational inertia is debatable and dependent on the required design criteria for a specific application. The most sublimely unique attribute of the hexagonally structured wheel in general is that the instantaneous pivot between wheel and ground regularly attunes to, by receiving in perfect harmony upon the dynamic of rolling, an extension of the axle centered geometry by reason of the infinite hexagonal potentiality in which the rim of the latter becomes the spoke of the former.

Hexagonal Wheel

Thus does torque pertain to the cubodal wheel. To deal with it, it was necessary to diverge from the loose analogy of force and torque and come to terms with their antithetical aspects of parallelism and perpendicularity, abstractions seeming to reflect the opposing existential metaphysics of karma and irony. So engaged I have found myself relating to this character from the film in 2001: A Space Odyssey.

 

 

 

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