The Co-planing Cubodal Wheel

The vertical-axis wind generator concluding the previous post is evocative of yet another application of the cubodal wheel – a free artifact in which the central hexagonal plane parallels the surface traveled upon.

Cubodal Disc Orientation

As such, the surface that this disc co-planes with may be as varied as subsea thermoclines, the water’s surface, atmospheric layers, and gravitational potentials of outer space. In its place, the cubodal disc is a dynamic entity in which its rotation is often only to change direction.

Because the disc is merely an orthogonally re-oriented wheel, its structure draws somewhat from a correspondingly re-oriented transport template. The critical difference is that its dynamism isn’t neutralized with a hexagonal shift and thus does not generally feature a hexagonal expansion.

Hexagonal Disc Expansions

However, the HXP may be employed internally if on either axial end, the cubodal structure meeting it is hexagonally shifted. The HXP may be also be implemented externally if it comes in even number sections as described in the last post. In either case, the dynamism of the disc as a whole is preserved.

The simplest application of the disc is a satellite. In such case the parallel surfaces are both natural and abstract ones: the orbital planes posed by the geocentric cuboda, or if you prefer – the macrocosmic wheel. So conceptualized, there are 2 polar and 2 subtropical planes.

The Basic Disc Satellite

Proof pending, I will go out on a limb and state that the disc orientation poses the maximum moment of inertia of the 4 basic axial possibilities, and is thus one of the 2 optimal orientations for attitude control. Please correct me via the contact info in the sidebar or in the comment section if I am wrong.

Of all the cubodal wheel applications, the satellite may conform most to the form’s hard-edged external framework and planes because 1) there is no human habitation, and 2) there is no fluid body like an atmosphere to contend with.

However, and this applies to all cubodal wheels, the disc may be manifested in other basic forms that express the cuboda in some way. As a double conical form, it can be shaped by the sloping triangles, squares, or the lines shared by them.

Conical and Ellipsoidal Disc ExpressionsAlternatively, there are ellipsoidal expressions. The fact their having 2 foci can be viewed as expressive of dynamism. Aside from that generality, of the 4 ellipses explored in the Rolling Transport post in which the ellipsoid was shaped by circles , there is one defined by circles inscribed in the outer triangular layer and circumscribing the cuboda’s central hexagonal that is specifically expressive of the cubodal wheel and disc.

I don’t like applying cubodal angles to elliptical curves as their key points are 0° horizontal and 90° vertical only. However, the angles can be applied – reasonably, I believe – from the focal points to the curve.

Cuboda-guided Elliptical Foci

As such, the common edge (that only has a defined angle with rotation) poses a kind of middle average and symmetry that can be manifested as an ellipse of modest eccentricity. The dual angles of the wheel’s sloping square and triangular planes yields a more pronounced ellipse that is expressive of the wheel’s asymmetry.

Use of both is made possible by the basic attributes of the ellipse: it is naturally bisected along either of its axes to allow vertical extension or transverse expansion or both with the extending or expanding forms being cylindrical.

Ellipsoidal Satellites

This feature allows the dueling ellipsoidal proportions to be used together in a hybrid. Alternatively, The 2 different half ellipsoids may be joined along their major axes to grant a preferred axial direction. Finally, the bisection allows for toroidal forms with elliptical cross-sections.

Of course ellipsoidal expressions may be structured with cubodal planes, and either or both the static housing and dynamic rotating element should manifest the cuboda’s asymmetric attribute to exhibit a dynamic resonance between the two.

Cubodal Wheel Asymmetric Rotor and Housing Resonance

I have digressed. This matter should have been addressed long ago in Rolling Transport when the cubodal wheel was introduced so that when it is referenced perplexity doesn’t ensue for curious visitors over an abstraction that at first glance isn’t circular and obviously won’t roll. But better late than never. Next up the cubodal disc is modified to give it a preferred direction.

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Fish and Fowl Friendly Electricity

In the last post regarding what the cuboda has to offer in the way of propeller design¸ it was noted that the available geometric elements, and the rules of their use, also applies to turbines in low-head hydroelectric dams.

Dam Geometric GuidanceA code-guided dam can be regarded as a combination of bridge and grid berm – and marine craft holding back and directing water. All these constructs can trace their design to the cubodal wheel, with the bridge guided by “path” and marine vessels by the transport template. Although these geometric conceptualizations employ the cuboda’s edge-up orientation, the bridge (and berm) aspect is generally oriented at a right angle to marine craft where viewed as aligned to the flow of water. As the dam has 2 sides facing entirely different situations, both orientations possess a clear opportunity for expression.

Like bridges guided by path, the code dam is oriented to either the Polar-rotational or diamond grid (or both if a 1/8 circular arch dam). In cross-section, one side takes its slope from path’s edge-up 19° or 35° angles which, being shallow slopes, would generally be earthen. In this sense the dam’s bridge aspect can morph into a wave-formed berm.

Cubodal Dam Cross-section

The other face uses the transport template’s leading 60° angle. This relatively steep angle would likely require concrete to hold its slope and a lot of it. Thus the angle would be ascribed to buttresses supporting vertical walls of a so-named dam. Without a seamless merger between the 2 orientations, their exposed disparateness requires a cube link to integrate them.

Recall that in deriving path geometry, the square-up oriented cuboda is obtained from simple rotation about the edge-up cuboda’s defining top edge. Thus the square-up’s 55° sloping triangle can also be used in lieu of the edge-up angles.

Square-up cuboda utilization

Regardless of such explicit manifestation, the essential lines of the square-up’s tetrahedron unifies the 2 edge-up orientations and as such may guide rebar placement schemes to tie them together.

If the main course of the dam is straight, its ends can be swept convexly or concavely, as with berms on the upstream and/or downstream side, in a straight sloping conical manner to join or constitute terrain-merging abutments.

Dam Rounding and Fish Ladders

Either path or template geometry can be used to construct fish ladders. Whatever structure is tried, additional planes should be appropriated to ensure fish land in the trough above instead of missing and skidding down the face of the dam.

As the penstock pertains to the dam’s dynamism, it is aligned with the transport template’s 60° slope, a line that also bears a line of spheres attuned to the component’s cylindrical cross-section.

Penstock and Spillway

Another component that can use the 60° slope is the spillway. In this case the angle is keyed to the maximum slope of a waveform to smoothly transport water from one level to another.

However the powerhouse is laid out in relation to the dam, the axial flow reaction turbine is vertically oriented and triangle-up and thus must be linked to the dam’s geometry. This can be done with a cube link placed on either the path or template profile triangle plane. But, for most every reason, a cube link placed on the square-up orientation would be superior.

Dam Turbine Accommodation

The spillway waveform presents an interesting possibility with the precisely 3 circular cross-sections allowed by the curvature which is directly related to the wave’s maximum slope. The bottom circle might represent a water chamber. In such case the chamber could supply the 6 hexagonally-arranged jets aimed tangentially at a Pelton-type impulse turbine in high head situations.

The basic cubodal turbine spinning on tangential flow would be characterized by the same asymmetric geometry as the cubodal wheel. This translates to 3 buckets oriented one way and interspersed with another 3 oriented oppositely.

Cubodal Impulse Turbines

A possibility exists for more buckets to be placed on the turbine using the 10.9° or 19.1° transformation angles . These will leave gaps, but ones arrayed hexagonally as a whole and led by the main bucket arrangement. In the above illustration, the simplest bucket configuration of a tetrahedron/square pyramid combo amply uses the vertex spherical rounding method.

Turbines using tangential flow can also employ the 6 radial planes of the hexagonal expansion, with a construct commonly viewed as a paddle wheel to propel a small boat.

Dynamic Hexagonal Expansion Turbines

But as the HXP is a static neutralized element, it must come in an even number of sections to become an asymmetric and therefore dynamic one. An ultrasimple configuration might also be effective in some run-of-the-river circumstances requiring no alteration of the waterway.

So specified, the radial planes of the HXP may also be employed as a more bird-friendly vertical axis wind generator, specifically the Savonius type. In such a construct, a set of opposing the planes may be curved with semi-circles or half ellipses. The code specifies these have a √3:1 proportion in their major and minor axes, and that they be centered on a common focal point. If a logarithmic spiral is used, its constant radial-to-tangential angle should be 60° which may also be used as a reference angle for the aforementioned transformation angles.

Vertical Axis Wind Generator

Another way to shape such a wind generator is to twist one plane-pair 60°, 120°, or 180° such that the minimum slope of the vertical wave projected from the helix formed is either the 55° or 70° slopes of a triangle-up cuboda. As such the cubodal geometry at the top, bottom, and middle of such a construct can be tapped to catch more wind or exhale it.

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The Transformative Fluid Dynamic Cuboda

Fluid Dynamic CubodaThe cubodal wheel has thus far seen little use of the internally interwoven hexagonal planes intrinsic to its geometry. But to move or be moved by gases or liquids, these planes of alternating equilateral triangles are essential. Since the last few posts have dealt with aircraft and marine vessels, axial flow will be addressed in this one.

As efforts to improve the conversion of rotary into orthogonal forward motion with propellers are ever continuing, the cubodal wheel joins the effort with a pattern from which a common 3-blade propeller may be fashioned – either from the central hexagon, its 3 identical though axially-biased mates, or a combination thereof.

The Cuboda's Internal Planes

The central hexagon might be viewed as something akin to the actuator disc of momentum theory, but it may also organize, bolster, and cup fluid in conjunction with the other 3 hexagons which will receive the focus for real propulsion. If representing a marine propeller, it is fairly easy to imagine how water contacting these rotating planes will be pushed out and (axially) away – but in opposing directions so that forward or backward vectors cancel each other.

Cubodal Propeller Blades

If only the portion of each hexagon on one side of the axis is employed, essentially a pair of triangles joined along a common edge, the push will only be in one direction, and in so doing, the simplest of propeller blades is signified.

Viewed in profile, the “blade” makes (about) a 19° angle with the axis of rotation. Keying this to the maximum slope of a wave projected from the helical form of a screw gives a 71° lead angle that is reflective of the propeller’s pitch.

Cubodal Propeller Pitch

This is pretty high, but fortunately the blade’s axis makes for pitch adjustment – all the way to zero with the vertical plane of the central hexagon.

For a monolithic screw however, modifications are in order. Aside from cutting along the lines of the central or skewed hexagons’ pattern where deemed advantageous, bending the planes along these same lines may proceed in a way consistent with cubodal wheel geometry.

Cubodal Propeller Blade Pattern

Specifically, the bend would be the 71°/109° dihedral for triangular-to-triangular planes, and the 55°/135° dihedral for triangular-to-rectilinear planes. Which way the pattern is biased is crucial in determining the allowable bends and their direction along any particular line, as well as being generally important for how the blade is shaped outward. With the central hexagon there are 2 choices: a tangential bounded by 2 radials and there parallels or 3 radials and there parallels.

Cubodal Blade Pattern Biases

The latter encourages length which would generally apply to aircraft propellers while the former’s tendency to width would be more suitable for marine vessels. Because much of the skewed hexagon’s pattern is cut away, there is only one choice with the one radial between the 2 intersecting skewed lines (and their parallels).

Such bending between planes – as well as in the planes themselves – may be rounded either circularly or by quarter waveforms in which maximum slopes and crest or trough tangents meld with lines or plane cross-sections. Blade curvatures may also be shaped elliptically with that form’s essentials keyed to the paired triangle √3 : 1 ratio.

Cubodal Blade Rounding

The ellipse, like waves, are naturally quartered but in an orthogonal manner that makes them suitable for rounding rectilinear planes. Applying the ellipse to the whole skewed blade is interesting as its projection to a direct view reverses the √3:1 ratio. Also interesting is how when the 3 blade configuration shares a focus and thus the same circle of curvature, tangents at their junctures intersect orthogonally.

Cube-linked template to orthogonal wheel fusionAnother way to shape the blade is found in breaking down the re-orientation of the cuboda from its vertex-on, edge-up template-guided travel direction to the direction facing the triangular propeller directly. In Polytechnic Integration this was attained in the static sense with only a cube link. But because the cuboda’s elements are arranged in recurring fashions of much less than 90°, it is at least interesting and quite possibly very useful to make the orthogonal transformation with a series of smaller steps involving more directions into something more akin to a continuum attuned to a fluid medium.

First considered is the profile of the template which represents the boat or plane, a view that also represents the cubodal propeller faced directly. To view the propeller in profile – the orientation necessary to propel the craft forward – a vertical axis is placed through top and bottom edges. Instead of turning this 90° to view the plane  edge-on, it is rotated only about 22° to view the next internal and identical hexagon as a line in which neither an edge nor a vertex of the plane is faced directly.

Initial Dynamic Propeller-toTemplate Transformation

At this point, one of 2 directions can be taken with an axis placed through opposing triangles orthogonal to that plane. A rotation of approximately 11° one way will result in a direct edge-on view which was the conclusion of the (corrected) flat plane derivation. If we simply rotated this through an axis centered on the middle of that edge, a vertically aligned hexagon and an orthogonal triangle would result. This is sort of what is sought, except that the template’s central hexagon is edge and not vertex-up, and the cubodal wheel’s triangle does not have athwart-ship symmetry as say an engine driving the propeller would have. The other option is to rotate the cuboda in the opposite direction about 19° such that a vertex of the hexagonal plane is faced directly.

Final Dynamic Propeller-to-Vessel Transformation

Now it only remains to rotate the cuboda about 28° to orient the cuboda edge-up and in the direction of travel. Thus is the dynamic transformation from one cubodal orientation made to incorporate a very dynamic component. All told, about 70° is divided in 3 rotations as opposed to the one hard 90° re-orientation of the wheel vector.

Application of this transformation should naturally be keyed to the blades’ corresponding geometric elements. Additionally, the angles should not be regarded as replacements for the cubodal wheel’s basic angles, but rather as enhancements to such – although in some cases vital ones. The transformation angles should reference the basic cubodal angles.

Triangular Axis-Vector Cross Product-Stress TensorThe angles can manifest between actual or projected lines; and as terminating tangents of circular, parabolic, or waveforms, or as the maximum slopes of the latter. Another possibility for how the angle swept by the opposing triangle axis may manifest arises from first viewing the axis as a vector cross-product, an entity that the late great Richard Feynman termed a “pseudo-vector.” In that spirit, might the axis/pseudo-vector also evoke a curvature-defining stress tensor responsive to the flows and pressures experienced by the propeller blade? I don’t know. My grasp of the physical meaning of tensors is dim and my skill in dealing with them mathematically is zero. It never felt right to receive an “A”  in a General Relativity course taken at UC Boulder decades ago because it seemed a gift not earned until I read that Albert Einstein didn’t understand tensors either before receiving special tutoring on them – with a specific application in mind.

Back to common sense geometry, one key reason for applying the cubodal wheel and not any of the 3 other cubodal orientations to propellers is its basic attribute of asymmetric motivation. Its presence in the 3 (swept-back) blade propeller might not be apparent until the blade tips are connected, with the connecting triangle on the fore end oriented oppositely to the triangle on the aft end. In profile one can see the non-matching alternation of squares and triangles.

Propeller Asymmetric Motivation

This attribute cannot be made to work with the opposing twin blades common to small aircraft as the differing geometries of the leading edges would tend to have the plane going backward and forward at the same time. In the area swept by the 3-blade prop there is plenty of space and geometric accommodation for another 3-blade construct to improve performance and efficiency with boss cap fins, vanes, or contra-rotating propellers.

Then there is the hub. A characteristic of cubodal wheel geometry is that its intrinsic spheres are not adjacent in the axial direction. If the form’s  pattern is extended in that direction however – vertically for ease of understanding in the illustration – there comes a point where 2 spheres do align – and does so with regularly.

Propeller Hub and Boss Cap Proportions

The cylinder that joins these spheres diametrically exhibits a length-to-radius ratio of √6 to 1. Thus the hub (or any other such cylinder) is proportioned as a whole number multiple of this basic ratio. So specified, the boss cap may take the shape of a cone, wave, or (half or cylindrically-centered) ellipsoid keyed to a √2:1 or 2√2:1 slope.

These ratios may also be keyed to the horizontally-aligned truncated conical form of a propeller duct which is advantageous in certain situations.

Fluid Flow Focusing

Another possibility is found in employing the same specified conical geometry to both the compression and turbine sections of a jet engine.

Many of the above considerations also apply to the propellers of Kaplan-type turbines used to generate electricity in low-head hydroelectric dams. And so are the rudiments of the cubodal wheel applied to axial flow.  How to apply all this is the job of the engineer and inventor, concerning whom the code is intended to spark and guide, not co-opt or preempt. Copyright pertains to the distribution of an expression of an idea and does not protect the idea itself.

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