The 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 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.
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.
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.
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.
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.
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.
Another 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.
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.
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.
The 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.
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.
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.
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.