Bubbles Aweigh!

The hard edges of the transport template’s planar convergences will likely suffice for a home-made dinghy intended for limited use. But the more a vessel sits and/or is engaged in moving through its intended medium, the more important rounding becomes in easing stresses caused by unruly waters and facilitating smooth efficient passage through them.

Spheres are appropriated extensively to guide design of GDCode-designed marine craft largely because of their omnipresent availability in the pattern of the transport template. This follows from the fact of spheres being the basic building unit of the code’s conceptual model, a choice influenced by nature’s spherical ubiquity from the innumerable electrons surfacing everything we see, touch, smell, taste, breath, and hear to stars to the detectable universe at large from any point of reference – including the place where you are right now, for example.

circle wave relationship

Aside from the sphere’s common familiarity, it is also the easiest to use, formulate, and program into rounding schemes whether realized by hand or computerized machinery. What’s more, the form that follows from locating a point rotated around a sphere’s plane cross-section according to its separate dimensions is the wave which also results from any disturbance to the  surface of water.

A rounding scheme for marine vessels framed according to the manner of the last post’s simplest of application examples requires some special considerations.

Dual boat template geometries

The first pertains to the separation of the template’s 2 infinitely expandable and divisible geometries – that of the cuboda and that of the triangular prism-based HXP which share common (hexagonal) planes and which correspond roughly to the vessel’s hull and superstructure.

If no deck extends outboard from the HXP superstructure, spheres of one radius may be appropriated for the rounding process. As with the elementary rounding method described in Rolling Transport, spheres are first conceptually centered on the vertices of the vessel’s framework.

spherical vertices of ship framework

If there is a separating deck, the cubodal hull and HXP superstructure can easily be rounded with spheres of differing radii, with the latter typically being smaller, and in some cases much smaller to the extent of nothing at all. As with rolling transport and aircraft frameworks, cylinders join the spheres and then planes paralleling those of the framework meld to the cylinders tangentially.

marine craft rounding

In viewing the bow head-on, the cubodal side caps are omitted from the superstructure to more clearly depict the rounding procedure. Once this is done, the resulting shell is sliced back along the original deck plane, the sealing planes, and up along the inside of the hull.

planar slicing of marine craft rounding

Slicing around the region of the gunnel can follow any of a few alternative directions consistent with template geometry. As always, spheres rolled along creases define concave curvatures where desired. As far as the superstructure is concerned, any one or two of the 3 plane sets comprising its HXP geometry may be sliced off similarly.

semi-rounded superstructure

The deck and other horizontal planes are not exactly flat but are shaped by very shallow wave forms with maximum slopes of about the 1.8° angle derived from the edge/triangle transformation.

template extension beyond rounding frameworkA couple of clarifications are in order regarding the “framework” of the rounding process. The region between the outside of the rounding framework and the inside of the rounded shell may be framed further by the template pattern as long as the pattern geometry is retained where contacting the shell. This should cause no problems where planes and their edges are concerned, but care for the spherical and cylindrical curves will be necessary. Meeting these curved surfaces perpendicularly with whatever is projected from the framework is a safe bet.

Framework extension is necessary in the next big rounding consideration – regaining a sharp bow and keel after the rounding. The simplest way to do this is to first extend the framework’s intrinsic plane past the shell.

planae keel and bow extension

This plane can be rounded in profile circularly or with half or quarter 60° max sloped waves. For the bow only, the wave can be keyed to the sphere radius and its maximum slope extended linearly as needed.

The extent of the central planar extension may be defined by twin spheres rolled along them in a manner similar to how concave curvature seams are dealt with.

cross-section keel and bow rounding

As viewed in cross-section the change in dihedral angle between matched squares and matched triangles means the extension of the bow from the hull will be about 40% more than that of the keel with the difference continuously smoothed around what is exposed of their common rounding spheres. If the triangle down template is employed the situation is reversed.

Design of submarine vessels centers around (spherically capped) cylinders to deal with pressure differentials.

Template-guided submersibles

The template supplies structural support, guidance in how spheres and cylinders are arranged, and exterior planes to protect them if necessary.

Indeed, one might look to rolling transport and aircraft approaches for other design possibilities. The simplified examples offered here don’t evoke broken speed records or fuel efficiency gains per se. But I believe one of the template’s potential virtues lie in economy of construction* that will advance efficiency when viewed wholistically. Another virtue is the potential for reusability of components before recycling; and yet another from a pattern as pliable to variation as there are species of fish in the sea.

*Volumes and areas for all geometric elements that comprise the template as employed in this and the previous post are accessible via those links.

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Crystalline Corks

Minus a bridge or a speedy winged craft by which to cross a divide, a floating construct is the ticket if the gap is flooded with water as nearly 3 quarters of the planet is.

Marine vessels fall into the realm of the last few months’ posts on Wheel Extrapolations because, even though the conventionally-oriented wheel of rolling transports (including aircraft) is mostly missing with modern boats and ships, the transport template remains quite applicable.

Essential Transport Template

This is largely so because the template’s geometric basis – the microcosmic wheel liberated from the greater earth-centered cuboda – naturally elongates with a vector aimed in the direction of travel in shuttling people, goods, and raw materials back and forth.

Regardless of whether the template’s matching tetrahedral triangles are oriented up or down, its geometry readily guides design of basic components such as a sharp-edged and well-angled bow for cutting through the water; the longitudinally-aligned vertical plane of the keel and rudder; and a winch structure.

Marine craft template

Less obvious are components guided by the template’s intrinsic spheres, whether these be centered continuously along a pattern line or sliced by its planes. Applications include basics like the winch drum itself, longitudinal drive shafts, spherically-capped fluid storage and stowage cylinders, portholes, alternatives like out-rigging pontoons, and rounded keels and rudders.

The template’s potentiality of the transverse triangular prism-based hexagonal expansion (HXP) will likely find greater explicit use with water going vessels than with aircraft, but the basic geometry meeting the water still derives from the pattern of the cuboda.

Nautical Hexagonal Shift

After a sharp bow is established, enveloping HXP geometry with the template’s predominant cubodal geometry must terminate at some point to allow exposure of such essentials as topside decking and a bridge with broad rectangular windows for looking out on the water ahead.

In making the transition between HXP and cubodal geometries, special attention is on how gaps are sealed from water and weather. Obviously this cannot be accomplished with mere struts but must utilize the planes to match up both geometries. The triangle-up template is used with an example focusing on the HXP’s horizontal rectilinear plane.

Ship Deck Sealing Planes

The longitudinal gap between the deck and the cubodal hull is filled by the 35° sloping rectilinear plane sweeping up from the deck. To address the gap extending from the deck plane’s athwart-ship edge, infinite lines paralleling those of the bow edges are run from one side to the other in a cross-stitching manner to create a 30° sloping plane as viewed in profile, or a √2:1 proportioned triangle viewed directly.

The gap between the longitudinal and forward planes is filled by a plane paralleling that of the template’s skewed triangle. This would be the end of it if only the deck had met the square/triangular juncture from the outset.

Partial HXP enveloping

But that wasn’t done in order to show what is required if the sealing comes up short. Fortunately, the last gap is filled by a plane paralleling the template’s top triangulated plane. Thus the 2 geometries may separate and meld as they should. As specified from the beginning, HXP sides – in this case the superstructure – are capped with a cuboda-patterned shell.

Aside from the 30° cross-stitched plane, the planes needed to seal the deck are found from the template’s parallel planes. This approach is also used in dealing with the inevitable narrowing of the rectilinear deck as it proceeds into converging bow or stern areas. In so doing, additional rectilinear planes can naturally be chiseled in the leftover regions which are identical to the roof structures of diamond grid architecture and can surely support useful constructs. Seagoing folks are possibly the best in making good use of odd spaces often because they must.

Deck sealing variations and accommodations

The cubodal planes conjoining the sealing planes can be used to change deck levels in conjunction with HXP’s hexagonal profile geometry. Struts following cubodal geometry (and likely already supporting the sealing junctures) can be joined to form planes that create pathways for water to wash back off deck. The bottom-most of these planes may serve as the sealing planes in the reverse case where the template’s triangle-down version is employed. Either way,  gaps between template geometries should generally be in the range of inches to yards.

The 30stitched plane is complemented by the 35° plane created by slicing along the vertically projected lines of the end pair of matched triangles which may define part or all of a transom.

transome and stairway guidance

As the 30° plane derives from the 19° sloping matched triangles  associated with diamond grid architecture, so the 35° matched squares are associated with wheel-based architecture. At any rate the 35° slope guides athwart-ship stairways while the HXP 60°  defines the slope of fore-and-aft stairways.

Other special considerations in using the template to guide design of marine vessels stem from the pattern’s lack of vertical lines and transverse (athwart ships) vertical planes. Fortunately, use of the methods and links of Polytechnic Integration solve these problems readily.

Vertical lines – and the cylindrical forms around them – are of course intrinsic to the cuboda’s vertex-up orientation. If a convenient hexagonal bulkhead presents itself, a vertically extended vertex-up cuboda is joined to that plane with the flat plate linking intermediary crafted to express both cubodal orientations and their common element – the circle. Minus a nearby plane, the link’s transversely-extended alternative  might be utilized.

Marine template mast incorporation

If a construct like a sailing mast or a rudder post requiring much bolstered stability is entailed, a spherically-enveloped tetrahedral link attached to the template’s cubodal structure may contribute reinforcement; or with the HXP, special links to a 2×2 array of cubes arranged around a common vertical edge that defines the vertical line and thus cylindrical form serves the same purpose. Sealing the ceiling penetration in either scheme is achieved with geometric harmony.

Transverse vertical planes for staterooms, bulkheads, and holds (to complement the packing efficiency of HXP geometry) are incorporated by a properly proportioned plated linking intermediary. Note the difference in how the circles are centered on this link compared to the vertex-up link. With this simple link, the familiar 3D rectilinear spaces formed are distinguished only by the relative size and spacing of the circular plates joined to the longitudinally extended hexagonal bulkheads.

HXP rectilinear accommodation

Hatches on hexagonal bulkheads are formulated as shown. Hatches on athwart-ship bulkheads formed by such a scheme may be straight or corner rounded rectilinear. The longitudinally-aligned rectilinear walls bear at their juncture with the HXP ceiling a link configuration identical to the aforementioned mast reinforcement. These links are more than mere cosmetic symbolism. They also aptly share in bearing the physical forces encountered transitioning from one geometric structure to the other.

Thus does the code apply itself to purposed as opposed to pleasure craft. With aircraft, there was some speculation that starting with cylindrical forms was the best design approach. Conversely, where code marine vessels are concerned, be they dinghies or supertankers, it would seem best to begin layout with HXP geometry in light of how both the internal 3D rectilinear adaptations and external sealing maneuvers described above are referenced from this structural pattern.

Rounding methods for vessels plowing through water also require special considerations and they will be addressed in the next post.

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Atmospheric Rounding

Transport template-guided aircraft design can and (to varying degrees) should use this pattern’s innate spheres and cylinders to round and streamline the structure’s hard angles.

To a large extent, streamlining code aircraft can follow the method described in Rolling Transport which first entailed centering spheres of equal size to each vertex of the vehicle’s framework. These are then joined cylindrically along the framework’s edges.

Elementary Options

Next, identical planes paralleling those of the framework join the cylinders tangentially such that convex surfaces are rendered with a continuous surface. Concave junctures are shaped by spheres rolled along creases to assure tangential melding and thus continuity.

Since the transport template is naturally extendible along the direction of travel (profile) without dependence on non-longitudinal lines, the design challenge centers around dimensions viewed from the front where transverse elements do depend on each other. In the examples to follow, hexagonal expansions are enveloped although this is not required.

Vertex-on Aircraft Options

So regarded, adjusting one transverse element usually necessitates a corresponding adjustment to the other element(s). Although there are multitudinous ways of doing this, the vertex-on view can be boiled down to 4 types: equal edge lengths; vertically-biased; horizontally biased; and any combination of the latter 2 in truncated configurations.

Another factor best assessed by the vertex-on view is the proportion of edge length to the radius of spheres centered on their endpoints, with the edge perhaps best represented by those sloping at 35° as these edges lies in the plane of visualization.

Nose Cone Configurations

For low and slow aircraft, smaller spheres should usually suffice, while for high and fast flyers, a proportionally larger radius is desirable and sometimes necessary. With the latter, the foremost vertex sphere extended forward – with a cylindrical form around the extended line – enables cones to be fitted un-encumbered by transverse rounding elements.

Further departure from the Rolling Transport rounding method transpires with the wings (and vertical stabilizers) as both sides of their planes are exposed. As such, wing are left out of the larger rounding method for special considerations later.

Independent GDCode Wing Treatment

In the meantime, the choice can be pondered on whether or not to slice off the bottoms of bottom spheres (in bottom wing configurations) and thereby extend the plane of each wing toward the center such that they can be joined smoothly there.

The very large ratio of sphere radius-to-edge length suggests a design method focusing on (spherically-capped) cylinders from the beginning. As these forms are intrinsic to the template, this approach is certainly allowable. These forms possess optimal area-to-material ratio and shape for dealing with pressure differentials. Another virtue – first noted in Polytechnic Integration – is that spheres and cylinders constitute universal internal linking intermediaries in which design components guided either by the other 3 cubodal orientations or non-code schemes may be incorporated. Such an attribute will almost certainly be necessary in jet engine design.

The cylinder of the fuselage (or that extended from such as described earlier) can easily be streamlined further with full 60° sloped cones or similarly sloped truncations capped with smaller semi-spheres. Alternatively, the major and minor axes of an ellipsoid can be proportioned by having one focus coincide with the spherical cap’s diametric extreme.

Nose Cone Configurations

The resulting √2 to 1 ratio of the ellipse’s major and minor axis conforms to the ratio of the template’s cuboda upon spinning it about its leading vertex-on led axis – a rotation that parallels and is enabled by the aircraft’s continual freedom and occasional need to roll about its longitudinal axis.

Beyond the fuselage and engine housings, there is probably little need for cylindrical forms. But if other applications are suitable – such as a vertically-aligned cylinder to smooth turbulence as is done with large ships – their cross-section connections should be consistent with template geometry.

Cylindrical Aircraft Assemblages

If radii of neighboring cylinders vary, either lines (struts) connecting their centers or tangentially-melding planes must conform to the given set of 19°, 35° (and 90°) angles. In lieu of properly-angled planes, spheres rolling in contact with both cylindrical surfaces assure continuous tangential smoothing – just as is the case with concave rounding.

Wing sections left out of the body rounding schemes have their own rounding method. The first step simply centers smaller spheres at the points of the wings’ structure such that cylinders connecting them and planes melding these form a smooth continuous surface.

simple wing rounding and symmetric airfoil

The result is a no-camber symmetric airfoil. Rounding the creases to join wing surfaces will likely have to wait for shaping the airfoils to maximize lift and minimize drag. The airfoil is shaped around the wing’s basic cubodal structure viewed edge-on. And built up as shown (not to scale) to support camber.  The cube-linked vertex-up cuboda allows vertical lines to be introduced and support the parabolic upper surface in a code-consistent manner by the same rationale used for parabolic bridges, and attune to lift vectors.

Code Airfoil Structures

The angle used for the camber parabola’s terminating tangent is about 4.6° so that of the upper surface is about doubled. Use of the ellipsoid for the leading edge follows the same reasoning as the nose cone.The chord line angle of attack is about 2°. Instead of applying the complexity of the edge-to-plane transformation angles of the last post, 4.6° was used for its simple universal representation of the difference between real and perfect earth-spheres – spheres that the atmosphere follows and can reasonably be regarded as part of by gravitational confinement and thus makes this angle speak to the omnidirectional potentiality of aircraft not restricted to grid roads.

Whatever slope is keyed to whatever airfoil element, the equilateral triangular planform assures an even progression of slopes in both directions from the high point at the root chord’s leading edge.

Triangular Planform Contouring Equivalence

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