Towering Cubodas

From rounded domes to pointed cones, the forms by which vertical cylindrical walls are capped have been exhausted as far as the code is concerned. However, a re-examination of one option introduced a few posts back will expand upward possibilities greatly.

With the toroidal form, expanded elements intrinsic to the cubodal pattern were spun about one localized axis and essentially acted on equally intrinsic spheres displaced transversely from that axis.

Parallel axis displaced cubodal forms

Simlarly, parallel cubodal elements regarded in isolation as distinct entities and displaced from the central axis may be spun about any of their own axes to form the same cones, hyperboloids, ellipsoids, and paraboloids generated from the local axis.

Flattened cuboda-derived forms

Once fashioned, these forms are spun about the central axis and joined laterally by reason of the horizontal planes paralleling the earth’s surface at the intersection of intrinsic projecting cylinders.

Vertical cubodal form stackingAll of this is to say that flat circular tops (and bottoms) are a legitimate alternative to rounded or pointed tops, and as such afford the bases by which cylindrical and other forms may be joined to build vertically extended structures in any way imaginable within the constraints of a particular cubodal orientation.

This brings us to another critical constraint. Derivations of cylindrical forms projecting from the earth were shown to be characterized by diameter-to-height ratios corresponding to the 4 cubodal orientations – with the edge, square, and triangle-up orientations diverging from the one-to-one proportion of the vertex-up. By reason of the natural line between vertex-aligned spheres and the point of contact between those spheres, the vertically-aligned cylindrical structure enveloping spheres so oriented may exhibit any diameter-to-height ratio.

Cylindrical tower proportioning

As the vertex-up orientation is invariably present in guiding the foundation pad’s outer grid interfacing slope as an expression of inter or intra-grid junctures, the other 3 orientations may possess cylindrical forms of any ratio up to the one-to-one ratio. Beyond that, vertical walls must be a whole number multiples of cylinders proportioned according to the orientation employed, meaning there is a range of proportions between one-to-one and each orientation’s ratio that is disallowed. This rule applies to both initial and stacked external cylindrical walls.

Another way of extending a structure upward is supplied by vertical waves in which the maximum slopes are invariably 90° and appropriate cubodal angles are keyed to the minimum slopes. Like horizontal waveforms, their vertical complements may come in an integral number of quarter max-to-min slope sections. If the waveform’s terminates with both max and min points, either may situate at the bottom (or top).

Towering Cubodas

Like horizontal waves, a different wave length may commence at another’s termination as long as it is keyed to the same (or inherent angled) slope. For vertical extension, triangle-up orientation supplies the steepest angles of about 55°and 71° to afford the maximum height-to-base area ratio. Like all other forms, vertical wave structure has its members pegged to the directions of the universal 24-point ring.

Thus are code designed towers guided to serve habitation, observation, mechanical (vertical axis wind turbines), hydraulic (water storage), and electromagnetic wave transmission purposes. As Part VII’s architectural phase is no complete, how the code’s conceptual model relates to electricity, magnetism, and light will be next.

 

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Extra-geodesic Domes

Radially centered dome structuresA couple of posts back, I specified requirements for (circular) window placements on spherical domes with little consideration for the underlying structure that needed to be avoided. What I had in mind for this were great circle radial members aligned according to the 24-point universal ring and converging on the dome’s apex. This seems to be the basic approach for large undertakings like the Superdome, but as 24 members are insufficient for larger domes, I have taken a cue from the method of specifying ellipsoidal domes which are projections of the arcs formed by intersecting the cuboda’s angled planes with a sphere.

The simplest way to proceed with is to view the universal 24-point ring from above with 12 pairs of opposing points signifying the great circle diameters.

Dome Base Parallels

Then, focusing on one diameter direction, parallel lines join opposing points equidistant from that diameter, with lengths shortening by (cosine of) 15° increments. These lines represent intrinsic plane intersections with the circle of the dome’s base. The planes also intersect with the dome’s curvature, with the particulars of the arcs formed being of prime interest. All but the triangle-up orientation manifest vertical planes – the easiest to picture and formulate in the context of the sphere from which the dome is horizontally sectioned.

vertical arc dome structure

 

Once this procedure is completed, the remaining 11 directions are treated similarly to obtain the final structure.

Although the triangle-up cuboda lacks vertical planes, it does inherently possess sloped ones (55° squares and 71° triangles) that may legitimately slice through the sphere to form arcs. Viewed directly from an orthogonal perspective, these are the ellipses concluding the last post. As with the vertical planes, the arcs descend incrementally in scale from the great circle diameter of the reference sphere; but as the angled plane orientations are swung completely around the dome’s circular base, mirrored arc slopes will result.

Spherical dome sloped arc structure

A 2- dimensional formulation of planes intersecting the reference sphere depicted in the illustration above garners the most vital (and useful) numbers of individual arcs but in order to locate arc intersections and determine curvature tangents for connectors and the triangular stability they afford, spherical coordinates would probably be the best way to proceed. Finally, angled slopes remain an option for the other 3 orientations.

EG domes “extra” is derived from the quantized departures of the arc radii from the great circle geodesics of the radial diameters. In light of the ease of finding particulars via a 2D approach, these structures might better be termed chord domes.

The advantages I see in such domes have to do with the great density of interwoven structure obtainable to support a more perfectly spherical covering with lighter structural members. If the bones do show through, they pose the same ellipses that are valid for the orientation specified. Lastly, this structural scheme is highly conducive to customizing – especially biasing density toward the base where the weight of everything above is borne.

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Code-keyed Structures

 

In contrast to the spheres and paraboloids of the last post, another class of forms employable as cylindrical wall caps are shaped at the outset according to the geometry of a particular cubodal orientation.

The simplest form in this class is the cone which, in the context of the cuboda, is shaped by a cubodal element’s straight slope rotated about the intersecting axis that signifies the orientation used.

cuboda-keyed cones

In the whole spectrum of allowable possibilities, the 7 standard cubodal angles are joined by their mirrored negatives. To use them implies that truncated cones are a valid and viable option that utilizes an innate sloping element displaced from the axis.

cone partitioning and access

The cone interior, like the paraboloid, may exhibit floor-to-ceiling cylindrical partitioning of any radius from the axis. Another possibility for cones using all but the triangle-up orientation (no vertical planes) are hyperbolic arches which may span pairs of alternate access points on opposing sides of the universal ring’s 24 directions. This is justified by reason of the hyperbola’s constant correspondence to the cone’s slope without regard for a terminating tangent.

hyperboloid dome

However, because they never do reach the prescribed guiding slope, hyperboloids employed as domed caps must terminate at a tangent angle that is both inherent to orientation used and, by necessity as no steeper angles do exist, and shallower than the asymptote defined by the orientation’s sloping element. With the ratio of the terminating tangent, its point in relation on an x,y coordinate system can be found with the derivative of y with respect x. Like paraboloids, hyperboloids are not an option for the triangle-up orientation because their are no vertical planes by which to slice the cone.

Another option for the cone is restricted to the edge-up orientation and its association with either grid type (but not both). If a rectilinear plane so-aligned penetrates the cone vertically, a hyperbola is formed at the juncture. Other (code) rectilinear schemes may thus build from the plane.

Conical section architectural options

The other option is to slice the cone on opposing sides with planes shallower than the cone slope to form ellipses. The juncture forms a ridge aligned to the grid.

This brings us to ellipsoids. These have already seen a few approaches – mostly using the triangle-up orientation for dynamic applications. First there was the speculative monolithic wheel using the cuboda’s central and outer layers to define an ellipse. Then in streamlining techniques, the diameter of a circular cross section was appropriated as the foci of an ellipse. In fluid dynamics, ellipses were keyed to a propeller’s matched triangle blade framework; and the hub’s streamlined half ellipsoids were proportioned to slopes of the cubodal turbine. Finally, ellipsoidal expressions of the disc were fashioned according to the cubodal angles arising from foci to the ellipses curvature.

For architectural purposes, any of these approaches have some validity, but finding one specializing for this function the is in order. One problem is that for each cubodal orientation, there is ambiguity as to the form’s exact proportions. The simplest approach is to key the ellipses major and minor axes to the cubodal slopes.

architectural ellipsoid

But a better way poses itself with spherical sections in which intrinsic spheres are sliced hemispheric-ally by intrinsic planes. In such instances the projections of those sections define the ellipses to be rotated 360° to form an ellipsoid. Either way, the ellipsoid is sliced horizontally such that its terminating tangent is always vertical.

architectural ellipsoid partitioning

Ellipsoidal interiors admit no concentric floor-to-ceiling cylindrical forms except for at a radius that coincides with the latus rectum. Otherwise, full partitioning along those lines radiating from the universal ring and attuned to the orientation’s grid layout are allowed.

 

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Invariant Dome Forms

Armed with the basics of conical sections, radially-projected cylindrical forms, and circular wave-formed foundation pads, this post will address the actual extra-topographic structures.

Basic Cylindrical WallFirst off, radial (vertical) cylindrical projections – typically structured with pillars centered on the universal 24-point ring – generally guide the constructs’ vertical walls. The form of their caps is about all that remains, at least as far as the cods is concerned. Naturally, such caps are more often than not the structures’ roofs, although they can potentially curve around to the realm of the walls to some degree.

However that may be, roof caps are invariably circular. The forms considered in this post are also invariant in the sense that they are what they are. The proportions of their shapes cannot be changed without changing the essence of what they are, and thus they admit no variation. For capping purposes, aside from scale, the only distinction that can be made with such forms lies in how they are sectioned to meet the cylindrical wall or pad proper, in which case the slope of the ring of uniform terminating tangents defines the section.

The first invariant form considered is the cuboda’s intrinsic sphere. To be employed as a cap, the sphere is sectioned horizontally with a plane paralleling the earth’s surface such that the boundary slope of the section created corresponds to an angle of a particular cubodal element in a particular the orientation, the one employed to guide the construct.

Spherical Domes

 

 

 

 

 

 

In all, aside from the cubodal 7, the terminating tangent angles include their vertically mirror angles and 90°. That’s basically all there is to it.

Toroidal Structure

There is one very distinguished variation of the sphere’s employment in which it is horizontally translated from the construct’s axis and spun to create toroidal forms set on the pad or wall structures of concentric cylindrical forms.

The other curvature in the class of forms that possess the attribute of invariance defined above does not possess the primal innateness of the sphere but is easily derived from the cuboda’s inherent nature. That derivation was described in the last post. In it the parabola was formed by the intersection of a cubodal cone and any intrinsic plane whose slope parallels the cone’s slope. The parabolic form is deemed invariant because regardless of cone slope, its curvature always approaches a line paralleling its axis of symmetry.

For purposes here, the axis of symmetry is vertical and its invariance means it can be formed by any vertical plane found in the cubodal orientation utilized. All but the triangle-up orientation manifest such planes.

Parabolic Dome

The parabola thus conceptually formed is spun about its axis of symmetry to form a paraboloid and all that remains is to locate the ring of the horizontally-aligned section that corresponds to the cubodal orientation used.

Although the paraboloidal dome is excluded from the triangle-up orientation, it offers much flexibility in the other 3 orientations with regard to internal partitioning. This follows from the parabola’s attribute of uni-directional reflection which was utilized 2-dimensionally in bridge design. With a vertically-aligned axis of symmetry all points on the parabolic dome’s curvature reflect vertically.

Paraboloid Interior Partitioning

Thus concentric cylindrical walls or rings of support pillars placed according to the universal 24-point ring otherwise have complete radial freedom. This is not so for spherical domes where floor-to-ceiling cylindrical walls may only manifest where they meet curvatures with terminating tangents inherent to the orientation employed.

Spherical Dome Partitioning

Otherwise, interior partitioning must conform to the orientation used.

Cubodal Dome Partitioning

The correspondence is: rectilinear for square and edge-up orientations; octagonal for the vertex-up orientation; and hexagonal for the triangle-up orientation.

Vents, skylights, and utility penetration panels are treated a bit differently for spheres and paraboloids. For the spherical dome, circular windows are centered anywhere along the planes sliced radially through the 24-point ring and their radii extend equidistantly to opposing curves made by planes sliced through the 48-point access ring.

Cubodal Dome Windows

For the paraboloidal dome, windows are basically truncated wedges bounded by curves made by planes sliced through the 48-point access ring and any pair of concentric rings.

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Cubodal Cones

In most recent posts, conic forms have found important roles in supplying access flexibility to mounded ET pads. And, as cones will see use in upcoming ET constructs,  I believe it worthwhile to explore just how intrinsic, or at least easily derived these forms are to the essence of the code’s geometric foundation – the cuboda.

For starters, the principle cone quality of a circular cross-section derived from a uniform slope spun about an axis and converging to a point has correspondence in the cuboda’s inherent axes – spanning opposing vertices or the immediately abstracted axes spanning midpoints of opposing pairs of its lines, triangles, and squares.

Cone and Cuboda commonalities

Innate or abstracted, each axis passes through the cuboda’s intrinsic center point. For viewing as well as practical purposes, these elements also define the cuboda’s 4 basic orientations by which the form’s internal structure (comprised of the same elements) are situated. Upon rotation, the slopes of these structural elements define cones.

cubodal cone

For example, the opposing triangles orientation poses a tetrahedron. Upon rotation its non-base triangles form cones sloped either according to their common edges, or to abstracted lines symmetrically bisecting the planes of those triangles. In such manner cones are formed that correspond to the 7 cubodal angles. In so doing, mirrored cones are formed and that brings us to another attribute of the cubodal cone.

intrinsic cuubodal circular conic section

What is especially interesting about the cubodal approach is how planes are inherently present to create other forms  – the conic sections and more. For instance, any of the infinite horizontally-aligned triangles of the tetrahedron’s intrinsic pattern supply the planes that slice through the cone (formed by same) to create a circle. What’s more, the circular intersection between cone and plane is characterized by the slope of the cone in the sense of the circle really being an infinitesimal band.

Instantaneous slope of the circular conic section

With the 2 non-planar orientations – edge and vertex-up – the bisecting plane may be supplied by gravitational potential fields parallel to and immediately above the surface of the earth – an abstraction applied in Part V to the maximum slopes of waveforms spun about crest or trough-pegged axes. In that derivation, the wave was merely keyed to, but not suggested by the cuboda by virtue of its vertical symmetry  corresponding to the waveform’s displaced vertical symmetry.

Another conic section – the parabola – may be formed by any of the innate planes paralleling the plane whose midline defined the cone. Like the circle, the parabola in its own plane is the same as the projected or any other parabola. The proportion of its precise shape is retained regardless of cone slope, i.e., the example below speaks for all parabolas.

parabolic conic section of the cuboda

Although it may appear to be flatter or narrower with varying cone slope, this is due to the part of curvature focused upon which is due to its projection guided by the cone slope. Note that this  abstraction requires the cuboda only – as opposed to the parabola suggested by the projections of the twin celestial cubes set on the geocentric cuboda.

Then there is the ellipse formed by an innate plane exhibiting a slope that is shallower than the slope of the cone, if there is one.

Cubodal generated Ellipse

 

For an example, the vertex-up cuboda’s 45° sloping square is bisected by the 35° sloping triangle to yield an ellipse. Happily, the most useful perspectives exhibit interesting major-to-minor axes proportions including a circle!

Conversely, the hyperbola is formed by planes steeper than the cone’s slope. This form has not yet found application in code constructs, and as of this writing, I still don’t see one forthcoming. But the special case of a vertical bisecting plane paralleling the cone axis holds some prospect.

Cubodal Hyperbola

The problem I find is where to cut the curve off as it approaches the slope of whatever cone is generated asymptomatically and never gets there. The same can be said of the parabola, but with a vertical asymptote aligned to gravity, its use is naturally justified.

There are a few legitimate ways of looking at parabolic, elliptical, or hyperbolic curves in order for them to find practical expression. If they are generated from a cone whose axis is aligned with the cubodal orientation, the conic section can either be projected onto the plane of profile visualization, or rotated to coincide with the orientation employed.

Practical perspectives of cubodal conic sections

Or the cone can be formed by non-vertical axes spanning the cuboda’s other opposing element sets in which the bisecting plane is vertical or parallel to the cubodal orientation. In such ways, variation is obtained that is reflective of the cubodal elements and their orientation determined slopes.

Thus does the cuboda generate a family of cones with their own specific conic sections – with the simplest circle yielding an overlooked yet quite practical piece of information. The conic sections go way back in the history of mathematics, yet now the connection seems to be treated as little more than an interesting curiosity with little detailed or formulated development between root and generated form. However, the connection is quite intrinsic to the cuboda in light of the form’s innate rotational dynamism; straight lines and planes to form the cones; and planes to slice the conic sections. As limited as these sections are by their quantized specifications, they constitute a more vital part of cubodal geometry.

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