Horizontal Grading

After all the waveform derivations, applications, variations, rules and options, one might guess that simply attaining a flat surface would be easy.  However, IMO, flat contouring actually poses the most complicated challenge for more than one reason.

Abstract and Natural Contouring BasesIn the context of the code, there are 2 definition of “flat.” First is the natural flat that is more or less perpendicular to lines of gravitational force, and as such is the local flat as determined by levels, plumb lines, and lasers. Then there is the abstract flat plane presented by the projection of the celestial co-cube. The discrepancy between the 2 flats arises from earth’s deviation from a perfect sphere, and as such varies with latitude.

A related problem posed by the quest for a flat surface arises from the question of what exactly is sought in the contouring. If one seeks conformity continuity by merging the tough lines and arcs of waveform constructs, flat contouring should be of the local natural works because such also happens to be the reference plane utilized in the construction of mounds and berms. However, this approach leads to water drainage problems in proportion to surface area graded thus.

Resolving these conflicts as simply and sensibly as possible entails using each plane definition according to its attributes; using the virtue of flat planes in general to afford a clean interface with natural terrain or intuitive freeform contouring; and formulating a kind of compromise that speaks to both flats.

As noted above, natural flat is the easiest to attain with conventional tools and methods. If the waveform construct does not have a vertical or max slope interface with natural terrain, it should be surrounded by this version of flat grading. The abstract flat on the other hand is restricted to one direction only, downward relative to the local flat toward the equator – due south in the northern hemisphere and due north in the southern hemisphere. Utilizing the abstract flat is not as easy to apply mounds and berms as it is with “flat” roof sections.

Water Drainage Contouring

The steepness of the abstract slope varies with latitude (Θ) according to the indicated formula. As with flat roofs, this slope provides some water drainage and can thus be applied to a larger area than the natural flat, but the slope is not that great (about 3 in 1000 in latitudes 30° to 60°) and should not be much larger. If both natural and abstract flats are used together, the interface is a line that coincides with latitude. Along the other 3 sides, the deepening break between the planes may be smoothed with intuitive freeform contouring. Otherwise, either flat plane provides a clean interface with natural terrain.

The reasoning behind the compromise formulation starts by keying the differing flat plane versions to the abstract square and natural triangle in the context of the vertex-up prism. This prism is apropos because its mundane pole affords an axis by which a circular area may be swept. Of course the area swept will also be disciplined by a right-angled rectilinear grid and for this reason an axis orthogonal to the mundane pole spanning the opposing edges is utilized. [Note: The above rationale for employing an orthogonal prism axis is a bit of a stretch. There might be a logical connection between orthogonal axes and rectilinear grid contouring, but if there is, I suspect the connection would have many steps. Fortunately there is a much simpler and sound connection afforded by the fact of the differing flats having their origins in oblate and perfect spherical earths. The flat cross-sections of either of these may conform to the other by rotation about either polar or orthogonal axes – RRW, 3-4-2015]. So established, attention is focused on the element common to the square and triangle – the edge.

Flat Contouring Derivation

At this point it is interesting to note that in all orthogonal (secondary) rotations of the cubodal shell, the edge is the only dimensional element, viewed directly in and of itself  regardless of what it borders, that has neither up/down or left/right symmetry. For the edge to participate in such the prism is rotated about the opposing edge axis until the vertically-aligned triangle is faced directly. The angular difference between the edge intersection of the 2 different planes – from symmetry to asymmetry – is about 1.5°.

This angle can then keyed to the tangent of an omnidirectional spherical segment or mono-directed cylindrical segment as with a football field whose maximum slope is about the same 1.5°. This angle can also be keyed to the maximum slope of a waveform used to create a mound or extended berm, either convexly upward or concavely downward.

Rectilinear areas can be reached with an approach that is the reverse of the inside corner contouring method.  The (half) wave’s size and proportion are kept equal by shifting the pivot point along a diagonal ridge. Thus may more effective water drainage be attained in the context of grid constructs.

Comprehensive Flat Contouring

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Outdoor Rooms

With rudimentary concave contouring techniques in place, ground design can now proceed to the areas immediately surrounding the abode which can be designed to attain the status of an outdoor room – a term I originally heard channel surfing and stumbling onto the Martha Stewart Show. The concept left an impression and more than a decade later, I began seeking ways to adapt code geometry to it.

Outdoor rooms are especially applicable to CBS architecture as its cosmic derivation paradoxically influences compactness. In that context, an outdoor room hypothetically presents the largest area for social gatherings, or for occupants to find more space for themselves.

The most immediate adaptation would come in the guise of a conventional courtyard of full or partial height walls. Such walls may adopt the same half waveform embanking as CBS walls – on both sides if part of a greater landscaping scheme, or on the inside only if it is the only landscaping scheme in a denser city dwelling setting for instance. Either way, the inside wall is always relevant. Although the optional CBS waveforms are available to the outside of the wall, the very nature of the space enclosed necessitates that the waveforms there only be of the relatively steeper 45° variety.

CBS Courtyards

Corners, if not swept by such waves, are 55° waves spun 90° the same as CBS inside corners may be. In either case, 55° is present (inherently with the 45° wave). This poses inherent tetrahedral linking to the same embankments built along the CBS wall. On the outside of the wall, if there is space, 35° embankments (subject to the same orthogonal wall prohibition as the house) pose an interesting complement to the 55° factor on the inside while their inherent 45° slopes match those on the inside of the wall.

Beyond the courtyard (or abode if there isn’t one), berms, owing to their innate vertical lines arising from their crests, may support, along with the flora, posts that support an open rectilinear latticework. Another possibility arising from berm or mound wave geometry, is to drop walls at the maximum slope level, and let an imaginary max slope extension determine the placement of seating in the carved out area.

Wave Posts and Furniture

On mounds or berm ends, the carved out spaces may be 90°, 180°, or 270°, and for grid junctures or integrating rings, the carvings may be further subdivided into properly aligned 45° integrals. By such carved out areas, berms, mounds (and embankments) serve as outdoor furniture – luxuriant furniture by reason of the flora growing on it.

To create areas of semi-private intimacy, berms can be swept concavely into tees, crosses, or u-shaped partitions against imaginary plane intersections. Another possibility is created by the 35° mound that usually serves as an intra-grid juncture in either grid.

Rectilinear Berm Intersections and Expansions

Since it has innate rectilinear extensions, the circle of the mound can be divided into quadrants which can then extended outward such that the crest is rectilinear wherefrom 4 straight sections may descend without rounding. By such maneuvers, the plateau can accommodate a sitting area accessed by steps rising along the 35° slope.

Depression Quadrant Wading PoolFinally, much of the above variations, options and maneuvers can be applied to concave depressions. In such case however, only the waveforms’ maximum slopes can be reasonably used to make mini-amphitheaters, conversation pits, and wading pools descending from a wide beach to cylindrical depths.

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Concave Contouring

Thus far, waveforms applied to ground constructs corresponding to grid elements have been convex, with waves bulging outward from their troughs and the earth; and where rounding occurs, the wave is spun around its crest. Convex curvature is the most prevalent waveform manifestation because it naturally follows the convexity of the whole cuboda as typically pictured–especially in its geocentric version.

However that may be, the cuboda has an internal pattern completely reflective of its exterior and because of this, indentations are easily carved out of the form. As such, concave waveforms may manifest such indentations and in so doing constitute a vital part of ground design that is worthy of special consideration.

Cubodal Pattern Concavity and Concave Waveform Contouring

In the simplest of applications, the crest of the wave coincides with some designated level and for rounding the wave is spun about an axis pinned to the wave’s trough. Such rounded (and extended) depressions  atop the plateaus of mounds (and berms) can meet the necessity of water retention for young plants.

Advancing a step, (convex) berms acquire a concave aspect if they are to make a 90° turn. To do so, the default is to spin them about the point of intersecting troughs. But in practice, the pivot point may be anywhere along a diagonal extended from the center of the turn such that the inside corner is also rounded.

Concave Berming Approaches

Applying the above maneuver to a 35° waveform poses an apparent contradiction to the rule for the architectural embanking option using the same angle. The reason for the rule of not allowing a continuous waveform to extend straight along two orthogonal walls was to avoid confusing the fact of orthogonal P-R grid lines being fundamentally different, i.e., polar versus rotationally aligned, especially where this is most conspicuous – the skewed bias of the CBS roof. Conversely, in allowing the 45° waveform to continuously follow any CBS wall, another reality is expressed in the obliviousness of the cuboda’s foundational square to any axial alignment with the equality of its sloping 45° edges from all basic directions relative to the P-R alignment. By such distinction in rule application, both relativistic nature and supernatural absolute realities have their place.

Separated-Continuous Grid BermsIn the greater P-R grid apart from the CBS abode, where gridline dualism grows less distinct, the 2 realities may be mixed, with orthogonal berms distinguished minimally by corner separation while continuous turns entail concave contours. The former is more suited for landscaping and farm fields while the latter might find more application in road and utility infrastructure although either has limited applicability for both.

Concave contouring allowed for 45° embanking at the CBS abode poses an alternative in dealing with inside corners. The (half-wave) embankment can make the turn in 2 ways, each of which may proceed by 2 alternatives. The simplest way is to make the concave sweep at the point where the orthogonal wave troughs intersect. Doing so will leave a flat arc in the corner which will be greater if the alternative concave sweep proceeds about a point along the corner bisecting diagonal out and away from the intersecting troughs.

Inside Corner Contouring

A sensible way to deal with the flat remainder poses the second way, which is to extend the length and height of the wave as the turn proceeds into the increased distance to the corner from the pivot point while maintaining the designated maximum slope. So done, the height at the corner will exceed that of the wall by a factor of √2, and the apparent slope of the larger waveform viewed along the wall will flatten to 2H (√2-1)/πL, or 22.5° for the 45° max slope embankment. (Interestingly, the height to length ratio of the total built-up corner mass equals the height-to-length ratio of the 55° wave designated specially for quarter turns). Such a technique can be applied to the imaginary inside corners of 3-way tees or 4-way intersections of orthogonal vertical planes.

Diamond Grid BiasAccording to the reasoning behind the separation/continuous dichotomy applied to P-R grid, the diamond grid would only manifest in a continuous manner without corner breaks. However, even though diamond grid lines are indistinguishable relative to the geocentric cuboda, the cubodal constructs following them exhibit individual length versus width biases, attributes that make them fusible to the inter-grid junctures. Therefore both separated and continuous berms are applicable to the diamond grid. The fundamental difference between the 2 grids lie in which prisms are used to guide their berms, and how they are used.

Grid Integrating RingIn addition to berms or embankments being swung concavely to make 90° turns, complete rings with no straight sections can be swung in the same manner – 360°. If such rings are properly centered, they cross both grid types at 90° angles and thus serve as grid integrators. As such, rings have the same slope as inter-grid junctures: 30°

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Embanking Variations

Previously, default embanking waveforms were shown to be guided by square and vertex-up terrestrial prisms with the waves’ identifiers  (maximum slopes) keyed to the prisms’ respective (45°) sloping edges and squares. As such, the 45° waveform is applicable to both straight line walls and corner turns, while the angle itself signifies the constant average of any and all CBS latitude-dependent roof slope combinations.

Aside from the universally applicable nature of the default, particular architectural elements may accommodate other waveforms  keyed to other prism features in a reasonable fashion. As previously observed, the vertical line delineated by the outside corner can be viewed as the mundane pole of a grid juncture. As such, the prism’s 30° sloping edge is spun 270° around the corner to shape a ¾ mound such.  By this option, the corner edge is associated with the typical mundane  pole manifestation – a tree trunk.

Outside Corner Embanking

Space permitting, one advantage of this option is more earth mass (for a given height) and thus greater moderation of temperature extremes (and less energy input requirements). Another virtue of the 30° mound is that as a grid juncture in limited space, a landscaping scheme can proceed directly to a diamond grid layout, an approach that can really open out a yard from the house and enable the roof more access to sun paths.

Special case waveforms for inside corners receive guidance from the square-up prism’s (55°) sloping triangle – the inherent angle of the 45° edge most appropriate for corners in general. In such case, the 55° maximum sloped wave is spun 90° to form a quarter mound, with orthogonal walls supporting the greater steepness.

Inside Corner Mounds

By engaging the fusion formula, 55º represents the solution to a slope of the same angle, a situation that essentially defines a tetrahedron. As this form diagonally bolsters the cube, the 55° inside corner mound is viewed as symbolically bolstering the orthogonal walls.

Finally, the 35° sloping square and triangle of the edge and triangle-up prisms are keyed to embankments in the same manner as with P-R grid berming because CBS architecture is of course attuned to that grid. Because the main attribute of this angle’s previous P-R application was posing the solution to the fusion formula for the default architectural slope, so its use as an embankment constitutes a direct fusion to the constant average of CBS roof variability.

Average Fusion Formula Embanking

Based on that consideration, the 35° half berm embankment along a wall must have a polar manifestation – the direction of the fusion. Additionally it may not be extended continuously around a corner and along orthogonal walls without a break as the natures of the orthogonal lines differs – polar versus rotational.

One last matter is worth addressing here. Where to get all the earth for these embankments if the site’s immediate vicinity does not provide a usable excess? Answer: Sink the structure into the ground, and use the dirt dug from that to make the embankments.

Embanked and Excavated CBS Home

Using this approach deepens the temperature moderating effects in a passive design approach. That about covers architectural embanking options. More will be offered in Part VI that pertain to the wheel-based annex structure.

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