Diamond Grid Hips

The code’s architectural style intended for the diamond grid is simple and conventional, although precisely specified. The simplicity of it stems from its host grid which is not dualistic like the P-R grid in that NE-SW and SE-NW directions are indistinguishable relative to the polar alignment.

However, what diamond grid structures lack in complexity is compensated by the added steps in maneuvering its parts into position. To start, the celestial co-cube is positioned much like it is for P-R grid structures, i.e., via primary and secondary rotations.

celestial co-cube positioning to diamond gridBut then the cube undergoes a 45° tertiary rotation about an axis spanning midpoints of foundation and opposing squares. From such position, the pattern of the co-cube is projected to earth where it guides floors, ceilings, and walls onto the pattern of the diamond grid.

To derive the roof, the cubodal shell (minus the cube) is spun about an axis passing through either pair of opposing vertices lying in a plane parallel to the skewed square. The rotation transpires until an edge aligns with the longitude of interest.

Macrocosmic wheel maneuvering to Diamond Grid

At this point the cuboda is the macrocosmic wheel. This then is rotated about its axis to locate the edge latitudinally. Next the wheel undergoes a hexagonal shift such that matching triangles slope from the edge that connects them.

A microcosmic representative of this arrangement, viewed at the ground, shows the paired triangles sloping at a pitch of 1: 2√2 ( ≈ 19°) which defines the pitch for roofs. Note the difference between this fixed tri-wing slope and the variability of tri-wing slopes characterizing the full CBS fusion in the P-R grid.

Diamond Grid Roofs

Viewed in profile, the roof’s isolated triangular pattern is extended to show how the roof tapers upward, with the end roof slope being 30°. Alas the result of all this maneuvering is a simple hip roof – although a highly specified one. Walls aligned to the triangular roof ridge are characterized by round windows to highlight the structure’s wheel-based nature.

The arrangement of the wall/roof pattern is easily chiseled from to suit the needs of a variety of applications. Indeed this is definitely the code’s most versatile style, being suited for realms beyond single or multiple unit residential purposes.

Diamond Grid Building Configurations

Its low slung  extendible nature makes the style conducive to agricultural, industrial, commercial, transportation, and institutional functions. For that matter the pattern may easily be sculpted in the vertical direction in the construction of towers. Although transverse extension is limited, clusters of diamond grid structures are facilitated by marking off zones for them with 30° inter-grid juncture mounds centered at the corners.

Diamond Grid Structure EmbankingDiamond grid embanking is also simple with the biggest distinction being the option of a 35°  maximum slope waveform extending along ridge-paralleling walls to underscore the wheel’s dynamism. If rounded, this should be done with 55° mounds. 45° berms along the end wall possess an extra-terrestrial 35° angle and the slope to which may be fused the 35° inherent angle of the 19° default. Three quarter mounds keyed 35° or 30° serve as intra-or inter-grid junctures on outside corners respectively. Inside corner mounds are maxed at 60° to express the turn of the wheel’s leading edge common to 19° and 35° slopes.

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WBA Embanking

With the derivation, options, and principles of wheel-based architecture now put forth, the style is not complete without the ground rules for its fusion to earth.

Because WBA and cube-based shelter share the same grid, much of the ground rules for the former follow those of the latter. The 45° CBS default slope keyed to embanking half-wave berms and mounds there mostly applies to WBA.

Where WBA departs from CBS rules, they do so because 1) the former is wheel-dominated and the latter is celestial cube (house) dominated, and 2) for the more subtle reason that each is directed to grid components that in their orthogonality they are characterized by very different vectors  –converging polar and parallel rotation, respectively.

The most striking difference in WBA embanking is an outgrowth of the stipulation that says 45° half berms extending along east or west walls must be rounded as such at outside corners, i.e., they must manifest in some way along north and south walls. In the zone prohibited them, 19° sloped embankments are permitted and even encouraged for the simple reason that this option – in conjunction with the opposing WBA 35° slope – aptly completes expression of WBA’s asymmetric wheel essence.

Special wheel-based shelter embanking

However, because a prime reason for assigning this slope to the diamond grid was to distinguish that grid as such, a condition is placed on its use in the P-R grid. Aside from direct contact with a P-R grid structure, the distinction is underscored by slicing the embankment cross-sectionally. End rounding, if any, proceeds with partial 35° mounds. Happily, this transition works hand in glove in light of how, for the same height or length, one slope vertically and horizontally coincides with the other at the maximum sloped half-way point. Otherwise, polar-directed walls are the domain of 35° slopes.

As with CBS embanking, 35° mounds are suitable for outside corners and as such serve as intra-grid juncture expressions. Alternatively, 30° mounds swung around outside corners constitute inter-grid junctures and thus enable direct transition to the diamond grid.

WBA (and CBS annex) embanking

A complementary (and parabolic) assignment to the 30° mound is the 60° max-sloped mound spun 90° around WBA inside corners – a more subtle departure from the 55° slope of the corresponding CBS corner. Inside corners posed by WBA/CBS junctures must be keyed to 45°. Otherwise embanking within CBS annexes follow CBS rules.

The reasoning behind 45° being keyed to the transition mound between WBA and CBS annexes begins with the fact of the vertex-up cuboda being shared by both architectural modes. Of that prism’s 3 angles, 45° also characterizes the square-up cuboda that the P-R  grid follows in supporting both styles. It is interesting to look at the transition mound from each perspective separately in the context of their common vertex-up prism.

WBA-CBS Annex juncture mounds

From the explicit 35° manifestation of the wheel (of WBA), 45° is inherent and thus fixed, and as such signifies the wheel’s resting place. Conversely, from the 45° average of CBS, 35° represents an extra-terrestrial angle and thus is dynamic and naturally evokes the life of the wheel. Such is the 35°/45° connection representative of the code’s most basic fusion in the context of the greater fusion to earth.

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Parabolic CBS

Before moving on to the ground rules for contouring wheel-based architectural embankments, I feel it important to report an interesting association between seemingly disparate ideas of recent posts that has just occurred to me.

The first concept pertains to the legitimate use of vertical lines extending from parabolic arches (or suspension cables) in bridge design, and the second entailed the separation of cube-based shelter roof projections to top WBA annexes. The link between the seemingly different geometries of these 2 realms is found in their identical qualities of reflection.

Parabola - CBS orthogonal reflection

In either case, any one line directed orthogonally to the form’s orientation and meeting the parabolic curve or CBS plane is ultimately reflected back in a parallel (but opposite) direction after bouncing off 2 planes (viewed edge-on in profile as lines) that are complementary in their alignments relative to each other.

The parabola’s complementary tangent line pairs essentially constitute an alternative definition of that form. By extending those tangents to their orthogonal convergence, a picture of the CBS roof profile presents itself.

Cube-based Shelter - Parabola Interchangeability

Conversely, in viewing the CBS profile, any vertical line converging with one roof plane (e.g., at a wall plane juncture), and reflected off the other roof (or its extension), together pose a complementary tangent pair onto which a parabola may be inscribed.

The parabola of course possesses infinite complementary pairs up to and exclusive of 90° as the curve never gets there.

Continuoum of CBS projections and Parabolic Tangents

Conversely, a potentiality of infinite complementary roof sets are posed over earth’s latitudes by reason of the fixed primary celestial cube’s projection onto earth’s ever changing curvature. The totality of this situation can be represented by a circle inscribed in a square.

It is interesting to compare the likeness and differences between the two expressions of the complementary relationship by nesting the circle in the parabola such that the circle’s center-point and the parabola’s focus coincide, and then circumscribing the circle with the square by which the complementary angles are provided.

Circle-square and parabolic reflection

The ultimate parallel reflection of the circle in square is with the radial line extending from the origin. Conversely, this line serves as the reflective intermediary to the parabola’s paralleling reflective pair and is not itself paralleled.

parabolic wheel-based architectureAs the cuboda in and of itself exhibits the quality of parabolic reflection, so does the geometry of the CBS derivation in which the prime celestial cube meets its co-cube partner. As this association arose from the separation of wheel-based shelter’s CBS planes suggestive of complementary parabolic tangents, so a more sublime virtue of WBA is its more pointed expression of the parabola.

In my view, such an attribute nails down the whole CBS reasoning – especially in light of how the parabola keeps popping up in the geometry of natural phenomena at the most basic levels. To the question of whether the reflection relationship shared between the parabola and the CBS circle/square representation depicted above has more profound implications I don’t know. If it is a mystery that hasn’t already been addressed elsewhere, it will have to wait for later as far as I am concerned.

Posted in Cube-based Shelter, Derivations, Rolling Transport, Wheel Extrapolations | 1 Comment