## The Tetrahedral Factor

The last post concluded with a simple abstract expression of a deeper 3D transformation between fundamentally different plane types – the square and the triangle. In either 2 or 3 dimensions, the transformations are attended by circular arcs, with such rotations intersecting in an equilibrium manifested by the equilateral triangle.

In the square (rectilinear) plane of the CBScase scenario of a fused gable delineating an equilateral triangle is depicted above by setting the angle omega () ; or for the slope of the gable . In either case, those angles are an identical: approximately 55°.

Conversely, one could set phi and delta equal to each other in the fusion formula to arrive at the same 55° -

The gist of both approaches is that upon fusing the tri-wings to a slope of 55°, the slope of the spread tri-wings slope is also 55°.  Again, the angle made by the tri-wing edges against the innate rectilinear lines of the slope fused to is the 60° of an equilateral triangle.

Recall that the tri-wings themselves – prior to their wall plane-conforming bisection – are a pair of equilateral triangles themselves, independent of latitude, always. Thus by the symmetry of the configuration, the remaining triangle projecting over and beyond the wall plane is also equilateral, and the 4 triangles meeting edge-to-edge is a tetrahedron.

The tetrahedron constitutes the alpha form, the first 3D form built in the original assemblage of spheres, and it was its not-so-obvious rectilinear aspects that guided the more deliberate rational accretion of spheres. After reaching fruition in the cuboda, exploration of that form’s unique intrinsic pattern revealed how innate octahedra are always bridged by tetrahedra, a condition meaning that each and every cubodal triangle (inside and out) represents an interface between the two forms.

With square/triangle fusion and transformation, the tetrahedron finds additional accommodation. If the cuboda is oriented to situate on any of its squares, the tetrahedron finds the 55° slope of an oriented triangle to fuse to, a matchup that forces the triangular interface to configure itself to satisfy the aforementioned condition.

In such case, orthogonal lines from each tetrahedron finds parallel lines in the other, a reality that presents the rudiments of a 3D rectilinear grid. Conversely, if the cuboda is oriented to situate on a triangle, the tetrahedron finds 55° fusing accommodation on the sloping square of the budding octahedron.

Like the octahedron’s square/triangle transformation, its matching form – the tetrahedron – finds its underlying rectilinear reality expressed of itself by the four 90° intersections characterizing the unique orientation that places its 2 essential lines in parallel with a plane of consideration, and by the apparent square delineated by the form’s 4 sphere center points projected onto that same plane.

## Planar Transformation

The last post dealing with the various dimensions of the transporter/house fusion gable concluded with the angle (omega) made by the gable’s tri-wing edge with intersecting lines intrinsic to the rectilinear CBS roof plane. This post will focus on what happens on that plane, what it means, and how planar transformation opens the door to 3D integration. Highlighting the relationship between omega (Ω) and that of the CBS roof slope (Δ) – with applicable ranges of 30° to 90° for the former and 0° to 60° for the latter - is the symmetry posed by the overlap centered at 45°,  the only angle in which omega and delta are equal. Over earth’s latitudes, the tri-wing edges move toward one set of CBS roof plane lines and away from the other antiparallel set of lines. In doing so, the triangular half wings comprising the gable’s roof planes delineate an infinite set of isosceles triangles superimposed onto the rectilinear plane of the CBS roof. In essence, the CBS roof plane presents an interface between the simplest of fundamental plane types: the even, parallel (and anti-parallel) symmetry of the square and the odd-numbered asymmetry of the triangle. In Geocentric Design Code, these plane types correspond to artifact types of fundamentally different functions: the planted symmetry of the abode, and the asymmetric motion of the transporter. Finding commonality between the 2 the plane types is very useful and essential to effecting polytechnic integration. Actually, one element in common to both planes has been noted in the line joining center-points of a rational accretion of spheres used in the very construction of the cuboda. With emergence of both plane types underlying the budding cuboda, the line constituted their shared border in a planar alternation that guided placement of subsequent spheres. With a complete cuboda, lines structuring the form’s interior also pose sides common to the form’s internal triangles and (nascent) right-angled squares. And in examining the cubodal pattern, we saw that completing these squares formed octahedra which led the growth of that potentially infinite pattern outward. The octahedron in and of itself can be viewed as a kind of seed form, the simplest polyhedral form possessing both square and triangles – with 8 of the latter encasing 3 interlocking of the former. Thus regarded, a thought experiment starring the octahedron yields a startlingly simple result. In the experiment, the structural octahedron (disregarding its planar reality) is enveloped by a sphere, and both are centered by a light source. So configured, the light is switched on and the 60° angle between the octahedron’s triangular edges become 90° separations at the very intersections of the edges’ arced shadows projected onto the inside surface of the sphere. In a sense, the octahedron’s internal squares have been externalized. Such a model has long been used to convey the idea of curved space and thus may be familiar to those grappling with general relativity. Here, the transformation is a kind of validation for a deeper commonality between otherwise disparate plane types. Returning full circle to planar reality and its expression, the square can be viewed as an expanded triangle or, conversely, the triangle can be viewed as a folded square – with each association united by circular arcs attending the transformation. As such, this commonality constitutes the basis of polytechnic integration rationales and procedures.

## Practical Cuts

For owner builders, carpenter/engineers, and even architects who risk being ostracized for sculpting from a socially unproven pattern as opposed to the pattern used since the days of flat earth consensus, the 3D roof fusion intended to house rolling transporters possesses useful trigonometric relationships.

Viewing the fusion from a polar perspective, the height (H) and half length (L) of the cross gable are related by the slope (Φ) which in turn is a function of the CBS roof slope (Δ) it is fused to according to the fusion formula: Φ = ArcSin (√3/3 Tan Δ)

The diagonal (C) that the gable presents is determined by the expressions indicated which quantify cuts to the gable’s face boards and the sheathing of its half-triangle “wings.”

Switching perspective to a top view, the gable’s ridge length (R) can be determined in 2 ways.

The diagonal (F) depicted in the profile is required for cutting CBS roof sheathing. The fusion angle (Ψ) from this perspective has no practical value that I can think of at the moment because it is an apparent angle, but it is useful for picturing the overall layout.

At this point it should be emphasized that the ridge is horizontal by reason of the wing’s fusion to the CBS roof and as determined by a level. With regard to the fusion option in which gable halves are separated at the ridge to form a flat plane between them, that plane should be tilted to correspond to the “horizontal” plane of the co-cube projection, with the angular difference (β) between natural and projected planes expressed by the equation:

β = Tan[Θ - ArcTan (.99345 Tan Θ)]

The difference varies by latitude (Θ). At latitude 30°, the tilt’s pitch (β) is .00284 to 1, or about 3/16″ over 62 inches. The purpose of the tilt is of course is to allow the “flat roof” to drain rain water.

More useful numbers are obtained by rotating the top view of the CBS roof (with fusion attached) about the axis of the (CBS) ridge such that the plane of the roof coincides with the plane of visualization.

So oriented, the diagonal (D), which is always twice the length of the ridge regardless of latitude, is required for beam cuts. Lastly, this perspective shows the angle (Ω) that the triangular fusion makes with the rectilinear plane. Omega is ultimately related to the latitude by Θ = Arctan (√3 Sin Φ). For CBS roof slopes ranging from 0° to 60°, Ω ranges from 30° to 90°. Interestingly, Ω is 45° for a slope of the same angle.

## CBS Roof Options

The nature of celestial cube-based shelter together with the cubodal wheel fusion poses the potential for 2 cross gables of differing slopes for any such house built in temperate zones between 30° and 60° (except at 45° where both gables slope at about 35°).

From that middle latitude, the difference between the gables increases – along with the CBS host roofs – until at 60° the polar gable reaches 90° to essentially vanish and the 30° gable settles at about 19° according to the fusion formula.

Φ = ArcSine [ (√3/3) TanΔ ]

Beyond 30° and 60° latitudes – in subtropical, tropical, arctic and subarctic realms – only one CBS roof is receptive to the fused gables - except at the poles and equator where any fusion whatsoever is precluded.

Such restricted variance coupled with the varied directions  of road access gives transporter accommodation the potential for being a convoluted undertaking. Fortunately, however, there are options available that are attuned to code geometry and philosophy.

The simplest option pertains to the steeper fused gables – the steeper the more applicable. In this option the 2 half wings  are separated such that the ridge becomes a 2-dimensional flat plane. In fusing to the roof’s prime cube projection according to the fusion formula, such a plane will be flat relative to the local horizontal shown by a level. To attain water drainage this plane can be tilted slightly according to the co-cube projection correction.

At this point, it should be stated that the steeper fused gable need not be for transporter housing. The gable can also be used for vents, windows, entrances, or bracing for vertical sun-blocking planes. If separated semi spherical skylights atop the resulting flat roofs are perfectly consistent with code geometry.

Another option applies to both shallow and steep gables and is a bit more complex but offers interesting configurations. Hybrid transporter housing combines the annexation approach with one gable slope.

With the hybrid roof, the annexation extends horizontally beyond the confines of the CBS roof such that its ridge is matched by that of the half gable. This configuration of course poses a corner approach to transporter housing.

The hybrid roof enriches the base CBS style with a combination of universal and variable elements. But because a certain complexity is also introduced, the code lays down rules for their implementation to avoid clutter and confusion.

1) an annexation can only extend past one CBS roof (north or south) so as not to obscure or undermine CBS expression of the juxtaposed celestial cube projections.

To construct hybrids using both gable angles (in temperate latitudes) they must be done separately on opposing (east and west corners).

2) separate (full) fused gables and annexations are not allowed on the same structure unless the 2 modes are informed by a hybrid roof.

With a particular sloped hybrid, either the annexation or the corresponding fusion or both may be constructed.

3) only full fused gables are allowed on mirror roofs, i.e., no annexations or hybrids. The reason for this is to avoid obscuring expression of the CBS celestially projected cubes. Mirror roofs already provide sufficient flexibility.

Such fusions are exempt from the rule requiring hybrids to be present for them and separate annexations on the same structure. If a hybrid is used, it must not be on the same side as the mirror roof to avoid confusing or obscuring CBS expression.

Again, the options and their rules are intended to afford variegated flexibility for particular circumstances while retaining intra-latitude identity and inter-latitude code integrity; and they lend the code’s CBS style – otherwise known as Humble Cosmic Architecture - more well rounded roundedness and pose a richer gem without diminishing its unique essence.