Methods to join artifacts employing differing orientations of the code's geometric model. Polytechnic incorporation via polyhedral integration. Begins with shelter and transporters.

Published on: **Mar 4, 2016**

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Engineering

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- 1. Geocentric Design Code Part IV Polytechnic Integration 1 Copyright © 2004-15 Russell Randolph Westfall Last Edited: January 17, 2015 Introduction: architectural and transporter design guidelines - based on the cuboda’s free and earth- centered planes – are fused by methods from which universal linking intermediaries are abstracted for the purpose of incorporating polytechnic functionalities via polyhedral integration. After detailing the method of integrating CBS architecture and cubodal wheel geometry, rules to discipline the flexibility of alternative options are followed by relevant measurements. Reconciliation of fundamental plane types involved lead to an abstract fusion of the triangular tetrahedron and as its 3D rectilinear aspects find correspondence in the cube, that form is deemed most useful as a linking intermediary. By centering spheres on cube corners, the link is posed to reorient the cuboda orthogonally such that axes of wheel-based constructs align to propel template guided transporters - or support cubodal shifts that enable wing accommodation. Circular based links support alternate hexagonal and cubical patterns and finally spheres of themselves serve as external and internal links to just about anything. ✨ Table of Contents ✨ Full CBS Fusion (2) - macrocosmic wheel; triangular wings / CBS roof; fusion formula; gable bisection Option Rules (3) – fused gable options; hybrid corner fusions; annexation and mirror roof rules Cuts and Angles (4) – direct fusion dimensions; top, profile, and CBS roof views; volumes and areas. Plane Transformation (5) - plane junctures; octahedral sphere projection, arced 2D transformations The Tetrahedral Factor (6) – abstract fusion; angular equivalence; tetrahedron formation; recti-linearity Orthogonal Linking (7) – tetrahedral inscription; cube link; orthogonal orientations; all cubodal positions Full Link Configurations (8) – reinforced cube links; corner spheres; radii ratios; unit link assemblages Vector Reorientation (9) – transport template linking; cubodal propulsion axis; vertical axis alignment The Cubodal Shift (10) – opposing pairs; tetrahedral bridging; rotor housing; existing plane shifts; wings Hexagonal Alternates (11) – orthogonal hexagons; circular links; crossed rectangle links; overlap radii Cubical Incorporation (12) – inside corner radii; abutting and separated lattices; hex-expansion links Universal Spheres (13) – cube and tetrahedral spheres; external and internal sphere linking ✨. ✨. ✨
- 2. Geocentric Design Code Part IV Polytechnic Integration 2 Copyright © 2004-15 Russell Randolph Westfall Last Edited: January 17, 2015 Full CBS Fusion In the Cube-based Shelter context, a transporter housing configuration is sought in which both cube projections are engaged, i.e. roof and walls. To begin, the earth-centered macrocosmic wheel is positioned such that any of its outer triangular edges parallel the longitudinally aligned surface tangent at a specified location (bL). Viewed from a local perspective (aC), the structure juxtaposed microcosmic representative is turned to a north/south orientation (aR). Regarded thus, the wheel undergoes a hexagonal shift to attain symmetry. Returning perspective to the profile (bL), the neutralize wheel’s top paired triangles are isolated, detached, and juxtaposed against the structure such that the ridge connecting the pair remains horizontal. Viewed directly (aR), the pair’s triangular “wings” are adjusted to fit the CBS roof in accordance with the fusion formula, where Φ is the angle of the tri-wing spread and ϴ equal to latitude or its complementary angle. The tri-wings are then bisected vertically such that they are flush with the north/south wall (bL), with the resulting cross gable serving as a canopy for wheeled accommodation while also manifesting latitude from the polar perspective (bR). Φ = Sin-1[(√3/3)Tanϴ] ϴ Φ ϴ = Tan -1 (√3SinΦ)
- 3. Geocentric Design Code Part IV Polytechnic Integration 3 Copyright © 2004-15 Russell Randolph Westfall Last Edited: January 17, 2015 Option Rules Options associated with the CBS/Wheel fusion pose sufficient complexity for implementation rules to be warranted. Primarily intended for transporter housing, the fused gable may also be used for entrances, vents, or windows (bL); but dormer structures may not be extended such that the gable roof geometry ceases to be triangular (bC). To harmoniously accommodate rectilinear elements under gables, truncation may follow the triangular pattern to effect hip style dormers (aR), an option required for overhanging extensions. An option applicable to steeper sloped gables entails splitting the ridge and extending the flat roof component transversely (bL). At CBS corners (bC), hybrid roofs join half fusion gables with wheel-based annexations to effectively juxtapose fixed and variable elements. In profile, hybrid annexations may be extended on one side only to avoid obscuring CBS expression (aR), but may be set on opposing corners (bL). Separate annexations and fusions may not exist on the same structure without a hybrid to inform both (bC). Mirror roof fusions are exempt from this rule, but supporting wall extensions may not bear either hybrids or annexations (bR). All such rules are intended to preserve intra-latitude identity and inter-latitude integrity. Annex Fusion Φ ≈ 35° No Corner Mirror Roof Fusion Top View No
- 4. Geocentric Design Code Part IV Polytechnic Integration 4 Copyright © 2004-15 Russell Randolph Westfall Last Edited: January 17, 2015 Cuts and Angles For builders, the following perspectives afford ways of determining critical measurements pertaining to the fused CBS gable. The direct longitudinal view above relates the fusion angle phi (Φ) with the half lateral dimension (L), height (H), and diagonal length (C). Both the top view gable angle (Ψ) and the direct view of the universally determined ridge length (R) aid crafting floor plans on both north and south sides, In profile (aR), the ridge is shown in relation to CBS slope delta (Δ), as well as the length (F) for sheathing cuts. A direct view of that dimension accompanies a direct view of the CBS roof plane (bL) - wherefrom the diagonal (D) and omega (Ω) are related to length, as well as the gable and CBS slopes. The extra material and space represented by a fused gable may be determined by the given formulas for area (A) and Volume (V). L Φ C H Fusion Formula: Φ = Sin -1 [(√3/3) Tan Δ] H / L = TanΦ Φ = Tan -1 [H/L] C = L / Cos Φ = H / Sin Φ = √ (H 2 + L 2 ) Ψ R R F Δ Ψ = Tan -1 ( R / L ) R = C / √3 = H / Tan Δ L F = H / Tan Δ Ω L D F D = 2R Ω = Cos -1 (L / D) = Tan -1 (F / L) = Sin -1 (1 / [2 Cos Δ]) = Sin -1 [ √ (1 + 3 Sin 2 Ф) / 2 ] A = H2 CosΦ [ 1 + 1 / (√3 SinΦCosΦ ) – 1 / √(1+3Sin2 Φ)] / SinΦ V = R3 SinΦ CosΦ Plane of Roof
- 5. Geocentric Design Code Part IV Polytechnic Integration 5 Copyright © 2004-15 Russell Randolph Westfall Last Edited: January 17, 2015 Plane Transformation From the perspective of the CBS roof plane viewed directly (bL), the edges of the microcosmic wheel’s triangular wings form another triangle whose proportions vary with latitude. The relevant CBS slopes range from 0° to 60° while the angle omega (Ω) of triangles superimposed onto CBS rectilinear planes vary from an overlapping 30° to 90°. Thus basic forms – the square and the triangle - are joined 2-dimensionally in the fusion of basic corresponding artifact types: shelter and transporter (aR). The common element between the square and triangle was first observed in the edges shared by the cuboda’s external alternating planes. Such commonality is also found internally (bL), and the octahedron is a natural growth of the innate pattern exhibiting that quality (bC). Being the simplest form possessing both squares and triangles (aR), the octahedron is centered by a light source and encased inside a sphere for a thought experiment. In it, 60° triangles become 90° arced shadows on the sphere to in effect externalize the squares (bL,bC). Expressed in a plane, the square can be regarded as an expanded triangle, or the triangle as a folded square, with transformations attended by circular arcs in either direction (bR). Plane of CBS Roof Ω (Δ) = Sin-1 [1 / (2 Cos Δ)] Ω 60° 90° Ω (0°) = 30° Ω (60°) = 90° Ω (45°) = 45°
- 6. Geocentric Design Code Part IV Polytechnic Integration 6 Copyright © 2004-15 Russell Randolph Westfall Last Edited: January 17, 2015 The Tetrahedral Factor In deriving the 3D plane transformation and expressing the result in 2 dimensions, the ultimate proportion of the triangle against the rectilinear background is equilateral (Ω = 60°, bL). To obtain this result in the context of an abstract fusion, the slope of the rectilinear plane is √2 : 1, or approximately 55° (bC). Alternatively, Ω = 60° is again the result with a tri-wing angle of Ф ≈ 55° (aR). This relationship is also arrived at upon equating Ф and Δ in the fusion formula. Such symmetry and the fact of both tri-wings being equilateral defines the form as a whole to be a tetrahedron (bL). In the cubodal context, the tetrahedral slope matches slopes of the cuboda oriented upon triangular or square bases (bR). In the latter case, the tetrahedron interfaces the octahedral square. The interface of the former case exhibits an octahedral overlap of 2 tetrahedra (bL). The parallel sets of orthogonal lines characterizing the pair thus joined (bC) suggest a 3-dimensional recti-linearity to the fusion. This quality is also exhibited by the form upon orienting it such that both essential lines lie in planes parallel to the viewer, with the intersection having 4 equivalent right-angled areas, and the external perimeter appearing as a square (bR). Ω = 60° @ Δ ≈ 55° & @ Ф ≈ 55° 55° = Sin -1 [(√3/3)Tan 55°] Ω = 60° Δ = 55° Ф = 55° Octahedron Tetrahedron T/O O/T
- 7. Geocentric Design Code Part IV Polytechnic Integration 7 Copyright © 2004-15 Russell Randolph Westfall Last Edited: January 17, 2015 Orthogonal Linking Approaching tetrahedral rectilinearly from the opposite direction, the 6 faces of the cube may be diagonally inscribed with a tetrahedron (aL). A second tetrahedron may then be formed by spanning the cube’s remaining corners (aC). Thus the abstract ideal but structurally weak cube is optimally reinforced. This quality - together with the form’s most economic expression of 3 dimensions (aR) - render the cube ideal as a linking intermediary (bL). By reason of planar transformation, the cube link reorients the cuboda to orthogonal positions. From a vertex-out perspective (aR), each primary orientation (edge, vertex, square, and triangle-up) is directly attained from any of the other orientations via (octahedral) square-to-square, triangle-to-triangle, or square-to-triangle interfacing. Triangularly interfaced cubodas may be further reoriented, as shown from the 2 perspectives below. Edge-up Triangle-up Square-up Vertex-up square-to-triangle square-to-square
- 8. Geocentric Design Code Part IV Polytechnic Integration 8 Copyright © 2004-15 Russell Randolph Westfall Last Edited: January 17, 2015 Full Link Configurations Cube links between constructs of differing cubodal orientations are distinguished by facially-inscribed reinforcing tetrahedra, if only symbolic (bL). Spheres - the common element between all cubodal orientations - are centered on link corners (bCl), then sliced radially to interface cubodal planes (Q1, Q2) and/or cube faces (bCr). In so doing, a sphere may contact more than one cubodal orientation, or more than one plane of a given orientation to harmonize and streamline constructs as well as disperse forces. Spheres originate only from cube corners (aR), but may overlap cubodal constructs anywhere. Default sphere radius is 1/2 cube edge length (bL), but may be as large as one full edge length, a configuration that may imply a triangle by the arc over the square face (bC). Space permitting, sphere radii may be crafted to reflect pattern interfaces such as the octahedral square (aR), or in an alternate way to express the inscribed tetrahedron (bL). In such cases, spheres may be absent but inferred by their complements or be present but may not overlap – even internally. Unit cubes may be assembled into any configuration if individually deducible (bC); while equal spheres along any edge may be joined cylindrically (bR). Q1 (square interface ) Q2 (triangle) no
- 9. Geocentric Design Code Part IV Polytechnic Integration 9 Copyright © 2004-15 Russell Randolph Westfall Last Edited: January 17, 2015 Vector Reorientations To apply the cube link schematically to a practical situation in general, it is interfaced with a edge-up cuboda orientation which, upon being hexagonally shifted, essentially represents the transport template (bL). Thus Interfaced, a rotating cubodal wheel-based construct is interfaced with the cube’s facing square. (bC). Viewed in profile (aR), the axis of such a wheel is aligned with the transport template’s direction of motion and may thus represent a propulsion vector pertaining to air and marine craft. Lines emanating from the cube link’s corners are intrinsic to and extended from the template’s pattern (bL). To house or frame a rotating element, accommodation design is guided by the h-shifted cubodal wheel for at-rest symmetry (bC). Alternatively, a cubodal wheel construct may be set on a template-linked cube’s top square (aR, bL). In such case the axis is vertical as manifested by seed broadcasters, ship’s radar, and helicopters. With vertical and motion-aligned axes, shaft accommodation necessitates omitting a line of unit cubes in a larger configuration (bCl). The problem of triangularly centering amid quantized squares is countered by freedom in varying sphere and cylinder radii (bR). Transport Template Transport Template Transport Template Transport Template Irrational Dimension
- 10. Geocentric Design Code Part IV Polytechnic Integration 10 Copyright © 2004-15 Russell Randolph Westfall Last Edited: January 17, 2015 The Cubodal Shift The transport template’s edge-up cuboda has other planes to which cube links may interface (aL). In one application, mirrored cube links interface triangle-up cubodas of opposing orientation on one hexagonal plane (aC). To enable such a cubodal shift, the pair’s linear juncture must align with the template’s motion direction (aR). Tetrahedral links bridge the pair’s opposing octahedral faces by virtue of identical slopes and planar transformation (bL). In the vertical axis context, hex-shifting and expanding both cubodas configures a construct suitable for housings or frameworks of rotating elements (aC). Such a configuration is characterized by an oscillatory geometric resonance between stationary and rotating elements (aR), while enabling precise shaft centering in both plane dimensions (bL). Another application of the cubodally-shifted pairing involves the template’s existing vertical planes (bC) In such case, one of the pairing is actually an h-shifted cubodal. As the juncture satisfies the motion-aligned requirement, the pair may interface cube links bridged by tetrahedra to accommodate edge-out hex-shifted cubodal constructs to provide a horizontally-oriented hexagonal plane for guiding wing design, etc. (aR). Transport Template Transport Template Top View Hexagonal Plane Tetrahedral Links
- 11. Geocentric Design Code Part IV Polytechnic Integration 11 Copyright © 2004-15 Russell Randolph Westfall Last Edited: January 17, 2015 Hexagonal Alternates Another transport template integration - applicable to its vertical planes (bL) - entails rotating the hexagonal orientation 30° to effectively obtain a pattern possessing vertical lines orthogonal to the travel direction (bC). Such an orthogonal shift (O-shift) is largely justified by planar transformation of squares and triangles reasoning (bR). As such, 2 orthogonal triangles - representative of hexagonal lattices by their innate patterns - share one arc of transformation. As circles are intrinsic to both orientations, these forms serve to model links between them (bL). For smaller O-shifted cubodal constructs abutting the template’s plane, a simple circumscribing circular plate serves as the link. For larger separated constructs, cylindrical extensions serve as the intermediary. Special links are derived from orthogonal rectangles proportioned to minimal hexagonal expressions (aR,aC). Radii of circles centered on outside corners are keyed to the rectangles’ overlap, and aligned circles join tangentially (bL). The links may manifest as plates or be extended transversely (bCl). They may also be joined end-to-end to economically align with vertical bias (bC). For abutting or separated O-shifted constructs, links are melded accordingly (bCr,bR). 30° 1xL 1xL √3xL [(√3-1)/2] x L
- 12. Geocentric Design Code Part IV Polytechnic Integration 12 Copyright © 2004-15 Russell Randolph Westfall Last Edited: January 17, 2015 Cubical Incorporation To integrate 3D rectilinear constructs with the transport template’s vertical cubodal planes, crossed rectangles keyed to minimal hexagonal expression again underlay the links, with circle radii equaling overlap but pegged to rectangular intersections (bL, bC). Configurations of such links are joined by the rectangles’ common squares (bR). For separated constructs, pre-existing rectangular melding is retained for the transversely extended links. The melding is omitted for abutting constructs as shown in the intermodal container scheme (bL). With regard to the triangular prism-based hex-expansion (bC), a spherical basis and transversely-directed squares are accommodative of square links to 3D rectilinear constructs. On such, corner-centered circles must have half-edge length radii (bR). Corner circles are joined by circles centered on the triangular convergence below them. Links join square-to-square and for separated constructs, link extensions meld tangentially along transverse lines, while overlap circles remain flat (bL). To encompass abutting constructs melding is omitted on flat plate circular patterns (bCl). Cube links to cubodal constructs bear half edge radius spheres and transverse cylindrical melding (bCr, bR). [(√3-1)/2] x L
- 13. Geocentric Design Code Part IV Polytechnic Integration 13 Copyright © 2004-15 Russell Randolph Westfall Last Edited: January 17, 2015 Universal Spheres Cube links, especially those purposed for the hex-expansion, evoke the alternate accretion of spheres that built the form. Similarly, vertex-centered spheres of the tetrahedron – the first 3D form created by spheres - may largely obscure the form in its role as a link, even without prohibited overlapping, as with the cubodal shift (bL). Upon employing a sphere radius of one edge length (aCl), the form is virtually hidden. In such case, the sphere is segmented along planes adjacent to, or opposite of the vertex/sphere center (aCr) to link square and triangle-up constructs (aR). Alternatively, the sphere comprises the whole link in joining vertically and horizontally-biased hexagonal lattice structures (bL). In practice, the sphere interfaces the ends of cylindrical tubes and/or rods (bCl). To avoid overlapping of structural members and attain hexagonally-consistent melding angles of 0, 30, 60, or 90°, relative sphere and member radii are quantified by the equation and corresponding expressions (aR). Because spheres are inherently omnipresent in the cubodal pattern and connected to it by a single line, such may serve (along with cylindrical extensions) as internal links to infinite cubodal orientations as well as non-code constructs. R r α α r / R = Cos α α = 0°, 30°, 60°, 90° @ - r = R, (√3/2)R, R/2, 0 Non- code