Wheel Extrapolations - Geocentric Design Code Part VI
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Transcript of Wheel Extrapolations - Geocentric Design Code Part VI
Geocentric Design Code Part VI Wheel Extrapolations - 1
Copyright © 2004-15 Russell Randolph Westfall Last edited: October 25, 2015
Introduction. The wheel interpretation of the cuboda is extended to embrace dynamic alternative
architectural styles, rolling infrastructure, and other realms of transportation that fully engage integration
methods to incorporate a range of functionalities applicable to mobility in general.
Overview: Part VI begins by appropriating the geocentric cuboda’s outer planes to guide wheel-based
shelter and diamond grid structures. From both grid and wheel orientations, path abstractions are made
for ground paths in the realm of roads as well as farm field furrows. The same geometry is shown be
equally applicable to code bridges or template aircraft of winged planar extensions to span gaps.
Wheel-based Shelter (p 2) – neutralized macrocosmic wheel; matched squares’ roof pattern; CBS annex
Diamond Grid Structures (3) – celestial co-cube D-grid rotation; wheel’s matched triangular roof pattern
Path Abstractions (4) – combined grid/wheel geometries; direction and traction; 2-point path fusion
Ground Paths (5) – grid berm paralleling; slopes; farm furrows; utility tubes; culverts; non-grid interfacing
Code Bridges (6) – square-up roadways; bracing; edge-up trusses; abutments; towers; parabolic arches
Template Aircraft (7) – runway; planar template extensions; angle of attack; HXP envelopment; wings
Air Streaming Methods (8) – line template extensions; cylindrical design rules; wing treatment; airfoils
Marine Vessels (9) – template hull; HXP deck gaps; vertical integration; dual rounding; submersibles
Fluid Dynamic Cubodas (10) – internal plane patterns; propeller element basics; dynamic transformation
Cubodal Turbines (11) – trifold dam geometry; impulse turbines; HXP dynamism; helicoid wind turbines
The Disc Orientation (12) – co-planing cubodal wheel; HXP stipulations; satellites and orbital planes
Directional Discs (13) – bow construction; stern square; orthogonal plane integration; docking schemes
After flight essentials are addressed with air streaming methods, the transport template is further applied
to designing marine vessels. The fluid dynamic cubodas used to propel such by utilizing internal planes
are further employed in cubodal turbines to generate electricity. Finally, the wheel is made to co-plane
with the disc orientation modified as a directional disc applicable to marine, air, and space craft.
Geocentric Design Code Part VI Wheel Extrapolations - 2
Copyright © 2004-15 Russell Randolph Westfall Last edited: October 25, 2015
Wheel-based Shelter
A polar-directed architectural style is guided by the macrocosmic wheel’s rotation such that any edge parallels a
location’s longitudinal tangent (bL). The wheel is then h-shifted such that 2 squares share the edge (bCl). Mirrored
rectilinear patterns posed by the microcosmic representative guide roofs set on co-cube projected walls (bC, bCr).
The slope is also applied to siding, internal bracing, stairs, etc. East and west walls paralleling the wheel’s central
plane are characterized by circular windows and vents (aR). WBS options include keying the slope to the terminating
tangents of rounded roofs; and to conical forms rounding or appended to polar ends (bL–C). The principle option sets
separated CBS roof planes upon co-cube projected annexes appended to north and south walls (bR).
Such annexes may serve as solar porches, bed-loft/bathrooms, etc. WBS embanking is 45° max-sloped half berms
which must manifest along north/south walls (bL). Along east/west walls, 19° berms express wheel asymmetry and
are corner-rounded with 35° mounds (bC). These and 30° mounds serve as intra- and inter-grid junctures at outside
corners. Inside corners key to the wheel’s profile angle, and WBS/CBS juncture mounds are keyed to 45°.
1:√2
≈ 35°
35°
E-W
35° Top View
N
35°
90°- α α
N - S
N-S
35°
19° 60°
45°
45°
45°
45°
35°
30°
60° 35°
Top View
E-W
35° 19°
Geocentric Design Code Part VI Wheel Extrapolations - 3
Copyright © 2004-15 Russell Randolph Westfall Last edited: October 25, 2015
Diamond Grid Structures
Diamond grid buildings begin with wall considerations. Upon location positioning, the celestial co-cube is rotated
about an axis spanning its foundation and opposing square midpoints to align with the diamond grid and project wall
guidelines (bL-C). The cubodal shell (minus the cube) is then rotated about either axis of opposing vertices (bCr).
After aligning an edge over location, the geocentric cuboda essentially becomes the macrocosmic wheel which is
hexagonally shifted such that the overarching edge is comprised of adjoining triangles (aR). From a microcosmic
ground perspective (bL), the triangular pair defines the slope and structure of roofs set on the D-grid aligned walls. In
profile, the triangular pattern is extendable and capped at each end with a √3:1 plane that joins triangular edges.
Ridge-aligned walls bear round windows. The style has the ability to extend vertically, along the ridge, and cluster in
corner mound-defined D-grid areas for retail, industrial, agricultural, and institutional buildings applicability (bL-C).
Ridge-aligned walls are embanked with 35° slopes corner-rounded with 55° slopes which otherwise use grid juncture
slopes. Inside corners are keyed to 60° and 45° expresses the grid’s square-up orientation (bR).
Macrocosmic Wheel Celestial Co-
cube
19° 1 : 2√2 ≈ 19°
NE – SW
or
NW - SE
30°
19°
30° N
19°
35°
60° 30°
45°
35° 30° Inter- grid juncture
mounds
E – W or N - S
N
55°
Geocentric Design Code Part VI Wheel Extrapolations - 4
Copyright © 2004-15 Russell Randolph Westfall Last edited: October 25, 2015
Path Abstractions
The geometry of that which the wheel rolls on - path - first follows the innate cubodal square lines that delineate either
grid type (bL). The integral tetrahedron integral to such lines of travel is minimally expressed by its orthogonal line
which corresponds to traction or torsion elements (bCl).
Conceptualized thus, any travel-directed line may serve as an axis about which the tetrahedron (and host cuboda) is
rotated to the edge-up position with a central vertically-aligned plane (aCr). Opposing rotations on either side of a
transversely extended path cross-section superimpose to constitute a symmetric path potentiality (aR). In profile,
such path mirrors the cubodal wheel to ground its asymmetric dynamism in a continual periodic resonance (bL-C).
Path is also viewed as the fusion of 2 points. A built-in fusion suggested by path’s 19° sloping triangle to its 30° host
element further evokes a vertex-up grid juncture with a vertical line (aR). When the wheel rolls to its vertex-up
position, rotation about an extended instantaneous vertical axis brings inherent angles of both wheel and juncture into
play (bL). As such, the 35° triangle adjoins to the deeper 45° slope in a dynamic intra-grid juncture fusion (bCl-C).
When the wheel’s vertex-up triangles pair with path’s edge-up sloping squares, resulting planar transformation arcs
signify dynamic areal traction (aCr). The square hosts the edge-up tetrahedron’s orthogonal vectors rotated 35° angle
to mirror the bend of the transport template’s hexagonal expansion which parallels path’s square-up orientation (aR).
Central (hexagonal)
Plane
Line of Travel
Orthogonal
Path Cross-section
Tetrahedron
2
30°
f (30°) =
sin-1
[(√3/3)tan30°]
≈ 19°
2 1 1
f (45°) ≈ 35°
35°
45°
30°
Intra-grid Juncture
35° Transport template 45°
35°
35°
Path cross-section
Grid Juncture
Geocentric Design Code Part VI Wheel Extrapolations - 5
Copyright © 2004-15 Russell Randolph Westfall Last edited: October 25, 2015
Ground Paths
Application of path geometry on the ground is limited and indirect. Path may be inferred by parallel grid berms, with
turns following rounded ends and/or grid junctures (aL-R), with farm field furrows expressing such parallelism fully.
Path is also inferred by straight or wave-formed slopes descending from each side and keyed to sloping square or
edge-up elements (bL). The road crown maximum slope contouring uses the plane-to-edge transformation angle.
Side slopes may also be keyed to the opposing edge-up angle of a particular grid’s berm slope to express wheel
asymmetry. For inter-grid turns, disparately sloped berms must have equal heights to facilitate berm switching at max
slope halfway points - or use 55° slopes common to both (aR). This universal grid angle may also be appropriate for
driveways, curbs, cuts, and unseen road bases in which culverts and utility conduits are positioned (bL).
Utility tubes are centered by the largest circles able to nest into the 55° maximum sloped wave. Wave height keys to
the number of road-width tetrahedra. Path supplies a 30° plane interface by which to overlap non-code roads (aR).
Shallower side slopes feather over the steeper, and the steeper crown is feathered over the shallower.
Parallel path cross-sections
Top View
Non-code road
Rebar Anchoring
Plane Interface
Culvert
Utilities # tetrahedra
Tetrahedron
H = √2/2 x W / #
N
≈ 19°
≈ 35° ≈ 55°
≈ 1.8°
35° 19°
19°
19°
55°
Cut
Curb Driveway
55°
Geocentric Design Code Part VI Wheel Extrapolations - 6
Copyright © 2004-15 Russell Randolph Westfall Last edited: October 25, 2015
Code Bridges
Abstract path is applied fully to grid-aligned bridges. The square-up cuboda defines roadway structure with its
essential tetrahedral lines whose connecting octahedral lines contribute some shear and sway bracing with (bL, bCl).
In profile, roadways set anywhere between the top and bottom members of edge-up hexagonal truss geometry (bCr).
Bracing also employs the edge-up cuboda’s transverse angles (aR, bL)). Such bracing joins truss to truss member or
abutment at a point that varies with road width (bCl). Bracing flexibility is increased with floor beam extensions, or by
vertical members introduced via path’s innate vertex-up grid junctures (bC). To join members of square and edge-up
orientations, semi-circular plates link intersecting lines, and cylinders join planes along their shared lines (bCr).
Towers and vertical abutments are structured with square-up octahedral stacking (aR). To such, edge-up bracing
infers vertical intra-grid junctures. Towers may support circular arches terminating at 30°, 60° or 90° tangents; and
parabolas at 30° or 60° (bL). As parabolas reflect verticals universally, they are allowed anywhere along their curve.
Cubodal angles reflect at 15°, 35°, 45°, 55° and 75°. Verticals outside the parabola terminate at the directrix (bC).
W
Abutment Profile
Truss
(√6/2)W
Square - to - edge-up
Cylinder 35°
Side Bracing
Focal Distance = 3H
L:H = 4√3 : 1
L:H = 4√3 : 3
Focal Distance = H/3
30°
60°
Directrix
Top View
Cross-section
Square-up bracing
Roadway
Geocentric Design Code Part VI Wheel Extrapolations - 7
Copyright © 2004-15 Russell Randolph Westfall Last edited: October 25, 2015
Template Aircraft
The transport template applies to rolling aircraft - with wings, stabilizers, etc., guided by triangular planes extended
therefrom (bL). The runway slab’s scoring and reinforcement is guided by path geometry. Viewed head-on, basic top
and bottom wing configurations generally exhibit dihedral and anhedral slopes of +/- 19° (bC).
The template pattern may also yield a wholistic angle of attack (aR). With bottom wings, lowering an endpoint is
attended by widening, and narrowing if raised; or the reverse with a top wing. The template’s hexagonal expansion
may be totally enveloped with cubodal geometry (bL-Cl). Transverse bracing meets intersecting lines of cubodal
planes - with struts extended from each HXP vertex and sheets extended from longitudinal edges where feasible.
The wing’s planform - with axes for ailerons, flaps, and elevators - is drawn from the pattern of the transversely
extended triangular plane (aCr). A version of the Part IV, p.10 wing may also be horizontally oriented via cylindrical
linking centered on the root chord (aR). Other options include extensions of the template’s rectilinear planes (bL); or
horizontally-oriented rectilinear planes of hexagonal extensions (bCr-R).
≈ 35°
Top View Fuselage
≈ 35°
HXT Planform
Fuselage
0° Dihedral
Triangle-up
Rebar 60°
Tetrahedra
Geocentric Design Code Part VI Wheel Extrapolations - 8
Copyright © 2004-15 Russell Randolph Westfall Last edited: October 25, 2015
Air-streaming Methods
Aircraft streamlining may proceed with the method of cylindrically joining spheres centered on vertices at either end of
a template-guided framework’s external edges. The proportion of sphere radii to transverse edge lengths is greater
for higher faster aircraft (bL). As longitudinal cylinders are template intrinsic, design may begin with these forms.
Where varied in radius, connecting cross-sections must manifest template angles via center-to-center lines, center-to-
tangent, or tangent-to-tangent planes (aCr). Template angles may also manifest in properly-sized concave rounding
spheres (aCr). Otherwise, creases are concavely melded with any sphere between sizes of spheres forming the
crease (aR). In profile, the leading sphere-cap defines the focus of an ellipsoidal nose and/or tail extension (bL).
As both wing (and stabilizer) sides are exposed, they are excluded from initial rounding and may meet centrally (aC).
A sphere situated at the root chord’s leading edge (aR) proportions an ellipse-led parabolic airfoil with angle of attack
keyed to the flat plane transformation; or that includes a drag-reducing waveform of matching slope below. The cube-
linked vertex-up cuboda supplies vertical lift-aligned structure. The bottom surface conforms to the template plane.
Rounding Spheres
Concave crease ≈ 19°
≈ 35°
90°
Cylindrical Cross-sections 35°
19°
≈19°
Cross-sectional structure Chord line
Airfoil
≈1.8°
1/4 wave parabola
LW
HW
HP
LP
HP
/ LP = ΠH
W / 4L
W
√2 : 1
Root Chord
Planform Leading Edge
Geocentric Design Code Part VI Wheel Extrapolations - 9
Copyright © 2004-15 Russell Randolph Westfall Last edited: October 25, 2015
Marine Vessels
Transport template cubodal geometry generally mirrors the classic hull framework (bL-Cl). The template’s hexagonal
expansion may also guide hull design, e.g. barges, but is more applicable to other elements of marine craft. To seal
rectilinear HXP decking, the gap between it and the hull requires special consideration.
From a deck’s fore-and-aft outboard edges and corners, cubodal planes are extended up to seal, and down for water
passage (aCr). Sealing the deck’s athwart-ship edge requires special cross-stitched planes (aR). Special planes are
also sliced vertically along bow or stern lines thus projected (bL). Vertical masts, etc. are integrated with bulkhead
plate links in conjunction with tetrahedral sphere or special cube/plate links according to overhead geometry (bCl-Cr).
Rectilinear athwart-ship bulkheads and HXP bulkhead hatches employ cubical links. Cubodal structure may extend
beyond a rounding framework to the hull. (aR). Spheres and cylinders are sliced radially along cubodal planes (bL).
The central plane may extend beyond the shell to regain a sharp keel which may be rounded spherically or with a
waveform (bCl-Cr). Submersibles’ spherically rounded cylinders are arranged according to aircraft methods (bR).
35°
Cuboda
HXP
Cross-stitched Plane
Deck
30°
Deck
19°
HXP deck, stowage, and
superstructure
Cubodal Hull
60°
Cubodal plane slice-offs
Sealing Plane
Drainage Plane
Deck Hull
Geocentric Design Code Part VI Wheel Extrapolations - 10
Copyright © 2004-15 Russell Randolph Westfall Last edited: October 25, 2015
Fluid Dynamic Cubodas
To move or be moved by fluids, the cubodal wheel’s 4 internal and interwoven hexagonal planes are engaged (aL).
For axial flow, the patterns of the skewed hexagons are cut so that rotation causes flow in one direction (aCl-Cr). The
3 paired triangles constituting propeller blade frameworks may be rotated about their common edge axes to conform
to the central hexagon (aR). In profile, a blade edge represents the maximum slope of a helix-projected wave (bL-Cl).
Propeller elements may be shaped outward from the central hexagon in 2 ways (aCr), with one set applicable to boss
cap fins or vanes. Elements may be bent along hexagonal lines; rounded circularly or with quarter waveforms (aR); or
centered on a common elliptical focus (bL). More curvature options are derived from a dynamic transformation in
which the cuboda is reoriented from the edge-up transporter mode to the triangle-up propeller (bCl-R)
Total rotation amounts to less than 70° in 3 rotation steps as opposed to one 90° re-orientation characteristic of a
simple static linking scheme. The transformation angles may be applied to the basic cubodal reference angle of a
propeller element’s corresponding geometric element
Tan-1
(√6/6)
≈ 22°
30° - Tan-1
(√3/9)
≈ 19°
Tan-1
√(14)/7
≈ 28° √3:1
≈ 71° √2Π x H
≈ 55° ≈ 71°
Geocentric Design Code Part VI Wheel Extrapolations - 11
Copyright © 2004-15 Russell Randolph Westfall Last edited: October 25, 2015
Cubodal Turbines
Cubodal propeller geometry also applies to hydro-electric reaction turbines. As such, dams possess bridge, berm,
and marine craft geometries, first manifested in the slopes of orthogonal edge-up cubodal orientations (bL), with walls
joined to buttresses by rounded cube links (bCl). The bridge’s square-up cuboda offers an alternative slope (bCr).
The square-up’s essential tetrahedral lines guide rebar linking both edge-up orientations. It may also base a cube-link
scheme to the triangle-up vertical axis turbine assembly (aR). An impulse turbine fed by hexagonally arrayed jets is
characterized by alternation of 6 oppositely-oriented cups (bL-Cl). More cups may be placed according to the planar
rotation angle of the dynamic transformation (bC). Penstock slope is keyed to the dynamic marine craft aspect (bCr).
The wave-formed spillway is keyed identically and may guide turbine chamber placement. With a straight grid-aligned
course, the dam may be curved convexly and/or concavely in the manner of berms (aR). Inner HXP planes present a
simple run-of-the-river turbine, but must be dynamically paired (bL). As a vertical axis wind turbine, 2 or 3 planes may
be variously curved (bCl-C). One plane set end may also be twisted and keyed to a helix-projected wave (bCr-R).
60°
19°, 35°
45°
r = (√6/4)l
55°
≈19°
HXP 60°
Top View
HXP
HXP
Top Views
60° = Tan-1
(1/b) b=√3/3
r = ae(√3/3)Θ
Logarithmic spiral 55°
Minimum slope
180° H = √2ΠW/2
60° Quarter waves
√3:1 half ellipses
Geocentric Design Code Part VI Wheel Extrapolations - 12
Copyright © 2004-15 Russell Randolph Westfall Last edited: October 25, 2015
The Disc Orientation
The cubodal wheel may be oriented such that its central bisecting hexagon parallels a surface of travel (bL). Co-
planing surfaces may be the water’s surface, subsea thermoclines, atmospheric layers, gravitational potentials, etc.
As such, the disc’s 2 motions are along the surface and about the axis perpendicular to both plane surfaces (bR).
Such rotation may simply be changing direction along the plane. As the disc is a complete dynamic construct, it may
not be hexagonally-shifted and expanded in the same way as template transporters are. However, HXP components
may be internally incorporated by h-shifting the cubodal pattern at one axis end (bL). HXP constructs may be crafted
externally also if paired to retain overall dynamism (bCl). A prime unmodified disc application is as a satellite (bCr-R).
Of the 4 cubodal axes orientations, rotational inertia is maximized with the disc for gyroscopic control and stability.
Surfaces of travel are the geocentric cuboda’s polar and subtropical orbital planes. Although unnecessary in non-
viscous space, the disc may manifest in forms of specified curvature below. The cone-forming line common to both
cubodal planes may key ellipsoidal forms to express cubodal asymmetry, or bias the axial line in hybridized forms.
≈ 55° Max slope
Tan-1
(√6/[3-√3])
≈ 63° a/b =
√ (3-√3)
a/b = 2+√3 /
√ (1+4√3) ≈ 55° ≈ 71°
Toroid
HXP
H-shifted pattern
Cubodal pattern
Geocentric Design Code Part VI Wheel Extrapolations - 13
Copyright © 2004-15 Russell Randolph Westfall Last edited: October 25, 2015
Direction-imbued Discs
The disc’s cubodal pattern can be extended by joining a square pyramid to a disc square to complete an octahedron.
The octahedral triangle adjoining the disc’s central hexagon is then matched by a tetrahedron to supply the disc with
a distinct vertex by which to lead non-shuttling marine, air, or space craft (bl-Cl).
Employing one square thus distinguishes its opposite as with the geocentric cuboda’s equatorial squares (aCr-R). A
3D rectilinear construct may thus be extended from the stern square (bL). If outward, the construct may be capped. If
in-ward, it may meld with an axially-aligned cubical structure - as do celestial co-cube projections – if properly-linked
(bCl). To integrate an orthogonal motion-directed plane, a circular plate link first enables an orthogonal shift (bCr).
With the resulting motion-aligned hexagon, the cubodal shift allowed hosts a cube-linking scheme (aR). Tetrahedral
linking bolsters one side, and an h-shifted edge-up cuboda is nested on the other to supply lines and planes for keels,
tail fins, or non-rotating propulsion (bL). Docking template-guided transporters utilizes tetrahedral links on cubical
extensions (bCl); or cube links on a hexagonal extensions or its cap (bCr-R) - with either also serving as conveyers.
Tetrahedral Link
H-shift
Top stern square
Bottom stern triangles
Tetrahedral Link