wheel extrapolations

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.



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°



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°



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



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°



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



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



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



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



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°



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



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



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

wheel extrapolations - geocentric design code part vi

Documents

wheel orientations

wheel interpretation

wheelbased shelter p

transport template

geocentric design code

integration methods

internal planes