Geocendtric Design Code (Patterns)

8/12/2019 Geocendtric Design Code (Patterns)

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Copyright does not protect ideas or methods - only distribution of their expressions. GDCode users are

allowed only those copies necessary for their personal or professional use. Laser printer recommended.

Introduction- Geocentric Design Code is a guiding framework of geometric patterns from which a broad

array of human constructs may be fashioned. Be they in the realm of architecture, mobile engineering, orland design, all artifacts are made integrable by reason of their common derivation from a universal model

keyed to earth. This document begins with basic concepts which advance sequentially to a maximum

midway complexity that informs the most far out designs concluding its 7 part organization:

Orientation - page 1: geometric construction; geocentric correspondence; polyhedral pattern generation

Cube-based Shelter - page 11: celestial cubes; projected architectural guidelines; design implications

Rolling Transport - page 21: cuboda wheel attributes; transport template; architectural accommodation

Polytechnic Integration - page 31: wheel/shelter fusion; orthogonalized cubodas; linking intermediaries

Ground Rules -page 41: surface grid integration; prismatic cuboda waves; applied criteria and options

Wheel Extrapolations - page 51: path abstractions; applied fluid dynamism; wheel-based architecture

Extra-topographic Guidelines - page 61: circular buildings; electromagnetic cuboda; rocket design

Orientation- for a conceptual foundation, Geocentric Design Code employs a universal model comprised

of an ideal geometric form matched to the real earth. As prescribed rules inform the various abstractions

of the geometry thus-oriented, this model serves as a virtual template for all GDCode applications:

Elements of Space - (p. 2) - sphere/point designation; line, plane, and 3 dimensional formation

Rational Accretion- (p. 3) - tetrahedral orthogonality; square formation; 6-sphere commonalities

The Cuboda- (p. 4) - common sphere/edge placements; hexagonal vacancies; the cuboctahedron

Cubodal Manifestations- (p. 5) - internal/externalities of spherical, structural, and planar expressions

Earth Alignment - (p. 6) - cubodal symmetries; geocentric keying; coaxial alignment; primary rotation

Universal Location - (p. 7) - opposing rotated elements; midpoint equatorial axes; secondary rotations

Octahedral Rectilinearity- (p. 8) - the octahedron; polyhedral packing; rectilinear pattern generation

Hexagonal Lattice - (p. 9) - tetrahedral perspective; alternating triangles; extendability; sub-divisibility

Indefinite Accretion - (p. 10) - intersections; sphere placements; triangle ambiguity; cubodal centrality

Part I commences by forming elemental geometric entities with reasoned sphere placements that build

the cuboctahedron. Spherical, structural, and planar aspects coaxially match the form!s innate centrality

to earth such that polar and equatorial rotations conceptually locate geometric elements to any latitude

and longitude - at any time. Thus locatable, rectilinear and hexagonal perspectives are explored to see

how infinite 3-dimensional generation bestows centrality upon any placed sphere.

Captions- bL, bCl, bC, bCr, bR & aL, aCl, aC, aCr, aR signify below Center left, above Right, etc.

Geocentric Design Code - 1 of 70

Copyright 2004-12 Russell Randolph Westfall Last Edited: March 24, 2012


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Construction of the GDCode model employs a process of rational accretionwhich designates the sphere as its

elemental building unit. As such, the sphere is regarded as an omnidirectional expansion of the dimensionless point

that traditionally begins the (geometric) analysis of space. Depicted or not, this (center) point is integral to the sphere.

With multiple spheres, different possibilities exist pertaining to relative position (separation, overlap, engulfing, etc.)

and size. In rational accretion, spheres are specified to be of equal size - with neighboring spheres positioned such

that their surfaces contact. The linejoining 2 neighboring spheres!center-points (bL) constitutes the 1st dimension.

A 3rd sphere placed in mutual contact with the paired spheres exhibits orbital freedom about the line joining that pair

(Ac). This type of placement - termed planar nesting- forms the 2nddimension as lines joining the 3 spheres!center-

points delineate the elementary plane of an equilateral triangle (aR).

Sphere 4 is deep nestedinto the 3-sphere cluster to form a stabile arrangement - the largest in which any given

sphere is in mutual contact with all remaining spheres. The 4-sphere cluster!s center-points are joined by 6 lines to

form 4 (equilateral) triangles and the underlying 3-dimensional solid termed tetrahedron(bR).



Elements of Space


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Before placing additional spheres, the tetrahedral cluster of spheres is regarded as 2 pairs of spheres. The underlying

tetrahedron is then rotated about one of its edges to illustrate how lines joining each pair effectively cross to suggest

the concept of right angles (orthogonality), manifested as 4 equivalent areas around an apparent intersection.

To form an actual right angle, sphere 5 is first planar-nested between 2 spheres, 1 from each pair. It is then rolled

between the 2 spheres to a position such that the line joining its center-point to that of the contacting sphere below itparallels the vertical sphere-pair line, and forms a right angle to the line joining the horizontal sphere-pair (bCL).

If sphere 6 is then placed similarly on the 5-sphere cluster!s right side (aCR), an underlying plane characterized by a

circuit of right angles and a minimum of parallel sides defines the square(aR). Below left, the new cluster is turned to

show that only 1 of its 6 spheres is in contact with all remaining spheres. Of those 5 spheres, the triangular and

square sphere-clusters show 2 spheres in common (bCR), as well as the line joining their center-points (bR).



Rational Accretion


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Employing the abstractions of the previous page, sphere 7 is deep-nested so as to contact the one common sphere

and the 2 spheres delineating a square line to form a new triangle and a new right angle (aL). Sphere 8 is then placed

so as to contact the common sphere and advance the pattern of alternating planes to form a new square (aR).

Spheres 9 and 10 are deep-nested between the common sphere and sphere-pairs of the new and original squares to

thereby continue the pattern of alternating squares and triangles (bL). Viewing the resulting cluster from the reverseside, the common sphere is shown to center 6 outer spheres to in effect form an underlying hexagon(bR).

The 6 vacancies created by the hexagonal cluster pose 2 accommodation possibilities of 3 additional spheres each.

Spheres 11, 12, and 13 are deep-nested according to the option in which the pattern of alternating planes followed

thus far is completed to form a cuboctahedron, shown below in its 3 prime manifestations. For ease of frequent

reference, cuboctahedron will henceforth be shortened to cuboda, with cubodalbeing the pertinent qualifier.



The Cuboda


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Each prime cubodal manifestation exhibits two aspects. The spherical manifestation is expressed externally by a

clusterof surfaces (aL); and internally by a constellationof arrayed center-points (aR). Below, the cuboda!s structuralmanifestation displays an external frame in which 4 lines join each of the 12 outer sphere center-points to their 4

outer neighbors; and a radialaspect in which 12 internal lines intrinsically join each of the outer points to the center.

The 12 internal lines and 24 external lines are all of equal length, an attribute unique to the cuboda. The cuboda!s

planar manifestation exhibits a facetedaspect in which 8 triangles alternate with 6 squares (bL) - with shared lines

becoming edgesthat span points in common termed vertices; and an internal aspect of 4 interlockinghexagonal

planes (bR). All expressions here find use in practice, either individually or by interacting with each other.



Cubodal Manifestations

Spherical

Structural

Planar


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In matching the foregoing geometric ideality to the real world, the central (common) sphere of the cuboda!s spherical

cluster corresponds to the earth sphere - obscured to some degree in each of the 4 prime cluster perspectives above

where underlying geometric elements (square, edge, vertex, or triangle) are brought to the fore. Amid the 12 outer

spheres (bL), the earth in actuality centers, bridges, and contacts 6 pairs of opposing spheres.

The 12 lines emanating from earth to outer sphere center-points can thus be regarded as 6 extended lines. As these

are essentially indistinguishable one from another, one line is singled out to coincide with the earth!s axis of rotation,

as depicted with the 3 foremost outer spheres removed for clarity (aR). Below, all outer spheres are removed to

reveal the cuboda!s faceted aspect encasing the coaxially-aligned earth.

In the context of this configuration, the cubodal shell revolves relative to the earth!s surface via primary rotationsuch

that prime geometric elements are positioned directly above the equator at any longitude - instantaneously. One

triangle tracks the rotation above.



Earth Alignment


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Each opposing set of triangles, vertices, edges, and squares that is equatorially oriented and positioned longitudinally

via primary rotation possess midpoints through which equatorial axesare introduced, as depicted with direct views in

the top row. Corresponding profiles of the imaginary axes (middle row) exhibit the potential for secondary rotation.

Following primary rotation, secondary rotation positions the cubodal shell longitudinally such that geometric elements

may be (instantaneously) located directly above any latitude, as suggested by the tracking triangles of the bottom 2

rows. Once set to a specific location, each geometric element presents a particular orientation of the cubodal pattern.



Universal Location

N N N N


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The structural pattern of the cuboda is depicted most completely by orienting it to face a triangular cluster (aL). Even

though the octahedral component is not complete, by examining the triangular perspective!s planar aspects the

cuboda!s internal polyhedral packing is evident (aR). Below, a (complete) octahedron and tetrahedron are isolated to

show how triangular faces are matched to generate the cubodal pattern!s hexagonallattice.

In contrast to the plane of the rectilinear lattice, a principal characteristic of the hexagonal lattice is that its single

component (equilateral triangle) has 2 (opposing) orientations. Below, the tetrahedron may be subdivided

infinitesimally by the same internal packing scheme that was shown with the octahedron, except that in this case

constituent tetrahedra occupy the larger tetrahedron!s (corner) vertices and edges.

Triangles not occupying vertices and edges signify octahedra bridging the gaps between constituent tetrahedra (aC).

Because polyhedral packing on any conceivable scale is also manifest from the tetrahedral perspective, arbitrarily

extending or subdividing the plane of hexagonal lattice is consistent with generating the cubodal pattern (aR).



Hexagonal Lattice

Tetrahedron Octahedron

Tetrahedra Octahedra

Tetrahedron


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An underlying characteristic of the cubodal pattern, whether dealing with hexagonal, rectilinear, or the (other 2) prime

orientations, is that any intersection of 2 lines constitutes a potential sphere center-point. Once the cubodal pattern is

established, such a sphere may be of any size. Conversely, employing spheres of a given set size, a full self-

contained cuboda may be constructed upon a base utilizing either plane type below.

Absent the guidance of a pre-existing cubodal pattern, a full one can be constructed by following sphere placement

rules, prescribed according to the plane orientation utilized above. With the hexagonal orientation (aR), there is some

ambiguity when deep-nesting spheres into triangular clusters. Even with an established pattern (bL), confusion

increases as to whether adjacent triangular clusters signal an underlying octahedron, or tetrahedron.

A sphere placed in a tetrahedral triangle terminates alternation of plane types - a key cuboda characteristic - and the

pattern is broken. Conversely, a sphere placed in an octahedral triangle continues the pattern. As such, every sphere

comprising it potentially centers 12 outer spheres with an arrangement identical to the original 13-sphere cuboda.



Indefinite Accretion

NoYes

Yes

Yes

yes

yes ?

yes no


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Part II - Cube-based Shelter- the cuboda!s geocentric orientation supplies foundations for opposing

celestial co-cubes that project uniquely intrinsic homogenous patterns onto earth!s surface from diverging

perspectives to guide design of basic architectural elements at any particular location.

Alternative Accretion- (p. 12) - random vs radical spheres; orthogonal planes and cube formation

Celestial Co-Cubes - (p. 13) - cubodal shell square sets; equatorial foundations; celestial cube qualities

Primary Projection- (p. 14) - prime cube; equatorial pattern projections; surface - latitude equivalence

Secondary Projection- (p. 15) - co cube positioning; direct latitudinal projections; polar-rotational grid

Architectural Reconstitution- (p. 16) - juxtaposed cube projections; conventionality and roof guidance

Precision Challenges- (p. 17) - sphere size issues; polar drift and wobble; oblate spheroidal correction

Exterior Considerations- (p. 18) - rotated cubes!circularity and perceptual effects; solar applications

Interior Design- (p. 19) - open floor/ceiling layouts; low-to-high walls; stairs and lofts; conventionalities

Building Options- (p. 20) - masonry walls; open beam schemes; top vent arcs; porches; roof mirroring

Part II addresses geometric duality by grafting cubical pattern uniformity onto the cuboda, the geocentricmanifestation of which bases a 2 cube set whose space filling quality is first projected longitudinally from

equatorial positions. To this, the co-cube adds and extends its essence upon further latitudinal positioning

and divergent celestial projections are then reunited on the ground to form the basis of an architectural

scheme in which precision meets earthly reality by a small-is-good building approach. External attributes

range from psychological lift to optimal solar applicability, and on the inside, conventionality in the context

of inherently open centralized layouts evoke various material, structural, and design possibilities.



S

N


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In an indefinite accretion of spheres (p. 10), sphere placement uncertainty arose from the inherent duality of the

cubodal form. Such ambiguity poses an even probability of placement error that easily leads to the random disorder

of uncontrolled growth (bL). Departure from this scenario entails grafting a uniform pattern order onto the cuboda.

To illustrate how pattern uniformity may be grafted onto the cuboda, the cubodal cluster is first oriented to face avertex perspective, wherefrom a square cluster is accentuated (aC). A 14thsphere is then placed on a corner sphere

in such a way that 2 right angles are made with both of that corner!s converging lines (aR). Sphere 15 placed

similarly on an adjoining corner forms a right angle between planes (bL).

With placement of sphere 16, all 3 dimensions are represented by mutually orthogonal square planes (aC). Sphere

17 placed on the remaining corner sphere completes formation of a cube (aR), as well as the final phase of rational

accretion. In contrast to the underlying disorder generated by indiscriminate placements (bL), the homogenous

pattern intrinsic to the cube fills and organizes space in a manner unique among geometric solids.



Alternative Accretion


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To utilize the special attributes of the cube in the context of the earth-centered cuboda, the cuboda!s 6 squares are

regarded as 3 opposing pairs of squares (bL). Though each pair is geometrically consistent within itself, each is non-

parallel with the others. Determining which pair to utilize is simplified by the orientation of the cubodal/earth axis.

Relative to such orientation, 2 pairs of squares are skewed and this sameness is evidenced by cubes appended to

them (aC). One problem with basing cubes on the skewed squares is that 45 swaths of earth!s latitude are excluded

from the projections of their lattices (aR). Another drawback to using the skewed cube-pairs is that in their sameness

relative to the axis, anti-parallelism relative to each other nullifies the homogenous space filling attribute of each (bL).

Conversely, the geocentric cuboda!s equatorially-oriented squares constitute one uniquely distinguished pair (aC). So

distinguished from the other pairs, cubes appended to them maintain (with parallel lines and planes) their attribute of

geometric uniformity, while their equatorial position ensures the earth full coverage of its extended projected pattern

(aR). Thus does the cuboda enable foundation for 2 celestial co-cubes and orient their position relative to earth.



Celestial Co-Cubes

1

2

3

2

2

3

3

!

!


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The geometric consistency of the celestial co-cubes is broken in order that each may serve a particular architectural

purpose. Such specialization entails differing ways of projecting the cubes!intrinsic patterns to the earth!s surface,

beginning with projection of the primecelestial cube, viewed directly and in profile below.

This cube is so qualified because it is positioned via primary rotation only, to face a specified location!s longitudewhile maintaining perpendicularity with the equator. Thus positioned, a narrow column of the prime cube!s pattern

corresponding to the location!s latitude is projected there, as in the lat 30 N example (aR). This circumstance is

magnified below, with the cube!s 3 essential square orientations labeled along with the parallel planes the project.

Of key importance in this illustration is how the projected column!s representative planes alight relative to the

curvature of the earth!s surface. Below, the circumstance is further magnified to the extent that the surface location

can be regarded as flat. Viewing the planes edge-on, equatorial-facing planes exhibit incident angles that are equal to

the latitude of the specified location, while angles made by polar-facing planes complement equatorial plane angles.



Primary Projection

NN

Earth

Polar Square (viewed edge-on)

Equatorial Square (viewed edge-on)

East/West RotationSquare

(facing page)

Polar PlanesEquatorial Plane

East/West Planes

Equatorial Plane

(edge-on)

Polar Plane

(edge-on)

60 ( = 90 - 30) 30


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The celestial co-cubeis also positioned to the longitude (or opposite longitude) of a specified location via primary

rotation (aL). Secondary rotation about the imaginary axis passing through opposing equatorial vertices positions the

co-cube directly above the location!s latitude (aR). So positioned, the co-cube projects a narrow column of its

rectilinear pattern from the center of its radial square to location on earth!s surface, as depicted below.

In the magnified view, the representative planes comprising the projected column simply meet the specified location

at perpendicular and tangential angles. Further magnification (bL) shows how flat ground corresponds to the farthest

projected plane representing the co-cube!s radial square. From an outer perspective of that square (bC), the pattern

projected coincides with conventional lines of latitude and longitude to constitute a polar-rotational or P-R grid (bR).



Secondary Projection

Celestial

Co-cube

Radial Square

(viewed edge-on)

Longitudinal Square(viewed edge-on)

East/West

Rotation Square

(facing page)

lat 30 N

Earth!s Surface

NN

S

W E


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Reconstitution of diverging celestial cube projections serve as an architectural concept template from which building

designers may abstract any planes and lines paralleling the projected patterns according to the requirements of their

vision. Walls, floors, and other latitude-independent elements are generally guided by the projected co-cube pattern.

Characteristic of both cube projections, co-planing rotational squares guide design of exterior east-west walls (aR).

Lines delineating these planes generally do not parallel, and those projected from the prime cube delineate profiles of

roof design, the guiding pattern for which is keyed to the prime cube!s polar and equatorial squares, according to

latitude. In the perspectives below, design components are shown to be guided by rectilinear guidelines.

Though doors, skylights, solar panels, windows, etc. do not necessarily co-plane, and wall and roof planes generally

do not parallel, lines delineating them appear so from these perspectives. With regard to the familiar conventionality

exhibited by these perspectives, Geocentric Design Code only specifies precise alignment to the polar-rotational grid.



Architectural Reconstitution

Prime Cube Projection

Co-Cube Projection

N S

Top ViewW - E

N


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A celestial cube projection column is specified to be narrow because the smaller its lateral dimensions, the closer its

vertically extended planes align with local verticals defined by gravity (bL). Another consideration is keying the P-R

grid!s alignment to drifting poles, although this occurs too slowly to have significance except in deepest Antarctica.

Additionally however, the poles meander in an irregular loop from an average position (aC). At 70 latitudes along the

Arctic rim and coastal Antarctica, this deviation is extremely slight, but nearing the South Pole, the cube-based shelterscheme is increasingly prone to discernible misalignment (aR). Of more universal relevance is earth!s deviation from

a perfect sphere (BL).

Although undetectable in real and idealized sphere representations (AC), the angular discrepancy between celestial

cube projection lines and local vertical lines (as determined by gravity-actuated measuring implements) maximizes

at .094 at 45 latitudes (aR), which translates to a " shift of one end of an 8!stud. In practice, the local vertical

should be used for structural reasons, and the correction factor should be built into the latitude-determined roof.

For example, at lat 40 N, the south-facing roof slope should be 40.093 while the complementary north roof should

be 49.907. If any correction is made, roof peak orthogonality and consistency within the same structure should be

maintained. Regarding all the above, building small increases precision, structural integrity, and perceptual harmony.



Precision Challenges

Average

North

Pole

0.01

Lat 70 N

70!Max

6/year

Lat 89 S

0.001

7900 mi

7926 miPlumb

Vertical

Latitude/

Projection Line

"

#= tan-1[-1.0033tan"]

= angular discrepancy due

to earth!s equatorial bulge(at "= latitude)

"+ #

90 - ("+ #)

Plumb Vertical

Corrected Roof

Projection

Discrepant

Roof

Projection

#

Plumb

Vertical

Co-Cube

Projection

Vertical


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A key abstraction drawn from the celestial cube projections is illustrated by co-planing squares representing the

rotational squares of each and viewed together from an east/west ground perspective (bL). Relative local rotation

manifests that of the celestial cubes, and is given expression with circular windows on east or west walls (bC).

The relative tilt of the squares (and the cubes they signify) bears both practical and psychological ramifications. Such

juxtaposition with conventionally oriented rectilinearity tends to incline perception away from the limited lateral bysupplying an upward vector toward the unlimited vertical (aR). Examples from southern and northern latitudes below

convey personal access (to infinity) zones, a concept reinforced by polar-rotational grid alignment.

Builders can play with the effect by adjusting the relative lengths of polar and equatorial-facing roofs according to

other factors including the immediate environment. On the practical side, equatorial facing walls assured by the P-R

grid alignment provides an optimally orientated passive solar (space) heating design framework; and thus aligned,

equatorially oriented roofs face the exact center of all sun positions manifest throughout each day of the year.

The Mazria-type diagram (aR) depicts the bounds of sun position azimuths and altitudes (in 15 increments) centered

by solar noon during either equinox at lat 40N. Thus the celestial cube-projected roof keyed to latitude constitutes

the optimal framework for photovoltaic, water heating, and albedo compensation applications - regardless of location.



Exterior Considerations

35, 5575, 15 65, 25

Solar Noon

Equinoxes

Summer

Solstice

WinterSolstice

South

Overhead

E W


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As with the exterior, the cube projections present a pattern to guide internal design. Cube-based shelter!s shape, the

origin of its derivation, implied compactness, and upward inclination suggest an open rectilinear floor plan adapted to

concentric zones radiating outward (bL), with overarching angularity providing paths for the house to breath (bC).

To facilitate breathing, an open ceiling grid frames movable panels for strategic air flow control (aR). In the profiles

below, the lower wall likely attending the tilted roof scheme naturally accommodates cozy areas not requiring full

headroom and partitioned for semi-privacy (bL). From there, the ceiling vaults steeply to a high ridge (bC), then

slopes more gently to the higher, natural light admitting wall that defines an area of airy spaciousness (bR).

To utilize upper space without detracting from interior openness, sleeping lofts are located under the roof ridge ends

of the house atop large closets, compact bathrooms, or bedrooms. Stairs accessing such lofts conform to the angles

of the roof and ceiling slopes (aC). The 3D co-cube projected pattern conforms to the conventional rectilinearity of

appliances, built-ins, fixtures, and most furniture (bL).

In many cases, GDCode geometry may be inapplicable to seating design and can only furnish its framework (aC).

Exceptions like cylindrical chairs, (quarter) rounded furniture, and round tables are consistent with GDCode geometry

(aR), by reason of spheres!role in the celestial co-cube projected. General circularity is derived in Part III.



Interior Design

Air Flow

Channels ( )

and Planes

Living/

Dining

Porch

7!

Equator

4!

Pole

Non-GDC

Upholstery


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Because the celestial cube-based scheme generally implies less interior walls (and more roof) than traditional

architectural design, exterior walls take more load and thus are often best constructed with block. In such cases, the

bottom end of the lower roof determines the height of the masonry wall upon which lumber is set (bL).

As with conventional design, wood best constructs cube-based roof design. Because this scheme implies an openbeam ceiling, necessary insulation can be accommodated by dividing the roof beam into components (aC). Such roof

construction hangs an open ceiling grid strung between the high sun wall and low-wall partitioning with geometrically-

consistent support pieces (bL). A central living/dining area construct (bC) provides mid-support to the roof structure.

The base of the central living construct provides seating around the framed space intended for a wood stove,

waterfall, hanging plants, etc. The reasoning behind circular windows on east or west walls applies to quarter circle

vents centered on the roof ridge (aR). Lines and planes projected from both cubes (bL) may be applied to porch

extensions and insets. Roof extensions sloped to mirror cube-projected roofs (bR) provide design flexibility.

The rationale for this option stems from the hypothetical reflection of a prime projection plane on the earth!s surface.

Because this option is not a substitution, expression of the celestial cube projection should not be blocked. Design

options of equal (or more) significance are derived directly from the cuboda, and are detailed in Parts III - VI.



Building Options

Insulation

1 X 2

1 X 6 Tongue

and Groove

Plywood

Sheathing

2 X 10

Up

32

Seasonal

InsertCeiling

GridSeating

$- $


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Part III - Rolling Transport - as architectural guidelines were based on the foundation supplied by the

earth-centered cuboda, the freed form!s full characteristic pattern reveals the essence of the wheel whose

detailed attributes and abstractions are drawn upon to guide design of mobile artifacts:

The Cubodal Wheel- (p. 22) - triangular sphere cluster; bisected circularity; hexagonal hub and spokes

Profile Abstractions- (p. 23) - co-spinning wheels; 2-wheel separation and direction; alternate pattern

Asymmetric Dynamism - (p. 24) - triangular drive; cuboda symmetry chase; dynamic wheel expression

Wheel Mechanics- (p. 25) - cubodal layer projections; hub to rim spoke torque transfer; lateral bracing

Static Symmetry- (p. 26) - hexagonal shift; symmetry lock positions; immobile transporter components

Transporter Template- (p. 27) - h-shifted wheel orientations; cubodal elongation; template abstractions

Hexagonal Expansion- (p. 28) - left/right frontal symmetry and pattern partition; transverse extension

Elementary Rounding- (p. 29) - body vertex spheres; edge-end sphere cylinders; planar continuity

Architectural Accommodation- (p. 30) - macro-wheel squares; slots; annexations; circular windows

Part III begins with bisection of the spherical cuboda!s central layer by its underlying hexagon to abstractthe patterned circularity, synchronous spin attributes, and rotation-reconciled asymmetry of the cubodal

wheel. After its geometry addresses torque considerations between hub and rim, full potential is realized

by a symmetry locking maneuver that leads to design guidance for transporter components at rest relative

to the dynamic wheel with the natural motion-aligned elongation of the resulting pattern. Design flexibility

arises from a pattern break-enabled transverse expansion and a method of streamlining the template!s

angularity, and finds architectural expression by abstracting information from the macrocosmic wheel.




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The cuboda orientation exhibiting a triangular cluster to the fore (bL) frequently signifies an essential wheel in profile.

Though focus is typically on the innermost sphere, the more general cubodal wheelentails the cuboda!s full spherical,

planar, and structural manifestations superimposed (and interacting) one upon the other (bC).

In the context of earth-centeredness, the cubodal wheel is a macrocosmic wheel, a term evoked in the design of

wheel-related constructs fixed to the earth (aR). Locally, the macrocosmic wheel is represented by a geometrically-

consistent microcosmic wheel. Unless so specified, cubodal wheels are regarded as free entities.

To examine essential characteristics, the 3 foremost spheres of the spherical cuboda!s cluster aspect are removed to

reveal the underlying hexagonal layer and the pattern of lines joining its spheres center-points. An extended

hexagonal plane generated by more spheres bisects one such sphere to define a circle.

Inside the circle, accentuation of the hexagonal plane!s structure exhibits essential wheel attributes of spokes

radiating from a central hub. Conceptually, these attributes manifest in the central sphere of the greater cubodal

wheel.



The Cubodal Wheel


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The greater cubodal wheel finds expression with concentric circles corresponding to the centers of layered sphere

clusters, as well as the inner and outer diameters defined by those clusters (bL). In the hexagonal cluster, the wheel

pattern underlying the center sphere extends to the outer 6 circles to imbue them with identical wheel attributes (bC).

A thought experiment supposes outer circles of such abstraction to be frictionally engaged with their neighbors, whilethe center circle is regarded frictionless (aR); By turning an outer circle, adjacent circles spin in opposite directions as

every other circle spins similarly. Joining hubs of nearest co-spinning wheels defines a line of travel(bL). Separation

between co-spinning circles relative to their diameters may find application in transporter design (bC).

Upon joining allco-spinning hubs directly, an alternately-oriented hexagonal pattern is formed (aR). Both pattern

orientations together constitute a full 2D pattern from which that aspect of a bicycle is drawn (bL). Circularity intrinsic

to the pattern adopts a co-spinning wheel separation of n = 1. Orthogonality of vectors drawn from the line joining co-

spinning wheels and that from its hexagonal pattern (aC) defines a plane expressed as a vector of rotation.



Profile Abstractions

3R 2%3/3R

R

2R

[(2%3/3)+1]R

[(2%3/3)-1]R

n=1

D SN= D(%3n-1)

n=2

S

Motion Vectors

Axial

Vector


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Manifested by the cubodal wheel hub, the axial vector extended orthogonally outward from the central plane of the

wheel passes through the midpoint of an outer 3-sphere cluster (bL) to constitute a rotation axis. From an edge-on

perspective (bC), the triangular cluster manifests as the left line of spheres, with the middle sphere to the fore.

As the axis passes through the 3-sphere cluster on the opposite (right) side, the middle sphere is behind the axis,

signifying an oppositely-oriented triangle. The triangles juxtaposed in profile (aR) suggest a wheel driven in an

alternating fashion, with torsional forces met by the triangles!innate stability.

From the edge-on perspective, minus the outer spheres (bL), the difference between the exposed left and right sides

characterizes the cubodal wheel!s dimension of width. One of 2 ways to reconcile the manifested asymmetry is by

achieving symmetry via dynamic rotation - with the wheel!s 4 primary positions regarded as time events (bC).

Events 1 and 3 mirror each other, as do events 2 and 4, though these will maintain their up/down asymmetry. With

the cubodal wheel attributes revealed thus far, 2-D expression of innate wheel dynamism is found in transverse

triangular ordering amid concentric circles corresponding to the greater wheel!s sphere centers and diameters.



Asymmetric Dynamism

Axis

of

Rotation

2

4

1

3


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Another way the dynamic wheel may be expressed derives from the angled lines joining the central hexagon to the

outer triangle of the cubodal wheel in profile, which parallel opposing lines on the back side (bL). Viewed vertex-on,

the transverse component vectors of these lines cancel (bCl), leaving their projections onto the central plane.

Such projections guide a parallel array of (6) radial spokes that join those derived from the central hexagon to form a

12-spoke wheel (aCr). This configuration provides radial strength only, and is thus applicable to free rolling wheelsonly. To meet special forces encountered by a powered wheel, the driving hub further utilizes cubodal geometry (aR),

with the hexagon perimeter supplying the tangential angles by which spokes may optimally transfer torque (bL).

With 12 spokes utilizing the outer hexagon (aC), 1 of 2 possibilities for their function divides the array into leading and

trailing spokes to meet either power or braking requirements through tension only. If spokes supply compression also,

each continually contributes to either requirement. In such case, an inner rim using the same geometry (aR) partially

shifts the angle toward a more advantageous spoke pull (and push), the tangent fully gained by circular arcs (bL).

Tangential spokes - or alternate radial projected-line spokes - are staggered transversely on the rim orthogonally from

hub flanges to provide and further enable lateral bracing (aCl). Internal and external lines comprising the cubodal

wheel!s dimension of width supplies guidelines (aCr) for such bracing at the hub, spoke crossings and at the rim (aR).



Wheel Mechanics

Leading

Spoke

Trailing

Spoke

Lateral

Bracing

@Hub

@Spoke

Crossing

@Rim


26/70

Aside from the dynamic of rotation, wheel asymmetry is fundamentally rectified by first considering the so interpreted

cuboda!s planar and structural manifestations from the vertex-on perspective (bL). Thus oriented, left and right halves

are separated along its central bisecting hexagonal plane (bC), along which the right half is rotated 60 degrees.

Upon rejoining the rotated half to its (left) counterpart, left/right symmetry is attained (aR). Because the wheel!s

dynamism is neutralized by this maneuver, the hexagonal shiftforms the basis of a 3-D pattern by which transporter

components at rest relative to the forward motion provided by the dynamic wheel may be designed. Symmetrypositions attained by subsequent rotation of the h-shifted wheel are depicted in the top row below.

The bottom row depicts corresponding h-shifted wheel positions in profile. Positions 2 and 4 can also represent top

(or bottom) views of 1 and 3, or visa versa if the alternate, vertically-aligned hexagonal orientations represent front (or

back) positions. Accentuated lines show how h-shifted geometry is most simply applied to the left/right symmetry

requirements of bicycle handlebar design below.



Static Symmetry

1 2 3 4


27/70

The design of transporter components at rest relative to the wheel!s forward motion receive their guiding pattern from

h-shifted wheel orientations exhibiting essential lines parallel to the direction of travel. Which orientation version is

selected depends on the transporter requirements. Triangle and square-up versions bear some geometric

consistency beyond parallel displaced planes. The triangle-up version below characterizes the transporter template.

The prime attribute of the transporter template is its natural elongation in the direction of intended motion. From this

(profile) perspective, the face of the template mirrors and thus eclipses its posterior side. In essence, the template

consists of the infinite lines, planes, and polyhedral forms comprising the cubodal pattern, interacting (or not) with the

ever present potentiality of intrinsic spheres. In the template context, dynamism of actual rolling wheels is inferred.



Transporter Template

Triangle Up Square Up


28/70

A profile of the square-up version (bL) is turned to a frontal view to illustrate the template!s characteristic left/right

symmetry (bC). From this perspective, all apparent planes angle away, while circles always signify spheres. The

cubodal pattern on any given side terminates at the central hexagonal plane, then resumes oppositely oriented.

One particular challenge posed to designers is dealing with polyhedral expansion (or contraction) attending the

transverse displacement of planes. Some flexibility is posed by the cubodal pattern break at the central vertically

bisecting plane (aR), wherefrom transverse hexagonal expansionis justified. In profile, such expansion manifests a

2D hexagonal pattern of alternating triangles (bL). The depth of the pattern extends perpendicularly into the page.

The faces of those depths are rectilinear (aC) and the unit forms comprising them are triangular prisms(aR). Along

transverse lines, the hexagon!s innate circles may follow to form cylinders (bL) for incorporation of axles, rotors, etc.

Transverse planes guide components like flooring, windshields, and seating frameworks (bC). Expansions can be off

center, or built onto any hexagonal base - provided they terminate with cubodal patterns of opposite orientation (bR).



Hexagonal Expansion

Triangular

Prism


29/70

The cubodal pattern of infinite lines infers infinite intersections, any of which may be seen as the center-point of a

potential sphere (of any size) - an attribute that can be used to round the hard angles of planar edges characterizing

the template-designed transporter (bL). So employed, spheres may be separated, touching, and/or overlapping.

To round a specific component (or the entire shell), spheres must be of equal size. Once specified, spheres centered

on each end point of a planar edge are first joined cylindrically (aCl). On a plane formed by 3 or more edges (aCr), a

matching parallel plane is melded to the cylinders joining the spheres. In practice, spheres are centered on all

vertices of 3 or 4 converging planes (aR), all of which are joined by equal-sized cylinders (bL).

Identical matching planes fitted to the cylinders forms a continuous surface in which all planar convergences are

convex (aR). To deal with the discontinuity resulting from planes converging concavely, designers can either go with

the manifest creases!lines and curves (bL), or they may bisect cylinders longitudinally along the shell!s template

guided planes (bR).



Elementary Rounding

Discontinuity


30/70

Rolling transporters find expression in cube-based shelter by employing the macrocosmic wheel - first oriented

longitudinally to a specified location via primary rotation. In actual accommodation schemes, a cubodal square of the

corresponding microscopic wheel is oriented latitudinally via secondary rotation (bL).

The wheel!s square is aligned vertically relative to the specified location such that when the square is faced

longitudinally (aC) to present an apparent rectangle of %(3/2) ratio. Integral numbers of such rectangles then guidedesign of transporter slots in north or south facing walls (aR). In the 3D accommodation scheme below, the cubodal

square is oriented such that its outer edge is aligned horizontallyto location (bL).

Viewing this square edge-on from a longitudinal perspective shows a 1:%2 slope (aC). So-sloped rectangles drawn

from the square!s innate rectilinear pattern guide design of roofs topping annexations to east or west walls of cube-

based buildings (aR). The architectural expression below draws on the macrocosmic wheel!s central plane whose

conceptual rotation is reflected locally by its microcosmic representative (bL).

This hexagonal plane joins the (relatively) rotated east/west facing square planes of cube-based shelter!s celestial

projections to imbue the previously derived round windows (p 18) with a wheel interpretation (aC)further validated by

the circular bisection of spheres by either rectilinear and hexagonal planes (aR).



Architectural Accommodation

EW

Ground Level

%(3/2)

E-W

1:%2 &35 E-W

N-S

+ = = +

N-S


31/70

Part IV - Polytechnic Integration - architectural design based on, and transporter guidelines drawn from

the geocentric and free cuboda!s external squares internal triangles are fused by methods that lead to

linking intermediaries which enable functional polytechnic accommodation via polyhedral integration.

Planar Reconciliation - (p. 32) - octahedral planar duality, spherical projection, circular arc commonality

Wheel/House Fusion- (p. 33) - macrocosmic h-shift; tri-wing adjustment; fusion formula; wheel ports

Tetrahedral Links- (p. 34) - 55 fusion; tri-wings!tetrahedron; matched orientation slopes; cubodal shifts

Orthogonalizing Cubes - (p. 35) - tetrahedral squares; diagonalized cube links; orthogonalized cubodas

Vector Alignment- (p. 36) - motion/axial vectors; dual symmetric linking; template incorporated functions

RadialTorques- (p. 37) - vertical axis couplings; cubodal shift symmetry; rotor and housing resonances

Link Configurations- (p. 38) - vertex-sphere placements; radii ratios; facial segmenting; cube link arrays

Template Options - (p. 39) - orthogonal shifts; link ratio; circular & cylindrical links; 3D rectilinear fusion

Universal Spheres- (p. 40) - tetrahedral links; unit edge/sphere options; ball joints; internal sphere links

Part IV begins by externalizing a deeper connection between cubodal planes via spherical projection toattain full 3D architectural transporter accommodation by a formula which derives the tetrahedron!s use

as an intermediary link. In turn, this form both suggests and fortifies the cube which orthogonalizes prime

cubodal orientations such that the transport template may thereby incorporate various functionalities by

geometrically aligning rotor axes with motion or vertical vectors. For both structural and aesthetic reasons,

necessary links express commonality between cubodal orientations with the use of segmented spherical,

circular, or cylindrical forms while full spheres are shown to constitute universal intermediaries.



!!


32/70

Thus far, guidelines for the design of mobile transporters have been abstracted from the cuboda!s asymmetric

triangularity, while earth-fixed shelter based its form on the cubodal square!s symmetric parallelism. To fully integrate

these fundamentals, first noted is the deeper commonality between respective plane types insidethe cuboda. There,

(partial) squares and triangles share common edges (bL) as they did externally in the cuboda!s construction.

The internal square was shown to be characteristic of the cuboda!s octahedral component (p. 8), with a full such form

being the simplest having both triangles and squares (aC). To externally manifest internal squares, the octahedron is

placed inside a sphere (aR), the equidistant extremities of the symmetric forms featuring inferred and integral centers,

respectively. Then, in a thought experiment, a light source is assigned to occupy the forms!common center (bL).

So illuminated, the octahedron!s triangular edges cast arced shadows onto the spherical surface, such that the 60

edge angles become 90 shadow juncture angles. On a plane, the square can be thought of as an expanded triangle,

or conversely, the triangle can be thought of as a stabilized square - each transformation attended by circular arcs.



Planar Reconciliation

60

90


33/70

The triangle/square transformation is utilized in a full 3-D wheel accommodation scheme involving both celestial cube

projections and the macrocosmic wheel. The latter is positioned via primary and secondary rotations to a specified

location, where its innate dynamism is neutralized by undergoing a hexagonal shift (p. 26) to attain symmetry (bL).

In particular, the symmetric wheel is positioned such that a pair of its matched triangles situate directly above location

(aCl). The matched triangular wings (of the microcosmic representative) are then isolated and juxtaposed onto thecube-projected roof such that their common edge remains horizontally orientated (aR), and the roof slope $signifies

either "or 90- "where "= latitude. Viewed above (bL), outer triangular wing tips coincide with a polar-facing wall.

Facing the polar-oriented wall, the triangular wings are adjusted to the roof slope expressed by the angle (')

determined by the fusion formula (aC). So fitted, the wings are then bisected flush with the polar-facing wall to in

essence extend the plane of that wall upward while leaving 2 equilateral triangle halves intact (aR). The resulting

cross gable thus expresses the latitudinal variance of the celestial cube-projected roof from a polar perspective (bL).

Applicable latitudes are 0 < $< 60 and in this range key gable dimensions possess useful relationships (aC). In

essence, the planar transformation concluding the previous page is manifested by arcing tri-wing edges

superimposed on the plane of the roof over earth curvature!s relevant range of latitudes (aR).



Wheel/House Fusion

$

'= arcsin[(%3/3)tan$]

'

Top View

P

T/2

T = tri-wing edge length

P = T/(2cos$)

L = (%3/2)Tcos'

H = T(tan$)/2 (= arctan[1/(%3cos'cos$)]

H

LRoof Plane

!

T


34/70

A special result is obtained upon solving the fusion formula with the 2 angles - 'and $- set equal to each other (aL).

The angle obtained (circa 55) is the tri-wing spread fusing to an identically angled slope (aC). The triangle formed on

the sloping plane by the tri-wing edges is an equilateral one (aR), and the 3D construct formed is a tetrahedron (bL).

This result and the unique angle are manifested in 2 prime cubodal perspectives.

Between the 2 perspectives, the triangular face of the cuboda!s innate tetrahedron (aC) matches the square of the

(internal) half-octahedron (aR). Thus an isolated tetrahedron positioned externally may function as a link between the

rectilinear faces of matched cubodal orientations (bL). The cubodal shift required for this entails a 60 rotation of a full

cubodal pattern component, as compared to the bisected half of the hexagonal shift.

The full shift occurs periodically between the dynamic wheel and the transport template!s static geometry (aC). Fixing

this resonance in a hexagonal pattern (aR) results in a symmetry orthogonal to that of an h-shift. The tetrahedron also

serves as a stand-alone link in other instances, and plays an abstract role in a more general linking intermediary.



Tetrahedral Link

Plane of

Slope

!= 60

Tri-wing edges

55

55

55 = arcsin[(%3/3)tan 55]

55

55

55

Tetrahedral

Link

(Fixed)

Transport

Template


35/70

The role served by the tetrahedron in a more generally applicable linking intermediary arises from how lines joining

the 4-sphere tetrahedral cluster project squares onto the plane faced. As such projection manifests in 2opposing

ways, the notion that these squares are imbued with a 3rd dimension is suggested. Approaching this relationship from

the other direction below, diagonals inscribed in each of a cube!s 6 squares meet to form a tetrahedron (bL).

Another tetrahedron may be inscribed onto the cube squares !remaining corners (aC). Together the 2 interlocked

tetrahedra joins the symmetry of the cube to reinforce structural stability with the geometry of diagonal bracing (aR).

This attribute in conjunction with the cube!s most economical representation of all spatial dimensions (bL) suggests

the cube serve as an orthogonalizing link - as depicted schematically in the diagram (bR).

Viewed from vertex-on perspectives in the diagram, each of the 4 prime cubodal orientations are joined directly to

each of the others via cube links. The 4 primes represent 4 pattern orientations from which the design of a variety of

constructs and components in the realms of mechanical, electrical, and fluid dynamical functionalities may be guided.



Orthogonalizing Cubes

T

T

TS

Triangle

Square

Edge

Vertex

S T

T

Edge


36/70

The cube link!s attribute of orthogonalizing cubodal orientations is exemplified by aligning the wheel!s axial vector

with that of the transport template!s innate motion vector (bL). This particular alignment is represented by the vertex/

triangle up coupling (bC) - 1 of 6 couplings depicted in the orthogonal linking diagram concluding the previous page.

To bring the vertex-up cuboda that signifies the transport template to a perspective in which the central vertical planeis viewed edge-on, the coupling is rotated about 55 (arctan%2) such that the vectors!parallelism is maintained (aR).

The coupling orientation in profile (bL) shows parallel vectors actually coinciding due to the ubiquity of the template!s

innate lines of motion even as non-axial lines of the incorporated wheel are skewed in relation to the template (bC).

If the incorporated wheel is dynamic and rotates, the skew is unimportant. If employed externally as a rotor housing,

the skew is balanced with a matched rotating component on the construct!s opposing side (aR), as with jet turbines.

Internally, proper orientation of the incorporated wheel is attained by first interfacing the cube link!s vertical motion-

aligned squares with the template!s vertical pattern orientation (bL).

The wheel triangle then interfaces the cube!s vertical transverse square (aC) in a symmetrical fashion. The so-

positioned cuboda may be h-shifted to frame rotors, screws, props, rotating cylinders, etc. Here, scales are distorted

to emphasize key aspects, and on page 38, the actual appearance of cube links are detailed.



Vector Alignment

TS

Top

(Transport

Template)

View

Directional

Vector

Axial

Vector

55

5 skew

(60 - 55)

Front

View

Front

Views


37/70

Another method of symmetrically framing, supporting, or housing a rotating component is illustrated by incorporating

vertically oriented axis wheels within the transport template. Such integration is theoretically enabled by 3 cubodal

triangle-up coupling possibilities with the edge-up cuboda generally representing the transport template (bL).

To fully signify the transport template, the edge-up cuboda is h-shifted so that cube links placed on its top trianglesinterface twin vertical-axis cubodal wheels (aC). Together, the 2 cubodas define a horizontally-oriented hexagonal

plane divided by the cubodal shift (aR), with the division joined 3-dimensionally by a tetrahedral link. Both (oppositely-

oriented) cubodas next undergo (horizontal) hexagonal shifts, which in turn enables hexagonal expansions (bL).

The shifts and expansions form a pattern framework by which the rotating component is housed, with both vertical

and horizontal symmetry. In the schematic (aR), the geometry of the (middle) rotating component resonates

periodically with that of the external housing. A top view of the arrangement depicts a tetrahedral linking configuration

accommodating the wheel axis (bL), and a frontal view depicts the scheme placed atop the transport template (bC).

Applications of the so-oriented wheel - radial in the context of an earth bound transporter - include steering wheels,

doppler radar, helicopter rotors, seed broadcasters, bush hogs, and trailer swivels, etc. Situated internally (aR), an

identical linking scheme accommodates fans, rotary engines, generators, and motors, etc.



Radial Torques

T

T

T

T

S S

Transport

Template


38/70


39/70

Links applicable to the transport template!s vertically-oriented hexagonal planes are based on triangles representing

alternate hexagonal orientations sharing a common circular arc inside the orthogonalizing square (bL). Vertically-

aligned cubodal patterns are thus incorporated with circular links to constitute orthogonal shifts (bCl).

Plane circular links circumscribe the alternate pattern, or are inscribed to lead cylinders transversely separating the

pattern. To incorporate vertically-extended components such as antennae, masts, etc., a special link is derived from

minimal hexagonal expression (aCr), with the ratio keyed to orthogonally-oriented rectangles (aR) and the overlapdetermining circle radius. To introduce the O-shift, circles are centered on corners and joined diametrically (bL).

This link may be extended transversely to interface surfaces of larger separated o-shifted patterns, while its stack-

ability enables vertically extended O-shifted pattern guided components to be incorporated (aR). Individual links

should be discernible, except when directly introducing O-shifted hexagonal expansions possessing transversely

vertical rectilinear planes. To link 3D rectilinear patterns, overlap circles are centered on rectangle intersections (bL).

Such cubical pattern links may interface abutting rectilinear components or those transversely separated from it, and

may also be adjoined along their square edges (aC). Individual link circles may be diametrically joined or not - as in

the space organizing alternative of a template-based intermodal container system (aR).



Template Options

%3

1

(%3-1)/2

Transport

Template

Vertical

(Motion-aligned)

Plane


40/70

Some linking situations are characterized by spheres with elevated roles. Sized to apparent ratios of vertex-on (bL) or

edge-on (bC) perspectives of a tetrahedron, they virtually obscure this link form - whether applied in conjunction with

cube links, substituting for them, or employed in a cubodal shift circumstance (bR).

Inline spheres found with tetrahedral links configured in arrays may be joined cylindrically as with cube links. Another

tetrahedral link configuration features a sphere of unit edge radius circumscribed around it to form a hemisphere (bL).In such case, the sphere is then segmented either along the tetrahedron!s adjacent or the opposite triangle to

introduce another hexagonal pattern orientation on planes thus exposed (bC).

With respect to cube links, considerable freedom exists in selecting radii of spheres centered on link corners,

although they generally may not exceed unit length unless the link is divided into smaller (equal-sized) cubes that can

be discerned by smaller, corner-centered spheres (aR). A sphere constituting an entire external link is in essentially a

ball joint utilized to fuse tubular forms derived from orthogonally oriented hexagonal patterns (bL).

Spheres may also function as interiorlinks by virtue of the cuboda!s spherical nature, and because any intersection of

lines represents a potential sphere center-point. A sphere placed anywhere in the cubodal pattern (and diametrically

extended along a line) internally accommodates non-GDCode artifacts or constructs of any cubodal orientation.



Universal Spheres

r = [(%3-1)/2]X

r = [(%2-1)/2]X

NGDC

NGDC


41/70


42/70


43/70

The cuboda!s surface geometric features slope symmetrically with a quantized set of angles upon vertical orientation

of the form!s opposing vertices, triangles, edges, and squares. Excluding 0 and 90, there are 7 angles: 19, 30,

35, 45, 55, 60, and 71. Of these, 30, 45, and 60 are exact, while 35, 45, and 55 each manifest twice.

Within a given orientation, angles other than an angle focused on or applied are referred to as inherent angles. For

example, 55 is an inherent angle when 45 of the square(up) orientation is the angle of interest. In a general sense,

the cuboda can be viewed as analogous to a prism splitting (white) light into specific wavelengths.

In the analogy, random terrain is shaped with topographic waveforms, each of which has a unique point of maximum

slope keyed to a cubodal symmetry angle. As cosine waves pose the smoothest transitions possible between one

level and another, they are used to imbue grid constructs with full 3-dimensionality.

Each waveform!s unique point of maximum slope is found precisely midway - in both horizontal and vertical senses -between minimum (0) slopes. Thus waves are naturally divided into ), *, and + waves. Wave height-to-lengthratios vary according to maximum slope angles by a factor of 2/,, where length is horizontal distance between levels.



Cubodal Prisms

60 (edge) 19 (triangle 35 (square) 45 (edge) 55

Vertex-up Orientation Triangle-up Orientation

Edge-up Orientation Square-up Orientation

35 (triangle) 45 (square) 30 (edge) 55 (square) 71

Maximum Slope

(&35)

Cubodal Angles &19 30 &35 45 &55 60 &71

Angle Slopes 1 / 2%2 1 /%3 1 /%2 1 / 1 %2 /1 %3 / 1 2%2 / 1

Height/Length Ratios 1 :%2, 2 :%3, 2 :%2, 2 : , 2%2 : , 2%3 : , 4%2 : ,


44/70

Criteria for matching cuboda-keyed waveforms to topographic constructs are: distinguishability of grid aspects and

types; harmonious connectivity of same; practicality; elegance; and symbolic parallels between grid constructs and

cubodal characteristics. E.G., the geocentric cuboda!s 8 remaining vertices that are neither polar nor equatorial (bL).

The 8 vertices are indistinguishable from one another, each representing convergences of identically positioned

squares and triangles. Following primary and secondary rotation to a specified location (aC), any such vertex signifies

a mundane pole, the following edge of which slopes at 30 as depicted locally (aR). This angle is next keyed to a

(half) waveform!s maximum slope (bL), and spun 360 about the pole to form a ring of maximum slope (bC).

As vertices represent junctures between cubodal square types, waveforms keyed to mundane pole edges serve

mainly as inter-gridjunctures (aR), but may also serve as intra-gridjunctures. Innate verticality of mounds formed

suggest bases for street lamps, wind generators, etc. Applied to the tree below, the waveform commences at the

outer surface of a vertically-oriented central cylinder allowing space for a water crater and anticipated trunk width.



Grid Junctures

30

Ground Level

Mundane

Pole

maximum slope

maximum slope

N

Top

Views

x

M = maximum

slope = H,/2L

H

30 = 1:%3

L

y = 2ML[x/L + cos(,x/L)]/,


45/70

Low walls characterized by the celestial cube roof projection (bL) suggest half cosine wave embankments that

complement the architectural scheme!s angularity and passively contribute to temperature swing moderation. Keying

the wave to a cubodal angle manifests the foundational earth form with 3D interplay between it and the celestial cube.

Keying such a berm!s maximum slope to the cuboda!s 45 angle constitutes a universal average of all latitude-dependent north/south roof combinations. Geometrically, 45 characterizes inherent lines that slope away from a

cubodal square serving as cube-based shelter!s foundation (bL). A house corner coinciding with the square!s center is

held constant as the cuboda is rotated 45 to align the square!s sloping lines with walls forming the corner (bC).

The alignment divides the square evenly into 4 spaces, the 3 not occupied by the house essentially filled by a half

octahedral pyramid. Next, the sloping 45 lines are matched with the 45 sloping skewed squares of the cuboda!s

vertex-up orientation to give the angle planar expression along a wall bounded vertically by the lines and horizontally

by the square-up!s inherent 55 sloped triangles (aR). Others cubodal angle!s are keyed to waveforms below.

Where an outside corner is regarded as a mundane pole, spinning a 30 edge 270 forms a 3/4 mound (aL). Inside

corner confinement supports steeper angled mounds keyed to 55 and spun 90. Finally, half berms keyed to 35

signify the average roof fusion, with angle selection dependent on available space, soil type, and viable ground cover.



Architectural Embanking

45

Top

View

45

45

55

Ground View

45

45

Top View

Top View

30 55


46/70

The 3-dimensionality of square grid lines is guided by the geometry of cubodal edges. In the P-R grid (bL), edges are

positioned via secondary rotation about an axis through opposing triangles to locate longitudinal edges; or about an

axis through opposing vertices to locate latitudinal edges. From the edge slopes a 20 triangle and a 35 square (bR).

Another geometric feature sloping symmetrically at 35 is the triangle of the vertex-up orientation. Such doublemanifestation suggests keying 35 to the maximum slopes of berms in the dualistic P-R grid (bL). There, the square!s

intrinsic parallelism naturally extends along the grid line of the berm (bC), then meets the sloping triangle rotated

about its vertex-up pole to round out each end.

The reasoning behind the sloping triangle also makes 35 applicable to mounds serving as intra-grid junctures (aR).

Otherwise, P-R grid berms meeting an inter-grid juncture mound keyed to 30 matches the vertex-up orientation!s

inherent 35 angle. In essence, 35 is keyed to full cosine waves extended continuously along polar or rotational lines

and spun 180 at each end to form full landscaping berms, the next shallower angle relative to the 45 house berm.

As with house embankments, the full 35 berm expresses the fusion solution to the average (45) cube-based roof

slope combination over all latitudes. Innate verticality of the integrated 35 triangle!s vertex-up orientation suggests

natural botanic verticality, making this topographic construct applicable to farm fields and landscaping schemes.



P-R Grid Berms

Ground Views

35

N

3035

35 &'@ $= 45E-W


47/70

Cubodal edges also frame the 3-dimensionality directed by diamond grid lines (bL). To so position such edges, the

diamond square is first positioned via primary and secondary rotations to a specified location (bCl). Thus oriented, the

required edge undergoes tertiary rotationabout an axis through opposing mundane pole vertices to location (bCr).

From a local diamond grid perspective on the ground, the applicable feature is the edge!s sloping triangle (aR). Its

19 slope is keyed to the maximum slope of berms designated for the diamond grid to distinguish it from the P-R gridwith a less steep option that begins a progression of slopes toward house embankments (bL). Transitioning between

grids, such berms!inherent 35 angles match the inherent triangle slopes of 30 inter-grid junctures (bC).

Diamond (intra-) grid juncture mounds are keyed to either 30 or 35 (aR). Merging natural terrain with GDCode

constructs is most simply implemented by hybrid forms like cylinder/wave mounds (bL), a method diametrically

elongated for berms or house embankments. Another option is enabled by the 1/4 waveform!s 0 and maximum

slope endpoints - which extended straight effects a distinguishable interface with natural terrain (bC).

Wave breaks include extending flat areas or plateaus as with terracing, which consists of a waveform and/or straight

maximum slope ascending from the upper level of same (aR). In such situations, waveforms need not be similarly

scaled. With regard to all options, lines and rings of maximum slope present valid planting loci.



Ground View

Topographic Options

Quarter Wave

19

30 Mound35 Mound


48/70

Options in which GDCode addresses outdoor expansion of the house begin with courtyards characterized by full (or

half) wall transitions. In the cross section (bL), the wall separates an outside (35) P-R grid embankment with an

inherent angle of 45 from a complementary inside embankment keyed to a 55 wave that minimizes area consumed.

The 55 embankment fuses geometrically to the 45 house embankment!s inherent 55 angle with a tetrahedronintrinsic to the polar-oriented octahedral square that delineates the P-R grid and bases the celestial cube foundation

(aC). Courtyard corners may be treated with separated waveforms, or swept concavely (aR). Beyond the courtyard,

mounds may be quartered and diametrically expanded (bL) to incorporate steps to elevated areas (bC).

Quarter sections may also be cut out of mounds, at the maximum-slope ring (aR). From the ring downward, a quarter

cylinder descends to the ground to facilitate incorporation of curved benches bounded by extended straight maximum

slope guides (bL). This scheme may be applied along a berm!s straight run to in essence provide landscaping

furniture to an outdoor area extending from a cube-based home patio (bC).

On corners, or in the middle of such an area, GDCode geometry is consistent with quarter sections being carved out

of the ground in a concave fashion to form arced steps that descend from a wide beach - and converge with

increasing depth to a small deep immersion pool (aR).



Landscaped Rooms

House

Top View

Courtyard

Wall


49/70

Thus far, waveforms have been spun about their crests to shape mounds and berms. Waves may also be spun about

their troughs (bL) to form inverse berms and mounds (bC); or sweep inside corners. Such sweeps may pivot off an

intersection of orthogonal berms (bCr), or off any point on a diagonal extended out from that intersection (bR).

In either case, the reach of the swept wave!s crest leaves a corner area untouched. To contour this area, the wave

extends in size from the trough point such that its length and height continually increases - to a factor of%2 at thecorner crest (bL). Viewed along a wall, the apparent height-to-length ratio of the extended wave is flattened by a

factor of%2-1 ( &.41), resulting in a 45 keyed wave appearing as a wave of 22.5 maximum slope (bCl).

At imaginary 4-corner settings, maximum slope and ridge lines of extended waves appear straight (aCr), compared to

the circular arcs of unextended corner sweeps (aR). With respect to concavity in general, a wave contouring guide is

posed by a useful mathematical relationship between the wave and the circle(s) it accommodates: the square of the

wave!s maximum slope equals the number of maximum-sized circles that can fit into the wave!s total height.

For examples, the largest circle able to fit into the crest (or trough) of a 35 keyed wave has a diameter relative to the

wave such that the wave holds (1/%2)2= *of such a circle, whereas a 55 wave holds (%2)2 = 2 circles. Circles signify

cross sections of spheres or cylinders, and reference concentricities that may guide placement of utility tubes, etc.



Concave Considerations

High

Ground

Depressions

L %2L

45

max

slope

H = (2/,)L

#H = (%2-1)(2/,)L

19

30

35

45

55

60

71

1 / 2%2 squared = 1/8

1 /%3 = 1/3

1 /%2 = 1/2

1 / 1 = 1

1 / 2%2 = 2

%3 / 1 = 3

2V2 / 1 = 8


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In addition to framing the 3 dimensionality of grid lines, the edge also represents the sole cubodal feature lacking both

vertical and horizontal symmetry when faced directly via secondary rotation (aL). Opposing edges also center the

equatorial axis by which an angular shift is made between the odd edge faced, and its orientation in the nearest

(vertical) symmetry perspective (aC). The angular shift of about 1.5 (aR) may be keyed to waves and circles (bL).

The equivalent slope of about 1:40 (2.5%) is in the realm of playing field contours. Keying 1.5 to a wave!s maximum

slope results in a height-to-length ratio of about 1:60. Aside from keying 1.5 to berms and mounds, this angle may

be keyed to waves extending into a corner by pegging their crests along a diagonal ridge to maintain equal wave

height and length (aR). Such method is employed on the rectangular portion of a quartered and terraced mound (bL).

The terracing scheme largely contours a flat area between a building embankment and (P-R) grid berms and mounds

to shed water. To shape flat areas not reachable by such methods, non-GDCode contouring best melds with grid

constructs on flat ground, and along vertical drops or extended maximum slopes of bounding waveforms (aR).



Shallow Contouring

&31.5

- 30.0

&1.5

31.5 &Tan -1%(3/2)/2 30

High (Ridge)

Contour

1.5 Maximum

Slope Contour

1.5 Keyed Waves

Low Contour

3 Circular Arc (1.5 + 1.5)

Mirrored 1:40 Max Slope (Top) Half Wave

Non-GDCode

Contouring

1.5 Max Slope(s)


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Part VI - Wheel Extrapolations- the essence of the wheel abstracted from the cuboda incorporated a

range of functionalities with integration methods that extended to fusing earth and shelter with ground

waves that base wheeled infrastructure, and enabled alternative realms of mobility and architecture.

Path Concepts- (p. 52) - cubodal wheel path; 19 & 35 fusions; path attributes; path/wheel relationship

Path Expressions- (p. 53) - symbolic path; parallel path; farm field furrows; road & utility paths; culverts

Waterway Constructs- (p. 54) - bridge design; canal, levee, & dam contouring; hydroelectric guidance

Fluid Dynamic Wheels- (p. 55) - inner cubodal wheel planes; turbine functions; tangential & axial flows

Wheel-less Transport- (p. 56) - template orientations; polytechnic functionalities; air & water transports

Disc Orientation- (p. 57) - cuboda co-plane dynamism; satellites & orbital planes; direction imbued disc

Disc Design Options- (p. 58) - interior integrations; distinguished square; guidance & docking schemes

P-R Wheel Shelter- (p. 59) - macrocosmic wheel roof; celestial annexations; polar sociability; hybrid roof

Diamond Grid Structures- (p. 60) - h-shifted wheel positioning; tri-wing roof geometry; hip roof clusters

Part VI begins by keying cubodal wheel geometry to established grid berms and intended rolling surfaceswith symbolic, parallel, and real guidelines for agricultural, utility, and road design - before extrapolation to

bridges and dams. With hydroelectric generation, cubodal turbines meet fluid dynamic functionalities that

oriented horizontally apply to air and water transport design or, when oriented vertically, to discs operating

in orbital or hydro planes. To such, control and other functional requirements are incorporated by linking

methods that extend to docking shuttles oriented by the transport template, which is also found applicable

to architectural schemes drawn from the macrocosmic wheel, each distinguishing P-R and diamond grids.




52/70

The cubodal wheel!s direct edge perspective best exhibited the attribute the asymmetric rotation (bL). In profile, the

edge corresponded to the wheel!s general motion vector, led orientation of full architectural integration, and in Part V

lent grid lines 3-dimensionality. Thus the edge naturally suggests framing the pathupon which rolls the wheel (bCl).

So conceptualized, the angle between path and the edge leading to parallel path (and common to declining squares

and triangles) is 60. To meet immobile symmetry requirements, path uses h-shifted wheel geometry (aCr), by which

paired 19 sloping triangles fuse to 30 sloping planes (aCr). Alternatively, matched squares do not pose integrablegeometry (bL); however, paired triangles sloping at 35 from the vertex-up position (bC) do fuse to a 45 plane.

Superimposing path!s identically-angled squares and triangles suggests the circular arcs linking the plane types (aR).

Together, wheel and path geometry pose optimal strength against deformation - the principal factor of rolling friction.

Below, rolling events between path and wheel depict a cycle of interfacing, twisting, and mirroring in relation to explicit

triangles and implicit squares toward a resonance of traction and rotation-maximized asymmetric instability.



Path Concepts

20

30

'&20

@

$= 30

2060

'= arcsin[(%3/3)tan $]

3535

'&35 @ $= 45

35


53/70

The concept of path can be manifested to varying degrees. In the most abstract sense, path may symbolically

connect grid constructs: (19) path fuses diamond grid berms to 30 grid junctures which in turn - by inherent 45

sloped squares - fuse (35) paths to both house berms and the inherent 45 slopes of P-R grid berms (bL).

Grid construct contouring guided explicitly by path geometry infers parallel and real path (aR). Depicted below is an

overt expression of path manifesting in P-R grid farm field furrows. Wave-shaped furrows keyed to 35 pose

sustainably steep slopes for dependable irrigation guidance; innate verticality; and economy of space between rows.

Path furrow shaping is facilitated by transport template-guided machinery.

Into such grid path/rolling artifact scheme, the maximum wave/circle relationship poses potential application. In actual

road design, path may be expressed by keying19 or 35 to the maximum transverse slopes on either side of a 1.5

max-sloped crown (bL). Because 19 or 35 slopes are separated by the crown, and lack the rounded ends of grid-

distinguishing berms, either angle may be applied to either grid, according to particular requirements and constraints.

Whatever the path!s slope angle, the other angle exists there inherently to both infer dynamic wheel resonance and

provide more flexibility in utility installation along and under shoulders according to the wave/circle relationship, with

pipe cross sections centered on circles. Cubodal path!s 60 angle logically finds application in culvert design (aR).



Path Expressions

35 Paths

19 Path

35

Road Crown (1.5 half-wave)

19 35

60


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A fuller expression of path may be found in bridge design. Because path geometry is limited to 2 grid orientations,

applicable circumstances where natural waterways are concerned restrict GDCode bridge possibilities compared to

those of roads. Where viable, such

Documents

Geocendtric Design Code (Patterns)