PANDORA’S SPHERE - A PARADOX

Jim Lehman

Original publication: Symmetry: Culture and Science Polyhedra 2 Part 2, Budapest, Hungary Revised and annotated Jan 17, 2016

PANDORA’S SPHERE - A PARADOX

Jim Lehman

Address: 3410 Harney St. Vancouver, WA 98660-1829, U.S.A.; Email: bcat@teleport.com

Pandora’s Sphere: a polyhedron whose exterior surface is completely covered with straight edges while the interior is filled with curved edges.

L1320

First published in the scientific journal Symmetry: Culture and Science, POLYHEDRA 2 Part2, A thematic special issueVol. 13, Nos 3-4, 287-310, 2002The journal of the SymmetrionP.O.Box 994, Budapest, H-1245 HungaryEditor: Gyorgy Darvas http://www.geocities.com/symmetrion/isa/symmetry-foundation.htm

On May 18, 2005 I receive the first copy of my article.

This paper is a revised updated version with added annotations.Published as a PDF Jan 17, 2016

This study began in 1961 in the era of do-your-own-thing. Because it was a new age, a time of new beginnings, I felt free to examine the basics of society and life. I posed the question: what is basic in the universe? I was intrigued by the theory that light was both a particle and a wave. I became fascinated with the wholeness which was underneath extremes. I felt overwhelmed by the complexity of everything I looked at or thought about and wanted to find a starting point. The tools at hand were pencil, paper, ruler, compass, scissors, and camera. I started with a graphic paradox by forming a set of contradictory graphic extremes on which to enclose space. I combined straight lines and circles in a flat all-space filling matrix where both could exist in harmony and still tile a surface. There were two graphic elements, the inner radial based chord and the outer circumferential 1/6 circular arc. The circular grid shown in L1525 was the result. As I viewed the overlapping circles I found that when I looked beyond the surface (what the artist calls free-viewing) of the paper, the circles merged into 3-D layers and the edges formed interwoven patterns.

Annotation: While this weaving in a 3D space did not give clues on forming enclosures, it did give a sense of 3D forming in the pattern.

L1525

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L1296 L1525

In the flat grid-work of straight sided equilateral triangles (L1525) there is a network of circular overlaps. The triangles with straight sides are the simplest shape to enclose space, the circles are the most complex, yet both create fields of enclosure in the same area and tile a flat surface. These enclosures are paradoxical since they are like apples and oranges, yet both form fields of enclosure. This graphic paradox focused my study.

Annotation: Why a paradox? At this time I found a fascination in things that were very different from one another but were cooperative in their differences.

L1296leaf L1297

Annotation: While the arcs are from the flat leaf shape they take on a new reality when bent to form the 3D pod. The pod bends the original flat faces resulting in the edges and faces forming 5-fold trajectories.

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L9154

My next step was to use flat leaf shapes to construct pods. Other shapes from the circular grid were used to fill in gaps between pods. I sidestepped a traditional mathematical approach and in its place constructed a series of models. It was already known that the straight sided triangles formed inside hexagons would build tetrahedrons and octahedrons and create the all-space filling matrix of R. Buckminster Fuller’s Isotropic Vector Matrix (IVM), a network of same length vectors ( L9154) connecting sphere centers in a twelve around one packing where all twelve spheres are in contact with the core sphere. This graphic shows a cubic section of the IVM.

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Annotation:

Annotation: The curvature has originated from a 1/6 circle segment. The pod bends the leaf such that a five fold arc is produced on the pod edge and there is a connection to a decagon as the pod face is followed to complete a circle. The following images explore this 5-fold trajectory.

All of the curved edges of the Curved Vector Matrix have 5-fold trajectories. The edges formed inside the pandora are 5-fold edges. There is no 5-fold activity outside the Pando-ras.

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The question I wanted to answer was this: could curved pieces create forms that would also fill all-space without requiring extra parts? I began to experiment and a multitude of parts emerged. I prepared my findings, and sent these to Fuller for his review. He then sent me a letter dated July 28, 1962 encouraging my further research. I met with Fuller several years later. I then realized that I was branching away from his direction and following my own interests, while allowing Fuller’s work to be a strong influence.

Extensive model making resulted in curved models that fit into one system that included a three-sided pod, a convex tetrahedron, a concave tetrahedron and a curved octahedron. Once I made a model, the negative space between the pods was defined by the models themselves. My role then, was to clearly see what was in front of me and render those relationships in solid form.

As I built the models, I found that the curved edges could be simplified by cutting facets on the edges. This made the assembly easier and faster and caused me think about the nature of curvature. The pi based curvature of circles has no ultimate resolution (irrational) and it creates virtual curves and arcs. The circles and arcs in my study only attain a virtual appearance of smooth curvature as resolution is increased, but the structural integrity is never lost.

Author model building in 1961, Eugene, Oregon.Explorations in magnetic bonding and dome building.Photos by Gary Meacham.

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A wasteland of models in 1961 pave the way to the present. There are many directions that now branch off the main trail of this journey, but the curved forms have been my primary focus over these many years. Using the CAD-like modeling program SketchUp, I fine tuned the fits and starts of physical model building. This helped me find how parts fit together. The physical models made from card stock lacked accuracy; the computer models were accurate to six decimals. All of the graphics in this paper were produced using this program.

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Annotation: The photos below shows the hump and dip that occur in the center of the Pandora as the two halves are parted. A sphere would have a great circle in place of the meandering edge on the Pandora.

Annotation: There are paths that swerve or meander around a grouping of forms. These are not random movements but are very predicatable.

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Annotation: My approach to pi is not conventional. While this study does not require the method shown here I have long considered allowing the radius and diameter to be based on the same curvature as the circumference and that curved pi is a rational 3.0 based on a primitave circle formed as a hexagon with a straight radius of 2. When the curved radius of 2 is divided into the circumference of 6 the result is 3.

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L9156

L9158

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Curvature

I define curvature as the accumulation of linear trajectories between vertices or node points. A hexagon has six vertices and a dodecagon has 12 vertices. As the nodes increase in number, they form chords and the appearance of curvature increases. Increasing the number of nodes only serves to make the curves appear smoother, but curvature is incremental and virtual. Convex and concave curvature begins with a dodecagon of twelve sides. This is the minimum condition that ”breaks” the edges of a hexagon. Curved edges must have a minimum of two chords to show curvature as described by the twelve sides of the dodecagon.

As the study progressed, I embedded my curved forms in Fuller’s IVM creating what I call the Lehman Vector Matrix (LVM), shown in the L9156 graphic. When using only the curved lines (L9158), I call the matrix a curved vector matrix (CVM). The linear building modules of the IVM require only radials of uniform length throughout. All of the arcs in the CVM have the same 1/6 circle length origin. The arc formation is created when the flat leaf shape is curved by the three sided pod. The position of the arc determines if it is outward bent towards convexity or inward bent towards concavity.

The CVM contains edges that originate from circles, unlike the IVM which contains radials that represent sphere center to sphere center unit distance. The forms maintain a kind of ”pattern memory” from the original flat curvature of the mother circle.

L9063

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Interestingly, the faces of all the curved forms contain flat equilateral triangles of various sizes that define one of the faces of an internal polyhedron. This flat spot allows the models to appear to have compound curvature, as seen in L411B, a concave tetrahedron. This flat triangle allows the curved faces to bend in three different directions, normally reserved for compound curvature, without distorting the form.

Annotation: Early on I referred to these internal forms as crystal centers. It is later that I realize these centers do not have any curved edge behavior. This separation between the body of the form and its extremedies becomes a major factor in the formation of an all space filling 3D matrix.

The edges and tips of the forms have been exploded from their center cores.

The crystal centers define the separation between the body of the form and its extrem-edies. The extremedies include the edges extending from the forms mid-point edge to its tip. These crystal centers are left behind as their extremedies embed in the Pandoras. The Pandora exists only as an invisible membrane between the tips of the tetrahedron and octahedrons as their bodies (crystal centers) are removed from their tips.

L411B

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The convex and concave surfaces of forms in the CVM order space in non-spherical terms. There are no verticals or horizontals related to arcs or global great circles, or axis of spin as in Fuller’s cosmology. In figures such as L464 there appears to be a circle but when this figure is viewed as a VRML (virtual reality modeling language) the circle is virtural and dependent on viewing angle.

L464L464

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In 2015 I begin to explore the use of silicon molds and resins to move more from the virtual world of drawings to more durable hands on models. Here I can drill and embed magnets to better feel the separation of the tips from the crystal centers. The Pandora moves from being an invisible membrane to a solid polyhedron. The pods can be cut and attached to plexiglass circles to better show their edge and face trajectories.

Annotation:

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Many years were involved in locating the exact shape of what I now call Pandora’s Sphere and which I earlier referred to as simply the egg. The realization that all the curvature in the CVM is contained in Pandora’s Spheres, came as a surprise. Just as classic spheres group twelve around one in spherical close packing, Pandora’s Spheres group in the same way. All curved edges of the pods, tetrahedrons and octahedrons are enclosed in these spherical clusters, while the internal bodies of the curved forms remain outside the spheres. The original graphic paradox of combining straight and curved lines in an all space filling matrix, now appears to emerge as a continuation of this paradox in volumetric form.

Figures L468 and L1302 represent two types of vector equilibrium. Pandora’s Sphere is a polyhedron that lies at the core of the Curved Vector Equilibrium (L468). Fuller’s Vector Equilibrium does not have a Pandora core, instead it has a 1/2 unit Vector Equilibrium core (L1302).

Pandora’s sphere

L1302

Pandora’s Sphere has 36 vertices, 38 faces and 72 edges. Euler: V (36)+F (38) = E (72)+2.

L1316A

L468

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L4007L1320

L427G

Polyhedrons are contained inside the curved tetrahedrons and octahedrons of the CVM and together with Pandora’s Sphere, fill space with no gaps or voids. Simultaneously, all of the curved forms fill space with no gaps or voids as shown in L427G, where a 4-frequency solid is formed by curved pods, tetrahedrons and octahedrons.

All of the curvature in the CVM is contained inside Pandora Spheres. The exterior of the Pandora is a polyhedron, but the interior contains curved edges and curved volumetric forms. Viewing Pandora’s Sphere as seen in L1320, demonstrates the curvature that lies inside. L4007 shows a cube with a Pandora core and Pandora Sphere’s centered on the cube mid-edges forming string-like clusters.

The exterior surface of Pandora’s Sphere is completely covered with straight edges while the interior is filled with curved edges.

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When the Pandora is synchronized and embedded in Fuller’s IVM they share the same center as the VE. This position is an important vector cross road where internal and external vectors are in balance. The IVM is an all-space filling framework of tetrahedron and octahedron edges.

An observer positioned at the center of a Pandora’s Sphere will only see curvature, shown in L1320, except for the outermost “sky” where flat constellations formed by the vertices of the Pandora would be apparent.

Pandora’s Spheres are readily apparent as clusters of twelve around one in L404. The spaces between the Pandora’s can be filled with polyhedrons formed in the centers of the curved tetrahedrons and octahedrons.

In this world of curvature and polyhedra, the curved edges and the tips (mid-edge to vertex) of all the forms are embedded in Pandora’s Spheres. The core or bodyof these curved forms are flat faced polyhedrons and do not enter Pandora’s sphere.

L404

Annotation: All of the curvature of the tips of the tetrahedrons and octahedrons termi-nate in the exact centers of Pandora’s.

L1320

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An observer on the outside of Pandora’s Spheres, seen in L501, will see only polyhedra and no curvature. The wire frame curvature shown here in L1320, fits inside Pandoras not yet added to this grouping.

L501

L1320

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L396

L1296

The graphics that follow show how the Pandora interacts with the curved forms.

Pandora’s Sphere is placed in a globe.

L396 shows the sphere-like nature of Pandora’s Sphere. The embedded globe is used for illustration purposes only to show how Pandora faces relate to the sphere.

The steps leading to the formation of Pandora’s Sphere start with this diagram of triangles and circles. The following graphics isolate different parts of the diagram.

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L1297 three leaf forms are grouped to form a three sided pod.

When the convex and concave tetrahedrons populate the IVM curved octahedrons are formed in the space between tetrahedrons. These curved octahedrons have convex and concave faces.

L1296concave

L466 Six pods form a convex tetrahedron.

L411B Four concave triangles form a concave tetrahedron.

L1296leaf L1296convex

L181A

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L1296triangle

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L1298 L9154

When a tetrahedron is formed in the IVM, its edges are linear and without volume or form. Edges in the CVM are pod enclosures with curved edges that can be added or removed from tetrahedrons in the matrix without the tetrahedron losing its identity.

L9158The pod is a structural element that occupies and encloses space. The pod is a basic volumetric structure that forms curved tetrahedra. Curved tetrahedra are formed by the pods. The CVM (L9158) is comprised of pod edges alone, or with pods as shown in L424. The curved forms, such as tetrahedrons and octahedrons within the CVM are determined by these relationships.

The pod shown in L1298 encloses a straight line that is unit length. This represents the radial used to form Fuller’s IVM (L9154). Just as the IVM consists of linear building struts, the CVM consists of pods with the linear struts of the IVM running inside them. Wherever there is a strut in the IVM there is a pod in the CVM and the linear strut and the pod exist in the LVM. The pod can be thought of as a 3D volumetric ‘line’ when compared to the linear elements of the IVM.

L424

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The external pod tips form a Vector Equilibrium and show the position of the Pandora inside the L505VE form.

L405L405 is a convex tetrahedron without pod edges.

L505 L505veTwelve pods are embedded in the Pandora with inner tips connecting at a common point in the center of the L505 form.

Pandoras dock one to another at the halfway point between the pod tips. Twelve Pandoras dock to a nuclear Pandora (L506).

L506

Twelve around one Pandoras are embedded in globes (L404) to illustrate the spherical close packing nature of the forms. Note that a vector equilibrium does not pack twelve around one.

L406

L404

A convex tetrahedron without pod edges (L406) is embedded in the irregular hexagonal face of a Pandora. Each Pandora accepts four convex tetrahedrons.

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L467 Four convex tetrahedrons are embedded in a Pandora.

L412 A concave tetrahedron is embedded in a Pandora (L412). The tip of the concave tetrahedron is embedded in the Pandora leaving the octahedron core outside.

L477

L468 Four convex tetrahedrons and four concave tetrahedrons are embedded in a Pandora L468), forming a CVE. Two types of vectors are involved: straight tip to tip vectors and curved trajectories forming pod edge vectors.

A curved octahedron is embedded in a Pandora L477).

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L469

A 2-frequency (double edge) curved octahedron is formed around a Pandora core. All 38 faces of the Pandora are filled with curved tetrahedrons and octahedrons and pods.

A 4-frequency curved octahedron is formed around a core of twelve around one Pandora (L427G).

L427G

As the edge lengths increase, the edges show undulating lines and the faces become wavy.

L469

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Pandoras (embedded in globes) around one.

L1299

L1299 shows the ball and joint nature of the Pandora as well as opposed convex and concave tetrahedrons.

L424 L424 shows a 3-frequency pod formed tetrahedron.

L1300

L1300 shows a two-frequency CVE formed around a sphere pack of twelve

Concave tetrahedron edges and pod edges have the same curve.

L464

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The leaf edges in this graphic are faceted into two segments and they lie inside the full curvature of the edge.

The circle formed of leaf shapes in L167 forms a twelve sided polygon, the minimum number of edge facets on a circle to meet my definition of primitive curvature. The minimum number of facets on a leaf shape to define a curved edge is two. Many of the graphics that follow use this primitive two facet edge. Model building and computer aided design is more efficient using this simplified edge.

L167

L168

L173

L134 L135L134 and L135 show two chords to illustrate primitive curvature on pod edges.

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L163

The curved forms of the CVM each have an internal polyhedron (except for the pod which has a polygon center). The convex tetrahedron without pod edges shown here (L163), contains an irregular truncated tetrahedron as a core. Pandora’s Sphere does not contain polygons or polyhedrons, only curved tips like those shown here as outlines. The term ’tips’ used in this paper, refers to that part of a form that extends from the curved edge mid-edge cross-section to the vertex.

L248 The core of the concave tetrahedron is a regular octahedron (L248). The tips of this form are contained in Pandora’s Sphere leaving the octahedron connected to one of the outer Pandora faces.

L488

The internal core of this curved octahedron (L488) forms two kinds of polyhedra. Both types are shown here combined. The curved tips shown in outline form extend into a Pandora when docked to a Pandora.

The L490 core is contained inside the concave features of the curved octahedron.

L490

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L489

Each of the four convex faces of the curved octahedron contains the same polyhedron as shown in the L489 graphic.

L1298

The only internal core polygon is the equilateral triangle, here shown as an invisible membrane inside the pod (L1298). If the mid-section (red triangle in L1298) of a pod which extends out of the Pandora were removed, the cross-section remaining on the surface would form an equilateral triangle on the Pandora. The Pandora materializes this triangle.

L505

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L501

Pandora’s Sphere surrounds (as a wire-frame image) the smaller VE that I call a VE core

Polyhedrons are connected to the face of a Pandora in L501 and L499.

L499

L1307 L1302

When the VE is cored (L1302), that is, when a unit edge (1.0) VE is sliced through the middle of each internal strut, a smaller half edge (.5) VE is formed. Note that Fuller does not have a need to divide the prime vector.

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L318L318 shows a relative size view of a Pandora inside a unit edge VE. Each vertex of a VE has a Pandora centered on it when the CVM is synchronized and embedded in the IVM.

L1312

L1312 shows a hollow invisible membrane Pandora socket that can accept the tips of pods, convex tetrahedra, concave tetrahedra and curved octahedra.

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Convex tetrahedron (without pod edges)

Curved Forms of the Curved Vector Matrix (CVM)

L405

Three sided pod

L491 Curved octahedron (concave faces)

L411BConcave tetrahedron

Curved octahedron (one of the four convex faces)

L494

L1297

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Curved Form Polyhedron Cores of the Curved Vector Matrix (CVM)

L1298 Equilateral triangle (invisible membrane) pod core (polygon)

L163

L490

Irregular truncated tetrahedron core of the convex tetrahedron

Core of the curved octahedron (concave faces)

L248

L489Core of the curved octahedron (one of the four convex faces)

Regular octahedron core of the concave tetrahedron

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Unusual features of the curved octahedron

L181AThe curved octahedron contains two different polyhedra. View L181A shows the combined convex and concave features.

L488L488 contains the cores for both the concave octahedron and the convex portion of the curved octahedron.

L491 L494

L491 shows the concave faces of the octahedron. L494 shows the convex faces. Four convex faces (L494) are needed to complete all four concave faces of the L181A octahedron.

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The following graphics (L1357, L327, L332, L333, L334, L335, L336, L337, L338) show nine steps in the formation of a 3-frequency octahedron with a curvedoctahedron core bounded by what I call a benchcube. The benchcube is a diagonal of two units and is used in this study as a familiar frame of reference.

L1357 L327 L332

L333 L334 L335

L336 L337L338

Formation of a 3-frequency octahedron

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L1323

Cubic Relationships :

L1324 L1322

L1327 L1326

L1325 L1332 L1331

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L1329 L1334 L1333

L1337 L1342

Curved stella octangula (duo-tet cube) in a benchcube.

L1354A

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L1338 L1336

Other Combinations:

L1340

L1339 L1346 L1344

L1350

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L1524L1523L1523 and L1524 are the inner and outer faces of circular planes in the CVM, and they compare to the hexagonal planes formed in the vector equilibrium. The equilateral triangles are the only flat areas.

The circular area inside the black tri-angles on the L1524 graphic show the area modeled below as a paper model.

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INSIDE-OUTThe following graphics show what happens when features that are normally contained inside a form are projected outward.

L1302C

L1302C is a 1/2-frequency Vector Equilibrium with inside vectors projected out. All vertices of the VE core, touch the outer VE face centers.

L1302D L1302B_1

L1493L1302AL1302A shows the half-octahedron faces of a 1/2-frequency VE with internal vectors extended outward.

L1302B Tetrahedron faces with internal vectors are extended outward.

Internal curved vectors of the Pandora are projected to the outside. Curved tips extend beyond the VE faces.

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The following images show different views of Pandora curvature projected outward. I was struck by contrast between the inner curved organic shapes and the outer flat faces of Pandora’s.

Pandora curvature projected outward.

L1494

L1489 convtetprofiles

L1489conctetview

L1489 conctetprofiles

L1489octview L1489octprofiles

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L1489threepodview

L1489tetview

L1494

L1489 convtetview

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OVERVIEW

What is basic in the universe?

I devise a paradox for exploring this issue using graphic shape enclosures.

Triangles, hexagons and circles are used to make three dimensional forms that tile a flat surface with no gaps.

Pods, curved tetrahedrons and octahedrons follow as shapes that are made from the flat paper tiles.

The edges are transformed into three dimensional edges that behave exactly in the same way as the edges of the forms except they have dimension. The forms now include curved tetrahedrons and octahedrons with removable edges.

The goal now is to see if these 3D forms will tile space with no gaps.

I start with models made from card stock and later use a computer aided 3D drawing program called SketchUp to bring my accuracy to six decimal places.

Earlier interest in Richard Buckminster Fuller's works brings to mind the isotropic vector matrix (IVM).

I merge the pods with all the vectors (lines) of the IVM to form a curved IVM or as I call it a Lehman Vector Matrix (LVM).

I define curvature as virtual and irrational and choose to define curvature in this study as rational based on the number of edges on a curve. A primitive circle I define as a hexagon. Increasing the appearance of curvature breaks the hexagon edges in two to form a "circle" composed of 12 edges. I develop curved pi as a constant rational 3.0 when using a curved radius and diameter.

As the curved pods, tetrahedrons and octahedrons form inside the LVM, Fuller's vector equilibrium (VE) is transformed into a different polyhedron that I call a Pandora's Sphere. There are no longer the great circles that occur in Fuller's work but swerves and meandering edges. All of the curved forms have their curved edges embedded inside Pandoras and the bodies or what I call crystal centers lie outside the Pandoras. There are no gaps or open spaces.

I use the card stock pods, tetrahedrons and octahedrons to make silicon molds to produce resin castings. With a series of five pods I attach them to a plexiglass circle to show the edge connection to the pentagon. In a similar way I connect 10 pods to a circle to show the pod face trajectory of the decagon. The edges of the pod have 5-fold symmetry. All of the edges in this study are contained inside Pandora's Spheres. The IVM has 4-fold symmetry. The LVM separates 5-fold and 4-fold symmetry. The 5-fold symmetry lies inside the Pandora's Spheres and as curved edges defining a 5-fold path. The 4-fold symmetry lies between the Pandoras.

The Pandora is an invisible membrane that separates the forms curved edges from its crystal centers.The triangle that lies in the middle of the pod is also an invisible mem-brane.

The pod is unusual in that it does not have a crystal center or body. The pod lies totally inside Pandoras. The pod is like a line that has been exploded at its center to form an object with three dimensions. The pod exists in a 3D space & is totally enclosed in a field of Pandora's that contain only 5-fold curvature.

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This brings to mind Fuller’s thinking about the formation of the prime vector used in his works. The prime vector is based on the line between two sphere centers. This study allows this line but divides it in half, idenfiying two realms, the inner and the outer realm. The inner is contained inside the spheres. The outer realm is formed outside the spheres. This creates a line that has its origin in the center of a sphere and ends with the exterior of the sphere then continues on to the center of the next sphere to complete the syncronization within the IVM. See the JL808 illustration.The edges of the tetrahedron are separated as a result of breaking the prime vector and a core “crystal center” is formed in the outer realm. When the tetrahedron shown below is constructed with curved edges (not shown here), the curved edges are contained in the inner realm.

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L427GThe polyhedrons contained inside curved forms, together with Pandora’s Sphere, fill space with no gaps or voids. Simultaneously, all of the curved forms fill space with no gaps or voids as shown in L427G, where a 4-frequency solid is formed by curved pods, tetrahedrons and octahedrons.

CONCLUSION:Externally the Pandora is a non-curved polyhedron formed from invisible membranes. Additiionally there are crystal centers of the curved forms on the outside of the Pandora. Internally the Pandora contains curved polyhedra. Therein lies the paradox of simultaneously filling all-space with both curved and non-curved polyhedra. All the curved forms that occur everywhere in the system are found inside the twelve around one Pandora clusters.

A three sided pod is formed when three of the leaf shapes are placed edge to edge. Just as the inside and outside arcs form the leaf shape by adding width to the leaf, assembly into a pod, creates volume. Building structures based on pods creates forms including Pandora’s Sphere and form a kind of ”pattern law” that make up the CVM.

Synchronizing the pod with Fuller’s IVM overlays his straight vectors, creating linear and non-linear trajectories. The tetrahedrons and octahedrons as well as internal forms such as the stella octangula (duo-tet cube) and the cube octahedron (Vector Equilibrium) are transformed into curved polyhedrons. Pods surround the vectors in the IVM and create volume where linear behavior existed. In effect, the line is expanded and becomes an object. Pods and IVM radials form a kind of ”pattern law” that make up the CVM.

Dividing the pod to form the Pandora, divides Fuller’s prime vector. This is a change to his basic premise. It is this division that allows the extremedies of the forms to separate from their crystal body centers and form Pandoras.

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The CVM formed with pods creates the Lehman Vector Matrix (LVM), which synchronizes the isotropic vector matrix (IVM) and curved vector matrix (CVM). Pod edges and the internal vector between tips of the pod are all that are required to form the LVM. The LVM can be populated with curved tetrahedrons and octahedrons as well as all of the polyhedra of the CVM and synchronized with Fuller’s cosmic hierarchy in his IVM.

I have taken a scientific approach rather than an artistic one in linking a flat pattern with its folded counterpart in the three dimensional world. After completing this study I found that others have been aware of this interplay between convex (positive) and concave (negative) flat patterns for centuries and have attached spiritual significance to this interplay beginning with the Egyptians and more recently Fuller.

The origins of the circle pattern that started this inquiry can be traced back 3,500 years to ancient Egypt and to Amenhotep IV better known as Akhenaten. This pattern has come to be known as The Flower of Life or Merkaba in the area of sacred geometry. The Flower of Life is said to be the most significant of all the symbols in sacred geometry and encoded within it is said to be the blueprint for all creation. Drunvalo Melchizedek in his book The Ancient Secret of the Flower of Life, says ”The code of the FLOWER OF LIFE actually contains all the wisdom found in the universe similar to the genetic code contained within our own DNA. This geometric code goes beyond ordinary forms of teaching and lies beneath the very structure of reality itself”.

While my interests in this pattern are not focused on sacred geometry, the highly generative nature of modules formed by this pattern have held my attention for many years. I know of no other study of this pattern where spatial tiling has been attempted. My long term interests in Fuller’s work provided the framework to make this study possible. In Fuller’s book Synergetics, paragraph 440.01, he states ”Equilibrium between positive and negative is zero. The vector equlibrium is the zero reference of the energetic mathematics. Zero pulsation in the vector equilibrium is the nearest approach we will ever know to eternity and god: The zerophase of conceptual integrity inherent in the positive and negative asymetries that propagate the differentials of consciousness.”

Fuller’s isotropic vector matrix provides the ”skeleton” to which this study adds the ”flesh” of curvature. Just as the Flower of Life pattern forms a dynamic, flowing regenerative flat tile matrix, the curved pod network of this study links this regenerative activity in three dimensional space, forming curved structures.

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FEATURE DIFFERENCES BETWEEN THE CVM & IVM: The CVM is non-linear and has convex and concave features. All tetrahedra are the same in the IVM (one kind), showing linearity.There are two kinds of tetrahedra in the CVM: convex and concave. The CVM is formed by the three sided pod, creating a 3 dimensional ”edge”. The edges and faces form a 5-fold trajectory.Pandora’s sphere is a polyhedron that contains all the curvature of the CVM.There are five different polyhedra inside the curved forms of the CVM as well as five different curved forms. The Pandora is an invisible membrane that lies outside the curvature and exists as a straight edged, flat faced polyhedron when grouped with the crystal centers of the curved forms. Non-geodesic trajectories are formed in the CVM.

Roger Baker: Constant support & encouragement over many years.John Braley: Contributions at Geojourney & Synergeo forums.John Brawley: Contributions at the Synergeo forum.Dr. Scott Childs: Chemist reviewed my 4-fold & 5-fold symmetry bridge.Russel Chu: Review of my models.Mark Curtis: Correspondence on DNA.Arnie Dyer: Revising, editing, & clarifying concepts.Rick Engle: Polymorph.Alan Ferguston: Collaboration on what I call the concave icosahedron.Dick Fishbeck: Frequent contributor at Geojourney.Rick Flowerday: Geometric toy designer and Synectics author.David Foster: Chairman art Dept. U of O (deceased). Foster was the catalyst that made this research possible.R. Buckminster Fuller: I reviewed my thinking with Fuller during a half hourmeeting and received a letter of encouragement.George Hart: Contributed to my understanding of concave dodecahedra.Steeg Lehman: Reviewed and contributed to my studies for many years.Roy Lewallen: Analog designer.Gary Meacham: Made early photos in my studio in Eugene.Ted Murphy: Helped formulate early modeling ideas.Kenneth O’Connell: Chairman Art Dept. U of O (retired).Rybo6: Frequent contact on the Synergeo & Geojourney forum.SketchUp: @Last Software and the SketchUp Software Team (www.sketchup.com).Kirby Urner: Webmaster of Synergetics on the Web; I Met with Kirby on several occasions to review models and my thinking. He posted on his site: The Lehman/Fuller Sculpture.Marion Walter: Geometry professor, University of Oregon.Steve Waterman: Founder of Waterman Polyhedra. Steffan Weber: A quasi-crystallographer who made an extensive review of atomic positions of regular pentagonal dodecahedrons in my models.Jack Wilkinson: Chairman Art Dept. U of O (deceased).

Other specialists in the field who have actively contributed to my research:

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Previous publication of major portions of this material was in the form of electronic self-publishing via a PDF file on 5/9/2003. Distribution consisted of but was not limited to a posting at the listserve fourm: http://groups.yahoo.com/group/GeoJourney/ and the file is located at: http://www.blackcatphotoproducts.com/pandora.pdf

The GeoJourney list is an open forum for discussion: http://groups.yahoo.com/group/GeoJourney/

Jim Lehman3410 Harney St.Vancouver, WA 98660-1829Phone: 360-750-7204Email: bcat@teleport.com

Copyright 1961-2016, Jim Lehman, All Rights Reserved

Annotation: The following pages relate to the original publication. They show the original cover of this publication and the contents pages of Part 1 and Part 2.

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Annotation: The following two pages of contents show the contributors for both publications.

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