Patent Publication Number: US-2007111634-A1

Title: Ball cluster construction system, basic construction unit and method of making

Description:
PRIORITY DATA  
      Non provisional Application for Provisional application titled “Ball Cluster Construction System”, filed Nov. 12, 2004 by Gary Lawrence Rosenzweig 2272 Snipe Ct. San Leandro, Calif. 94579-2763 (510) 895-2636 (home/fax) (510) 918-6065 (cellular) Class/subclasses, inclusive 446/85-127, plus classes 24, 52, 403. with various subclasses to be named later. All subclasses thoroughly reviewed within each stated class. Dates written: Oct. 11 to Nov. 11, 2004 Modified Nov. 14, 2005 11:33 PM by George M. Steres. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       FIG. 1 . BCU Basic Cluster Unit in oblique view from slightly above and to one side.  
       FIG. 2 . Plan view of BCU. Looking straight down on end balls, through them to the center ball. Preferred end ball colors are given.  
       FIG. 3 . Nest Bond, 2 BCU. Side view, slightly oblique. First, lower BCU  100  on bottom, holds second, upper BCU  100 ′ in its pocket, rotated 45 degrees to the first.  
       FIG. 4 . Interdigit, 1st position bond, 2 BCU. Side view.  
       FIG. 5 . Interdigit, 2nd position bond, 2 BCU. Same view as  FIG. 4 , but now pressed together more.  
       FIG. 6 . Super-Interdigit bond, 4 BCU. Same view as  FIGS. 4 and 5 , now with one BCU added to each side. This configuration has multiple conformers.  
       FIG. 7 . Like bond, slow helix, 2 BCU. Side/oblique views of empty pocket-lower BCU  100  and upper BCU  100 ′, with end ball pair (R′,B′) above and parallel with end ball pair (R,B) of BCU  100 .  
       FIG. 8 . Like bond, fast helix, 2 BCU. Same views as  FIG. 7 , but upper BCU  100 ′ is rotated farther, to extreme position, which has end ball pair (B′,R′) to the side and perpendicular with end ball pair (B,R) of lower BCU  100 .  
       FIG. 9 . Like chain, slow helix, 8 BCU. Side-oblique view. Single helix is formed. Same ball from each pair is taken into the pocket of the next, upper BCU  100 ′ each iteration. R is always between (W,W).  
       FIG. 10 . Like, z-helix ring chain. 8 BCU. Side partial oblique view. An alternate end ball from each pair is taken on each iteration, first B is between (W,W), then R is between (W,W), canceling the left/right helix each time; instead, a ring or helix on the “z-axis” is formed.  
       FIG. 11 . Unlike bond, 2BCU. Side partial oblique view. Empty-pocket BCU  100  is shown behind occupied-pocket BCU  100 ′.  
       FIG. 12 . Unlike chain, 12 BCU. Side/oblique. Same end ball, B, is donated to parallel pairs of end balls (W,W) each iteration, making a straight chain that has a twist along its axis.  
       FIG. 13 . Unlike, z-helix ring chain, 6 BCU. Top/oblique view. Alternate end ball, first R then B, etc. is donated to parallel end ball pairs (W,W) to occupy pockets each iteration.  
       FIG. 14 . Straddle bond, 2 BCU. Side-oblique view. So named because BCU 100′ in the pocket of BCU 100 straddles the R ball, as shown. The acceptor pair on  100 ′ is (B′,W′).  
       FIG. 15 . Straddle bond DNA-type double helix chain, 10 BCU. Side-oblique view. Same set up as  FIG. 14 , (B′,W′) straddles R, each iteration. Ten BCUs make one turn of the double helix chain, similar to nature.  
       FIG. 16 . Pair of BCUs held together on either end of center strut, or center-ball strut,  200 . Side/partial oblique view.  
       FIG. 17 . One type of end strut, or end ball strut,  200 , shown in pure side view, with BCU  100 . End ball strut is socket type, screwed to standard strut body,  210 . Sockets attach to strut bodies with threads and attach to balls with snap fits, with at least 90 degrees of freedom.  
       FIG. 18 . Mini-ball-strut and BCU, with orifice,  132  for specialized strut with mini-ball,  240 . Shown in side-oblique view. Not yet connected, but in close proximity. 
    
    
     SUMMARY  
      A Basic Construction Unit (BCU) of the present invention includes a Central body, a plurality of spaced apart longitudinal Stalks and an equal plurality of separate End bodies. A construction set consists of a multiplicity of identical BCUs where the number, N, of stalks and end bodies for a is a single fixed number: two, three, four, or five. Each Central body is joined to each of the N separated End bodies by a respective one of said Stalks. The Stalks each have opposite ends: a proximal 1st end and distal 2nd end, with each 1st end proximally fixed to a respective spaced apart location on its Central Body and with its 2nd free end fixed to a respective End body. The stalks intermediate section between the two ends are semi-rigid, laterally resilient, longitudinal cantilever members of equal length. The Central body and the End bodies are preferably rotationally symmetric about their respective axes, having respectively, identical solid symmetry. Such bodies include identical polyhedra, discs, wheels and spheres being symmetrical about a center point or body axis. The End bodies of each BCU are fixed and oriented at the distal end of the stalks with a proximal pole of their symmetry axis coaxial with the free end of the stalk. The stalks and end bodies are oriented an distributed in symmetric order about the BCU central axis, where the central axis passes through a 1st point of symmetry defined by said the center of the Central body and a 2nd central axis point disposed equidistant from the respective end bodies points of symmetry defined by the respective End bodies. The End bodies are disposed in said symmetric distribution such that the Stalks extend between their fixed and free ends along their respective axes oriented at equal divergence angles between each stalk and the central axis.  
      The dimensions and angles of the bodies and the stalks are arranged so that one of the balls of another BCU will be constrained within a limited cage region defined axially between the center body and the end bodies, longitudinally between the stalk fixed ends and the end bodies, and laterally between the central axis and the surrounding stalks. The stalks and end balls are spaced, oriented and distributed so that the cage region is no smaller than that permitting any one of the bodies of another identical BCU to be retained within it with the retained ball no closer than touching the surrounding stalks and bodies, and no larger than enough to keep the retained body from being removed from the cage region or inserted into it without bending at least one stalk (cantilever) away from an adjacent stalk. Pairs of BCUs can be connected and disjointed in different configurations. For a given body type, and plurality number, the useful arrangements of stalks and bodies are characterized by dimensions and angles which are parameters that constrain the types of pair configurations that can be assembled from identical BCUs. For a sphere, the relevant dimensions are: the ball diameters, the stalk length and diameter and the stalk divergence angles (the angle at which a stalk diverges from the central axis, from its fixed end on the center body to the respective end body), a center axis intercept point (where the stalk axes intercept the central axis) and whether there are two, three, four or five end bodies. The most useful center body shape is an oblate spheroid with a central axis axial diameter slightly smaller than a lateral diameter.  
     PHYSICAL DESCRIPTION 1    
     Preferred Embodiment: FIG.  1 .  
      A preferred embodiment of the present invention is an educational construction system with a set of identical basic construction units (BCU)  100 .  
      The BCU of  FIG. 1  is a plurality of five spheres, or balls, four of which are identical and connected to one single, similar ball. The basic construction unit, or BCU,  100 , connects to other copies of itself, analogous to chemistry, where a single monomer unit connects to other identical ones to make polymers. The BCU has a supporting center ball,  110 , and in the preferred embodiment four stalks,  120 , which connect the four end balls,  130  to said center ball. The center ball is a sphere truncated with the top and bottom ends,  112 , removed. This is accomplished indirectly by creating a through hole at the same points,  112 , from the bottom to the top of a solid ball. A hollow center ball will have the same diameter holes as a solid center ball. This missing material or hole acts to truncate the center ball along a longitudinal axis,  140 , and also provides a connection point for struts,  200 , which may be threaded, to connect to the center ball. The center ball has attachment points,  122 , for stalks,  120 , at axially evenly spaced points just above the equator. If the center ball&#39;s truncation were continued to its maximum, the center ball would take the shape of a fully flattened ball, or disk of nominal thickness just enough to hold the stalks. The shortening or truncation of the center ball effectively lengthens the stalks, giving them a longer “reach”, allowing them to connect to other balls and pull the other BCUs closer in.  
      The central longitudinal axis,  140 , passes through the BCU&#39;s middle. The lines described by the stalks,  142 , lean out as they proceed from their origin,  114 , at an angle from the central axis of approximately 7 to 15 degrees, through the bottom of the center ball and exiting at the upper attachment point,  122 , then proceeding for a distance, d to the end ball attachment point,  124 .  FIG. 1  shows these angled lines,  142 , radiating up and out from the origin,  114 . These pairs of lines representing stalks oppose each other with an included angle of 14-20 degrees, with a preferred included angle of 15 degrees.  
      For BCUs with the preferred embodiment of four end balls, the axial spacing between stalks is 90 degrees at their attachments,  122 . The four stalk projection lines converge at their origin, 1.5 d below the bottom of the center ball, and 2 d below the center of the center ball. These lines,  142 , extend up and out, longitudinally and axially, as described, from the center ball to the end balls,  130 , for distance of exactly d in the preferred embodiment as measured from the inner side of the stalk, facing the central longitudinal line,  142 . Since they are sphere-mounted, the stalks have a greater measurement on their outer sides than their inner sides. The inner stalk side is greater than the outer side by a factor of d/5, or 1.2 d. Shorter stalks, shorter than d, still have utility in the ball cluster system, only incrementally less so for each smaller length tested, to a minimum of approximately 0.375 d, where it will only perform a single longitudinal connection. At its maximum length d, it performs many connections indeed, some of which will be described below. Table 1 shows how all connections between BCUs are possible for stalk length=d, and how fewer bonds are possible as l=line length shortens to l=0.375 d.  
      The stalks,  120 , of length d, insert into their respective end balls,  130 , at end ball insertion points  124 . Each identical end ball can be at its preferred maximal longitudinal axis circular size of diameter d, and just as with the center ball, can be made smaller along the stalk axis, so that it becomes progressively flattened, until it becomes finally, a disk of diameter d, with nominal thickness, enough to hold the stalk. As with the stalk length, however, when the end ball takes this minimal disk shape, it only is able to make one single longitudinal connection, and is therefore of limited utility. This is analogus to the truncation to disk of the center ball.  
      When all parts are together, and indeed the BCU is ideally molded in as much a unitary body as possible so that small parts do not come off and present choking hazards, the BCU has an overall height of 3 d. The space called the pocket,  170 , shown in  FIG. 1  is that volume inside the aggregate undersides of the end balls, the aggregate inner sides of the stalks, and the top side of the center ball. The pocket is large enough to hold one ball, plus a little more, such as a stalk. The pocket allows the BCUs to be snap fittingly added on to each other in various ways, call connections, or bonds, and these bonds are organized in a hierarchy of ever more complicated, interwoven, useful configurations, which help define the ball cluster construction system. In addition, struts are added to center balls and to end balls to allow many more structures to be built, including many skeletal Platonic and Archemedian solids, plus more esoteric shapes. These structures may be static, or, given the nature of ball-socket connections, they may be dynamic, allowing struts and balls to swing through arcs of at least 90 degrees for each connection, allowing the collapsing and folding entire structures.  
      While the primary task of a stalk is end ball support, it is capable of movement in a limited fashion, and it imparts this to the end balls to which they are connected. It is made of strong, stiff, elastic material, preferably nylon, or similar tough, flexible plastic elastiomer, such as polyurethane, including the toughest foams, or silicone rubber and the like. Relatively small guage spring tensioned metals may be used straight or embedded in the plastics as an alternate, plus many other materials have been tested. The stalks must “give” slightly when a ball passes between them, and then must snap back into their original place, once the ball is through and occupies that pocket. Once the pocket is occupied, the stalks are ideally unstressed, though they may be stressed if geometry requires it, as they may remain a little offset. If stalks are kept in this stressed position for an extended period, they could appear to permanently deform, a situation known as creep. Nylon is generally thought of as a most creep-resistant material, and this physical characteristic is of prime importance for stalks. Fortunately, nylon can also reverse creep by being bent in the opposite direction to the creep, thereby canceling out the effect, straightening out again. This is also a critical stalk material characteristic: if it creeps, it must be reversible. Stalks may be most any shape, but are generally cylindrical, with a rounded profile, though any profile will do as long as it fulfills the requirements of elasticity, stiffness, creep resistance, and creep reversibility, plus the geometric considerations of being attached to the center and end balls, and the ability to make, keep, and release the connections repeatedly. They may be smooth or contoured (such as threaded, grooved, etc.)  
      Alternate Numbers of End Balls  
      The number of stalks and end balls is of prime importance for the ball cluster construction system, and while four of each is the preferred embodiment, two, three, and five end balls have also been successfully utilized and thoroughly explored. Beyond these limits, one or six end balls are not within the scope of the invention. If six end balls were mounted on a center ball, the geometry would create an effective passage where the pocket should be, canceling its purpose. And while two end balls have no pocket either, a simple 45 degree offset can be created when two units face each other so that the basic longitudinal connection is easily created, but it is the only one possible. Two end ball BCUs, when connected in this fashion can also be said to link together in a hinge line, so that a certain degree of freedom is attainable with them. A one end ball BCU, however, has no mate as such, and is not considered.  
      Three end ball BCUs are quite useful. Structures made from them are quite similar to four end ball BCU systems, but three end ball systems have a limitation that four end ball systems don&#39;t have: spiral or helical structures. Helical structures are a basic function of four end ball BCU systems, but not three end ball BCU systems. Three end ball BCUs might make circular structures, and certainly make straight line structures, but they do not spiral from one BCU to the next, like four end ball systems do, and so have more limitations than four end ball BCUs. Helices are important because they are closer analogues of biological, chemical, and nano-structures in real life systems. Three end ball BCUs, however, are highly desirable when basic, simple structures are called for. This simpler-to-make connection allows for faster to build and more compact structures.  
      Five end ball BCUs can actually build more structural configurations than the chosen preferred embodiment of four end ball BCUs. The drawback is noticed when actually using them: they are less comfortable to use. They are simply not as highly ergonomic as the preferred embodiment. They can make most structures that the preferred BCUs make, and more; they are just over-crowded. They are, however, most useful in conjunction with end ball struts because they allow for the simplest construction of five-fold symmetry: a basic geometric function of pentagons, pentagrams, and shapes built up from them. Five fold structures look the same when rotated 360/5, or 72 degrees. Five fold systems are attainable with four end ball BCUs when more than two or more four-end ball BCUs are combined on a single vertex, so they take a bit more work for such structures, though it is well worth the effort, for they reward the builder with a multitude of easily built pleasing configurations.  
      How to Build Ball Cluster Units  
      One method of creating BCUs of the preferred embodiment is to begin with five ½″ balls of polypropylene or polyethylene or other thermoplastic, thermoset, or other plastic or elastiomer, or other materials such as metal, wood, or combination thereof, or layered material. Metal is called for if magnetic attraction is desired. Metal end balls or a metallic coating in or on end balls could be the only needed alteration for this case. A useful range for balls is from 0.25″ to 1.5″ but these are by no means absolute limits. In fact, tiny or huge spheres might be quite interesting and useful, depending on the application. Chose one as center ball and insert into chuck. Set stops at 0, 90, 180, and 270 degrees. For other embodiments: Two end balls, 0, 180 degrees; Three end balls, 0, 120, 240; Five end balls, 0, 72, 144, 216, 288 degrees. Drill/mill the proper number of holes and tap, using in the preferred embodiment 7/64 drill and 6-32 tap, then chamfer. Cut holes as close to ball periphery as feasible, allowing for tapping. Cut center longitudinal hole, tap, then chamfer same size as others. Drill and tap all four end balls, and place headless screws into them and tighten. Take each end ball-stalk sub-unit and connect to each center ball hole and screw in until proper ½″ length i.d. is reached for the stalk and is as tight as possible. Trim off bottom of stalk below center ball if it protrudes. BCU is finished. Alternate end ball configurations have orifices at the top of each ball for mini-ball-strut access, as shown in  FIG. 18 . Here solid or hollow end balls may be used. If solid, holes are drilled at top, in line with the stalk mount at the bottom. It is enlarged at the top just enough for the mini-ball to snap in and snap out, yet move easily while connected. Hollow end balls may use a nut inside for securing the end ball to the stalk and more importantly, for acting as a high stop for the mini-ball so that it does not get bound up deep in the hollow.  
      The pouring, injection, blow, rotation, extrusion, or other molding method of construction would be preferable for higher production. In this case, a simple single material such as nylon might be most efficacious. If hollow balls are used, a combination molding system might prove best. Combination molding is also possible for staking metal end balls or similar material end balls to stalks, so that they are securely molded into them either held top and bottom, or preferably by forming a ball or plug of plastic borrowed from the stalk on the interior of the ball with the same effect.  
      Connections: Bond Types and Chains  
      A BCU&#39;s pocket holds a ball from an adjacent BCU, in figures designated BCU-prime. More than two of one type of connection (or more) is called a chain. If it also folds back on itself and connects head to tail-like, it is termed a ring chain, or ring. A typical chain is best viewed with six or more BCUs, or units, but two units show connections, or bonds most clearly. There are several types of bonds which are described, including the Nest, Interdigit, Like, Unlike, and Straddle. With these five bonds, dozens of different types of chains can be created. These five connections form the basis of the ball cluster construction system, and each one has many conformers.  
      Like/Unlike Orientation  
      BCUs are put together so that a first unit is pointing in a like or unlike direction from a second unit. All chains are therefore termed Like or Unlike. When both are the same direction, as defined by which direction the center ball “points”, the bond is Like; when they alternate direction, the bond is Unlike.  
      Left/Right Handed  
      Every bond can be made left or right handed, in addition to Like/Unlike. This virtually doubles the number of possible chains that can be built. Further, a left handed chain might be connected to a right handed one, and how it looks depends on if the bond between them is left or right so for example, a first left handed chain can connect, left handed, to a second left handed chain.  
      Colors: End Ball Identification  
      To differentiate between end balls and chain types, the end balls need to be told apart from each other. One method of accomplishing this on plastic is to color/dye the material, another is to create different textures. Three colors are required for four end balls. This is best seen when chains are built, because it turns out that often, in the main feature of a chain, such as the “backbone” of a helix, two adjacent end balls are contributed by each BCU. Therefore two adjacent end balls are colored the same, and the next two will have different colors, so that the labels of the colors for them are a, b, c, and d. When viewed in plan,  FIG. 2 , the numbering starts where the end balls,  130 , have letters: c, d, a, b.  
     Preferred Embodiment  
      a=R=red  
      b=B=blue  
      c=W=white  
      d =W =white, or W, W 
          R, B.        

     Second Embodiment:  
      B, B,  
      W, R, is also useful.  
      Other ideas for coloring include different color layers, such as an inner darker color, and an outer translucent color in a clear plastic end ball. End balls might be illuminated with an LED, given enough room for power and electronics. If the LED were in the center ball, the light could travel through stalks acting as fiber optics to the end balls which would appear backlit.  
      Nest Bond  
      The first bond a builder is likely to make is called the Nest. See  FIG. 3 . It is a type of Like chain: both BCUs  100  and  100 ′ point in the like direction. The pocket of a lower, first BCU  100  is occupied by the center ball of a higher, second BCU  100 ′ in the like direction. If more BCUs are added on using the same bond, a chain is created. Hold the first BCU,  100 , by pinching the center ball,  110  in one hand. Hold the second BCU,  100 ′ in the other hand by the end balls and push the center ball of this BCU through the nexus,  138  of the lower BCU into the pocket of BCU  100 . The chain created by the Nest requires eight units until it repeats a full turn. Since it needs eight ( 8 ) units to go 360 degrees, each step turns through 45 degrees. The right/left handedness is created by choosing which way it “turns” by orienting an end ball, say R′, on the second BCU,  100 ′, to the left of R on the lower BCU,  100 , and each one higher will be one step farther to the left, similarly, B′ is to the left of B by 45 degrees. Alternately, choosing B′ to go one step to the right of B, and so on causes rotation in a helix of the opposite rotation. The ninth B will line up with the first, beginning the second turn. The builder can experiment to see how many units in the Nest are needed to create a ring ( 20 ).  
      Interdigit Bond, First Position  
      The interdigit bond is an Unlike connection,  FIG. 4 . The second BCU,  100 ′, points in the unlike direction with respect to the first BCU,  100 . This is a bit more complicated than the Nest and exactly the opposite. Hold the first BCU  100  in the “pinch” by its center ball,  110 , then push it down into the fist of the same hand, cupping the stalks. Now bring the second BCU  100 ′ over the first BCU, holding this one by its center ball,  110 ′, in the “pinch”. Place the end balls of BCU  100 ′ over the detents of the first BCU&#39;s  100  end balls,  130 , so that they are offset 45 degrees. Push down. The BCU  100  in the fist is allowed to rise up toward the pinched BCU,  100 ′ as it is pushed down and the bond is initiated, and the end balls interdigitate. This is the Interdigit, first position,  FIG. 4 . Note how for both BCUs, B from one BCU is between R,B from the other BCU.  
      Interdigit, Second Postion  
      See  FIG. 5 . The first position Interdigit converts quickly to the second position Interdigit. Hold the BCUs  100  and  100 ′ together in the first position with the ends, center balls  110  and  110 ′, between the thumb and the fingers, and squeeze together. The center balls move together and occupy the opposing BCU&#39;s pocket. This is the second position,  FIG. 5 . The stability of the second position Interdigit is useful for certain basic structures.  
      Interdigit Second Position Chain  
      From the second position Interdigit, the user can next move in one of two directions. First, if another second position Interdigit is placed above this, aligning end balls as described for making the first position Interdigit, a chain of second position Interdigits of any length can be formed. It is strong, yet pliable, and can rotate along its axis in a limited, but interesting way. When long enough, one end can connect to the other to form a strong ring. The user experiments to find out how many units are required to form the ring (15 sets, or 30 BCUs). The amount of center ball truncation is found when making this chain, as all space between center balls is collapsed and the optimum, ideal reach of the stalks is made.  
      Super Interdigit Bond  
      From the second position Interdigit, one more BCU is placed on each end. The structure created when these two additional units are snapped on, making a total of four BCUs, is named the Super Interdigit. See  FIG. 6 . The Super Interdigit can collapse and bend; it can also be enlarged and expanded in several different ways, thus the name “super”. The collapsing of the super Interdigit is similar to the previous one, where the center balls&#39; truncation makes it possible. When made and built right, the Super Interdigit configuration is virtually indestructible and interesting due to the many sizes and conformations it can take. These attributes, among others, may have implications in other fields as well, on large and very small scales. Note how the end balls at the top and bottom of the structure is the same as in  FIG. 5 .  
      Like Bond  
      The Like bond,  FIGS. 7 and 8 , is a connection where the first and second BCUs are oriented in the same, or like direction, where the end ball of a first BCU  100  is in the pocket of a second BCU,  100 ′. There are two final conformations in which they can mutually lie: a fast helix and a normally preferred slow helix. If the upper second BCU  100 ′ is to the left of the lower, first BCU,  100 , a fast helix is attainted by rotating the upper one further left, until it can turn no more. The slow helix is most easily identified by feel by turning the upper BCU back to the first position. It will be seen that the slow and fast helices look different as well, when the end ball pairs are compared. In the fast helix the end ball pairs (R′,B′) and (R,B) are perpendicular with each other; in the slow helix, the same end ball pairs are parallel to each other, which happens to be more like all other useful configurations. Comparison of  FIGS. 7 and 8  shows this. The fast helix will require  4  units to make one helical turn, and is quite steep, repeating every 5 BCUs. Each step is 90 degrees. The more useful slow helix,  FIG. 7 , makes one helical turn after 15 units, the 16th lining up over the first. Fifteen steps to go 360 degrees means that each step is 24 degrees. Both are examples of single helices. Schematically, the  FIGS. 7 and 8  are represented: (R--like--&gt;W,W)  
      Both left and right handed helices are created with the ball cluster construction system. Since handedness is by convention where a right handed screw appears to turn into the ground when rotated clockwise/right (tightening), testing a single helix with 5 or more BCUs by rotating a chain in this manner and seeing if agrees with the screw, will determine handedness.  FIG. 9  is made in the exact same fashion as the bond above, (R--like--&gt;W,W), for  8  units. This chain would in fact turn into the ground when rotated clockwise, so it follows that it is RH, right handed. It follows that (B--like--&gt;W,W) will be LH, left handed.  
      Z-Helix, Like Bond  
      The like bond can be experimented with by changing the end ball that occupies the previous BCU&#39;s pocket. Instead of always using the same end ball, R, for example, an adjacent one, B, is used on alternate steps. This creates a fascinating “z-axis” helix,  FIG. 10 , where its rotation is in a plane different from the that found in the previous examples of left/right handed helices. Here the representation is: (R--like--&gt;W,W), (B--like--&gt;W,W) and normally a large bracket would cover both lines to show that both operations are done before they begin again at R, ensuring R,B,R,B, etc between W,W. The builder then brings the last BCU around, over the top of the chain in this case, to the first BCU, connecting it the same. There must be enough units in the chain to allow it to complete the ring, and this is part of the challenge and mystery of the system.  
      Unlike Bond  
      The unlike bond,  FIG. 11 , is where each additional BCU points in one direction and then another opposite direction, and the end ball of the first BCU,  100  occupies the pocket of the second BCU  100 ′. This has properties similar to the Like bond, where two basic types of connections are created depending on whether the same end ball or alternate end balls are chosen. Pointing in unlike directions, if the same end ball is chosen for each bond iteration, a slow helix is created where each successive BCU is part of a different helix,  100  is part of one helix,  100 ′ is part of the second helix, so that a first type double helix is created.  FIG. 11  only shows two BCUs, one from each helix. For  FIG. 11 : (B--unlike--&gt;W,W). The bond is highly flexible, much more so than the Like bond, yet fairly robust chains can be built and when enough units are used, ten or more, a ring can be formed where the tail connects to the head.  FIG. 12  shows the Unlike chain. Schematically, it is: (R--unlike--&gt;W,W). Note how the first few units repeat. The Unlike bond double helix requires 30 BCUs to go one full turn and begin the next, twice as many units as the single helix needs to go one turn, which was given as 15 BCUs. Since it takes twice as many units for one turn, and each successive unit in the chain faces the opposite direction, the individual helices actually require the same number of units to get through a single turn, which is 15, so that each step per helix per BCU on the same helix is still 24 degrees, just as with the Like single helix.  
      Z-Helix Unlike Chain  
      A second type of double helix is created when successive BCUs utilize alternating adjacent end balls. Here, instead of a slowly spiraling long chain, a double spiraling ring is created, termed z-helix, because just as with the Like example, this Unlike example curves when alternate end balls are chosen (R, B, R, etc.) each iteration. See  FIG. 13 . As with the Like Z chain, the Unlike Z chain is able to create right or left handed forms. Schematically, (R--unlike--&gt;W,W) (B--unlike--&gt;W,W) and again a bracket covering both lines would show that both operations must be done before repeating, in order that the proper chain is created.  
      Many different and more interesting chains and rings can be made utilizing different combinations of end balls with this Unlike Bond.  
      Straddle  
      The most useful and interesting connection is the straddle,  FIG. 14 . First a target, or donor end ball is chosen on the first BCU,  100 , and a pair of acceptor end balls is chosen for on the second BCU,  100 ′, which will capture the donor end ball from  100 . Choose the donor ball as R for the first BCU,  100  and in this case, choose B,W as the acceptor pair from the second BCU,  100 ′. Hold BCU  100  in the “pinch” with the thumb on the R stalk. Drop the second BCU,  100 ′ through the nexus,  138  of BCU  100 , with the pair B,W aligned over R. Now the acceptor pair of end balls B,W is pushed over the donor end ball, R. Once a snap is felt, the target end ball is temporarily in the lower BCU&#39;s pocket; (This is an intermediate step, and there are multiple directions to go from here, though only one will be given) a further push ends in a second light snap, pushing the target end ball through and out of the pocket and past it so that the second, upper BCU  100 ′ is straddling the target end ball&#39;s stalk, just like  FIG. 14 . There are much quicker ways to make this bond with practice: as long as the end ball donor and acceptors are kept in mind, and the BCUs are appropriately rotated, the configuration can be made in a single, deft move. This is shown schematically as, (BW--straddle--&gt;R).  
      As with the previous two bonds, Like and Unlike, more than one type of chain is created depending on whether identical or alternating end balls are chosen on successive BCUs. In fact, the utility of this connection far exceeds the others, and a family of choices are available, creating dozens of different chains and rings. But it is the simplest Straddle bond utilizing a single designated end ball for each iteration that is one of the most interesting chains: a DNA type double helix,  FIG. 15 . And just like with the real DNA, it repeats every ten units. The more common DNA helices tend to be right handed. And as with the Like and Unlike Bonds, the Straddle Bond may also be made to be right or left handed. It is quite a straight forward task to make a right handed double helix, with one turn repeated after ten units, just like in nature.  FIG. 15  is just like  FIG. 14  schematically: (BW--straddle--&gt;R), which is RH, and is possible to verify in the figure by observing the helix running from lower left to upper right; if it turned clockwise, it would tighten/go in the ground. A left handed double helix would use the inverse end balls, which are in this case: (RW--straddle--&gt;B).  
      The 360 degrees turned for ten units should mean that each bond turns through 36 degrees. However, since adjacent BCUs are on different helices, the number of degrees per bond between units on the same helix is actually 72 degrees. This is true five fold symmetry: where structures appear the same when turned 360/5, or 72 degrees. Ten BCUs are used to build one double helix, and the angle between units on each helix is 72 degrees. Five units do make one turn, but in this case, five units make two half turns (half helices); ten units make two full turns or one double helix. Additionally, the five-fold symmetry of the double helix may have implications for the nanotechnology field, as it apparently has not been explored in this geometric fashion (perhaps) because it has not been physically available to manipulate in this way.  
      For building different structures, the selection of alternately different end balls on successive iterations creates a series of rings of increasing complexity. Many, many interesting experiments can be made with the Straddle bond.  
      Strut Bodies and Sockets  
      The previously mentioned chains and rings only included BCUs to this point. Struts,  200 , are added in different ways, see  FIG. 16, 17 , and  18 . Struts for center balls are called center struts or center ball struts. They plug or screw into the center balls; one BCU can be at each end of one strut, for a total of two BCUs per center strut, as shown in  FIG. 16 . Here, the center strut,  200 , interchangeably acts as a center strut and a strut body. Strut bodies are part of center struts and end struts, but end struts have sockets, as shown in  FIG. 17 . The strut body in  FIGS. 16 and 17  are the same. The sockets,  220 , screw or plug into the strut body,  210 . The socket has an opening,  222 , where the balls enter and exit, and an orifice, which can be threaded, for accepting strut bodies. Note that the socket shown in  FIG. 17  can also be snapped onto center balls, if a rotation-only connection is desired. Alternately, the socket can be removed and the strut body can be connected to the center ball in a more permanent fashion. This threaded connection is special in that it allows full rotation without problems often associated with it.  
      A strut body is utilized where the threads do not exceed the rod diameter, but rather are matched to the diameter of the rod. There is no shoulder or transition on the rod at the threaded-not threaded interface on the rod. Since there is no shoulder, there is no stop, and no stop means that the rod will not “bottom out” inside the center ball, but rather, continues to rotate all the way through, since the center ball has a through hole and threads. In this way the struts will never strip threads and will last indefinitely. They simply continue to rotate on through when turned “too” far, and can be simply rotated back to their proper length, as needed. Sockets simply screw onto these same threads to convert to end ball struts. The sockets have the same female threads as center balls with through holes to eliminate stripping. This feature is unique to the Ball Cluster System.  
       FIG. 17  shows a generic end ball strut with a screw-on socket body,  220 . The socket body,  220 , is generally made of a pliable, tough, threadable thermoplastic or similar material like PETG, where clarity, formability and toughness are important, though it can be of wood, metal, or other natural or synthetic material. A strut in the form of a tube or rod could have the socket formed directly from it, from a material like nylon for strength, where a one piece rod with sockets at both ends is feasible. Normally the strut is in two main parts: strut body and socket, where the socket is for end balls, but can be for center balls, too. For rods, the body&#39;s threads are essentially the same diameter as the strut body proper, and it is highly unlikely to break with no abrupt change in diameter. In one preferred embodiment, a nylon (for its properties of strength, creep resistance and reversibility, ability to hold threads, and flexibility) strut body with a one eighth inch (⅛″, 0.125″) diameter is matched with 6-32 threads, with a high (maximal) thread diameter of approximately one eighth inch, hence no breakage. These fit directly into center balls, or into end ball sockets.  
      Mini-Ball-Strut  
       FIG. 18  shows a mini-ball-strut. The mini-ball-strut uses normal strut bodies,  210 , as previously described, but in this case, small balls, or mini-balls,  240 , are screwed onto them. Naturally, they may also be molded on, or secured otherwise. The mini-balls fit into an orifice,  132 , molded or cut into the end of each end ball, which is the same diameter as the mini-ball. It is a somewhat tight fit, so that the ball stays in the orifice when untouched, yet is free enough to move in the orifice the same as a good ball-socket connection, approximately 90 degrees. In this case, however, the situation is reversed from the one already mentioned: The socket is in the end ball instead of the strut; and the ball is a miniature ball at the end of the strut.  
      Table of Constructions  
               TABLE I                          Some constructions enabled by BCUs with different ratios of stalk length Ls to ball diameter d.                     ID#   Configuration                                                         (1)   Type   LS = d   LS = .875d, .75d   LS = .75d   LS = .67d, .625d, .5d   LS = .625d   LS = .5d   LS = .375d       (2)   1 st  position, interdigitated   Yes   Yes   Yes   Yes   Yes   Yes   Yes       (3)   2 nd  position, interdigitated   yes   Yes   Yes   Yes   Yes   Yes   No       (4)   2 nd  position, crushed   Yes   No   No   No   No   No   No       (5)   Super node (interdigitated) - normal   Yes   Yes   Yes   Yes   Yes   Yes   No       (6)       (7)       (8)   Hinge (3-position: normal,   Yes   Yes   Yes   Yes   Yes   Yes   No           1 st  &amp; 2 nd  ortho)       (9)   Nest pair   Yes   Yes   Yes   Yes   Yes   Yes   No       (10)   Like connection   Yes   Yes   Yes   Yes   Yes   Yes   No       (11)   Unlike connection   Yes   Yes   Yes   Yes   Yes   Yes   No       (12)   Straddle connection   Yes   Yes   Yes   no   no   no   no