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BACKGROUND OF THE INVENTION
U.S. Pat. Nos. 4,942,700 and 5,024,031, hereby incorporated by reference as
if fully disclosed herein, disclose a method for constructing reversibly
expandable truss-structures in a wide variety of shapes and the teachings
therein have been used to build structures for diverse applications
including architectural uses, public exhibits and unique folding toys.
In accordance with the teaching of the '700 patent, the resulting
structures comprise substantially linear, but angulated, strut elements
and smaller hub elements that are pivotally connected. The angulated
struts always have three pivot points, one central pivot point and two
terminal pivot points, and they lie in planes that are essentially
orthogonal to the surface of the structure. Utilizing the methods taught
in the '700 patent, one may construct foldable structures in a wide
variety of shapes. However, certain shapes are more practical to construct
in order to maintain a reasonable part count, have good structural
integrity and ease of movement. In particular, the method is better suited
to structures whose shape has a gentle curvature, rather than sharp
corners. Also, the parts that make up a given structural shape will, in
general, be unique to that particular shape. Therefore, it is not a simple
matter to make a kit of interchangeable parts that may be used in
different shaped structures.
SUMMARY OF THE INVENTION
In accordance with the present invention reversibly expandable structures
are formed from loop assemblies comprising interconnected pairs of
polygonal shaped links which lie essentially on the surface of the
structure or parallel to the plane of the surface of the structure. The
polygon links in the loop assembly have at least three pivot joints. At
least some of the polygon links however, have more than three pivot
joints. One of the pivot joints on each link is a center pivot joint for
connecting to another link to form a link pair. Each link also has at
least one internal pivot joint and one perimeter pivot joint. The internal
pivot joints are used for interconnecting adjacent link pairs to form the
loop assembly. Finally, loop assemblies can be joined together and/or to
other link pairs through the perimeter pivot joints to form structures.
In one preferred embodiment of the present invention link pairs may be
connected to adjacent link pairs to form a loop assembly through hub
elements that are connected at the respective internal pivot joints of the
two link pairs. Similarly hubs elements can be used to connect loop
assemblies together or loop assemblies to other link pairs through the
perimeter pivot joints to form structures. In yet another embodiment of
the present invention the pivot joints can be designed as living hinges as
described more fully below.
Structures built in accordance with the subject invention have specific
favorable properties, including: a) The ability to use highly rigid
materials rather than bending or distortion of the mechanical links,
allowing for a smooth and fluid unfolding process; b) The use of compact,
structurally favorable and inexpensive joints in the form of simple
pivots; c) Retaining the strength and stability of the structure during
folding and unfolding since all movement in the structure is due to the
actual deployment process, without floppiness in the structure; d) A wide
range of geometries; e) Inexpensive manufacture of structures with
flexible hinges that are formed continuously with the links themselves; f)
Convenient assembly of structures of many different shapes through kits of
the necessary parts; and g) The ability to create a `space-filling`
structure by arranging linkages in a three-dimensional matrix.
BRIEF DESCRIPTION OF THE DRAWINGS
The invention will be further described with reference to the accompanying
drawings wherein:
FIG. 1 is a plan view of the basic polygon link element of the invention.
FIGS. 2-3 are plan views of a linked pair of polygon links.
FIGS. 4-6 are plan views of one type of two dimensional loop assembly of
polygon links in accordance with the present invention, shown in three
positions: retracted, partially expanded and fully expanded, respectively.
FIGS. 7-9 are plan views of a second type of two dimensional loop assembly
of polygon links in accordance with the present invention shown in three
positions: retracted, partially expanded and fully expanded, respectively.
FIGS. 10-12 are perspective views of a three dimensional loop assembly of
polygon links in accordance with the present invention, shown in three
positions.
FIGS. 13-15 are perspective views of a three dimensional reversibly
expandable structure of polygon links in accordance with the present
invention, shown in three positions: retracted, partially expanded and
fully expanded, respectively.
FIG. 16 is a plan view showing an alternate embodiment of a polygon link
assembly.
FIGS. 17-19 show plan views of a two dimensional embodiment of the
invention using a pair of the polygon link assemblies of FIG. 16, shown in
three positions: partially expanded, fully expanded and retracted,
respectively.
FIGS. 20-21 are perspective views of a cylindrical assembly of polygon
links in accordance with the present invention shown retracted and
expanded, respectively.
FIGS. 22-24 are perspective views of a three dimensional reversibly
expandable structure of the present invention using polygon links, having
an icosahedral shape and shown in three positions: retracted, partially
expanded and fully expanded, respectively.
FIG. 25A shows a polygon link.
FIG. 25B shows a link pair.
FIG. 25C shows a loop assembly.
FIG. 26 shows the structure 900 in a folded position.
FIG. 27 shows the structure 900 in a fully unfolded position.
FIG. 28 shows a link pair comprised of a single piece of material.
FIG. 29 shows a loop assembly consisting of eight link pairs.
FIG. 30 shows a structure 1000 consisting of thirty-two polygon link pairs.
FIG. 31 shows structure 1000 in a fully unfolded position.
FIGS. 32-34 shows a loop assembly 1200 in a folded position, a partially
unfolded position and in a fully unfolded position, respectively.
FIGS. 35A and 35B show an alternative embodiment in which separate hub
elements are replaced with a ball and socket arrangement.
FIGS. 36-37 show front views of an alternate embodiment of the invention, a
triangle loop assembly having perimeter corner pivots that are themselves
pivotally connected to polygon links.
FIGS. 38-39 show perspective views of this embodiment of the invention in
its closed and opened states.
FIG. 40 shows a detail of the perimeter corner joints.
FIGS. 41-42 show front views of a square loop assembly in its closed and
unfolded states.
FIGS. 43-44 show perspective views of the square loop assembly.
FIGS. 45-50 show how loop assemblies having a special perimeter corner
joint may function as elements of a "snap-together" kit for making
reversibly expandable structures.
FIGS. 51-52 show a prism-shaped structure in its closed and opened state.
FIGS. 53-54 show another element in a kit for making reversible expandable
structures, a scissor pair that may be attached to loop assemblies.
FIGS. 55-56 show a prism-shaped structure that incorporates scissor pairs
in its closed and opened state.
FIGS. 57-59 show means to attach two loop assemblies in a stacked
arrangement, and further shows means to attach separate hub elements onto
loop assemblies to provide extra attachment points.
DETAILED DESCRIPTION OF THE INVENTION
The present invention relates to a new reversible expandable loop assembly
formed by connecting at least three link pairs, and reversibly expandable
structures which are created from multiple interconnected loop assemblies
and/or link pairs. Each link pair comprises two links i.e., polygon links,
each having a polygonal profile with three or more corners, a central
joint and a corner pivot joint proximate to at least two of the three or
more corners. The central joint is used to connect the two links together.
The corner pivot joints comprise at least one internal corner pivot joint
and at least one perimeter corner pivot joint. To form the loop assembly
each link pair is connected to at least two adjacent link pairs through at
least one of its internal corner pivot joints.
When the loop assembly stands alone, the perimeter corner joints of the
links are not connected to anything. The perimeter corner joints, however,
are used to connect loop assemblies together and/or loop assemblies to
link pairs to form expandable structures.
The polygon links of the present invention can be made from any suitable
material, ascertainable by one skilled in the art. Examples of suitable
material include metal, plastic and wood.
Loop assemblies formed in accordance with the present invention can expand
and retract. In many cases the geometry of the perimeter outline of the
loop assembly will remain constant in all positions, with only a change in
size. Each loop assembly can be identified by a ring of line-segments
formed by intersecting the perimeter corner joints of the link pairs. This
property is a result of constructing the loop assembly such that the angle
formed between any two line-segments corresponding to a particular two
link pairs in a given position of the loop assembly, is the same as the
similarly formed angle between the line segments corresponding to the same
two link pairs for any other position of the loop assembly.
There are two aspects to finding the correct location of pivot points such
that this particular property is obtained.
First, an arrangement of links must be found such that the loop-assembly
does fold freely, that is, it does not lock up. This ability to fold is
not guaranteed. For example, by applying the equation to determine the
degrees of freedom of a typical planar loop assembly, the result will be
negative, indicating a over determined (i.e. locked) condition.
Therefore, the ability to fold is dependent on particular geometric
conditions. When constructing a planar loop-assembly, an aid to
determining possible location of pivot points, is to draw a four sided
shape that connects the center joint from one link-pair to two of its
interior corner joints, and then in turn connecting those corner joints to
the center joint of its neighboring link-pair. According to a typical
construction, all such quadrilaterals similarly drawn within a
loop-assembly should be parallelograms.
If all these parallelograms are similar (have identical angles) the
loop-assembly will definitely fold. However, it is possible to construct
foldable loop assemblies with dissimilar parallelograms, and indeed to
form foldable loop-assemblies where the quadrilaterals, and indeed to form
foldable loop-assemblies where the quadrilaterals are not parallelograms
at all. These alternative constructions require other symmetric
arrangements that may be discovered through deeper study and inquiry.
Once a foldable loop-assembly is constructed, the location of the perimeter
corner joints must be considered. The goal is to ensure that line segments
drawn through paired perimeter corner joints maintain a constant angle
relative to one another as the loop assembly is folded.
In a similar fashion to finding rules for constructing foldable
loop-assemblies, we can find rules for locating perimeter corner joints
that will always work. If, for example, each link-pair in a given foldable
loop-assembly is comprised of two polygon links having identical relative
locations of their perimeter and interior corner-joints, the angles
between line segments will remain constant. Generally, paired polygon
links that are similar in shape, but different in size will have this
property as well. However, there are alternative arrangements that exist
as well.
As explained above the position of the pivot joints are critical to the
function of the loop assemblies and structures of the present invention.
The profile of the links, however, are less critical and more design
related. It will be apparent to one of ordinary skill in the art that so
long as the pivot holes are the same, the links can have most any
geometry. The selection of geometries thus is primarily one of creative
design choice. However, it will also be obvious to one skilled in the art
that certain polygon shapes may restrict the ability of the structure to
reach a fully expanded or fully retracted position.
The loop assemblies and structures in accordance with the present invention
have many applications including: medical devices, toys, architectural
design and displays.
Referring now more particularly to the drawings, shown in FIG. 1 is a link
10, which has a triangular shape and four pivot holes. Pivot hole 2 is in
the central region of the link hereinafter the "center joint," and pivot
holes 4, 6, and 8 are proximate to the corners of the link. A dashed line
25 is drawn connecting the center of the three corner-pivot holes 4, 6,
and 8 hereinafter "corner joints," forming a triangle.
Referring to FIG. 2 the polygon link 10 of FIG. 1 is linked to a second
polygon link 20 by center joint 2 to form a link pair. Links 10 and 20
have essentially the same profile and pivot hole locations. A dashed line
24 is shown passing through the center of paired corner joints 4, 14.
Similarly, dashed line 26 passes through 6, 16, and dashed line 28 passes
through 8, 18. The triangle 30 formed by lines 24, 26, and 28 has
essentially the same shape as dashed line triangle 25 shown in FIG. 1.
FIG. 3 shows the link pair 10, 20 in a new position having been rotated
relative to each other about their center joint. Three dashed lines 34, 36
and 38, are shown passing through paired corner joints 4, 14; 6, 16; and
8, 18, respectively. The angle formed between dashed lines 34 and 36 is
the same as the angle formed between dashed lines 24 and 26 shown in FIG.
2. Likewise the angles formed respectively between dashed lines 36, 38 and
38, 34 are the same as those angles formed respectively between dashed
lines 26, 28 and 28, 24 shown in FIG. 2. Thus triangle 35 has the same
shape as triangle 30 shown in FIG. 2, but larger in size.
Referring to FIG. 4 the expanding right triangle is extended to an
expanding hexagon by forming a loop assembly 38 consisting of 12 polygon
links 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90 and 95. These polygon
links are respectively joined by center joints 41, 51, 61, 71, 81 and 91
into 6 link pairs 49, 59, 69, 79, 89 and 99. The loop assembly 38 is
formed by joining the internal corner joint of each top layer to the
adjacent internal corner joints of the two adjacent lower polygons on both
sides. The internal corner joints are easily seen with reference to FIG.
5.
Thus, referring to FIG. 5 loop assembly 38 is shown unfolded into a
different position while maintaining the overall hexagon shape defined by
edges drawn between the outer joints of each polygon, as discussed below.
In this new position, it is more readily noticeable how adjacent polygon
links are connected. For example, link pair 49 is connected to link pair
59 by the two corner joints 42 and 43. These corner joints are referred to
as internal corner joints since they are located on the interior portion
of the loop assembly 39. Likewise link pairs 59 and 69 are connected to
each other by internal corner joints 52 and 53. Similarly link pairs 69,
79; 79, 89; 89, 99; and 99, 49 are connected by internal corner joints 62,
63; 72, 73; 82, 83; and 92, 93, respectively.
A dashed line 44 is shown passing through corner joints 46 and 48. These
corner joints are located near the outer edge of loop assembly 38 in its
unfolded position. These joints are the perimeter corner joints of the
loop assembly. Likewise a dashed line 54 is shown passing through
perimeter corner joints 56 and 58 and dashed lines 64, 74, 84 and 94 are
shown passing through perimeter corner joints 66, 68; 76, 78; 86, 88; and
96, 98, respectively. These dashed lines through the perimeter corner
joints define the edges of the expanding hexagon 100, mentioned above.
Referring to FIG. 6, loop assembly 38 is shown unfolded further, into yet a
different position while maintaining the overall hexagonal shape. Dashed
lines 47, 57, 67, 77, 87 and 97 are shown passing through perimeter corner
joints 46, 48; 56, 58; 66, 68; 76, 78; 86, 88; and 96, 98 respectively,
forming hexagon 105. The commonality between hexagons 100 and 105 is that
the angle formed between dashed lines 47 and 57 is the same as the angle
formed between 44 and 54 shown in FIG. 5. Likewise the angles formed
between dashed lines that correspond to any two link pairs as shown in
FIG. 6 are identical to those angles similarly formed corresponding to the
same two link pairs, as shown in FIG. 5.
Referring now to FIG. 7 a different triangle loop assembly 108 is shown
consisting of 6 polygon links 110, 120, 130, 140, 150 and 160. These
polygon links are respectively joined by center joints 114, 134 and 154
into three link pairs 119, 139 and 159. Links 110, 120 are joined at 134
to form pair 119. Links 130, 140 are joined at 114 to form pair 139 and
links 150 and 160 are joined at 154 to form pair 159. In FIG. 8 loop
assembly 108 is shown unfolded into a different position and as with FIGS.
4-6, the overall triangle shape is maintained. Link pair 119 is connected
to link pair 139 by internal corner joints 131 and 141, link pair 139 is
connected to link pair 159 by internal corner joint 161 and link pair 159
is connected to link pair 119 by internal corner joints 121 and 151.
The triangular ring perimeter outline 170 of the loop assembly as shown in
FIG. 8 comprises line segments 114, 134 and 154. Dashed line 114 is shown
passing through perimeter corner joints 115 and 125 and dashed lines 134
and 154 are shown passing through perimeter corner joints 135, 145 and
155, 165 respectively.
In FIG. 9 the loop assembly 108 is shown in a further unfolded position.
Dashed lines 117, 137 and 157 are shown passing through paired perimeter
corner joints 115, 125; 135, 145; and 155, 165 respectively, thereby
forming triangular ring 180 which is larger in size than ring 170 of FIG.
8.
The angle formed between dashed line-segments 117, 137 is the same as the
angle formed between 114, 134 in FIG. 8. Similarly, the angles formed
between dashed lines 137, 157 and lines 157, 117 are the same as those
angles formed between lines 134, 154 and lines 154, 114 in FIG. 8.
The loop assemblies shown to this point were all formed by joining adjacent
link pairs directly at internal pivot points. The result was a loop
assembly with all link pairs lying on parallel planes. It is also possible
to add relative dimension to the loop assembly by introducing hub elements
between the internal corner pivot joints of adjacent link pairs. As seen
in loop assembly 208 of FIG. 10, hub elements such as 299 are used to
pivotally connect each link pair to its neighbor. In addition, using hub
elements, more than two link pairs can be joined at a single connection
point.
Other than the hub elements, loop assembly 208 is similar to the other loop
assemblies discussed above. Indeed, it will be recognized that the loop
assembly of FIGS. 10-12 is similar to that of FIGS. 4-6. In FIGS. 4-6
there are no hub elements and the link pairs lie in parallel planes. In
FIGS. 10-12 the hub elements position the same link pairs into non
parallel planes. Loop assembly 208 contains polygon links and each has
three cover joints and one center joint through which they are paired into
link pairs, 249, 259, 269, 279, 289 and 299.
A hub element can be any linking material with at least two separate pivot
points that are not coaxial with each other. The hub element could have an
angle or it could be straight. The axes of the hub pivot points could be
parallel, perpendicular or from some other angle therebetween. Each of
these variations will impact on the creative design element of the loop
assembly including its range of motion.
The size of the hub element and the material chosen for its construction
will also impact on the durability of the loop assembly.
FIG. 11 shows loop assembly 208 unfolded into a different position while
the lines crossing the perimeter joints of the polygon links maintain the
same polygon shape. Link pair 249 may be seen to be pivotally connected to
two hub elements 252 and 253 which connect in turn to link pair 259.
Likewise link pair 259 is connected to link pair 269 via hub elements 262
and 263. Similarly, link pairs 269, 279; 279, 289; 289, 299; and 299, 249
are successively connected by hub elements 272, 273; 282, 283; 292, 293;
and 242, 243, respectively. As explained above, these hub elements
introduce angles between the planes of adjacent link pairs.
The dashed lines 344, 354, 364, 374, 384 and 394 lie in the planes of their
corresponding link pairs, 349, 359, 369, 379, 389 and 399 respectively,
and form a three dimensional ring 400. These lines cross through the
perimeter corner joints of their respective links: 240, 245 for link pair
249; 250, 255 for link pair 259; 260, 265 for link pair 269; 270, 275 for
link pair 279; 280, 285 for link pair 289; and 290, 295 for link pair 299.
In FIG. 12 the loop 208 is shown further unfolded into a different
position. The dashed lines 444, 454, 464, 474, 484 and 494 drawn
respectively through the perimeter corner joints of the polygon links 240,
245; 250, 255; 260, 265; 270, 275; 280, 285; and 290, 295. As with the
other loop assemblies described above, these line segments form a ring 450
that is larger in size than ring 400 shown in FIG. 11. However, the angle
formed between dashed line 444 and 454 is the same as that angle formed
between lines 344 and 354 of FIG. 11. Likewise the angles formed between
dashed lines that correspond to any two adjacent link pairs as shown in
FIG. 12 are identical to those similarly formed angles corresponding to
the same two link pairs as shown in FIG. 11. Perimeters may be left open
or used to connect to another assembly or polygon link pair.
As described above, loop assemblies formed in accordance with the present
invention can be used in forming three dimensional closed structures. In
some instances it will be sufficient to connect two or more loop
assemblies together. Other cases may require additional link pairs
connected to the loop assemblies to close the structure.
Generally, the loop assemblies and/or link pairs are connected together at
the perimeter pivot joints described above. It will not always be
necessary to use all available perimeter pivot joints. However, the
interconnections may only use perimeter corner joints. The
interconnections between loop assemblies will generally involve hub
elements, although direct pivotal connections are possible, as well as
living hinges, as described below.
It is important to note that reference to perimeter corner joints has
meaning only with respect to a given loop assembly. Once a structure is
assembled the perimeter outline of the loop assembly can be drawn with any
arbitrary selection of link pairs due to the symmetry inherent in the
structure.
Referring to FIG. 13 a structure 500 is shown in a folded position.
Structure 500 consists of 20 link pairs in interlocking loop assemblies,
each link pair comprised of two polygon links. One such loop assembly 510,
within structure 500, consists of five link pairs 520, 530, 540, 550 and
560. Link pair 520 is pivotally connected to link pair 530 by two hub
elements 522 and 523. Similarly link pairs 530, 540, 550, and 560 are
successively joined together by hub elements 532, 533; 542, 543; and 552,
553, respectively. Link-pair 560 is connected to link-pair 520 by hub
elements 562 and 563. One may recognize that the loop assembly 510 is
similar to that shown in FIGS. 10-12 except that only five link pairs are
used and the hub elements have different angles.
A structure constructed in accordance with the present invention can
include as a creative design element, the formation of a continuous
surface. As shown in FIG. 13, in the folded position, structure 500 forms
a substantially closed and continuous surface. The degree of continuity
will depend on the polygon profile of the links, the number of links in
the loop assembly and the angle in the hub elements.
FIG. 14 shows structure 500 unfolded into a larger position. Dashed line
524 passes through the perimeter corner joints of link pair 520. Similarly
dashed lines 534, 544, 554 and 564 respectively pass through the perimeter
corner joints of link pairs 530, 540, 550 and 560. Dashed line segments
524, 534, 544, 554 and 564 form a five-sided ring 570.
In FIG. 15 the structure 500 is again further unfolded. The dashed lines
526, 534, 546, 556 and 566 pass respectively through the perimeter corner
joints of link pairs 520, 530, 540, 550 and 560, thus forming a five sided
ring 580 which is larger in size than ring 570 in FIG. 14. The angles
formed between dashed lines that correspond to any two adjacent link pairs
in FIG. 15 are identical to those similarly formed angles corresponding to
the same two link pairs in FIG. 14. In its fully unfolded position,
another creative design element resulting from the polygon links that make
up structure 500 may be seen. Namely the link pairs separate and create
openings that are pentagonally shaped.
In addition to the simple pivots shown above for the inter-link
connections, either hub or direct, connections can also comprise living
hinges. A living hinge is a flexible portion of a material, continuous
with, and connecting two or more stiff portions of the material. A change
in dimension from the stiff portion gives rise to the flexible portion.
FIG. 16 shows a sheet of material 601 that consists of triangular stiff
regions of material that act as polygon links, which are connected by
thinner flexible regions of material that act as corner joints. While many
materials are suitable for living hinges to be used in accordance with the
present invention, and those skilled in the art will be readily able to
determine the same, polypropylene and nitemol are believed to be
especially suitable materials for forming living hinges.
FIG. 17 shows a flat structure 600 which consists of two sheets of material
601 as above, and 602 which is the mirror image of 601. Sheet 601 is
joined to sheet 602 by thirty-six pivot joints to create thirty-six link
pairs. The folded position of this structure is shown in FIG. 19. These
link pairs are arranged in interlocking loop assemblies. One such loop
assembly 605 consists of six link pairs 610, 620, 630, 640, 650 and 660.
Dashed line 615 passes through the perimeter corner joints of link pair
610. Similarly dashed lines 625, 635, 645, 655 and 665 respectively pass
through the perimeter corner joints of link pairs 620, 630, 640, 650 and
660.
While FIG. 18 shows living hinges used at internal corner pivot joints, it
is also possible to use living hinges at the center pivot joint. An
example of a link pair with a living hinge center pivot joint is shown
below in connection with FIG. 28.
In FIG. 18 the structure 600 is shown unfolded into a larger position.
Dashed line 616 passes through the perimeter corner joints of link pair
610. Similarly dashed lines 626, 636, 646, 656 and 666 respectively pass
through the perimeter corner joints of link pairs 620, 630, 640, 650 and
660. The angle formed between dashed lines 616 and 626 is identical to the
angle formed by dashed lines 615 and 625 shown in FIG. 17, however, unlike
the loop assemblies shown in prior FIGS., the shape of the loop assembly
changes with folding and unfolding since the size of the edges do not
change proportionally. Similarly the angles formed respectively between
dashed lines 626, 636; 636, 646; 646, 656; and 656, 666 are identical to
those angles formed respectively by dashed lines 625,635; 635,645;
645,655; and 655, 665 shown in FIG. 17.
Structure 700 shown in FIG. 20 also consists of two sheets of material 701
and 702. Similar to sheets 601, 602 shown in FIG. 16, sheets 701, 702 are
comprised of triangular stiff regions of material acting as polygon links
that are connected by thinner flexible regions of material acting as
corner joints. Sheets 701 and 702 have been joined together by a plurality
of center pivot connections and are formed into a cylindrical shape.
The cylindrical structure can be formed by joining the opposite, parallel
edges of a loop assembly much like that of FIGS. 17-19. Alternatively, two
cylinders can be formed from a continuous cylindrically shaped material
with links cut out much like FIG. 16. One cylinder can be placed over and
around a second cylinder joined by center pivot joints. Yet, a third
method would be to cut out link pairs from a single cylindrical material
with living hinge center pivot joints. Other embodiments will become
apparent to those skilled in the art and fall within the scope and spirit
of this invention.
In its folded position, the polygon links that make up structure 700 may be
seen to form a continuous surface much as described in connection with
FIG. 13. Six dashed lines 710, 720, 730, 740, 750 and 760 are shown to
pass through the perimeter corner joints of six of the link pairs.
FIG. 21 shows the structure 700 in an unfolded position in which it
maintains its overall cylindrical shape. Six dashed lines 715, 725, 735,
745, 755 and 765 pass through the perimeter corner joints of six link
pairs. The angle formed between dashed lines 715 and 725 is identical to
the angle formed between dashed lines 710 and 720 shown in FIG. 20.
Similarly, the angles formed between dashed lines that correspond to any
two adjacent link pairs as shown in FIG. 21 are the identical to those
similarly formed angles corresponding to the same two link pairs as shown
in FIG. 20.
FIG. 22 shows yet another structure 800 comprised of interconnected loop
assemblies, in a folded position. This structure is comprised of 20 loop
assemblies, one of which is loop assembly 810 which is similar to loop
assembly 108 of FIG. 8.
FIG. 23 shows the structure 800 in a partially unfolded position. Loop
assembly 810 may be seen to be comprised of three link pairs 820, 830 and
840. Dashed line 825 passes through the perimeter corner joints of link
pair 820 while dashed lines 835 and 845 respectively pass through the
perimeter corner joints of link pairs 830 and 840.
FIG. 24 shows structure 800 in a fully unfolded position, with dashed line
826 passing through the perimeter corner joints of link pair 820 and
dashed lines 836 and 846 respectively passing through the perimeter corner
joints of link pairs 830 and 840. The angle formed between dashed lines
826 and 836 is identical to the angle formed by dashed lines 825 and 835
shown in FIG. 23. Likewise the angles formed between the other adjacent
dashed lines shown in FIG. 24 are identical to those similarly formed
angles shown in FIG. 23.
FIG. 25A shows a polygon link 901, which has a center pivot joint 957, two
interior pivot joints 954 and 956, and a perimeter pivot joint 955.
FIG. 25B shows a link pair 903 consisting of two polygon links 901 and 902
which share the center pivot joint 957. Also shown are the interior pivot
joints for polygon links 901 and 902, respectively 952, 956, 958 and 959.
Finally, the perimeter pivot joints for 902 and 903 are shown, being
respectively 954 and 955.
FIG. 25C shows a loop assembly 910 in a partially unfolded position. Loop
assembly 910 consists of four link-pairs 903, 913, 923 and 933, each
link-pair comprised of two polygon links. A dashed line 906 passes through
perimeter joints 954 and 955 which belong to link-pair 903. Similarly
dashed lines 916, 926 and 936 pass through perimeter joints 964, 965 and
974, 975 and 984, 985 respectively, forming a four-sided shape.
Loop-assembly 910 (FIG. 26) shows an alternative arrangement for the
connection of link-pairs to one another. Rather than all interior
corner-joint connections being made between adjacent link-pairs, some
interior corner joints are connected to link-pairs that are non-adjacent.
Specifically, link-pair 903 (FIG. 27) is connected to adjacent link-pair
913 by its interior corner joint 958, and likewise to adjacent link-pair
933 by interior corner joint 956. However, in addition link-pair 903 is
connected to non-adjacent link-pair 923 by two interior corner-joints 952
and 959.
FIG. 26 shows the structure 900 in a folded position. This structure is
comprised of 6 loop-assemblies, one of which is loop-assembly 910.
FIG. 27 shows the structure 900 in a fully unfolded position. Dashed line
907 passes through the perimeter corner-joints of link-pair 903. Likewise
dashed line 917, 927 and 937 respectively pass through the perimeter
corner-joints of link-pairs 913, 923 and 933. The angle formed between
dashed lines 907 and 917 is identical to the angle formed by dashed lines
906 and 916 shown in FIG. 25C. Similarly, the angles formed between dashed
lines that correspond to any two adjacent link-pairs as shown in FIG. 27
are identical to those similarly formed angles corresponding to the same
two link-pairs as shown in FIG. 25C.
FIG. 28 shows a link-pair 1001 that is comprised of a single piece of
material, cut to form two polygon links 1002 and 1003. Center-joint 1004
is comprised of a region of flexible material which is formed in a
continuous manner with links 1003 and 1004. Thus link 1003 can rotate
relative to link 1004 by flexing the center-joint 1004.
In FIG. 29 is shown the loop assembly 1005 consisting of eight link pairs
1011, 1021, 1031, 1041, 1051, 1061, 1071 and 1081. Similar to link-pair
1001 shown in FIG. 28, each link pair is formed of two polygon links that
are connected by a center-joint comprised of a flexible region of material | | |
