Note: Descriptions are shown in the official language in which they were submitted.
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BACKGRO~IND OF T~-IE INV TION
Since the invention of the geodesic dome by R. Buckminster Fuller
in 1951, the use o~ domed enclosures has become widespread and there is now
extensive technical and popular literature in the field. The primary engineer-
ing advantage of a domed building enclosure is that a sphere enclosed the most
volume with the least surface area of any geometrical shape and the sphere is
the strongest shape against internal and radial pressure. There are also other
advantages of these sphere like dome building enclosures; a dome is essentially
self supporting; all structural members are adjacent to or part of the faces or
the skin of the dome; and the triangular pattern of the faces or the skin grid
results in desirable load distribution and rigidity of the overall structure.
The term geodesic relates to a method of subdividing the surFace of a
sphere into a triangular grid pattern of intersecting great circle arcs and
utilizing struts which are chords of these arcs to form the skeletal Framework
for a generally spherical enclosure. The resulting structure is actually a poly-
hedron, a multi-sided, spherically symmetrical figure which more closely
approximates a sphere, as the frequency of subdivision increases. If the poly-
hedron is to have faces which are all identical, equilateral ~ triangles, ~he
largest figure which can be constructed is the icosahedron, a twenty sided shape
which has twelve identical corners, each of which is the apex of a pentagon, and
thirty edges, all of equal length. Mathematicians have also proved that each oF
the twenty equilateral icosahedron triangles can be further divided into six
identical triangles having 30-60-90 angles and that this is the largest number of
identical triangles into which a sphere can be subdided. ~ This type of geodesic
dome is known as a two frequency icosahedron and has 120 identical faces~
Greater subdivision results in higher frequency figures which more closely
approximate a sphere but also require a larger number of strut lengths and a
variety of different apex connectors.
Although a geodesic dome structure has many advantages over con-
ventional building structures, it also has disadvantages. A hemispherical dome
has a large amount of space near its perimeter, which has only limited use,
because the dome arches toward the ground resulting in a very low ceiling. Also
as the size of the enclosure increases, greater frequency of subdivision results
in an increasing number of non-interchangeable parts. Moreover, it is difficult
,
to divide or add onto the basic hemispherical shape of a geodesic dome structure.
Although there are an infinite number oF multi-sided building construc-
tions which can be assembled From identical panels, the basic structure of the
present invention, has sixty identical equilateral triangular -faces. This basic
polyhedral structure was invented because its shape results in less unusable
enclosed space than does a sphere and because it is also easily expanded or
subdivided, while retaining a great deal of rigidity and strength. The symmetry
properties of this basic polyhedron structure can be expanded upon to generate
a variety of domed structures which have these advantages and additional advan-
tages further described herein. These expanded polyhedral domed structures are
systematically designed and constructured, so the number of new components,
such as faces and struts~ which are employed are kept to a minimum.
SUMMA F THE INVENTION
These polyhedral domes building structures are designed to have
many advantages over conventional geodesic or other geometric domed building
structures. A family of unique polyhedrons defines the many overall domed
shapes of the many illustrated and possible embodiments of these polyhedral
structures. In all dwelling embodiments, there is an orientation of struts and
their connecting hubs to form a dome gridwork, which ultimately corresponds to
the edges and vertices of a plurality of panels which collectively form the faces
of the polyhedra structure.
These polyhedra dome structures, after assembly, are rigid and self
supporting, can be oriented to be built with substantially vertical side walls,
creating a great deal of usable space within these dome structures which will
accommodate, for example, conventional furniture. The polyhedron family of
embodiments of these dwelling structures is also selected to maximize usage of
identical, interchangeable parts and also to be easily constructed with economical
building materials.
The basic poiyhedron domed building structure from which all embodi-
ments are derived, has 90 equal lengthed edges meeting at 32 vertices to form 60
equilateral triangular Faces. This basic embodiment is derived during actual
construction by first creating two parallel hexagonal sides, each Formed with six
triangular faces, which are thereafter joined at their edges, by six identical
side sections, each consisting of a pair of pentagonal pyramids which have two
:
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triangular faces in common. A hexagonal prism has six planes of reflectional
symmetry which share a common axis passing through the center of the two
planar hexagons and a seventh plane of symmetry which is perpendicular to this
axis and bisects the prism. This polyhedron retains these planes of symmetry.
Further, this polyhedron with Faces of equilateral triangles can exist in more
than one form that retains this symmetry, and many forms that have other
symmetry patterns. The form shown in Figure 1 is chosen because it encloses
the most volume and has the best structural pattern.
A systematic method of varyin~ the strut lengths, further described
and illustrated herein, may be accomplished in a way that preserves the symmetry
of the basic shape, and hence preserves also the maximum number oF identical
parts. The vertex pattern arrangement thus formed can have curvatures such
as parabolic or geodesic.
A method of expansion of the basic polyhedron is accomplished by
folding the faces into three planar sections and then adding groups of additional
faces and refolding the faces to form enlarged polyhedra. The polyhedron can
be expanded indefinitely while adding parts which are nearly identical and
interchangeable. Conventional geodesic expansion, on the other hand, would
require an increasing number of strut lengths and Form many different triangular
faces as the frequency of subdivision increases.
Another advantage of the polyhedron shapes used in this building
structure is that subdivision is possible along any one of several series of
connected struts to leave a substantially planar face which may be a base on
which the truncated structure can be supported. The planar trucation could also
be used as a side wall or as a junction for connection to another similar enclo-
sure or to a conventional rectangular building. At least nine substantially
planar truncat7ons of the basic polyhedron are defined by series of struts which
are connected end to end and completely circumscribe the polyhedron.
Rigid concave-convex hubs having curvature which tends to round
off the corners of the polyhedron and smooth load paths, may be used at each
vertex of the building structure to connect struts in the required angular align-
ment .
DESCR!F'TION OF PF~EFERRED EMBODIMENTS
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Basic Embodiments
The building structures clescribed herein are derived from the
polyhedron lO shown in Figures l, 2, 3 and 4 which is also a preferred
embodiment of the invention. This polyhedron is constructed with sixty
identical, equilateral triangular faces 14, defined by ninety equal lengthed
struts 12. The polyhedron lO has two paralle, planar, hexagonal sides 30, 32
each formed by six triangular faces 14. Although one advantage of this struc-
ture is that many orientations of the polyhedron building framework are possible,
the alignment shown in Figure l will be referred to herein and the two planar
hexagons designated as the top 30 and base 32 hexagons.
This basic polyhedron lO has a hexagonal top 30 and base 32,
connected by six segments, each consisting of eight equilateral, triangular
faces forming a pair of pentagonal pyramids 18 which have two faces in common,
Each of twelve pentagonal pyramids l8 in the polyhedron lO, has one edge in
common with an edge of either, the top 30 or base 32 planar hexagon. The six
pentagonal pyramids l 8, which have edges in common with and depend from the
top planar hexagon 30 are paired with the corresponding six pentagonal pyramids
l8 which have edges in common with the bottom planar hexagon 32, such that each
pentagonal pyramid l8 shares two of its five triangular faces with its paired
pentagonal pyramid 18.
These shared triangular faces from the circumferential side band
46 of the dome that is perpendicular to the two hexagons forming the top 30 and
base 32 of the polyhedron 10, thereby forming a vertical sidewall.
Subdivision and Truncation of the Basic Embodiment
This polyhedron dome 10 may be readily subdivided along any one oF
numerous series of connected edges or struts 48 which circumscribe the poly-
hedron and lie substantially within a plane. Hexagonal apex 16 is the inter-
section of three such series of co-planar edges. Each edge 26 of the planar
hexagon 30, shown in Figure 2 is part of three series of substantially planar
co-terminous struts 48 that are almost coplanar, and many series that have
greater variation from a co-planar condition. Subdivisions of the polyhedron lO
along any of these series of struts 48, leaves a surface which can be a base or
sidewall or the junction with another structure. Additional planes of truncation
which require faces l4 to be subdivided are aiso possible.
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\/aria~ons of t_e Basic Embodiment
While the rquilateral properties of the dome 10 minimi~e the number
of different components, there are other aclvantages that result when other
triangle types are used. For example, the two planar hexagonal apexes 16
form the weakest parts of the structure as six struts intersect, all Iying within
a plane. These struts may be increased in length to result in triangulation of
the forces at a peaked hexagonal apex, greatly increasing the load bearing
strength of the now non-planar top 30 of !the polyhedron. ~nother consideration
is the fact that covering materials are commonly manufactured in Four by weight
foot rectangles. These rectangles can be cut diagonally to cover an iscosceies
triangle without waste, when the sides of the triangle are chosen suitably.
The important criteria in considering variations of basic dome 10 is
that many identical interchangeable parts result in symmetry of the polyhedronO
Conversely, if a polyhedron is constructed from a small number of different
parts by the operations of rotation about an axis and reflection in a plane then it
will be symmetric and have a large number of identical parts. The dotted lines
in Figure 2 show where six planes 71, 72 bisect the polyhedron 10, a theoretical
line passing through the two points 16, being their common intersection and axis.
A seventh plane 73 is perpendicular to this axis and also bisects the polyhedron
as shown in Figures 3 and 4. The whole polyhedron may be generated by
reflecting the four triangles 80, 81, 82, 83 in these seven planes. 3n Figure 31
these four triangles are shown as a flat planar net, the dotted lines indicating
how they are cut by the planes of reflection symmetry 71 and 73. In Figure 32
this planar net is shown in a modified form where triangles 80, 81, 82, 83 are
iscoscles. This modified net can be refolded in space so that the vertices 22
and 24 lie in the planes 72 and the vertices 20 line in the planes 71. The four
triangles thus oriented can be reflected through the seven planes of symmetry
to produce a variation which preserves the symmetry of the basic embodiment
10. A variation of this kind is called a ~symmetric embodiment" of the invention
and is depicted in Figure 33.
1~ triangles 80, 81, 82 and 83 are made congruent isosceles triangles,
when this net is foided and reflected as described above, the resulting polyhedron
is called a 1'symmetric isosceles embodiment~ of the invention. It is made from
identical isosceles triangles.
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Planes of symrnetry can be selectively r emoved to produce further
modiFications which contain less identical parts. For example, the horizontal
plane 73 can be removed to produce ~Ivertically symmetric embodiments". Since
the halves above and below the hori~ontal plane 73 have different triangles and
di-fferent nets, symmetry in this plane has been sacri-Ficed.
These examples show how a variety oF embodiments may be constructed
with different types of symmetry, the number of identical parts decreasing with
the successive removal of the planes oF symmetry. It is possible to construct
embodiments with no symmetry at ali, termed "asymmetric embodiments~l.
All of the variations considered above come under the following
definition.
A Comprehensive Description of Stair's Basic Po~Lh dron
"A polyhedron with sixty triangular faces, ninety edges, and thirty two
vertices. "
"The thirty-two vertices are of four kinds:
1. Twelve vertices where five edges meet, called 'pentagonal
vertices ~.
(Identified in Figure 1 as 20.~
2. Twenty vertices where six edges meet, called 'hexagonal
vertices~.
The twenty hexagonal vertices are of three kinds:
3. Two vertices having a common edge with six other hexagonal
vertices, called ~cap vertices'. (Identified in Figure 1 as 16.)
4. Twelve vertices having an edge in common with two pentagonal
vertices and an edge in common with four hexagonal verticles,
called ~meridian vertices~. (Identified in Figure 1 as 22.)
5. Six vertices having an edge in common with two hexagonal
vertices and four pentagonal vertices, called lequatorial vertices'.
(Identified in Figure 1 as 24.)"
Systematic Expa~nsion of the Basic Polyhedron
Figures 5 through 8 illustrate a method by which the sixty faced
polyhedron 10 may be systematically expanded to form larger building structures.
In Figure 5, x, y, and z axes are designated to be used as a frame of reference
for such expansions. The z axis passes through the two planar hexagonal
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vertices 16 and is perpendicular to the two planar hexagons 30, 32. The x axis
is oriented parallel to any strut which is a spoke of the planar top hexagon 30.
In Figure 6 the polyhedron 10 is shown folded into three sections. A circumfer-
ential band of triangles 46 surrounds the polyhedron 10, comprising the shared
faces of the six pairs of pentagonal pyramids 18, and forming a vertical side wall
of the polyhedron 10. This band of triangular faces 46 is shown unfolded into a
flat pattern in Figure 11. Two identical flat gridwork patterns 56, 58 are formed
by folding the remaining triangular faces into the planes of the two planar hexagons
30, 32, as illustrated in Figure 6. As shown in Figure 7, the flat gridwork
pattern 56 is hexagonal in shape and includes the original planar hexagon 30 plus
an additional arcade or band of eighteen triangular faces. This band of eiyhteen
triangular faces along with the twelve triangular faces in the circumferential band
46, when assembled in the polyhedron 10 of Figure 5, form the thirty faces of
six pentangonal pyramids 18.
With the polyhedron 10 illustrated in Figures 1 through 5 dismantled
into two flat gridwork patterns 56, 58 as shown in Figure 7, plus the band of
triangular faces ~6 shown unl~olded in Figure 11, it is possible to add identical
triangular faces to expand the structure while retaining symmetry of the dome
shape.
Expansion Alon~the X and y and Z Axes
Figure 8 illustrates the addition of two incremental bands 63 and 65
of eight triangles each expanding the flat pattern 56 in the x direction. When
both the top 56 and bottom 58 flat gridwork patterns are expanded in this way, a
lengthened circumferential band 96 as shown in Figure 1S is used which includes
four square panels 54 and twelve triangular panels 1~. Thus an expansion by two
increments in the x direction would result in a domed builcling structure having
ninety-two equilateral triangular faces and four square faces.
Expansion of the gridwork flat patterns 56, 58, is also possible in
the y direction as shown in Figure 9, where expansion of two increments 67, 68
is illustrated. Figure 10 combines the expansion in the x direction as shown in
Figure B and the y direction expansion shown in Figure 9. Whenever the domed
building structure is expanded in either the x or y direction, faces must also be
added to the circumferential band of panels as shown in Figures 14, 15 and 16.
Figure 14 illustrates the circumferential band 9~ of sixteen equilateral triangles
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14 which is required in a dome having an expansion of one increment in the y
di rection.
Expansions in the z direction are illustrated in Figures 12, 13, 17
and 18, where the circumFerential band 46 is enlarged by the addition of faces to
widen it.
Thus three distinct directions of expansion are possible as are
combinations of such expansions. The breakdown of the dome structure into
three flat components 46, 56, 58 provides a convenient starting point from which
systematic, symmetrical expansions may be made, enlarging the polyhedron
indefinitely.
Examp~d Domes
In Figure 20 a dome 14 having seventy-eight faces is shown which is
derived from the sixty sided polyhedron 10 by expansion by one increment in the
x direction. A lengthened circumferential side 94, illustrated in Figure 157 is
required which has two square faces 54 and twelve triangular faces 14. The
shape of this expanded polyhedron 15 is more complicated than the polyhedron 10
shown in Figure 1, however, a gridwork of equal lengthened struts 12 is still
formed and identical, equilateral triangular Faces 14 are defined with the
exception of the two square faces 54 as shown in Figures 15 and 20. (~onven-
tionally shaped windows or doorways can be framed within these square faces
54. This expanded dome building structure 15 can be constructed from two
identical flat patterns 56, 58 as illustrated in Figure 8 and the extended cir-
cumferential side 94 shown in Figure 15.
The polyhedron 10 expanded by two increments in the z direction is
illustrated in Figure 21. Figure 7 il lustrates the Flat gridwork pattern of the
hexagonal top 30 and bottom 32 used to construct this dome. A circumferential
band of panels 90, expanded by two increments in the z direction used in this
dome is shown folded flat in Fi~ure 13. Thus two gridwork panels as shown in
Figure 7 and one as shown in Figure 13 can be assembled to form the dome
illustrated in Figure 21, having eighty-four equilateral triangular Faces.
A third embodiment of an expanded dome structure is illustrated in -
Figures 22, 23, 24, and 25. This dome 19 is derived from the basic polyhedron
10 shown in Figure 5 by expanding one increment in the x direction, one increment
in the y direction and two increments in the z direction. The circumferential
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side panel a8 is shown Folded flat in i=igure 18.
Variations of ExE~nsion
As with the basic embodiment, variations of the strut lengths can
be used to satisfy specific application needs such as: alter load distribution
characteristics; accommodate various material sheet sizes; and/or insure
water runoff.
Modu I ar Extensions
.
The remarkable structural property of the present invention, which
is shared also by other triangulated structures such as the geoclesic dome, is
that it will hold its shape even with flexible vertex connectors if the struts are
rigid. A polyhedron which does not have this property is the rectangular prism,
which is the basic geometric shape on which the overwhelming majority of modern
buildings are structured. A major reason for the widespread appeal of such a
non-structural form is the ease with which a rectangular prism may be sub-
divided into units which are multiples oF the whoie, as well as being able to be
extended in the same way, combined with the ability of such a simple modular
system to adapt itself to mass production~ The embodiments of the invention
share this modular expansion property of the rectangular prism while retaining
the considerable structural advantages of the geodesic dome. Polyhedral units
consisting of standard identical triangles and their modular subdivisions can be
fastened to basic embodiments of the invention to create structures with greater
numbers of equal parts that also have more vertical wall space and in Fact are
even closer approximations to rectangular prisms, but still \,vithout their struc-
tural disadvantages.
In Figure 33, a variation of the basic embodiment is shown~ It is
constructed from two different isosceles triangles 92, 93. In Figure 3~, two
possible rhombic base pyramids 95, 97 are constructed so that their Faces are
formed from triangles 92, in the case of pyramid 97, and pyramid 95 is con-
structed from triangles 92 and 99 where 99 is obtained from 92 by bisection.
These pyramids can be attached to the structure in Figure 33 to produce a
structure as shown in Figure 35 which consists only of triangles 92 and their
subdivision 99. This structure has much more vertical wall space and usable
floor area than the embodiment shown in Figure 33. It may be covered with 30
uncut sheets of 4 x 8 foot material and 21 sheets cut on the diagonal. Similarly,
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modular adclitions can be added to any embocliment discussed herein.
DESCRIPTION OF DRAWING5
Figure 1 illustrates the gridwork of the basic polyhedron domed
building structure 10;
Figure 2 is a top view oF Figure 1;
Figures 3 and 4 are side views of Figure 1;
Figure 5 is a perspective view of the basic domed building structure,
with x, y and z axes designated as a frame of reference;
Figure 6 illustrates the domed building structure folded into two
planar flat patterns, leaving a circumferential band oF faces;
Figure 7 is a top view of the flat pattern shown in Figure 6;
In Figure 8, the flat pattern has been expanded by two increments
in the x direction;
Figure 9 illustrates a planar flat pattern expanded by two increments
in the y direction;
In Figure lQ, a flat pattern is shown expanded by two increments in
the x direction and two increments in the y direction;
In Figure 11, the circumferential band of faces of Figure 6 is shown
severed and Folded flat;
Figure 12 illustrates a circumFerential band 69 of panels, used in a
dome which has been expanded by one increment in the z direction;
Figure 13 illustrates a circumferential band of panels 90 used in a
dome which has been expanded by two increments in the z direction;
Figure 14 illustrates the extended circumferential band 91 of panels
for a dome which has been expanded by one increment in the y direction;
In Figure 15, a circumferential band of panels is shown For a dome
which has been expanded by one increment in the x direction; :
Figure 16 illustrates a circumferential band 96 after a second
incremental expansion in the x direction;
Figure 17 shows a widened circumferential band 98 of panels expanded
by two increments in the z direction and one increment in the x direction;
The circumFerential band of panels shown in Figure 18 is used in a
dome which has been expanded once in the x direction, once in the y direction
and twice in the z direction;
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Figure 19 shows how a rectangular sheet o-F material may be cut to
make an equilateral triangle. If the sheet is a 4 x 8 foot sheet, the resulting
waste 74 is approximately one percent;
Figure 20 illustrates a dime gridwork which has been expanded by
one increment in the x direction;
Figure 21 illustrates a dome gridwork which has been expanded
by two increments in the z direction;
Figure 22 illustrates a dome gridwork which has been expanded by
one increment in the x direction, one increment in the y direction and two
increments in the ~ direction;
Figure 23 illustrates a top view of the dome shown in Figure 22;
Figures 24 and 25 illustrate side views of the dome shown in
Figure 22;
Figure 26 illustrates a hub connecting five struts at a pentagonal
vertex in the dome gridwork;
Figure 27 is a partial cross sectional view of a double hub assembly;
Figure 28 is a view of the interior side of a hub;
Figure 29 is a cross sectional view of a single hub assembly;
Figure 30 is a perspective view of a hub 36 used at a hexagonal
vertex in the dome gridwork;
Figure 31 shows the four generating triangles as a planar net;
Figure 32 shows a modified form of the net shown in Figure 31;
Figure 33 shows a variation of the basic embodiment constructed
from two different isosceles triangles;
Figure 34 shows two rhombic pyramids constructed so that the four
base edges will coincide with four edges of Figure 33; and
Figure 35 shows one way the rh~3mbic pyramids of Figure 34 may be
attached to the polyhedron of Figure 33.
LQAD TR~ANSMlTT[NG Cl:~UP INGS
In Figure 26, a perspective view illustrates a one piece hub 38 used
to connect struts 12 of these building structures at a pentagonal vertex 20. A
second, inner hub 76 may be connected to the inside of each vertex of the building
: .
structure as illustrated in Figure 27 to provide a particularly strong structure
having improved sealing and insulating properties. A more economic and light-
weight dome gridwork utilizes single outer hubs 38 as shown in Figure 29. In
either the single or double hub conFigurations, nuts 60 and bolts 62 are used to
fasten the hubs and strut ends. The hub 38 illustrated in Figure 26 through 29
has strut receiving slots 44, 49 designed to receive tubular struts 64 or standard
framing lumber 66. Strut receiving slots 49 are provided to secure additional
struts which may be used to subdivide dorne Faces 14.
For each type of vertex required for a particular dome, a hub is
designed with the proper geometrical alignment of strut receiving slots and
radius of curvature to connect the gridwork of struts. Four different hub con-
figurations are used in the sixty -Faced dome 10. A larger number of different
hub shpes are required for expanded dome configurations. In any of its embodi-
ments, the cast or molded one piece hub 38 has a concave-convex shape to
integrate the hubs with the strut gridwork, resulting in smooth overall curvature
and even load distribution. The convex outer surface 40 of the hub also provides
a continuous interface between triangular panels or other covering skin 59 of the
dome, as illustrated in Figures 27 and 29. ~ sealant material 70 is shown at the
hub-skin interface and also covering bolts 62 which are exposed to the dome
exterior.
SUMMARY OF ADVANTAGES
The dome space enclosing structure which is described herein has
many advantages over conventional rectangular buildings and also has features
which are improvements over geodesic or purely spherical dome structures.
1. This building structure is not purely spherical and several
possible orientations of the structure are possible.
2. A high degree of symmetry allows a plurality of identical light-
weight components, resulting in a building structure that may be easily and -
quickly assembled or dismantled.
3. By symmetrically altering strut lengths, the vertex pattern can
have parabolic curvature, which contributes to a large ground area being
covered with a small amount of structural weight.
4. The triangulation of the skin gridwork of the structure results
in a frame rigidity and good load distributionO
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5. A greater amount of usable space is enclosed than under a
purely spherical dome, particularly around the periphery of the enclosed area.
6. The basic polyhedron building structure may be systematically
expanded as additional faces are added.
7. Numerous identical, interchangeable parts are used in this
building structure which may be assembled to form many embodiments with all
struts of equal lengths and with equilateral triangular face panels.
8. The basic polyhedron shape has numerous series of connected
edges through which substantially planar truncations may be made, and as the
polyhedron is expanded the number of difFerent orientations and truncations
increases resulting in architectural flexibility.
9. These domes are more compatible with conventional building
materials and methods than is a hemispherical dome.
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