Polyhedral annular structures and blanks for forming same

Claim 2

A polyhedral structure which is radially substantially symmetrical about a central axis, said structure comprising a number of planar triangles hinged together at their sides so as to form a continuous multiplanar toroidal web having two edges, which structure can be rotated, about its core, into at least six different stable configurations each of which is radially substantially symmetrical about said central axis, said structure comprising congruent plane triangles arranged in at least four rows of adjacent triangles, each of said rows being a ring of single triangles arranged in alternation so that each triangle has a hinged side in common with one of its two neighbors of its row and has an apical point in common with the other of its neighbors of its row, said rows interfitting so that each triangle of each row has a side in common with a triangle of the adjacent row, the common sides within a row being infold hinges arranged within planes which radiate from and include said axis and the sides which adjacent rows have in common being outfold hinges, said apical points being situated on the intersections of infold and outfold hinges, the apical angles of said triangles being at least about 108*, each of said rows having at least 18 triangles, the structure, when in each of said stable configurations, having two sets of axially spaced apical points situated approximately at said central axis.

Claim 3

A structure as in claim 1 in which two of said rows are adjacent to said edges of said structure and the others of said rows are arranged between said edge-adjacent rows, the triangles of said edge-adjacent rows each have a side which is not common to another triangle of said rows, and said non-common sides of adjacent triangles of said rows are connected by a folded web having an infold hinge in one of said radial planes, with outfold hinges at said non-common sides.

Claim 4

A structure as in claim 1 in which there are only four of said rows, said structure being rotatable to only six different stable positions, said apical angle being about 108* .

Claim 5

A structure as in claim 1 in which there are five of said rows, said structure being rotatable to seven different stable positions, said apical angle being at least about 114* .

Claim 6

A structure as in claim 1 in which said congruent triangles are isosceles triangles.

Claim 7

A structure as in claim 1 in which the three sides of each of said triangles are unequal.

Claim 8

A structure as in claim 1 in which said hinges are integral with said triangles, said hinges and triangles being of foldable sheet material.

Claim 9

A structure as in claim 2 in which each of said folded webs comprises a pair of equal triangles and said structure is a nonagonal polyhedron and is which there are only four of said rows, said apical angle is about 108*, said structure being rotatable to only six said different stable positions, and each of said rows has only 18 triangles.

Claim 10

A structure as in claim 3 in which each of said rows has only 18 triangles.

Claim 11

A polyhedral structure which is radially substantially symmetrical about a central axis, said structure comprising a number of planar triangles hinged together at their sides so as to form a continuous multiplanar toroidal web having two edges which structure can be rotated, about its core, into at least five different stable configurations each of which is radially substantially symmetrical about said central axis, said structure comprising at least three rows of adjacent congruent plane triangles, each of said rows being a ring of single triangles arranged in alternation so that each triangle has a hinged side in common with one of its two neighbors of its row and has an apical point in common with the other of its neighbors of its row, said rows interfitting so that each triangle of each row has a side in common with a triangle of the adjacent row, the common sides within a row being infold hinges, arranged within planes which radiate from and include said axis and the sides which adjacent rows have in common being outfold hinges, said apical points being situated on the intersections of infold and outfold hinges, the three sides of each of said triangles being unequal.

Claim 12

A polyhedral structure which is radially substantially symmetrical about a central axis, said structure comprising a number of planar triangles hinged together at their sides so as to form a continuous multiplanar toroidal web having two edges which structure can be rotated, about its core, into at least five different stable configurations each of which is radially substantially symmetrical about said central axis, said structure comprising at least three rows of adjacent congruent plane triangles, each of said rows being a ring of single triangles arranged in alternation so that each triangle has a hinged side in common with one of its two neighbors of its row and has an apical point in common with the other of its neighbors of its row, said rows interfitting so that each triangle of each row has a side in common with a triangle of the adjacent row, the common sides within a row being infold hinges, arranged within planes which radiate from and include said axis and the sides which adjacent rows have in common being outfold hinges, said apical points being situated on the intersections of infold and outfold hinges, the three sides of each of said triangles being unequal.

Claim 13

A structure as in claim 10 in which said hinges are integral with said triangles, said hinges and triangles being of foldable sheet material.

Claim 14

A polyhedral structure which is radially substantially symmetrical about a central axis, said structure comprising a number of planar triangles hinged together at their sides so as to form a continuous multiplanar toroidal web having two edges, which structure can be rotated, about its core, into at least five different stable configurations each of which is radially substantially symmetrical about said central axis, said structure comprising at least three rows of adjacent congruent plane triangles, each of said rows being a ring of single triangles arranged in alternation so that each triangle has a hinged side in common with one of its two neighbors of its row and has an apical point in common with the other of its neighbors of its row, said rows interfitting so that each triangle of each row has a side in common with a triangle of the adjacent row, the common sides within a rows being infold hinges arranged within planes which radiate from and include said axis and the sides which adjacent row have in common being outfold hinges, said apical points being situated on the intersections of infold and outfold hinges, said structure being of cardboard, said cardboard having, at least at said hinges thereof, an upper layer of thermoplastic polymer which has been heat set to bias said hinges in the folded direction.

Claim 15

A polyhedral structure which is radially substantially symmetrical about a central axis, said structure comprising a number of planar triangles hinged together at their sides so as to form a continuous multiplanar toroidal web having two edges, which structure can be rotated, about its core, into at least five different stable configurations each of which is radially substantially symmetrical about said central axis, said structure comprising at least three rows of adjacent congruent plane triangles, each of said rows being a ring of single triangles arranged in alternation so that each triangle has a hinged side in common with one of its two neighbors of its row and has an apical point in common with the other of its neighbors of its row, said rows interfitting so that each triangle of each row has a side in common with a triangle of the adjacent row, the common sides within a rows being infold hinges arranged within planes which radiate from and include said axis and the sides which adjacent row have in common being outfold hinges, said apical points being situated on the intersections of infold and outfold hinges, said structure being of cardboard, said cardboard having, at least at said hinges thereof, an upper layer of thermoplastic polymer which has been heat set to bias said hinges in the folded direction.

Claim 16

A structure as in claim 12 in which said thermoplastic layer is up to 15 mils thick.

Claim 17

A structure as in claim 12 in which said thermoplastic layer is a polyvinyl chloride film.

Claim 18

A flat blank of sheet material, said blank having means, including score lines at which said blank can be folded and edges adapted to be secured together to form a continuous web after said blank is folded at said score lines, for converting said blank to a polyhedral structure which is radially substantially symmetrical about a central axis, said structure comprising a number of planar triangles hinged together at their sides so as to form a continuous multiplanar toroidal web having two edges, which structure can be rotated, about its core, into at least six different stable configurations each of which is radially substantially symmetrical about said central axis, said structure comprising congruent plane triangles arranged in at least four rows of adjacent triangles, each of said rows being a ring of single triangles arranged in alternation so that each triangle has a hinged side in common with one of its two neighbors of its row and has an apical point in common with the other of its neighbors of its row, said rows interfitting so that each triangle of each row has a side in common with a triangle of the adjacent row, the common sides within a row being infold hinges arranged within planes which radiate from and include said axis and the sides which adjacent rows have in common being Outfold hinges, said apical points being situated on the intersections of infold and outfold hinges, the apical angles of said triangles being at least 108*, each of said rows having at least 18 triangles, the structure, when in each of said stable configurations, having two sets of axially spaced apical points situated approximately at said central axis, said score lines being situated so that, on folding, they form the boundaries of said triangles.

Claim 19

A flat blank of sheet material, said blank having means, including score lines at which said blank can be folded and edges adapted to be secured together to form a continuous web after said blank is folded at said score lines, for converting said blank to a polyhedral structure which is radially substantially symmetrical about a central axis, said structure comprising a number of planar triangles hinged together at their sides so as to form a continuous multiplanar toroidal web having two edges, which structure can be rotated, about its core, into at least six different stable configurations each of which is radially substantially symmetrical about said central axis, said structure comprising congruent plane triangles arranged in at least four rows of adjacent triangles, each of said rows being a ring of single triangles arranged in alternation so that each triangle has a hinged side in common with one of its two neighbors of its row and has an apical point in common with the other of its neighbors of its row, said rows interfitting so that each triangle of each row has a side in common with a triangle of the adjacent row, the common sides within a row being infold hinges arranged within planes which radiate from and include said axis and the sides which adjacent rows have in common being Outfold hinges, said apical points being situated on the intersections of infold and outfold hinges, the apical angles of said triangles being at least 108*, each of said rows having at least 18 triangles, the structure, when in each of said stable configurations, having two sets of axially spaced apical points situated approximately at said central axis, said score lines being situated so that, on folding, they form the boundaries of said triangles.

Claim 20

A flat blank of sheet material, said blank having means, including score lines at which said blank can be folded and edges adapted to be secured together to form a continuous web after said blank is folded at said score lines, for converting said blank to a polyhedral structure which is radially substantially symmetrical about a central axis, said structure comprising a number of planar triangles hinged together at their sides so as to form a continuous multiplanar toroidal web having two edges which structure can be rotated, about its core, into at least five different stable configurations each of which is radially substantially symmetrical about said central axis, said structure comprising at least three rows of adjacent congruent plane triangles, each of said rows being a ring of single triangles arranged in alternation so that each triangle has a hinged side in common with one of its two neighbors of its row and has an apical point in common with the other of its neighbors of its row, said rows interfitting so that each triangle of each row has a side in common with a triangle of the adjacent row, the common sides within a row being infold hinges, arranged within planes which radiate from and include said axis and the sides which adjacent rows have in common beeing outfold hinges, said apical points being situated on the intersections of infold and outfold hinges, the three sides of each of said triangles being unequal, said score lines being situated so that, on folding, they form the boundaries of said triangles.

Claim 21

A flat blank of sheet material, said blank having means, including score lines at which said blank can be folded and edges adapted to be secured together to form a continuous web after said blank is folded at said score lines, for converting said blank to a polyhedral structure which is radially substantially symmetrical about a central axis, said structure comprising a number of planar triangles hinged together at their sides so as to form a continuous multiplanar toroidal web having two edges which structure can be rotated, about its core, into at least five different stable configurations each of which is radially substantially symmetrical about said central axis, said structure comprising at least three rows of adjacent congruent plane triangles, each of said rows being a ring of single triangles arranged in alternation so that each triangle has a hinged side in common with one of its two neighbors of its row and has an apical point in common with the other of its neighbors of its row, said rows interfitting so that each triangle of each row has a side in common with a triangle of the adjacent row, the common sides within a row being infold hinges, arranged within planes which radiate from and include said axis and the sides which adjacent rows have in common beeing outfold hinges, said apical points being situated on the intersections of infold and outfold hinges, the three sides of each of said triangles being unequal, said score lines being situated so that, on folding, they form the boundaries of said triangles.

Claim 22

A blank as in claim 15, said polyhedral structure being a nonagonal polyhedron in which two of said rows are adjacent to said edges of said structure and the others of said rows are arranged between said edge-adjacent rows, the triangles of said edge-adjacent rows each have a side which is not common to another triangle of said rows, and said non-common sides of adjacent triangles of said rows are connected by a folded web having an infold hinge in one of said radial planes, with outfold hinges at said non-common sides, each of said folded webs comprises a pair of equal triangles, there are only four of said rows, said apical angle is about 108said structure being rotatable to only six said different stable positions, and each of said rows has only 18 triangles.