Minimum inventory maximum diversity building system

Claim 2

A triangulated convex dome structural module erected upon a planar base comprising: a. a first regular hexagon in a first plane having sides equal to unit length A; b. a series of six isosceles triangles, the sides of which meet in pairs at the vertices of said first regular hexagon, said sides of said first hexagon defining the base sides of said triangles, the apices of said triangles falling at points which define the vertices of a second, not necessarily planar, hexagon positioned above said first regular hexagon, the centers of said hexagons being vertically aligned; the sides of said triangles being equal in length to A√6/4, the sides of said second hexagon being equal to A√3/3; and c. six additional members connecting a central apex that is directly above said vertically aligned centers of said first and second hexagons to the said second hexagon.

Claim 3

A triangulated convex dome structural module as in claim 1, above, combined with six identical such hexagonal structural modules to define and enclose a full truncated octahedron.

Claim 4

A triangulated convex dome structural module as in claim 1, above, combined with six other hexagonal structural modules to define and enclose a full truncated octahedron.

Claim 5

A structural configuration comprising the dome of claim 1, above, wherein said dome is substantially self-supporting, said dome being formed from panels.

Claim 6

A structural configuration, comprising the dome of claim 1, above, wherein said dome is substantially self-supporting, said dome being formed by linear strut members.

Claim 7

A triangulated convex dome structural module erected upon a planar base comprising: a. a first regular hexagon in a first plane having sides equal to unit length A; b. a series of six isosceles triangles, the sides of which meet in pairs at the vertices of said first regular hexagon, said sides of said first hexagon defining the base sides of said triangles, the apices of said triangles falling at points which define the vertices of a second, not necessarily planar, hexagon positioned above said first regular hexagon, the centers of said hexagons being vertically aligned; the sides of said triangles being equal in length to A√2/2, the sides of said second hexagon being equal to A√3/3; and c. six additional members connecting a central apex that is directly above said vertically aligned centers of said first and second hexagons to said second hexagon.

Claim 8

A triangulated convex dome structural module erected upon a planar base comprising: a. a first regular hexagon in a first plane having sides equal to unit length A; b. a series of six isosceles triangles, the sides of which meet in pairs at the vertices of said first regular hexagon, said sides of said first hexagon defining the base sides of said triangles, the apices of said triangles falling at points which define the vertices of a second, not necessarily planar, hexagon positioned above said first regular hexagon, the centers of said hexagons being vertically aligned; the sides of said triangles being equal in length to A√2/2, the sides of said second hexagon being equal to A√3/3; and c. six additional members connecting a central apex that is directly above said vertically aligned centers of said first and second hexagons to said second hexagon.

Claim 9

A triangulated convex dome structural module as in claim 6, above, combined with six identical such hexagonal structural modules to define and enclose a full truncated octahedron.

Claim 10

A triangulated convex dome structural module as in claim 6, above, combined with six other hexagonal structural modules to define and enclose a full truncated octahedron.

Claim 11

A structural configuration comprising dome of claim 6, above, wherein said dome is substantially self-supporting, said dome being formed from panels.

Claim 12

A structural configuration comprising the dome of claim 6, above, wherein said dome is substantially self-supporting, said dome being formed by linear strut members.

Claim 13

A triangulated convex dome structural module erected upon a planar base comprising: a. a first regular hexagon in a first plane having sides equal to unit length A; b. a series of six isosceles triangles, the sides of which meet in pairs at the vertices of said first regular hexagon, said sides of said first hexagon defining the base sides of said triangles, the apices of said isosceles triangles falling at points which define the vertices of a second, not necessarily planar, hexagon, positioned above said first regular hexagon, the centers of said hexagons being vertically aligned; the sides of three of said six isosceles triangles being equal in length to A√6/4 and the sides of three other of said isosceles triangles being equal in length to A√2/2, the sides of said second hexagon being equal to A√6/4; and c. six additional members connecting a central apex that is directly above said vertically aligned centers of said first and second hexagons to said second hexagon.

Claim 14

A triangulated convex dome structural module erected upon a planar base comprising: a. a first regular hexagon in a first plane having sides equal to unit length A; b. a series of six isosceles triangles, the sides of which meet in pairs at the vertices of said first regular hexagon, said sides of said first hexagon defining the base sides of said triangles, the apices of said isosceles triangles falling at points which define the vertices of a second, not necessarily planar, hexagon, positioned above said first regular hexagon, the centers of said hexagons being vertically aligned; the sides of three of said six isosceles triangles being equal in length to A√6/4 and the sides of three other of said isosceles triangles being equal in length to A√2/2, the sides of said second hexagon being equal to A√6/4; and c. six additional members connecting a central apex that is directly above said vertically aligned centers of said first and second hexagons to said second hexagon.

Claim 15

A triangulated convex dome structural module as in claim 11, above, combined with six identical such hexagonal structural modules to define and enclose a full truncated octahedron.

Claim 16

A triangulated convex dome structural module as in claim 11, above, combined with six other hexagonal structural modules to define and enclose a full truncated octahedron.

Claim 17

A structural configuration comprising the dome of claim 11, above, wherein said dome is substantially self-supporting, said dome being formed from panels.

Claim 18

A structural configuration comprising the dome of claim 11, above, wherein said dome is substantially self-supporting, said dome being formed by linear strut members.

Claim 19

A triangulated convex dome structural module erected upon a planar base comprising: a. a first regular hexagon in a first plane having sides equal to unit length A; b. a series of six isosceles triangles, the sides of which meet in pairs at the vertices of said first regular hexagon, said sides of said first hexagon defining the base sides of said triangles, the apices of said isosceles triangles falling at points which define the vertices of a second, not necessarily planar, hexagon, positioned above said first regular hexagon, the centers of said hexagons being vertically aligned; the sides of two of said six isosceles triangles being equal in length to A√6/4, the sides of four other of said six isosceles triangles being equal in length to A√2/2, two of the sides of said second hexagon being equal to A√3/3, four of the sides of said second hexagon being equal to A√6/4; and c. six additional members connecting a central apex that is directly above said vertically aligned centers of said first and second hexagons to said second hexagon.

Claim 20

A triangulated convex dome structural module erected upon a planar base comprising: a. a first regular hexagon in a first plane having sides equal to unit length A; b. a series of six isosceles triangles, the sides of which meet in pairs at the vertices of said first regular hexagon, said sides of said first hexagon defining the base sides of said triangles, the apices of said isosceles triangles falling at points which define the vertices of a second, not necessarily planar, hexagon, positioned above said first regular hexagon, the centers of said hexagons being vertically aligned; the sides of two of said six isosceles triangles being equal in length to A√6/4, the sides of four other of said six isosceles triangles being equal in length to A√2/2, two of the sides of said second hexagon being equal to A√3/3, four of the sides of said second hexagon being equal to A√6/4; and c. six additional members connecting a central apex that is directly above said vertically aligned centers of said first and second hexagons to said second hexagon.

Claim 21

A triangulated convex dome structural module as in claim 16, above, combined with six identical such hexagonal structural modules to define and enclose a full truncated octahedron.

Claim 22

A triangulated convex dome structural module as in claim 16, above, combined with six other hexagonal structural modules to define and enclose a full truncated octahedron.

Claim 23

A structural configuration comprising the dome of claim 16, above, wherein said dome is substantially self-supporting, said dome being formed from panels.

Claim 24

A structural configuration comprising the dome of claim 16, above, wherein said dome is substantially self-supporting, said dome being formed by linear strut members.

Claim 25

A triangulated convex dome structural module erected upon a planar base comprising: a. a first regular hexagon in a first plane having sides equal to unit length A; b. a series of six isosceles triangles, the sides of which meet in pairs at the vertices of said first regular hexagon, said sides of said first hexagon defining the base sides of said triangles, the apices of said isosceles triangles falling at points which define the vertices of a second, not necessarily planar, hexagon, positioned above said first regular hexagon, the centers of said hexagons being vertically aligned; the sides of two of said six isosceles triangles being equal in length to A√6/4, the sides of four other of said six isosceles triangles being equal in length to A√2/2, two of the sides of said second hexagon being equal to A√3/3, four of the sides of said second hexagon being equal to A√6/4; and c. six additional members connecting a central apex that is directly above said vertically aligned centers of said first and second hexagons to said second hexagon.

Claim 26

A triangulated convex dome structural module erected upon a planar base comprising: a. a first regular hexagon in a first plane having sides equal to unit length A; b. a series of six isosceles triangles, the sides of which meet in pairs at the vertices of said first regular hexagon, said sides of said first hexagon defining the base sides of said triangles, the apices of said isosceles triangles falling at points which define the vertices of a second, not necessarily planar, hexagon, positioned above said first regular hexagon, the centers of said hexagons being vertically aligned; the sides of two of said six isosceles triangles being equal in length to A√6/4, the sides of four other of said six isosceles triangles being equal in length to A√2/2, two of the sides of said second hexagon being equal to A√3/3, four of the sides of said second hexagon being equal to A√6/4; and c. six additional members connecting a central apex that is directly above said vertically aligned centers of said first and second hexagons to said second hexagon.

Claim 27

A triangulated convex dome structural module as in claim 21, above, combined with six identical such hexagonal structural modules to define and enclose a full truncated octahedron.

Claim 28

A triangulated convex dome structural module as in claim 21, above, combined with six other hexagonal structural modules to define and enclose a full truncated octahedron.

Claim 29

A structural configuration comprising the dome of claim 21, above, wherein said dome is substantially self-supporting, said dome being formed from panels.

Claim 30

A structural configuration comprising the dome of claim 21, above, wherein said dome is substantially self-supporting, said dome being formed by linear strut members.

Claim 31

A triangulated convex-concave dome structural module erected upon a planar base comprising: a. a first symmetrical irregular nonagon in a first plane, having sides equal to arbitrary unit length A with three alternating collinear pairs of sides and three additional sides between said collinear paired sides forming nine interior angles, respectively: 120°, 180°, 120°, 120°, 180°, 120°, 120°, 180°, and 120°; b. a series of 9 isosceles triangles, the sides of which meet in pairs at the vertices of said first nonagon in said first plane, the sides of said first nonagon defining the base sides of the triangles, the sides of said triangles being equal in length to A√6/4; c. three respective sets of three vertices each being defined by the apices of said isosceles triangles which are common to three corresponding vertices of three respective regular, not necessarily planar hexagons said three hexagons being arranged around a common point vertically positioned above the center of said irregular nonagon in said first plane, sharing three common edges emanating from said common point, said common edges terminating at three respective vertices shared by respective pairs of said hexagons, the sides of said hexagons being equal in length to A√3/3; d. three members connecting the respective vertices common to each of said three alternating collinear pairs of sides of said first nonagon, to said three respective vertices shared by respective pairs of said hexagons arranged around a common point, the length of said three members being A √6/4; and e. eighteen additional strut members in three respective groups of six each, respectively erected upon said set of three hexagons and arranged around a common point, such that three hexagonal pyramids are formed, the apices of which fall vertically above the respective centers of said hexagons, arranged around a common point.

Claim 32

A triangulated convex-concave dome structural module erected upon a planar base comprising: a. a first symmetrical irregular nonagon in a first plane, having sides equal to arbitrary unit length A with three alternating collinear pairs of sides and three additional sides between said collinear paired sides forming nine interior angles, respectively: 120°, 180°, 120°, 120°, 180°, 120°, 120°, 180°, and 120°; b. a series of 9 isosceles triangles, the sides of which meet in pairs at the vertices of said first nonagon in said first plane, the sides of said first nonagon defining the base sides of the triangles, the sides of said triangles being equal in length to A√6/4; c. three respective sets of three vertices each being defined by the apices of said isosceles triangles which are common to three corresponding vertices of three respective regular, not necessarily planar hexagons said three hexagons being arranged around a common point vertically positioned above the center of said irregular nonagon in said first plane, sharing three common edges emanating from said common point, said common edges terminating at three respective vertices shared by respective pairs of said hexagons, the sides of said hexagons being equal in length to A√3/3; d. three members connecting the respective vertices common to each of said three alternating collinear pairs of sides of said first nonagon, to said three respective vertices shared by respective pairs of said hexagons arranged around a common point, the length of said three members being A √6/4; and e. eighteen additional strut members in three respective groups of six each, respectively erected upon said set of three hexagons and arranged around a common point, such that three hexagonal pyramids are formed, the apices of which fall vertically above the respective centers of said hexagons, arranged around a common point.

Claim 33

A triangulated convex-concave dome structural module as in claim 26. above, in combination with three hexagonal structural modules and three pentagonal structural modules to form and enclose an equatorial space unit of a truncated octahedron positioned atop a polar space unit of a truncated octahedron.

Claim 34

A structural configuration comprising the dome of claim 26, above, wherein said dome is substantially self-supporting, said dome being formed from panels.

Claim 35

A structural configuration comprising the dome of claim 26, above, wherein said dome is substantially self-supporting, said dome being formed by linear strut members.

Claim 36

A triangulated convex-concave dome structural module erected upon a planar base comprising: a. a first symmetrical irregular nonagon in a first plane, having sides equal to arbitrary unit length A with three alternating collinear pairs of sides and three additional sides between said collinear paired sides forming nine interior angles, respectively: 120°, 180°, 120°, 120°, 180°, 120°, 120°, 180°, and 120°; b. a series of 9 isosceles triangles, the sides of which meet in pairs at the vertices of said first nonagon in said first plane, the sides of said first nonagon defining the base sides of said triangles, the sides of said triangles being equal in length A √2/2; c. three respective sets of three vertices each being defined by the apices of said isosceles triangles which are common to three corresponding vertices of three respective regular, not necessarily planar hexagons, said three hexagons being arranged around a common point vertically positioned above the center of said irregular nonagon in said first plane, sharing three common edges emanating from said common point, said common edges terminating at three respective vertices shared by respective pairs of said hexagons the sides of said hexagons being equal in length to A √3/3; d. three members connecting the respective vertices common to each of said three alternating collinear pairs of sides of said first nonagon, to said three respective vertices shares by respective pairs of said hexagons arranged around a common point, the length of said three members being A √2/2; and e. eighteen additional strut members in three respective groups of six each, respectively erected upon said set of three hexagons and arranged around a common point, such that three hexagonal pyramids are formed, the apices of which fall vertically above the respective centers of said hexagons arranged around a common point.

Claim 37

A triangulated convex-concave dome structural module erected upon a planar base comprising: a. a first symmetrical irregular nonagon in a first plane, having sides equal to arbitrary unit length A with three alternating collinear pairs of sides and three additional sides between said collinear paired sides forming nine interior angles, respectively: 120°, 180°, 120°, 120°, 180°, 120°, 120°, 180°, and 120°; b. a series of 9 isosceles triangles, the sides of which meet in pairs at the vertices of said first nonagon in said first plane, the sides of said first nonagon defining the base sides of said triangles, the sides of said triangles being equal in length A √2/2; c. three respective sets of three vertices each being defined by the apices of said isosceles triangles which are common to three corresponding vertices of three respective regular, not necessarily planar hexagons, said three hexagons being arranged around a common point vertically positioned above the center of said irregular nonagon in said first plane, sharing three common edges emanating from said common point, said common edges terminating at three respective vertices shared by respective pairs of said hexagons the sides of said hexagons being equal in length to A √3/3; d. three members connecting the respective vertices common to each of said three alternating collinear pairs of sides of said first nonagon, to said three respective vertices shares by respective pairs of said hexagons arranged around a common point, the length of said three members being A √2/2; and e. eighteen additional strut members in three respective groups of six each, respectively erected upon said set of three hexagons and arranged around a common point, such that three hexagonal pyramids are formed, the apices of which fall vertically above the respective centers of said hexagons arranged around a common point.

Claim 38

A triangulated convex-concave dome structural module as in claim 26, above, in combination with three hexagonal frame modules and three pentagonal structural modules to form and enclose an equatorial space unit of a truncated octahedron positioned atop a polar space unit of a truncated octahedron.

Claim 39

A structural configuration comprising the dome of claim 30, above, wherein said dome is substantially self-supporting, said dome being formed from panels.

Claim 40

A structural configuration comprising the dome of claim 30, above, wherein said dome is substantially self-supporting, said dome being formed by linear strut members.

Claim 41

A triangulated convex-concave dome structural module erected upon a planar base comprising: a. a first symmetrical irregular nonagon in a first plane, having sides equal to arbitrary unit length A with three alternating collinear pairs of sides and three additional sides between said collinear paired sides forming nine interior angles, respectively: 120°, 180°, 120°, 120°, 180°, 120°, 120°, 180°, and 120°; b. a series of 9 isosceles triangles, the sides of which meet in pairs at the vertices of said first nonagon in said first plane, the sides of said first nonagon defining the base sides of said triangles, the sides of six of said 9 isosceles triangles being equal in length A √2/2, the sides of three other of said 9 isosceles triangles being equal in length to A √6/4; c. three respective sets of three vertices each being defined by the apices of said isosceles triangles which are common to three corresponding vertices of three respective regular, not necessarily planar hexagons said three hexagons being arranged around a common point vertically positioned above the center of said irregular nonagon in said first plane, sharing three common edges emanating from said common point, said common edges terminating at three respective vertices shared by respective pairs of said hexagons, the sides of said hexagons being equal in length to A √6/4; d. three members connecting the respective vertices common to each of said three alternating collinear pairs of sides of said first nonagon, to said three respective vertices shares by respective pairs of said hexagons arranged around a common point, the length of said three members being A √2/2; and e. eighteen additional strut members in three respective groups of six each, respectively erected upon said set of three hexagons and arranged around a common point, such that three hexagonal pyramids are formed, the apices of which fall vertically above the respective centers of said hexagons arranged around a common point.

Claim 42

A triangulated convex-concave dome structural module erected upon a planar base comprising: a. a first symmetrical irregular nonagon in a first plane, having sides equal to arbitrary unit length A with three alternating collinear pairs of sides and three additional sides between said collinear paired sides forming nine interior angles, respectively: 120°, 180°, 120°, 120°, 180°, 120°, 120°, 180°, and 120°; b. a series of 9 isosceles triangles, the sides of which meet in pairs at the vertices of said first nonagon in said first plane, the sides of said first nonagon defining the base sides of said triangles, the sides of six of said 9 isosceles triangles being equal in length A √2/2, the sides of three other of said 9 isosceles triangles being equal in length to A √6/4; c. three respective sets of three vertices each being defined by the apices of said isosceles triangles which are common to three corresponding vertices of three respective regular, not necessarily planar hexagons said three hexagons being arranged around a common point vertically positioned above the center of said irregular nonagon in said first plane, sharing three common edges emanating from said common point, said common edges terminating at three respective vertices shared by respective pairs of said hexagons, the sides of said hexagons being equal in length to A √6/4; d. three members connecting the respective vertices common to each of said three alternating collinear pairs of sides of said first nonagon, to said three respective vertices shares by respective pairs of said hexagons arranged around a common point, the length of said three members being A √2/2; and e. eighteen additional strut members in three respective groups of six each, respectively erected upon said set of three hexagons and arranged around a common point, such that three hexagonal pyramids are formed, the apices of which fall vertically above the respective centers of said hexagons arranged around a common point.

Claim 43

A triangulated convex-concave dome structural module as in claim 34, above, in combination with three hexagonal frame modules and three pentagonal structural modules to form and enclose an equatorial space unit of a truncated octahedron positioned atop a polar space unit of a truncated octahedron.

Claim 44

A structural configuration comprising the dome of claim 34, above, wherein said dome is substantially self-supporting, said dome being formed from panels.

Claim 45

A structural configuration comprising the dome of claim 35, above, wherein said dome is substantially self-supporting, said dome being formed by linear strut members.

Claim 46

A triangulated convex-concave dome structural module erected upon a planar base comprising: a. a first symmetrical irregular nonagon in a first plane, having sides equal to arbitrary unit length A with three alternating collinear pairs of sides and three additional sides between said collinear paired sides forming nine interior angles, respectively: 120°, 180°, 120°, 120°, 180°, 120°, 120°, 180°, and 120°; b. a series of 9 isosceles triangles, the sides of which meet in pairs at the vertices of said first nonagon in said first plane, the sides of said first nonagon defining the base sides of said triangles, the sides of six of said 9 isosceles triangles being equal in length A √6/4, the sides of three other of said 9 isosceles triangles being equal in length to A √2/2; c. three respective sets of three vertices each being defined by the apices of said isosceles triangles which are common to three corresponding vertices of three respective regular, not necessarily planar hexagons said three hexagons being arranged around a common point vertically positioned above the center of said irregular nonagon in said first plane, sharing three common edges emanating from said common point, said common edges terminating at three respective vertices shared by respective pairs of said hexagons, the sides of said hexagons being equal in length to A √6/4; d. three members connecting the respective vertices common to each of said three alternating collinear pairs of sides of said first nonagon, to said three respective vertices shared by respective pairs of said hexagons arranged around a common point, the length of said three members being A √2/2 and e. eighteen additional strut members in three respective groups of six each, respectively erected upon said set of three hexagons and arranged around a common point, such that three hexagonal pyramids are formed, the apices of which fall vertically above the respective centers of said hexagons arranged around a common point.

Claim 47

A triangulated convex-concave dome structural module erected upon a planar base comprising: a. a first symmetrical irregular nonagon in a first plane, having sides equal to arbitrary unit length A with three alternating collinear pairs of sides and three additional sides between said collinear paired sides forming nine interior angles, respectively: 120°, 180°, 120°, 120°, 180°, 120°, 120°, 180°, and 120°; b. a series of 9 isosceles triangles, the sides of which meet in pairs at the vertices of said first nonagon in said first plane, the sides of said first nonagon defining the base sides of said triangles, the sides of six of said 9 isosceles triangles being equal in length A √6/4, the sides of three other of said 9 isosceles triangles being equal in length to A √2/2; c. three respective sets of three vertices each being defined by the apices of said isosceles triangles which are common to three corresponding vertices of three respective regular, not necessarily planar hexagons said three hexagons being arranged around a common point vertically positioned above the center of said irregular nonagon in said first plane, sharing three common edges emanating from said common point, said common edges terminating at three respective vertices shared by respective pairs of said hexagons, the sides of said hexagons being equal in length to A √6/4; d. three members connecting the respective vertices common to each of said three alternating collinear pairs of sides of said first nonagon, to said three respective vertices shared by respective pairs of said hexagons arranged around a common point, the length of said three members being A √2/2 and e. eighteen additional strut members in three respective groups of six each, respectively erected upon said set of three hexagons and arranged around a common point, such that three hexagonal pyramids are formed, the apices of which fall vertically above the respective centers of said hexagons arranged around a common point.

Claim 48

A triangulated convex-concave dome structural module as in claim 38, above, in combination with three hexagonal frame modules and three pentagonal structural modules to form and enclose an equatorial space unit of a truncated octahedron positioned atop a polar space unit of a truncated octahedron.

Claim 49

A structural configuration comprising the dome of claim 38, above, wherein said dome is substantially self-supporting, said dome being formed from panels.

Claim 50

A structural configuration comprising the dome of claim 38, above, wherein said dome is substantially self-supporting, said dome being formed by linear strut members.

Claim 51

A triangulated convex concave dome structural module erected upon a planar base comprising: a. a first irregular, symmetrical octagon in a first plane having sides equal to unit length A with two parallel and opposed sets of collinear paired sides, and with two opposed sets of obtuse paired sides, forming eight interior angles respectively: 120°, 120°, 180°, 120°, 120°, 120°, 180°, and 120°. b. a series of eight isosceles triangles, the sides of which meet in pairs at the vertices of said first octagon in said first plane, such that the sides of said first octagon form the base sides of said isosceles triangles, the sides of said isosceles triangles being equal in length to A √6/4; c. two respective sets of four vertices each which are common to four corresponding vertices of two respective, contiguous regular, not necessarily planar, hexagons positioned above said first plane, being defined by the apices of said isosceles triangles, said regular hexagons sharing a common edge and a pair of vertices at the ends of said common edge, the sides of said regular hexagons being equal in length to A √3/3; d. two strut members connecting the respective vertices common to each of said two parallel and opposed sets of collinear paired sides of said first octagon in said first plane to the respective vertices at the ends of said common edge shared by said contiguous pair or regular hexagons positioned above said first plane, the lengths of said two strut members being A √6/4; and e. twelve additional strut members in two groups of six each, respectively erected upon said contiguous pair of regular hexagons positioned above first plane, forming two hexagonal pyramids, the apices of which fall vertically above the respective centers of said regular hexagons.

Claim 52

A triangulated convex concave dome structural module erected upon a planar base comprising: a. a first irregular, symmetrical octagon in a first plane having sides equal to unit length A with two parallel and opposed sets of collinear paired sides, and with two opposed sets of obtuse paired sides, forming eight interior angles respectively: 120°, 120°, 180°, 120°, 120°, 120°, 180°, and 120°. b. a series of eight isosceles triangles, the sides of which meet in pairs at the vertices of said first octagon in said first plane, such that the sides of said first octagon form the base sides of said isosceles triangles, the sides of said isosceles triangles being equal in length to A √6/4; c. two respective sets of four vertices each which are common to four corresponding vertices of two respective, contiguous regular, not necessarily planar, hexagons positioned above said first plane, being defined by the apices of said isosceles triangles, said regular hexagons sharing a common edge and a pair of vertices at the ends of said common edge, the sides of said regular hexagons being equal in length to A √3/3; d. two strut members connecting the respective vertices common to each of said two parallel and opposed sets of collinear paired sides of said first octagon in said first plane to the respective vertices at the ends of said common edge shared by said contiguous pair or regular hexagons positioned above said first plane, the lengths of said two strut members being A √6/4; and e. twelve additional strut members in two groups of six each, respectively erected upon said contiguous pair of regular hexagons positioned above first plane, forming two hexagonal pyramids, the apices of which fall vertically above the respective centers of said regular hexagons.

Claim 53

A structural configuration comprising the dome of claim 42, above, wherein said dome is substantially self-supporting, said dome being formed from panels.

Claim 54

A structural configuration comprising the dome of claim 42, above, wherein said dome is substantially self-supporting, said dome being formed by linear strut members.

Claim 55

A triangulated convex concave dome structural module erected upon a planar base comprising: a. a first irregular, symmetrical octagon in a first plane having sides equal to unit length A with two parallel and opposed sets of collinear paired sides, and with two opposed sets of obtuse paired sides, forming eight interior angles respectively: 120°, 120°, 180°, 120°, 120°, 120°, 180°, and 120°. b. a series of eight isosceles triangles, the sides of which meet in pairs at the vertices of said first octagon in said first plane, such that the sides of said first octagon form the base sides of said isosceles triangles, the sides of said isosceles triangles being equal in length to A √2/2; c. two respective sets of four vertices each which are common to four corresponding vertices of two respective, contiguous regular, not necessarily planar, hexagons positioned above said first plane, being defined by the apices of said isosceles triangles, said regular hexagons sharing a common edge and a pair of vertices at the ends of said common edge, the sides of said regular hexagons being equal in length to A √3/3; d. two strut members connecting the respective vertices common to each of said two parallel and opposed sets of collinear paired sides of said first octagon in said first plane to the respective vertices at the ends of said common edge shared by said contiguous pair or regular hexagons positioned above said first plane, the lengths of said two strut members being A √2/2; and e. twelve additional strut members in two groups of six each, respectively erected upon said contiguous pair of regular hexagons positioned above first plane, forming two hexagonal pyramids, the apices of which fall vertically above the respective centers of said regular hexagons.

Claim 56

A triangulated convex concave dome structural module erected upon a planar base comprising: a. a first irregular, symmetrical octagon in a first plane having sides equal to unit length A with two parallel and opposed sets of collinear paired sides, and with two opposed sets of obtuse paired sides, forming eight interior angles respectively: 120°, 120°, 180°, 120°, 120°, 120°, 180°, and 120°. b. a series of eight isosceles triangles, the sides of which meet in pairs at the vertices of said first octagon in said first plane, such that the sides of said first octagon form the base sides of said isosceles triangles, the sides of said isosceles triangles being equal in length to A √2/2; c. two respective sets of four vertices each which are common to four corresponding vertices of two respective, contiguous regular, not necessarily planar, hexagons positioned above said first plane, being defined by the apices of said isosceles triangles, said regular hexagons sharing a common edge and a pair of vertices at the ends of said common edge, the sides of said regular hexagons being equal in length to A √3/3; d. two strut members connecting the respective vertices common to each of said two parallel and opposed sets of collinear paired sides of said first octagon in said first plane to the respective vertices at the ends of said common edge shared by said contiguous pair or regular hexagons positioned above said first plane, the lengths of said two strut members being A √2/2; and e. twelve additional strut members in two groups of six each, respectively erected upon said contiguous pair of regular hexagons positioned above first plane, forming two hexagonal pyramids, the apices of which fall vertically above the respective centers of said regular hexagons.

Claim 57

A structural configuration comprising the dome of claim 45, above, wherein said dome is substantially self-supporting, said dome being formed from panels.

Claim 58

A structural configuration comprising the dome of claim 45, above, wherein said dome is substantially self-supporting, said dome being formed by linear strut members.

Claim 59

A triangulated convex concave dome structural module erected upon a planar base comprising: a. a first irregular, symmetrical octagon in a first plane having sides equal to unit length A with two parallel and opposed sets of collinear paired sides, and with two opposed sets of obtuse paired sides, forming eight interior angles respectively: 120°, 120°, 180°, 120°, 120°, 120°, 180°, and 120°. b. a series of eight isosceles triangles, the sides of which meet in pairs at the vertices of said first octagon in said first plane, such that the sides of said first octagon form the base sides of said isosceles triangles, the sides of four of said eight isosceles triangles being equal in length to A √6/4 the sides of four other of said eight isoceles triangles being equal in length to A √2/2; c. two respective sets of four vertices each of which are common to four corresponding vertices of two respective, contiguous regular, not necessarily planar, hexagons positioned above said first plane, being defined by the apices of said isosceles triangles, said regular hexagons sharing a common edge and a pair of vertices at the ends of said common edge, the sides of said regular hexagons being equal in length to A √6/4; d. two strut members connecting the respective vertices common to each of said two parallel and opposed sets of collinear paired sides of said first octagon in said first plane to the respective vertices at the ends of said common edge shared by said contiguous pair of regular hexagons positioned above said first plane, the lengths of said two strut members being equal in length respectively to A √2/2; and e. twelve additional strut members in two groups of six each, respectively erected upon said contiguous pair of regular hexagons positioned above first plane, forming two hexagonal pyramids, the apices of which fall vertically above the respective centers of said regular hexagons.

Claim 60

A triangulated convex concave dome structural module erected upon a planar base comprising: a. a first irregular, symmetrical octagon in a first plane having sides equal to unit length A with two parallel and opposed sets of collinear paired sides, and with two opposed sets of obtuse paired sides, forming eight interior angles respectively: 120°, 120°, 180°, 120°, 120°, 120°, 180°, and 120°. b. a series of eight isosceles triangles, the sides of which meet in pairs at the vertices of said first octagon in said first plane, such that the sides of said first octagon form the base sides of said isosceles triangles, the sides of four of said eight isosceles triangles being equal in length to A √6/4 the sides of four other of said eight isoceles triangles being equal in length to A √2/2; c. two respective sets of four vertices each of which are common to four corresponding vertices of two respective, contiguous regular, not necessarily planar, hexagons positioned above said first plane, being defined by the apices of said isosceles triangles, said regular hexagons sharing a common edge and a pair of vertices at the ends of said common edge, the sides of said regular hexagons being equal in length to A √6/4; d. two strut members connecting the respective vertices common to each of said two parallel and opposed sets of collinear paired sides of said first octagon in said first plane to the respective vertices at the ends of said common edge shared by said contiguous pair of regular hexagons positioned above said first plane, the lengths of said two strut members being equal in length respectively to A √2/2; and e. twelve additional strut members in two groups of six each, respectively erected upon said contiguous pair of regular hexagons positioned above first plane, forming two hexagonal pyramids, the apices of which fall vertically above the respective centers of said regular hexagons.

Claim 61

A structural configuration comprising the dome of claim 48, above, wherein said dome is substantially self-supporting, said dome being formed from panels.

Claim 62

A structural configuration comprising the dome of claim 48, above, wherein said dome is substantially self-supporting, said dome being formed by linear strut members.

Claim 63

A triangulated convex-concave dome structural module erected upon a planar base comprising: a. a first irregular, symmetrical pentagon in a first plane, having sides equal to unit length A, with one collinear pair of sides and three additional sides forming five interior angles, respectively: 120°, 60°, 180°, 60°, and 120°; b. a series of five isosceles triangles, the sides of which meet in pairs at the vertices of said pentagon in said first plane, the sides of said pentagon defining the base sides of said isosceles triangles, the sides of said isosceles triangles being equal in length to A √6/4 two pairs of said isosceles triangles sharing respective vertices which occur at said 60° interior angles of said pentagon, a common edge and a common apex for each of said pair of isosceles triangles forming a first and second apex, a third apex being defined by a fifth said isosceles triangle; c. a member joining said third apex to the vertex common to said collinear pair of sides of said pentagon, said member being equal in length to A √6/4; and d. two additional members joining respectively said first apex with said third apex and said second apex with said third apex, said third apex being common to both additional members, and said additional members being equal in length to A √3/3.

Claim 64

A triangulated convex-concave dome structural module erected upon a planar base comprising: a. a first irregular, symmetrical pentagon in a first plane, having sides equal to unit length A, with one collinear pair of sides and three additional sides forming five interior angles, respectively: 120°, 60°, 180°, 60°, and 120°; b. a series of five isosceles triangles, the sides of which meet in pairs at the vertices of said pentagon in said first plane, the sides of said pentagon defining the base sides of said isosceles triangles, the sides of said isosceles triangles being equal in length to A √6/4 two pairs of said isosceles triangles sharing respective vertices which occur at said 60° interior angles of said pentagon, a common edge and a common apex for each of said pair of isosceles triangles forming a first and second apex, a third apex being defined by a fifth said isosceles triangle; c. a member joining said third apex to the vertex common to said collinear pair of sides of said pentagon, said member being equal in length to A √6/4; and d. two additional members joining respectively said first apex with said third apex and said second apex with said third apex, said third apex being common to both additional members, and said additional members being equal in length to A √3/3.

Claim 65

A structural configuration comprising the dome of claim 51, above, wherein said dome is substantially self-supporting, said dome being formed from panels.

Claim 66

A structural configuration comprising the dome of claim 51, above, wherein said dome is substantially self-supporting, said dome being formed by linear strut members.

Claim 67

A triangulated dome structural module as in claim 51, above, joined with two similar structural modules, a hexagonal dome module and a base plane to define a polar space unit of a truncated octahedron.

Claim 68

A triangulated dome structural module as in claim 51, above, joined with two identical modules, a nonagonal dome module and a base plane to define the equatorial space unit of the truncated octahedron.

Claim 69

A triangulated dome structural module as in claim 51, above, combined with two additional such modules, a base plane and four hexagonal dome modules to define a polar space unit of a truncated octahedron positioned atop an equatorial space unit from a truncated octahedron.

Claim 70

A triangulated dome structural module as in claim 51, above, combined with two substantially identical structural modules to define secondary wall regions, an irregular, symmetrical honagonal dome structural module defining an upper roof region, an irregular symmetrical octagon dome module defining primary wall regions and a base plane, whereby a triangulated dome structure is formed.

Claim 71

A triangulated convex-concave dome structural module erected upon a planar base comprising: a. a first irregular, symmetrical pentagon in a first plane, having sides equal to unit length A, with one collinear pair of sides and three additional sides forming five interior angles, respectively: 120°, 60°, 180°, 60° and 120°; b. a series of five isosceles triangles, the sides of which meet in pairs at the vertices of said pentagon in said first plane, the sides of said pentagon defining the base sides of said isosceles triangles, the sides of said isosceles triangles being equal in length to A √2/2, two pairs of said isosceles triangles sharing respective vertices which occur at said 60° interior angles of said pentagon, a common edge and a common apex for each of said pair of isosceles triangles forming a first and second apex, a third apex being defined by a fifth said isosceles triangle; c. a member joining said third apex to the vertex common to said collinear pair of sides of said pentagon, said edge module being equal in length to A √2/2; and d. two additional members joining respectively said first apex with said third apex and said second apex with said third apex, said third apex being common to both additional members and said additional members being equal in length to A √3/3.

Claim 72

A triangulated convex-concave dome structural module erected upon a planar base comprising: a. a first irregular, symmetrical pentagon in a first plane, having sides equal to unit length A, with one collinear pair of sides and three additional sides forming five interior angles, respectively: 120°, 60°, 180°, 60° and 120°; b. a series of five isosceles triangles, the sides of which meet in pairs at the vertices of said pentagon in said first plane, the sides of said pentagon defining the base sides of said isosceles triangles, the sides of said isosceles triangles being equal in length to A √2/2, two pairs of said isosceles triangles sharing respective vertices which occur at said 60° interior angles of said pentagon, a common edge and a common apex for each of said pair of isosceles triangles forming a first and second apex, a third apex being defined by a fifth said isosceles triangle; c. a member joining said third apex to the vertex common to said collinear pair of sides of said pentagon, said edge module being equal in length to A √2/2; and d. two additional members joining respectively said first apex with said third apex and said second apex with said third apex, said third apex being common to both additional members and said additional members being equal in length to A √3/3.

Claim 73

A structural configuration comprising the dome of claim 58, above, wherein said dome is substantially self-supporting, said dome being formed from panels.

Claim 74

A structural configuration comprising the dome of claim 58, above, wherein said dome is substantially self-supporting, said dome being formed by linear strut members.

Claim 75

A triangulated dome structural module as in claim 58, above, joined with two similar structural modules, a hexagonal dome module and a base plane to define a polar space unit of a truncated octahedron.

Claim 76

A triangulated dome structural module as in claim 58, above, joined with two identical modules, a nonagonal dome module and a base plane to define the equatorial space unit of the truncated octahedron.

Claim 77

A triangulated dome structural module as in claim 58, above, combined with two additional such modules, a base plane and four hexagonal dome modules to define a polar space unit of a truncated octahedron positioned atop an equatorial space unit from a truncated octahedron.

Claim 78

A triangulated dome structural module as in claim 58, above, combined with two substantially identical structural modules to define secondary wall regions, an irregular, symmetrical nonagon dome structural module defining an upper roof region, an irregular symmetrical octagon dome module defining primary wall regions and a base plane, whereby a triangulated dome structure is formed.

Claim 79

A triangulated convex-concave dome structural module erected upon a planar base comprising: a. a first irregular, symmetrical pentagon in a first plane, having sides equal to unit length A, with one collinear pair of sides and three additional sides forming five interior angles, respectively: 120°, 60°, 180°, 60°, and 120°; b. a series of five isosceles triangles, the sides of which meet in pairs at the vertices of said pentagon in said first plane, the sides of said pentagon defining the base sides of said isosceles triangles, the sides of four or said five isosceles triangles being equal in length to A √6/4, two pairs of said isosceles triangles sharing respective vertices which occur at said 60° interior angles of said pentagon, a common edge and a common apex for each of said pair of isosceles triangles forming a first and second apex, a third apex being defined by a fifth said isosceles triangle, the sides of said fifth isosceles triangle being equal in length to A √2/2; c. a member joining said third apex to the vertex common to said collinear pair of sides of said pentagon, said member being equal in length to A √2/2; and d. two additional members joining respectively said first apex with said third apex and said second apex with said third apex, said third apex being common to both additional members, and said additional members being equal in length to A √6/4.

Claim 80

A triangulated convex-concave dome structural module erected upon a planar base comprising: a. a first irregular, symmetrical pentagon in a first plane, having sides equal to unit length A, with one collinear pair of sides and three additional sides forming five interior angles, respectively: 120°, 60°, 180°, 60°, and 120°; b. a series of five isosceles triangles, the sides of which meet in pairs at the vertices of said pentagon in said first plane, the sides of said pentagon defining the base sides of said isosceles triangles, the sides of four or said five isosceles triangles being equal in length to A √6/4, two pairs of said isosceles triangles sharing respective vertices which occur at said 60° interior angles of said pentagon, a common edge and a common apex for each of said pair of isosceles triangles forming a first and second apex, a third apex being defined by a fifth said isosceles triangle, the sides of said fifth isosceles triangle being equal in length to A √2/2; c. a member joining said third apex to the vertex common to said collinear pair of sides of said pentagon, said member being equal in length to A √2/2; and d. two additional members joining respectively said first apex with said third apex and said second apex with said third apex, said third apex being common to both additional members, and said additional members being equal in length to A √6/4.

Claim 81

A structural configuration comprising the dome of claim 65, above, wherein said dome is substantially self-supporting, said dome being formed from panels.

Claim 82

A structural configuration comprising the dome of claim 65, above, wherein said dome is substantially self-supporting, said dome being formed by linear strut members.

Claim 83

A triangulated dome structure module as in claim 65, above, joined with two similar structural modules, a hexagonal dome module and a base plane to define a polar space unit of a truncated octahedron.

Claim 84

A triangulated dome structural module as in claim 65, above, joined with two identical modules, a nonagonal dome module and a base plane to define the equatorial space unit of the truncated octahedron.

Claim 85

A triangulated dome structural module as in claim 65, above, combined with two additional such modules, a base plane and four hexagonal dome modules to define a polar space unit of a truncated octahedron positioned atop an equatorial space unit from a truncated octahedron.

Claim 86

A triangulated dome structural module as in claim 65, above, combined with two substantially identical structural modules to define secondary wall regions, an irregular, symmetrical nonagon dome structural module defining an upper roof region, an irregular symmetrical octagon dome module defining primary wall regions and a base plane, whereby a triangulated dome structure is formed.

Claim 87

A triangulated convex-concave dome structural module erected upon a planar base comprising: a. a first irregular, symmetrical pentagon in a first plane, having sides equal to unit length A, with one collinear pair of sides and three additional sides forming five interior angles, respectively: 120°, 60°, 180°, 60°, and 120°; b. a series of five isosceles triangles, the sides of which meet in pairs at the vertices of said pentagon in said first plane, the sides of said pentagon defining the base sides of said isosceles triangles, the sides of four or said five isosceles triangles being equal in length to A √2/2, two pairs of said isosceles triangles sharing respective vertices which occur at said 60° interior angles of said pentagon, a common edge and a common apex for each of said pair of isosceles triangles forming a first and second apex, a third apex being defined by a fifth said isosceles triangle, the sides of said fifth isosceles triangle being equal in length to A √6/4; c. a member joining said third apex to the vertex common to said collinear pair of sides of said pentagon, said member being equal in length to A √6/4; and d. two additional members joining respectively said first apex with said third apex and said second apex with said third apex, said third apex being common to both said additional members, and said additional members being equal in length to A √6/4.

Claim 88

A triangulated convex-concave dome structural module erected upon a planar base comprising: a. a first irregular, symmetrical pentagon in a first plane, having sides equal to unit length A, with one collinear pair of sides and three additional sides forming five interior angles, respectively: 120°, 60°, 180°, 60°, and 120°; b. a series of five isosceles triangles, the sides of which meet in pairs at the vertices of said pentagon in said first plane, the sides of said pentagon defining the base sides of said isosceles triangles, the sides of four or said five isosceles triangles being equal in length to A √2/2, two pairs of said isosceles triangles sharing respective vertices which occur at said 60° interior angles of said pentagon, a common edge and a common apex for each of said pair of isosceles triangles forming a first and second apex, a third apex being defined by a fifth said isosceles triangle, the sides of said fifth isosceles triangle being equal in length to A √6/4; c. a member joining said third apex to the vertex common to said collinear pair of sides of said pentagon, said member being equal in length to A √6/4; and d. two additional members joining respectively said first apex with said third apex and said second apex with said third apex, said third apex being common to both said additional members, and said additional members being equal in length to A √6/4.

Claim 89

A structural configuration comprising the dome of claim 72, above, wherein said dome is substantially self-supporting, said dome being formed from panels.

Claim 90

A structural configuration comprising the dome of claim 72, above, wherein said dome is substantially self-supporting, said dome being formed by linear strut members.

Claim 91

A triangulated dome structural module as in claim 72, above, joined with two similar structural modules, a hexagonal dome module and a base plane to define a polar space unit of a truncated octahedron.

Claim 92

A triangulated dome structural module as in claim 72, above, joined with two identical modules, nonagonal dome module and a base plane to define the equatorial space unit of the truncated octahedron.

Claim 93

A triangulated dome structural module as in claim 72, above, combined with two additional such modules, a base plane and four hexagonal dome modules to define a polar space unit of a truncated octahedron positioned atop an equatorial space unit from a truncated octehedron.

Claim 94

A triangulated dome structural module as in claim 72, above, combined with two substantially identical structural modules to define secondary wall regions, an irregular, symmetrical nonagon dome structural module defining an upper roof region, an irregular symmetrical octagon dome module defining primary wall regions and a base plane, whereby a triangulated dome structure is formed.

Claim 95

A repeating triangulated planar structure comprising: a. a repeating planar array of equilateral triangles having sides equal to unit length A; b. a repeating array of equal tripods constructed upon each of said equilateral triangles such that a repeating array of coplanar triangular pyramids is formed, the apices of said pyramids being vertically above the centers of said equilateral triangles, each pyramid being bounded by three equal isosceles triangular faces, the sides of which are equal in length to A √6/4; and c. a plurality of equal length members joining the apices of said pyramids for forming an array of copolanar regular hexagons, three of said members meeting at each apex of said pyramids, and the length of said members being A √3/3.

Claim 96

A repeating triangulated planar structure comprising: a. a repeating planar array of equilateral triangles having sides equal to unit length A; b. a repeating array of equal tripods constructed upon each of said equilateral triangles such that a repeating array of coplanar triangular pyramids is formed, the apices of said pyramids being vertically above the centers of said equilateral triangles, each pyramid being bounded by three equal isosceles triangular faces, the sides of which are equal in length to A √6/4; and c. a plurality of equal length members joining the apices of said pyramids for forming an array of copolanar regular hexagons, three of said members meeting at each apex of said pyramids, and the length of said members being A √3/3.

Claim 97

A structural configuration comprising the structure of claim 79, above, wherein said structure is substantially self-supporting, said structure being formed from panels.

Claim 98

A structural configuration comprising the structure of claim 79, above, wherein said structure is substantially self-supporting, said structure being formed by linear strut members.

Claim 99

A repeating triangulated planar structure comprising: a. a repeating planar array of equilateral triangles having sides equal to unit length A; b. a repeating array of equal tripods constructed upon each of said equilateral triangles such that a repeating array of coplanar triangular pyramids is formed, the apices of said pyramids being vertically above the centers of said equilateral triangles, each pyramid being bounded by three equal isosceles triangular faces, the sides of which are equal in length to A √2/2; and c. a plurality of equal length members joining the apices of said pyramids for forming an array of coplanar regular hexagons, three of said members meeting at each apex of said pyramids, and the length of said members being A √3/3.

Claim 100

A repeating triangulated planar structure comprising: a. a repeating planar array of equilateral triangles having sides equal to unit length A; b. a repeating array of equal tripods constructed upon each of said equilateral triangles such that a repeating array of coplanar triangular pyramids is formed, the apices of said pyramids being vertically above the centers of said equilateral triangles, each pyramid being bounded by three equal isosceles triangular faces, the sides of which are equal in length to A √2/2; and c. a plurality of equal length members joining the apices of said pyramids for forming an array of coplanar regular hexagons, three of said members meeting at each apex of said pyramids, and the length of said members being A √3/3.

Claim 101

A structural configuration comprising the structure of claim 82, above, wherein said structure is substantially self-supporting, said structure being formed from panels.

Claim 102

A structural configuration comprising the structure of claim 80, above, wherein said structure is substantially self-supporting, said structure being formed by linear strut members.

Claim 103

A repeating triangulated planar variable depth framework structure comprising: a. a first repeating hexagonal frame defined by a planar array of regular hexagons in a first plane, said hexagons having sides equal to A √3/3 where A is a unit length; b. a second, repeating hexagonal frame defined by a planar array of said regular hexagons in a second plane, said second repeating hexagonal frame being vertically aligned over and parallel to said first repeating hexagonal frame; c. a plurality of parallel strut members joining respective vertices of said first repeating hexagonal frame with respective vertices of said second repeating hexagonal frame, said parallel strut members being perpendicular to said first and second repeating hexagonal frames, and said parallel strut members being equal in length to A √6/6; d. a plurality of first inclined strut members arranged in groups of six, radiating from common apices to form a set of first hexagonal pyramids which are erected upon said first repeating hexagonal frame, the apices of said set of first hexagonal pyramids defining a common plane parallel and intermediate to said first and second repeating hexagonal frames, the length of said first inclined strut members being equal to A √6/4; and e. a plurality of second inclined strut members arranged in groups of six, radiating from common apices to form a set of second hexagonal pyramids which are erected upon said second repeating hexagonal frame, the apices of said set of second hexagonal pyramids defining a common plane parallel and intermediate to said first and second repeating hexagonal frames, the respective apices of said set of second hexagonal pyramids exactly coinciding with respective apices of said set of first hexagonal pyramids, the length of said second inclined strut members being equal to A √6/4.

Claim 104

A repeating triangulated planar variable depth framework structure comprising: a. a first repeating hexagonal frame defined by a planar array of regular hexagons in a first plane, said hexagons having sides equal to A √3/3 where A is a unit length; b. a second, repeating hexagonal frame defined by a planar array of said regular hexagons in a second plane, said second repeating hexagonal frame being vertically aligned over and parallel to said first repeating hexagonal frame; c. a plurality of parallel strut members joining respective vertices of said first repeating hexagonal frame with respective vertices of said second repeating hexagonal frame, said parallel strut members being perpendicular to said first and second repeating hexagonal frames, and said parallel strut members being equal in length to A √6/6; d. a plurality of first inclined strut members arranged in groups of six, radiating from common apices to form a set of first hexagonal pyramids which are erected upon said first repeating hexagonal frame, the apices of said set of first hexagonal pyramids defining a common plane parallel and intermediate to said first and second repeating hexagonal frames, the length of said first inclined strut members being equal to A √6/4; and e. a plurality of second inclined strut members arranged in groups of six, radiating from common apices to form a set of second hexagonal pyramids which are erected upon said second repeating hexagonal frame, the apices of said set of second hexagonal pyramids defining a common plane parallel and intermediate to said first and second repeating hexagonal frames, the respective apices of said set of second hexagonal pyramids exactly coinciding with respective apices of said set of first hexagonal pyramids, the length of said second inclined strut members being equal to A √6/4.

Claim 105

A repeating triangulated planar variable depth framework structure comprisng: a. a first repeating hexagonal frame defined by a planar array of regular hexagons in a first plane, said hexagons having sides equal to A √3/3 where A is a unit length; b. a second, repeating hexagonal frame defined by a planar array of said regular hexagons in a second plane, said second repeating hexagonal frame being vertically aligned over and parallel to said first repeating hexagonal frame; c. a plurality of parallel strut members joining respective vertices of said first repeating hexagonal frame with respective vertices of said second repeating hexagonal frame, said parallel strut members being perpendicular to said first and second repeating hexagonal frames, and said parallel strut members being equal in length to A √6/4; d. a plurality of first inclined strut members arranged in groups of six, radiating from common apices to form a set of first hexagonal pyramids which are erected upon said first repeating hexagonal frame, the apices of said set of first hexagonal pyramids defining a common plane parallel and intermediate to said first and second repeating hexagonal frames, the length of said first inclined strut members being equal to A √6/4; and e. a plurality of second inclined strut members arranged in groups of six, radiating from common apices to form a set of second hexagonal pyramids which are erected upon said second hexagonal frame, the apices of said set of second hexagonal pyramids defining a common plane parallel and intermediate to said first and second repeating hexagonal frames, the respective apices of said set of second hexagonal pyramids exactly coinciding with respective apices of said set of first hexagonal pyramids, the length of said second inclined strut members being equal to A √2/2.

Claim 106

A repeating triangulated planar variable depth framework structure comprisng: a. a first repeating hexagonal frame defined by a planar array of regular hexagons in a first plane, said hexagons having sides equal to A √3/3 where A is a unit length; b. a second, repeating hexagonal frame defined by a planar array of said regular hexagons in a second plane, said second repeating hexagonal frame being vertically aligned over and parallel to said first repeating hexagonal frame; c. a plurality of parallel strut members joining respective vertices of said first repeating hexagonal frame with respective vertices of said second repeating hexagonal frame, said parallel strut members being perpendicular to said first and second repeating hexagonal frames, and said parallel strut members being equal in length to A √6/4; d. a plurality of first inclined strut members arranged in groups of six, radiating from common apices to form a set of first hexagonal pyramids which are erected upon said first repeating hexagonal frame, the apices of said set of first hexagonal pyramids defining a common plane parallel and intermediate to said first and second repeating hexagonal frames, the length of said first inclined strut members being equal to A √6/4; and e. a plurality of second inclined strut members arranged in groups of six, radiating from common apices to form a set of second hexagonal pyramids which are erected upon said second hexagonal frame, the apices of said set of second hexagonal pyramids defining a common plane parallel and intermediate to said first and second repeating hexagonal frames, the respective apices of said set of second hexagonal pyramids exactly coinciding with respective apices of said set of first hexagonal pyramids, the length of said second inclined strut members being equal to A √2/2.

Claim 107

The triangulated polyhedral structures comprising truncations of a complete polyhedral structure of eighty-four quadrilateral faces, the complete polyhedral structure comprising: a. twelve first rhombic faces, the sides of which are equal in length to A √6/4, the long diagonal of said first rhombic face being equal to unit length A and the short diagonal of said first rhombic face being equal in length to A √2/2, said long diagonal dividing said first rhombic face into a pair of identical isosceles triangles which share a common base of unit length A, and said short diagonal dividing said first rhombic face into a pair of identical isosceles triangles which share a common base of length A √2/2; b. forty-eight, second rhombic faces, the sides of which are equal in length of A √6/4, the long diagonal of said second rhombic face being equal in length to A √42/6, and the short diagonal of said second rhombic face being equal in length to A √3/3, said long diagonal dividing said second rhombic face into a pair of identical isosceles triangles which share a common base of length A √42/6, and said short diagonal dividing said second rhombic face into a pair of identical isosceles triangles which share a common base of length A √3/3; and c. twenty-four trapezium faces each having two sides equal in length to A 6/4 and two sides equal in length to A 34/8, the long diagonal of said trapezium face being equal to unit length A, and the short diagonal of said trapezium face being equal in length to A 5√2/8, said long diagonal dividing said trapezium into a pair of first and second isosceles triangles which share a common base of unit length A, said first isosceles triangle having sides of A √6/4, and said second isosceles triangle having sides of A √34/8, and said short diagonal dividing said trapezium face into a pair of identical but reversed scalene triangles sharing a common side of 5√2/8, and having two other sides of A √6/4 and A √34/8.

Claim 108

The triangulated polyhedral structures comprising truncations of a complete polyhedral structure of eighty-four quadrilateral faces, the complete polyhedral structure comprising: a. twelve first rhombic faces, the sides of which are equal in length to A √6/4, the long diagonal of said first rhombic face being equal to unit length A and the short diagonal of said first rhombic face being equal in length to A √2/2, said long diagonal dividing said first rhombic face into a pair of identical isosceles triangles which share a common base of unit length A, and said short diagonal dividing said first rhombic face into a pair of identical isosceles triangles which share a common base of length A √2/2; b. forty-eight, second rhombic faces, the sides of which are equal in length of A √6/4, the long diagonal of said second rhombic face being equal in length to A √42/6, and the short diagonal of said second rhombic face being equal in length to A √3/3, said long diagonal dividing said second rhombic face into a pair of identical isosceles triangles which share a common base of length A √42/6, and said short diagonal dividing said second rhombic face into a pair of identical isosceles triangles which share a common base of length A √3/3; and c. twenty-four trapezium faces each having two sides equal in length to A 6/4 and two sides equal in length to A 34/8, the long diagonal of said trapezium face being equal to unit length A, and the short diagonal of said trapezium face being equal in length to A 5√2/8, said long diagonal dividing said trapezium into a pair of first and second isosceles triangles which share a common base of unit length A, said first isosceles triangle having sides of A √6/4, and said second isosceles triangle having sides of A √34/8, and said short diagonal dividing said trapezium face into a pair of identical but reversed scalene triangles sharing a common side of 5√2/8, and having two other sides of A √6/4 and A √34/8.

Claim 109

A structural configuration comprising the structure of claim 87, above, wherein said structure is substantially self-supporting, said structure being formed from panels.

Claim 110

A structural configuration comprising the structure of claim 87, above, wherein said structure is substantially self-supporting, said structure being formed by linear strut members.