Method for producing polyhedral nets

A method of producing a plurality of polyhedron nets for a truncated polyhedron, the method comprising the steps of:

• providing a flat piece of foldable material;
• defining an orthogonal coordinate system on the material, the coordinate system having a horizontal axis denoted (a), a vertical axis denoted (α), and an origin denoted (1);
• selecting a value for a distance (X) that is greater than zero;
• selecting a value for a constant (Y) that is greater than zero;
• selecting values equal to or greater than zero and equal to or less than one for the truncating constants (Z1), (Z2), (Z3), (Z4), (Z5), (Z6), (Z7), (Z8), (Z9), (Z11), (Z12), (Z13), (Z14), (Z15), (Z16), (Z17), (Z18), (Z19), (Z20), (Z21), (Z22), (Z23), (Z24), (Z25);
• the value for the truncating constant (Z7) being less than or equal to:
• one minus the truncating constant (Z6);
• the value for the truncating constant (Z11) being less than or equal to:
• one minus the truncating constant (Z8);
• the value for the truncating constant (Z13) being less than or equal to:
• one minus the truncating constant (Z2);
• the value for the truncating constant (Z14) being less than or equal to:
• one minus the truncating constant (Z9);
• the value for the truncating constant (Z16) being less than or equal to:
• one minus the truncating constant (Z1);
• the value for the truncating constant (Z17) being less than or equal to:
• one minus the truncating constant (Z5);
• the value for the truncating constant (Z19) being less than or equal to:
• one minus the truncating constant (Z12);
• the value for the truncating constant (Z20) being less than or equal to:
• one minus the truncating constant (Z4);
• the value for the truncating constant (Z22) being less than or equal to:
• one minus the truncating constant (Z15);
• the value for the truncating constant (Z23) being less than or equal to:
• one minus the truncating constant (Z21);
• the value for the truncating constant (Z24) being less than or equal to:
• one minus the truncating constant (Z18);
• the value for the truncating constant (Z25) being less than or equal to:
• one minus the truncating constant (Z3);
• selecting a value for an angle (D) that is between, but not including, 0 and 180 degrees;
• drawing a line along (a) from the origin (1) to a new point (2) that is the distance (X) from the origin (1), whereby forming a line (1)-(2);
• continuing the line from the point (2) at an angle of (180−D) degrees from (α) to a new point (3) that is the distance (X) from the point (2), whereby forming a line (2)-(3);
• continuing the line from the point (3) at the angle of (D) degrees from (α) to a new point (4) that is the distance (XY) from the point (3), whereby forming a line (3)-(4);
• continuing the line from the point (4) at an angle of (−[180-2D]) degrees from (α) to new point (5) that is the distance (XY) from the point (4), whereby forming a line (4)-(5);
• continuing the line from the point (5) at an angle of (D) degrees from (α) to a new point (6) that is the distance (X) from the point (5), whereby forming a line (5)-(6);
• continuing the line from the point (6) at an angle of (2D) degrees from (α) a new point (7) that is the distance (XY) from the point (6), whereby forming a line (6)-(7);
• continuing the line from the point (7) at an angle of (180−D) degrees from (α) to a new point (8) that is the distance (XY) from the point (7), whereby forming a line (7)-(8);
• continuing the line from the point (8) at an angle of 180 degrees from (α) to a new point (9) that is the distance (X) from the point (8), whereby forming a line (8)-(9);
• continuing the line from the point (9) at an angle of (−D) degrees from (α) to a new point (10) that is the distance (XY) from the point (9), whereby forming a line (9)-(10);
• continuing the line from the point (10) at an angle of 180 degrees from (α) to a new point (11) that is the distance (XY) from the point (10), whereby forming a line (10)-(11);
• continuing the line from the point (11) at an angle of (−[180−D]) degrees from (α) to a new point (12) that is the distance (X) from the point (11), whereby forming a line (11)-(12);
• continuing the line from the point (12) at an angle of 0 degrees from (α) to a new point (13) that is the distance (XY) from the point (12), whereby forming a line (12)-(13);
• continuing the line from the point (13) at an angle of (−[180−D]) degrees from (α) to a new point (14) that is the distance (XY) from the point (13), whereby forming a line (13)-(14);
• continuing the line from the point (14) at an angle of (−D) degrees from (α) to the origin (1) that is the distance (X) from the point (14), whereby forming a line (14)-(1);
• connecting the point (14) to the point (3), whereby forming a fold line (3)-(14);
• connecting the point (13) to the point (4), whereby forming a fold line (4)-(13);
• connecting the point (13) to the point (10), whereby forming a fold line (10)-(13);
• connecting the point (10) to the point (7), whereby forming a fold line (7)-(10);
• connecting the point (4) to the point (7), whereby forming a fold line (4)-(7);
• marking a point (15) along line (14)-(1), the point (15) being a distance (Z1X) from the point (14);
• marking a point (19) along line (11)-(12), the point (19) being the distance (Z1X) from the point (12);
• marking a point (16) along the fold line (3)-(14), the point (16) being a distance (Z2X) from the point (14);
• marking a point (17) along the line (13)-(14), the point (17) being a distance (Z3XY) from the point (14);
• marking a point (18) along the line (12)-(13), the point (18) being the distance (Z3XY) from the point (12);
• marking a point (23) along the fold line (7)-(10), the point (23) being a distance (Z4X) from the point (7);
• marking a point (24) along the fold line (4)-(7), the point (24) being a distance (Z5X) from the point (7);
• marking a point (28) along the line (6)-(7), the point (28) being a distance (Z6XY) from the point (7);
• marking a point (29) along the line (7)-(8), the point (29) being the distance (Z6XY) from the point (7);
• marking a point (27) along the line (6)-(7), the point (27) being a distance (Z7XY) from the point (6);
• marking a point (20) along the line (7)-(8), the point (20) being the distance (Z7XY) from the point (8);
• marking a point (31) along the line (5)-(6), the point (31) being a distance (Z8X) from the point (6);
• marking a point (32) along the line (2)-(3), the point (32) being the distance (Z8X) from the point (2);
• marking a point (33) along the line (8)-(9), the point (33) being a distance (Z9X) from the point (8);
• marking a point (34) along the line (1)-(2), the point (34) being the distance (Z9X) from the point (2);
• marking a point (59) along the line (5)-(6), the point (59) being a distance (Z11X) from the point (5);
• marking a point (35) along the line (2)-(3), the point (35) being the distance (Z11X) from the point (3);
• marking a point (36) along the line (3)-(4), the point (36) being a distance (Z12XY) from the point (3);
• marking a point (37) along the line (4)-(5), the point (37) being the distance (Z12XY) from the point (5);
• marking a point (38) along the fold line (3)-(14), the point (38) being a distance (Z13X) from the point (3);
• marking a point (39) along the line (1)-(2), the point (39) being a distance (Z14X) from the point (1);
• marking a point (41) along the line (8)-(9), the point (41) being the distance (Z14X) from the point (9);
• marking a point (42) along the line (9)-(10), the point (42) being a distance (Z15XY) from the point (9);
• marking a point (43) along the line (10)-(11), the point (43) being the distance (Z15XY) from the point (11);
• marking a point (44) along the line (11)-(12), the point (44) being a distance (Z16X) from the point (11);
• marking a point (45) along the line (14)-(1), the point (45) being the distance (Z16X) from the point (1);
• marking a point (46) along the fold line (4)-(7), the point (46) being a distance (Z17X) from the point (4);
• marking a point (47) along the fold line (4)-(13), the point (47) being a distance (Z18X) from the point (4);
• marking a point (48) along the line (3)-(4), the point (48) being a distance (Z19XY) from the point (4);
• marking a point (49) along the line (4)-(5), the point (49) being the distance (Z19XY) from the point (4);
• marking a point (51) along the fold line (7)-(10), the point (51) being a distance (Z20X) from the point (10);
• marking a point (52) along the fold line (10)-(13), the point (52) being a distance (Z21X) from the point (10);
• marking a point (53) along the line (9)-(10), the point (53) being a distance (Z22XY) from the point (10);
• marking a point (54) along the line (10)-(11), the point (54) being a distance (Z22XY) from the point (10);
• marking a point (55) along the fold line (10)-(13), the point (55) being a distance (Z23X) from the point (13);
• marking a point (56) along the fold line (4)-(13), the point (56) being a distance (Z24X) from the point (13);
• marking a point (57) along the line (12)-(13), the point (57) being a distance (Z25XY) from the point (13);
• marking a point (58) along the line (13)-(14), the point (58) being the distance (Z25XY) from the point (13);
• connecting the point (15) to the point (16), whereby forming a fold line (15)-(16) and a triangle (14)-(15)-(16);
• connecting the point (16) to the point (17), whereby forming a fold line (16)-(17) and a triangle (14)-(16)-(17);
• connecting the point (56) to the point (58), whereby forming a fold line (56)-(58) and a triangle (13)-(56)-(58);
• connecting the point (55) to the point (56), whereby forming a fold line (55)-(56) and a triangle (13)-(55)-(56);
• connecting the point (55) to the point (57), whereby forming a fold line (55)-(57) and a triangle (13)-(55)-(57);
• connecting the point (18) to the point (19), whereby forming a fold line (18)-(19) and a triangle (12)-(18)-(19);
• connecting the point (43) to the point (44), whereby forming a fold line (43)-(44) and a triangle (111)-(43)-(44);
• connecting the point (52) to the point (54), whereby forming a fold line (52)-(54) and a triangle (10)-(52)-(54);
• connecting the point (51) to the point (52), whereby forming a fold line (51)-(52) and a triangle (10)-(51)-(52);
• connecting the point (51) to the point (53), whereby forming a fold line (51)-(53) and a triangle (10)-(51)-(53);
• connecting the point (41) to the point (42), whereby forming a fold line (41)-(42) and a triangle (9)-(41)-(42);
• connecting the point (20) to the point (33), whereby forming a fold line (20)-(33) and a triangle (8)-(20)-(33);
• connecting the point (23) to the point (29), whereby forming a fold line (23)-(29) and a triangle (7)-(23)-(29);
• connecting the point (23) to the point (24), whereby forming a fold line (23)-(24) and a triangle (7)-(23)-(24);
• connecting the point (24) to the point (28), whereby forming a fold line (24)-(28) and a triangle (7)-(24)-(28);
• connecting the point (27) to the point (31), whereby forming a fold line (27)-(31) and a triangle (6)-(27)-(31);
• connecting the point (37) to the point (59), whereby forming a fold line (37)-(59) and a triangle (5)-(37)-(59);
• connecting the point (46) to the point (49), whereby forming a fold line (46)-(49) and a triangle (4)-(46)-(49);
• connecting the point (46) to the point (47), whereby forming a fold line (46)-(47) and a triangle (4)-(46)-(47);
• connecting the point (47) to the point (48), whereby forming a fold line (47)-(48) and a triangle (4)-(47)-(48);
• connecting the point (36) to the point (38), whereby forming a fold line (36)-(38) and a triangle (3)-(36)-(38);
• connecting the point (35) to the point (38), whereby forming a fold line (35)-(38) and a triangle (3)-(35)-(38);
• connecting the point (32) to the point (34), whereby forming a fold line (32)-(34) and a triangle (2)-(32)-(34);
• connecting the point (39) to the point (45), whereby forming a fold line (39)-(45) and a triangle (1)-(39)-(45);
• verifying that the fold line (24)-(28) has a distance of √[(Z6XY)2+(Z5X)2−2(Z5Z6X2Y cos(D))];
• verifying that the fold line (23)-(29) has a distance of √[(Z6XY)2+(Z4X)2−2(Z4Z6X2Y cos(D))];
• verifying that the fold line (23)-(24) has a distance of √[(Z5X)2+(Z4X)2−2(Z4Z5X2 cos(D))];
• modifying a triangle selected from the group consisting of the (7)-(23)-(24) triangle, the (7)-(23)-(29) triangle, and the (7)-(24)-(28) triangle;
• verifying that the fold line (46)-(47) has a distance of √[(Z17X)2+(Z18X)2−2(Z17Z18X2 cos(180−D))];
• verifying that the fold line (46)-(49) has a distance of √[(Z17X)2+(Z19XY)2−2(Z17Z19X2Y cos(180−D))];
• verifying that the fold line (47)-(48) has a distance of √[(Z19XY)2+(Z18X)2−2(Z18Z19X2Y cos(D))];
• modifying a triangle selected from the group consisting of the (4)-(46)-(47) triangle, the (4)-(46)-(49) triangle, and the (4)-(47)-(48) triangle;
• verifying that the fold line (37)-(59) has a distance of √[(Z12XY)2+(Z11X)2−2(Z11Z12X2Y cos(D))];
• verifying that the fold line (36)-(38) has a distance of √[(Z12XY)2+(Z13X)2−2(Z12Z13X2Y cos(180−D))];
• verifying that the fold line (35)-(38) has a distance of √[(Z13X)2+(Z11X)2−2(Z11Z13X2 cos(180−D))];
• modifying a triangle selected from the group consisting of the (5)-(37)-(59) triangle, the (3)-(36)-(38) triangle, and the (3)-(35)-(38) triangle;
• verifying that the fold line (27)-(31) has a distance of √[(Z7XY)2+(Z8X)2−2(Z7Z8X2Y cos(180−D))];
• verifying that the fold line (20)-(33) has a distance of √[(Z7XY)2+(Z9X)2−2(Z7Z9X2Y cos(180−D))];
• verifying that the fold line (32)-(34) has a distance of √[(Z8X)2+(Z9X)2−2(Z8Z9X2 cos(D))];
• modifying a triangle selected from the group consisting of the (6)-(27)-(31) triangle, the (8)-(20)-(33) triangle, and the (2)-(32)-(34) triangle;
• verifying that the fold line (39)-(45) has a distance of √[(Z14X)2+(Z16X)2−2(Z14Z16X2 cos(180−D))];
• verifying that the fold line (43)-(44) has a distance of √[(Z15XY)2+(Z16X)2−2(Z15Z16X2Y cos(180—D))];
• verifying that the fold line (41)-(42) has a distance of √[(Z14X)2+(Z15XY)2−2(Z14Z15X2Y cos(D))];
• modifying a triangle selected from the group consisting of the (1)-(39)-(45) triangle, the (11)-(43)-(44) triangle, and the (9)-(41)-(42) triangle;
• verifying that the fold line (15)-(16) has a distance of √[(Z1X)2+(Z2X)2−2(Z1Z2X2 cos(D))];
• verifying that the fold line (16)-(17) has a distance of √[(Z3XY)2+(Z2X)2−2(Z2Z3X2Y cos(D))];
• verifying that the fold line (18)-(19) has a distance of √[(Z1X)2+(Z3XY)2−2(Z1Z3X2Y cos(D))];
• modifying a triangle selected from the group consisting of the (12)-(18)-(19) triangle, the (14)-(16)-(17) triangle, and the (14)-(15)-(16) triangle;
• verifying that the fold line (55)-(57) has a distance of √[(Z25XY)2+(Z23X)2−2(Z23Z25X2Y cos(180−D))];
• verifying that the fold line (56)-(58) has a distance of √[(Z25XY)2+(Z24X)2−2(Z24Z25X2Y cos(180−D))];
• verifying that the fold line (55)-(56) has a distance of √[(Z24X)2+(Z23X)2−2(Z23Z24X2 cos(D))];
• modifying a triangle selected from the group consisting of the (13)-(55)-(57) triangle, the (13)-(55)-(56) triangle, and the (13)-(56)-(58) triangle;
• verifying that the fold line (51)-(53) has a distance of √[(Z22XY)2+(Z20X)2−2(Z20Z22X2Y cos(180−D))];
• verifying that the fold line (51)-(52) has a distance of √[(Z21X)2+(Z20X)2−2(Z20Z21X2 cos(180−D))];
• verifying that the fold line (52)-(54) has a distance of √[(Z21X)2+(Z22XY)2−2(Z21Z22X2Y cos(D))];
• and
• modifying a triangle selected from the group consisting of the (10)-(51)-(53) triangle, the (10)-(52)-(54) triangle, and the (10)-(51)-(52) triangle.

A method of producing a three dimensional polyhedra, the method comprising the steps of:

• providing a flat piece of foldable material;
• defining an orthogonal coordinate system on the material, the coordinate system having a horizontal axis denoted (a), a vertical axis denoted (β), and an origin denoted (1);
• selecting a value for a distance (X) that is greater than zero;
• selecting a value for a constant (Y) equal to 0.8;
• allowing truncating constants (Z1) and (Z2) to equal 0.53;
• allowing truncating constants (Z3), (Z4), (Z5), and (Z6) to equal 0.6;
• allowing truncating constants (Z7), (Z8), (Z9), (Z11), (Z12), (Z13), (Z14), (Z15), (Z16), (Z17), (Z18), (Z19), (Z20), (Z21), (Z22), (Z23), (Z24), and (Z25) to equal zero;
• selecting a value for an angle (D) equal to 80 degrees;
• drawing a line along (a) from the origin (1) to a new point (2) that is the distance (X) from the origin (1), whereby forming a line (1)-(2);
• continuing the line from the point (2) at an angle of (180−D) degrees from (α) to a new point (3) that is the distance (X) from the point (2), whereby forming a line (2)-(3);
• continuing the line from the point (3) at the angle of (D) degrees from (α) to a new point (4) that is the distance (XY) from the point (3), whereby forming a line (3)-(4);
• continuing the line from the point (4) at an angle of (−[180−2D]) degrees from (α) to new point (5) that is the distance (XY) from the point (4), whereby forming a line (4)-(5);
• continuing the line from the point (5) at an angle of (D) degrees from (α) to a new point (6) that is the distance (X) from the point (5), whereby forming a line (5)-(6);
• continuing the line from the point (6) at an angle of (2D) degrees from (α) a new point (7) that is the distance (XY) from the point (6), whereby forming a line (6)-(7);
• continuing the line from the point (7) at an angle of (180−D) degrees from (α) to a new point (8) that is the distance (XY) from the point (7), whereby forming a line (7)-(8);
• continuing the line from the point (8) at an angle of 180 degrees from (α) to a new point (9) that is the distance (X) from the point (8), whereby forming a line (8)-(9);
• continuing the line from the point (9) at an angle of (−D) degrees from (α) to a new point (10) that is the distance (XY) from the point (9), whereby forming a line (9)-(10);
• continuing the line from the point (10) at an angle of 180 degrees from (α) to a new point (11) that is the distance (XY) from the point (10), whereby forming a line (10)-(11);
• continuing the line from the point (11) at an angle of (−[180−D]) degrees from (α) to a new point (12) that is the distance (X) from the point (11), whereby forming a line (11)-(12);
• continuing the line from the point (12) at an angle of 0 degrees from (α) to a new point (13) that is the distance (XY) from the point (12), whereby forming a line (12)-(13);
• continuing the line from the point (13) at an angle of (−[180−D]) degrees from (α) to a new point (14) that is the distance (XY) from the point (13), whereby forming a line (13)-(14);
• continuing the line from the point (14) at an angle of (−D) degrees from (α) to the origin (1) that is the distance (X) from the point (14), whereby forming a line (14)-(1);
• connecting the point (14) to the point (3), whereby forming a fold line (3)-(14);
• connecting the point (13) to the point (4), whereby forming a fold line (4)-(13);
• connecting the point (13) to the point (10), whereby forming a fold line (10)-(13);
• connecting the point (10) to the point (7), whereby forming a fold line (7)-(10);
• connecting the point (4) to the point (7), whereby forming a fold line (4)-(7);
• marking a point (15) along line (14)-(1), the point (15) being a distance (Z1X) from the point (14);
• marking a point (19) along line (11)-(12), the point (19) being the distance (Z1X) from the point (12);
• marking a point (16) along the fold line (3)-(14), the point (16) being a distance (Z2X) from the point (14);
• marking a point (17) along the line (13)-(14), the point (17) being a distance (Z3XY) from the point (14);
• marking a point (18) along the line (12)-(13), the point (18) being the distance (Z3XY) from the point (12);
• marking a point (23) along the fold line (7)-(10), the point (23) being a distance (Z4X) from the point (7);
• marking a point (24) along the fold line (4)-(7), the point (24) being a distance (Z5X) from the point (7);
• marking a point (28) along the line (6)-(7), the point (28) being a distance (Z6XY) from the point (7);
• marking a point (29) along the line (7)-(8), the point (29) being the distance (Z6XY) from the point (7);
• connecting the point (15) to the point (16), whereby forming a fold line (15)-(16) and a triangle (14)-(15)-(16);
• connecting the point (16) to the point (17), whereby forming a fold line (16)-(17) and a triangle (14)-(16)-(17);
• connecting the point (18) to the point (19), whereby forming a fold line (18)-(19) and a triangle (12)-(18)-(19);
• connecting the point (23) to the point (29), whereby forming a fold line (23)-(29) and a triangle (7)-(23)-(29);
• connecting the point (23) to the point (24), whereby forming a fold line (23)-(24) and a triangle (7)-(23)-(24);
• connecting the point (24) to the point (28), whereby forming a fold line (24)-(28) and a triangle (7)-(24)-(28);
• verifying that the fold line (24)-(28) has a distance of √[(Z6XY)2+(Z5X)2−2(Z5Z6X2Y cos(D))];
• verifying that the fold line (23)-(29) has a distance of √[(Z6XY)2+(Z4X)2−2(Z4Z6X2Y cos(D))];
• verifying that the fold line (23)-(24) has a distance of √[(Z5X)2+(Z4X)2−2(Z4Z5X2 cos(D))];
• verifying that the fold line (15)-(16) has a distance of √[(Z1X)2+(Z2X)2−2(Z1Z2X2 cos(D))];
• verifying that the fold line (16)-(17) has a distance of √[(Z3XY)2+(Z2X)2−2(Z2Z3X2Y cos(D))];
• verifying that the fold line (18)-(19) has a distance of √[(Z1X)2+(Z3XY)2−2(Z1Z3X2Y cos(D))];
• marking a point (25) along the fold line (24)-(28) at a distance of 0.6(X) from the point (24);
• marking a point (26) along the line (5)-(6) at a distance of 0.86(X) from the point (5);
• connecting the point (25) to the point (26), whereby creating a fold line (25)-(26);
• marking a point (22) along the fold line (23)-(29) at the distance of 0.6(X) from the point (23);
• marking a point (21) along the line (8)-(9) at the distance of 0.86(X) from the point (9);
• connecting the point (21) to the point (22), whereby creating a fold line (21)-(22);
• marking a point (50) at a midpoint of the fold line (1)-(3);
• connecting the point (2) to the point (50), whereby creating a line (2)-(50) which is perpendicular to the fold line (1)-(3);
• marking a point (2f) along the line (2)-(50), the point (2f) being the distance of 0.86(X) from the point (1) and the distance of 0.86(X) from the point (3);
• creating a new triangle, (1)-(2f)-(3), with the points (1), (2f), and (3);
• folding the fold line (18)-(19);
• separating the polyhedron net from excess foldable material by cutting along the
• following lines:
• (1)-(2f)-(3)-(4)-(5)-(6)-(7)-(8)-(9)-(10)-(11)-(19)-(12)-(18)-(13)-(14)-(1);
• cutting the line (14)-(16);
• folding the polyhedron net along the fold lines (23)-(29), (24)-(28), (21)-(22), (25)-(26), (15)-(16), (16)-(17), (18)-(19), (23)-(24), with all of the fold lines being located on an exterior surface of the polyhedron;
• securing the line (25)-(26) to the line (21)-(22);
• securing the line (9)-(10) to the line (10)-(11);
• securing the line (4)-(5) to the line (3)-(4);
• securing the line (13)-(18) to the line (13)-(17);
• securing the line (2f)-(3) to the line (5)-(26);
• securing the line (1)-(2f) to the line (9)-(21);
• securing the line (1)-(15) to the line (11)-(19);
• securing the line (7)-(28) to the line (7)-(29);
• securing the line (7)-(23) to the line (22)-(23);
• and
• securing the line (7)-(24) to the line (24)-(25).

A method of producing a polyhedron net for a truncated polyhedron, the method comprising the steps of:

• providing a flat piece of foldable material;
• defining an orthogonal coordinate system on the material, the coordinate system having a horizontal axis, a vertical axis, and an origin;
• selecting values for line distances that are greater than zero;
• selecting values for angles between lines having the line distances, the angles being between 0 and 180 degrees;
• selecting values between zero and one, inclusive, for truncating constants;
• drawing the lines, points, and fold lines on the foldable material in accord with the line distances and angles;
• defining the value of each truncating constant being less than or equal to one minus another truncating constant on a respective line that will form a single edge of the polyhedron;
• marking new points along the lines at distances defined by the truncating constants;
• connecting the new points, whereby forming new fold lines and new triangles;
• verifying the distances of the new fold lines mathematically;
• and
• modifying the new triangles to define a perimeter.