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ANNEX 2: Summary of symmetrical 2- and 3-dimensional forms

Needs Communication: viable need patterns and their identification (Part #13)

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Symmetrical 2- and 3-dimensional forms
A : 2-DIMENSIONS (circular symmetry)	without tension elements	B: 3-DIMENSIONS (spherical symmetry)
Stabilite : If a square of polygon is made from a series of struts which define its edges, and if those struts are connected by flexible joints, the resulting figure can be distorted and is therefore unstable. To be stable a shape must have its faces composed of triangles. If triangulation is done with tension elements, the shape cannot be distorted in 2-dimensions, but it is unstable if lifted off the plane surface.		Stability : If a cube or polyhedron is made from a series of struts which define its edges, and if those edges are connected by flexible joints, the resulting figure can be distorted (and is therefore unstable).unless all the faces are triangular (as in the tetrahedron, octahedron or icosahedron) . certain countracting configurations of struts and tension elements (tensegrity structures) are stable without triangular faces. The resulting network of tension elements outlines the polyhedral form on which the tensegrity structure is based.
1. Struts linked end-to-end in a ring pattern; N struts enclose an area of the forrn of a regular polygon. N = 3, triangle 4. square 5. pentagon 6. hexagon 7. heptagon etc.		1. Strut end linked to M other ends; N struts enclose a volume. 1.1 Equal faces forming 5 regular polyhedra N = 6. tetrahedron (4 triangles) 12. octahedron (8 triangles) 12. cube (6 squares) 30. icosahedron (20 triangles) 30. dodecahedron (12 pentagons)
		1.2 Equal face arrangement around each vertex 1.2.1 forming 13 semi-regular polyhedra N = 18. truncated tetrahedron 24. cuboctahedron 36. truncated octahedron 36. truncated cube 48. small rhombicuboctahedron 60. icosicodecanedron 60. snub cube 72. great mombicuboctahedron 90. truncated icosahedron 90. truncated dodecahedron 120, small rhombicosidodecahedron 150. snubdodecahedron 180. great rhombicosidodocahedron
1.2.3 Struts linked end-to-end in several overlapping (or interweaving) ringpatterns enclosing an area of the form of a regular polygon N= 6, triangles (2) 8. squares (2) 9. triangles (3) 10. pentagons (2), etc.		1.22 forming facially regular prisms (i.e. not spherically symmetrical) N = 9. triangular prism 12. square prism (i.e. cube) 15. pentagonal prism 21. heptagonal prism, etc,
		1.23 forming facially regular antiprisma (i.e. not spherically symmetrical) N = 6, triangular antiprism (i.e. octahedron) 16. square antiprism 20. pentagonal antiprism 24. hexagonal antiprism, etc.
1.4. Concave regular forms 1.4.1 Elaboration central symmetry by stellation 1.4.2 Elaboration of central symmetry by faceting.		1.3 Unequal face arrangement (regular face only) 1.31 Portions of 1.1 or 1.2.1 (14 forms) 1.3.2 Joining polyhedra from 1.1 Joining polyhedra from 1.1 or 1.2.1 to those from 1.3.1 (15) 1.3.3 Joining polyhedra to those from 1.2.2 (26) 1.3.4 Joining polyhedra to those from 1.2.3 (11) 1.3.5 Special cases (6) 1.3.6 Joining polyhedra from 1.3.1 and from 1.3.5 (18) (N.B. These are not spherically symmetrical).
		1.4 Concave regular forms 1.4.1 Elaboration of central symmetry by -stellation (equal regular laces only) N = 30. small stellated dodecahedron 30. great stellated dodecahedron 1.4.2 Elaboration of central symmetry by faceting (equal regular faces only) N = 30. great dodecahedron 30. great icosahedron

Symmetrical 2- and 3-dimensional forms
A : 2-DIMENSIONS (circular symmetry)	with tension elements	B: 3-DIMENSIONS (spherical symmetry)
2. All struts pass (approximately) through centre point, ends do not touch and are linked by tension elements (outlining a regular polygon). N = 2. square outlined 3. hexagon outlined 4. octagon outlined, etc.		2. All strut centres pass (approximately) through centre point, ends do not touch but are linked by tension elements (outlining a regular polyhedron) N = 3. octahedron outlines) 4. cube outlined, etc.
3. Strut ends overlap (but are only connected via tension elements), enclosing an area in the form of a regular polygon. N = 3 triangle 4. square, etc.		3.1 Tensegrity diamond pattern with struts enclosing 3 volume; external tension elements outline a regular polyhedron N = 8. octahedron 12, cuboctahedron, etc. 3.2 Tensegrity zig-zag pattern with struts enclosing a volume; external tension elements outline a regular polyhedron. N = 6. tetrahedron 12. octahedron 30, icosahedron 36. cube, etc. 3.3 Tensegrity prism, with struts not enclosing a volume (i. e. not spherically symmetrical) N = 3, triangular prism
4. Strut ends linked together to form a regular polygon; tension links from vertices to a common central point N = 3. triangle 4, square, etc.		4. Square prism, etc. 4. Strut ends linked to form a regular polygon with a single strut passing at right angles through the centre point of the olane. Vertices linked to the ends of the single strut. (N. B. not spherically symmetrical) N = 4. triangular polygon 5. square polygon, etc.
5. Strut ends linked together with struts overlapping: vertices linked by tension elements. Forming continuous circuit (for N odd) N = 5. pentagram 7. heptagram. etc.		5. Strut ends linked together with struts interweaving: vertices Iinked by tension elements. 5.1 Forming a continuous tensegrity circuit pattern. 5.2 Forming a tensegrity made up of several independent interweaving circuit pattern of struts (each forming a regular polygon N = 9. triangular circuits (3) : cuboctahedr. 12. square circuits (3) 15. pentagon circuits (3). 5.3 Forming a tensegrity made up of several independent interweaving circuit patterns of struts (each forming a polyhedron) N= 12. tetrahedra (2) 18. tetrahedra (3)
5.2 Forming independent overlapping (or interwaving) circuits N = 6 (2 triangles) 8 (2 squares), etc.
6 Regular polygon strut patterns linked together (e.g. as tesselations) 6.1 Same polygonal shapes 6.2 Same polygonal shape arrangement about each vertex		6. Regular polyhedral (or tensegrity) forms linked together (e.g as cylindrical masts, arrays, etc.) (N.B. The compound form may be spherically symmetrical if the constituent polyhedral forms are approoriatety chosen and linked) 6.1 Same polyhedral forms. 6.2 Same polyhedral form-arrangment about link points.
6.3 Various polygonal shape arrangement about each vertex
7. Strut ends linked so as to nest one regular polygon within another the two polygons are linked by tension elements.		7. Strut ends linked so as to form regular polyhedra(or tensegrities) nested one within the other, the two structures are linked by tension elements.

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