TensegrityWiki - User contributions [en]

Mapping solid polyhedra to tensegrity structures

2026-04-24T02:14:55Z

Unhandyandy:

Read here about a proposal by Lutz Golbs to assign names of solid bodies from classical geometry to tensegrity structures.

=Introduction=

Many (man-made) tensegrity structures correlate to the shape of 'solid' bodies described originally in Greek geometry. Lutz Golbs suggests here a mapping of well-known names for this shapes to Class 1 tensegrity structures. This kind of mapping might make it easier to transfer knowledge about the behaviour and properties of certain structures to physical models.

Skelton and Oliveira proposed a wider definition of tensegrity systems, which distinguishes different classes of these structures:

"A tensegrity configuration that has no contacts between its rigid bodies is a class 1 tensegrity system, and a tensegrity system with as many as k rigid bodies in contact is a class k tensegrity system."

Regular polyhedral structures can be described by the number of vertices, edges and faces and their relation to each other. In idealised form, they are often represented by wireframe drawings, showing all edges (and therefore, showing the vertices as well, yet leaving the faces transparent). In the physical universe, the objects deviate from this idealised description. A dice has a cubical shape, although most of them have rounded corners, so technically they represent a stellated (truncated) cube. It still makes sense to 'ignore' the stellation while considering the overall behaviour of the dice, the 'cube-ness' supersedes the 'stellated-ness'.

By considering the rounded corner of a dice as the vertex of an 'ideal' cube we intellectually 'zoomed out' of the object, by making the measurement more coarse the complex shape of the rounded corner transforms for practical purposes into a vertex where equally long edges meet in the 'same' point in space.

=Deriving a construction plan from a geometric shape=

Lutz Golbs found so far two methods to 'tensegrify' regular, symmetrical polyhedra. One seems universally applicable, the other works only for the icosahedron (v=12, e=30, f=20) and the dodecahedron (v=20, e=30, f=12).

'''Method 1: Connecting non-adjacent vertices with struts. '''This method requires v/2 struts, ie 6 for the icosahedron and 10 for the dodecahedron. The tendons of a 10-strut dodecahedron share the topology of the edges of the Platonic Solid with the same name, while a tensegrity icosahedron lacks 6 edges, so that 12 trigonal faces merge into 6 rhombic faces. All vertices in a tetrahedron are adjacent to each other, which prevents building a 4-strut class 1 tensegrity.

[[file:10_strut_dodecahedron_Marcelo_Pars_tns015.jpg|thumb|500px|left|10 strut dodecahedron by Marcello Pars]]

[[file:6 strut model, expanded octahedron From Tensegrity Structures by JÃ¡uregui.png|thumb|500px|right|6 strut icosahedron]]

'''Method 2: Converting edges into struts. '''This method worked with any regular shape so far, and means to stellate all corners. The tendon then outline a stellated version of the Platonic Solid. All Platonic Solids can be build with the method. Like the rounded corner in a dice, the vertices lose their 'idealised' state and transform into a tension loop. The struts rotate around the imaginery vertex in either clockwise or counter-clockwise direction. Each strut has exactly three tendons on each end, two of which connect to the vertex loop, and one to another vertex loop. The connections from loop to loop have the same topology as the original Platonic Solid.

||~ Platonic Solid ||~ faces ||~ edges ||~ vertices ||~ Edge per vertex ||~ Struts ||~ 'Loop' tendons ||~ 'Edge' tendons ||~ Minimal tendons ||
|| Tetrahedron || 4 || 6 || 4 || 3 || 6 || 4x3=12 || 6 || 18 ||
|| Cube || 6 || 12 || 8 || 3 || 12 || 8x3=24 || 12 || 36 ||
|| Octahedron || 8 || 12 || 6 || 4 || 12 || 6x4=24 || 12 || 36 ||
|| Dodecahedron || 12 || 30 || 20 || 3 || 30 || 20x3=60 || 30 || 90 ||
|| Icosahedron || 20 || 30 || 12 || 5 || 30 || 12x5=60 || 30 || 90 ||

Cube and octahedron are 'duals', as are dodecahedron and icosahedron, ie the vertices of one shape correlate the faces of the other. Tetrahedron, cube and dodecahedron have triangular corner loops, octahedra square loops, and icosahedra pentagonal corner loops. The stellation changes the original shape of each face as well, triangular faces become hexagonal, squared faces octagonal, pentagonal faces become decagonal. The graph consisting of the network of tendons has then one additional face for each vertex of the Platonic Solid. That means duals will have the same number of struts, tendons and faces on the graph derived from the tension network.

The number of edges connecting to a vertex distinguishes duals, and the shape of faces distinguishes physical models of these special set of tensegrities. A 12-structure with 6 square faces (plus 8 hexagonal) is therefore a (stellated) octahedron, a structure with the same number of tendons and strut yet 8 triangular face (plus 6 octagonal) represents a cube.

The method to convert edges into struts works for non-Platonic Solids as well, eg Cuboctahedron, rhombic dodecahedron and prismatic shapes. Maximal symmetry requires that the struts cross each other in each corner in the same direction (chirality). Structures with mixed chirality behave much less symmetric and balance less stable.

I suggest considering tensegrity systems build with above method which show the same chirality along its faces as '''Atomic tensegrity representation of the Platonic Solids''' or shorter, '''Tensegrity Atoms.''' Obliviously, and similar to modern physics, these atoms are made of small units. Accordingly, the minimal tensegrity structure with 3 struts and 9 tendons can be described as '''tensegrity electron''', which can be positively or negatively charged, depending on the spin. The chirality of a structure means that two physical structures represent each Platonic Solid (and the tensegrity electron).

Besides introducing the ideas of tensegrity atoms and tensegrity electrons, and following their mapping to the Platonic Solids, Lutz Golbs suggests neglecting the constructive necessity to truncate corners, and follow basically the established naming for untruncated solids. Objects constructed by converting edges into struts all fit into this scheme, complex objects can be either described by identifying its module, or still be named along the imagination of the builder.

=Links and References=

[[Category:Portal to polyhedra]]
[[Category:polyhedra]]

Mapping solid polyhedra to tensegrity structures

2026-04-23T15:29:18Z

Unhandyandy: /* Links and References */

Read here about a proposal by Lutz Golbs to assign names of solid bodies from classical geometry to tensegrity structures.

=Introduction=

Many (man-made) tensegrity structures correlate to the shape of 'solid' bodies described originally in Greek geometry. Lutz Golbs suggests here a mapping of well-known names for this shapes to Class 1 tensegrity structures. This kind of mapping might make it easier to transfer knowledge about the behaviour and properties of certain structures to physical models.

Skelton and Oliveira proposed a wider definition of tensegrity systems, which distinguishes different classes of these structures:

"A tensegrity configuration that has no contacts between its rigid bodies is a class 1 tensegrity system, and a tensegrity system with as many as k rigid bodies in contact is a class k tensegrity system."

Regular polyhedral structures can be described by the number of vertices, edges and faces and their relation to each other. In idealised form, they are often represented by wireframe drawings, showing all edges (and therefore, showing the vertices as well, yet leaving the faces transparent). In the physical universe, the objects deviate from this idealised description. A dice has a cubical shape, although most of them have rounded corners, so technically they represent a stellated (truncated) cube. It still makes sense to 'ignore' the stellation while considering the overall behaviour of the dice, the 'cube-ness' supersedes the 'stellated-ness'.

By considering the rounded corner of a dice as the vertex of an 'ideal' cube we intellectually 'zoomed out' of the object, by making the measurement more coarse the complex shape of the rounded corner transforms for practical purposes into a vertex where equally long edges meet in the 'same' point in space.

=Deriving a construction plan from a geometric shape=

Lutz Golbs found so far two methods to 'tensegrify' regular, symmetrical polyhedra. One seems universally applicable, the other works only for the icosahedron (v=12, e=30, f=20) and the dodecahedron (v=20, e=30, f=12).

'''Method 1: Connecting non-adjacent vertices with struts. '''This method requires v/2 struts, ie 6 for the icosahedron and 10 for the dodecahedron. The tendons of a 10-strut dodecahedron share the topology of the edges of the Platonic Solid with the same name, while a tensegrity icosahedron lacks 6 edges, so that 12 trigonal faces merge into 6 rhombic faces. All vertices in a tetrahedron are adjacent to each other, which prevents building a 4-strut class 1 tensegrity.

[[file:10_strut_dodecahedron_Marcelo_Pars_tns015.jpg|thumb|500px|left|10 strut dodecahedron by Marcello Pars"]]

[[file:6 strut model, expanded octahedron From Tensegrity Structures by JÃ¡uregui.png|thumb|500px|right|6 strut icosahedron"]]

'''Method 2: Converting edges into struts. '''This method worked with any regular shape so far, and means to stellate all corners. The tendon then outline a stellated version of the Platonic Solid. All Platonic Solids can be build with the method. Like the rounded corner in a dice, the vertices lose their 'idealised' state and transform into a tension loop. The struts rotate around the imaginery vertex in either clockwise or counter-clockwise direction. Each strut has exactly three tendons on each end, two of which connect to the vertex loop, and one to another vertex loop. The connections from loop to loop have the same topology as the original Platonic Solid.

||~ Platonic Solid ||~ faces ||~ edges ||~ vertices ||~ Edge per vertex ||~ Struts ||~ 'Loop' tendons ||~ 'Edge' tendons ||~ Minimal tendons ||
|| Tetrahedron || 4 || 6 || 4 || 3 || 6 || 4x3=12 || 6 || 18 ||
|| Cube || 6 || 12 || 8 || 3 || 12 || 8x3=24 || 12 || 36 ||
|| Octahedron || 8 || 12 || 6 || 4 || 12 || 6x4=24 || 12 || 36 ||
|| Dodecahedron || 12 || 30 || 20 || 3 || 30 || 20x3=60 || 30 || 90 ||
|| Icosahedron || 20 || 30 || 12 || 5 || 30 || 12x5=60 || 30 || 90 ||

Cube and octahedron are 'duals', as are dodecahedron and icosahedron, ie the vertices of one shape correlate the faces of the other. Tetrahedron, cube and dodecahedron have triangular corner loops, octahedra square loops, and icosahedra pentagonal corner loops. The stellation changes the original shape of each face as well, triangular faces become hexagonal, squared faces octagonal, pentagonal faces become decagonal. The graph consisting of the network of tendons has then one additional face for each vertex of the Platonic Solid. That means duals will have the same number of struts, tendons and faces on the graph derived from the tension network.

The number of edges connecting to a vertex distinguishes duals, and the shape of faces distinguishes physical models of these special set of tensegrities. A 12-structure with 6 square faces (plus 8 hexagonal) is therefore a (stellated) octahedron, a structure with the same number of tendons and strut yet 8 triangular face (plus 6 octagonal) represents a cube.

The method to convert edges into struts works for non-Platonic Solids as well, eg Cuboctahedron, rhombic dodecahedron and prismatic shapes. Maximal symmetry requires that the struts cross each other in each corner in the same direction (chirality). Structures with mixed chirality behave much less symmetric and balance less stable.

I suggest considering tensegrity systems build with above method which show the same chirality along its faces as '''Atomic tensegrity representation of the Platonic Solids''' or shorter, '''Tensegrity Atoms.''' Obliviously, and similar to modern physics, these atoms are made of small units. Accordingly, the minimal tensegrity structure with 3 struts and 9 tendons can be described as '''tensegrity electron''', which can be positively or negatively charged, depending on the spin. The chirality of a structure means that two physical structures represent each Platonic Solid (and the tensegrity electron).

Besides introducing the ideas of tensegrity atoms and tensegrity electrons, and following their mapping to the Platonic Solids, Lutz Golbs suggests neglecting the constructive necessity to truncate corners, and follow basically the established naming for untruncated solids. Objects constructed by converting edges into struts all fit into this scheme, complex objects can be either described by identifying its module, or still be named along the imagination of the builder.

=Links and References=

[[Category:Portal to polyhedra]]
[[Category:polyhedra]]

Mapping solid polyhedra to tensegrity structures

2026-04-23T15:27:12Z

Unhandyandy:

Read here about a proposal by Lutz Golbs to assign names of solid bodies from classical geometry to tensegrity structures.

=Introduction=

Many (man-made) tensegrity structures correlate to the shape of 'solid' bodies described originally in Greek geometry. Lutz Golbs suggests here a mapping of well-known names for this shapes to Class 1 tensegrity structures. This kind of mapping might make it easier to transfer knowledge about the behaviour and properties of certain structures to physical models.

Skelton and Oliveira proposed a wider definition of tensegrity systems, which distinguishes different classes of these structures:

"A tensegrity configuration that has no contacts between its rigid bodies is a class 1 tensegrity system, and a tensegrity system with as many as k rigid bodies in contact is a class k tensegrity system."

Regular polyhedral structures can be described by the number of vertices, edges and faces and their relation to each other. In idealised form, they are often represented by wireframe drawings, showing all edges (and therefore, showing the vertices as well, yet leaving the faces transparent). In the physical universe, the objects deviate from this idealised description. A dice has a cubical shape, although most of them have rounded corners, so technically they represent a stellated (truncated) cube. It still makes sense to 'ignore' the stellation while considering the overall behaviour of the dice, the 'cube-ness' supersedes the 'stellated-ness'.

By considering the rounded corner of a dice as the vertex of an 'ideal' cube we intellectually 'zoomed out' of the object, by making the measurement more coarse the complex shape of the rounded corner transforms for practical purposes into a vertex where equally long edges meet in the 'same' point in space.

=Deriving a construction plan from a geometric shape=

Lutz Golbs found so far two methods to 'tensegrify' regular, symmetrical polyhedra. One seems universally applicable, the other works only for the icosahedron (v=12, e=30, f=20) and the dodecahedron (v=20, e=30, f=12).

'''Method 1: Connecting non-adjacent vertices with struts. '''This method requires v/2 struts, ie 6 for the icosahedron and 10 for the dodecahedron. The tendons of a 10-strut dodecahedron share the topology of the edges of the Platonic Solid with the same name, while a tensegrity icosahedron lacks 6 edges, so that 12 trigonal faces merge into 6 rhombic faces. All vertices in a tetrahedron are adjacent to each other, which prevents building a 4-strut class 1 tensegrity.

[[file:10_strut_dodecahedron_Marcelo_Pars_tns015.jpg|thumb|500px|left|10 strut dodecahedron by Marcello Pars"]]

[[file:6 strut model, expanded octahedron From Tensegrity Structures by JÃ¡uregui.png|thumb|500px|right|6 strut icosahedron"]]

'''Method 2: Converting edges into struts. '''This method worked with any regular shape so far, and means to stellate all corners. The tendon then outline a stellated version of the Platonic Solid. All Platonic Solids can be build with the method. Like the rounded corner in a dice, the vertices lose their 'idealised' state and transform into a tension loop. The struts rotate around the imaginery vertex in either clockwise or counter-clockwise direction. Each strut has exactly three tendons on each end, two of which connect to the vertex loop, and one to another vertex loop. The connections from loop to loop have the same topology as the original Platonic Solid.

||~ Platonic Solid ||~ faces ||~ edges ||~ vertices ||~ Edge per vertex ||~ Struts ||~ 'Loop' tendons ||~ 'Edge' tendons ||~ Minimal tendons ||
|| Tetrahedron || 4 || 6 || 4 || 3 || 6 || 4x3=12 || 6 || 18 ||
|| Cube || 6 || 12 || 8 || 3 || 12 || 8x3=24 || 12 || 36 ||
|| Octahedron || 8 || 12 || 6 || 4 || 12 || 6x4=24 || 12 || 36 ||
|| Dodecahedron || 12 || 30 || 20 || 3 || 30 || 20x3=60 || 30 || 90 ||
|| Icosahedron || 20 || 30 || 12 || 5 || 30 || 12x5=60 || 30 || 90 ||

Cube and octahedron are 'duals', as are dodecahedron and icosahedron, ie the vertices of one shape correlate the faces of the other. Tetrahedron, cube and dodecahedron have triangular corner loops, octahedra square loops, and icosahedra pentagonal corner loops. The stellation changes the original shape of each face as well, triangular faces become hexagonal, squared faces octagonal, pentagonal faces become decagonal. The graph consisting of the network of tendons has then one additional face for each vertex of the Platonic Solid. That means duals will have the same number of struts, tendons and faces on the graph derived from the tension network.

The number of edges connecting to a vertex distinguishes duals, and the shape of faces distinguishes physical models of these special set of tensegrities. A 12-structure with 6 square faces (plus 8 hexagonal) is therefore a (stellated) octahedron, a structure with the same number of tendons and strut yet 8 triangular face (plus 6 octagonal) represents a cube.

The method to convert edges into struts works for non-Platonic Solids as well, eg Cuboctahedron, rhombic dodecahedron and prismatic shapes. Maximal symmetry requires that the struts cross each other in each corner in the same direction (chirality). Structures with mixed chirality behave much less symmetric and balance less stable.

I suggest considering tensegrity systems build with above method which show the same chirality along its faces as '''Atomic tensegrity representation of the Platonic Solids''' or shorter, '''Tensegrity Atoms.''' Obliviously, and similar to modern physics, these atoms are made of small units. Accordingly, the minimal tensegrity structure with 3 struts and 9 tendons can be described as '''tensegrity electron''', which can be positively or negatively charged, depending on the spin. The chirality of a structure means that two physical structures represent each Platonic Solid (and the tensegrity electron).

Besides introducing the ideas of tensegrity atoms and tensegrity electrons, and following their mapping to the Platonic Solids, Lutz Golbs suggests neglecting the constructive necessity to truncate corners, and follow basically the established naming for untruncated solids. Objects constructed by converting edges into struts all fit into this scheme, complex objects can be either described by identifying its module, or still be named along the imagination of the builder.

=Links and References=

Lutz Golbs: [http://smart-at.info/drupal/ website]; he is also a user of the [http://alextech.wikia.com/wiki/User:Lutz_Golbs Alexander Technique Wiki].
Martin Friedrich Eichenauer and Daniel Lordick: [https://pdfs.semanticscholar.org/334a/04b729632f0108cea7f3a02514b5c834555f.pdf]

[[Category:Portal to polyhedra]]
[[Category:polyhedra]]

Icosahedron

2026-04-22T17:48:11Z

Unhandyandy:

Read here about the icosahedron, one of the Platonic polyhedra, and the dual of the dodecahedron. Some of the most significant tensegrity structures are argued to be congruent to the icosahedron.

=Overview=

The icosahedron polyhedron has 30 edges, 20 faces and 12 vertexes. A tensegrity that outlines this polyhedron has 6 struts and 24 tensile vectors.

This tensegrity structure is one of the few tensegrities that exhibit mirror symmetry. Burkhardt wrote, its "tendon network would mark out a cuboctahedron if the (non-planar) quadrilaterals in which the struts are nested were changed to squares. The struts are inserted as the diagonals of these squares so each strut is parallel to the strut in the opposite square and so no strut shares a vertex with another strut. This tensegrity is classified as a “diamond” type because each strut is surrounded by a diamond of four.

[[file:icosahedron_by_Burkhardt.png| frame | left | Tensegrity structure of 6 struts and 24 tensile vectors that conforms to the classic icosahedron. From Practical Guide to Tensegrity Design 2nd edition © 2008 by Burkhardt.]]

The icosahedron-like tensegrity structure is argued to be the most significant of all the structures, particularly by researchers in bio-tensegrity of the fascia and muscle system. For example, see the right upper human extremity modelled as a sequence of interconnecting icosahedral tensegrities with compression struts of different lengths, below.

[[file:Right_upper_extremity_icosahedral_model_by_Flemons.png|frame|right|
Right upper extremity modeled as a sequence of interconnecting icosahedral tensegrities with compression struts of different lengths. Model: Graham Scarr as reproduced in Simple Geometry in Complex Organisms 2010. http://www.tensegrityinbiology.co.uk/publications/geometry/.]]

=Comparing the icosahedron to other tensegrities=

Motro published a detailed comparison between the polyhedron and its associated tensegrity structure. “The two geometries can be compared on basis of the ratio between the length of struts “s” and the distance between two parallel struts “d”. For the icosahedron this ratio is equal to approximately 1.618 (that is the “golden” ratio), for the associated tensegrity system it is equal to exactly 2. This resulting tensegrity system can be seen as the expansion of an octahedron, since there are at the end eight triangles of cables (the same as the number of triangular faces for an octahedron), and the three pairs of struts can be understood as the splitting of the three internal diagonals.”

[[file:Comparison_of_icosahedron_and_tensegrity_with_phi_by_Motro.gif|frame|left|
Comparison between the icosahedron polyhedron and its associated tensegrity structure. The blue lengths are the strut length ''s'', and the red lengths are the distance ''d''. In a regular icosahedron, ''s''/''d'' = phi, the golden ratio; in a tensegrity icosahedron, ''s''/''d'' = 2. From Structural Morphology Of Tensegrity Systems by Motro.]]

[[file:Comparison_of_icosahedron_tensegrity_and_polyhedron_by_Tibert.gif|frame|right|
Comparison of a tensegrity icosahedron and a polyhedron with which it conforms. Side view, left, and top view, right. The polyhedron is drawn with dashed lines. From Deployable Tensegrity Structures for Space Applications by Tibert.]]
William Brooks Whittier also discusses this tensegrity, calling it a T-6. It has 6 struts and 24 tensile vectors. It outlines an icosahedron with 30 edges, 20 faces, and 12 vertexes.

[[file:6_strut_icosahedron_By_Whittier.gif|frame|left|
Tensegrity structure that outlines an icosahedron. In order from left to right two perspective views and a top view perpendicular to one strut pair. From Kinematic Analysis of Tensegrity Structures By William Brooks Whittier.]]
[[Gómez Jáuregui, Valentín|Gómez Jáuregui]] published a photo of a tensegritoy model he built that conforms with a truncated icosahedron.

[[Motro, René| Motro ]]discusses a tensegrity structure related to the icosahedron. He wrote, “Since it is not possible to design a regular icosahedron with six equal struts, we tried to build one with six struts, one of them being greater than the five others. The basis of this design is a prismatic pentagonal system; a central strut is placed on the vertical symmetry axis. This axis becomes a rotation axis. The lengths of the struts and of the cables are calculated in order to reach an equilibrium state which is characterized by the fact that the twelve nodes occupy the geometrical position of the apices of an icosahedron. The name is chosen by reference to this axis of rotation and to the icosahedron. This system can be classified as a “Z” like tensegrity system according to the classification submitted by Anthony Pugh. There are only two cables and one strut at each node, except for the central strut.

[[file:Spinning_icosahedron_by_Motro.gif|frame|left|
Spinning icosahedron tensegrity structure, perspective and plane view. This can be classified a “Z” tensegrity system in the Pugh system. From Structural Morphology Of Tensegrity Systems by Motro.]]

=Discussion of the importance of the icosahedron=

==Is the icosahedron tensegrity the critical, primitive building block of biological tensegrity?==
[[Pars, Marcelo|Pars]] wrote: The icosahedron-tensegrity has unique characteristics that tensegrities normally never have: The struts are exactly parallel or in straight angle to eachother. This is very rare for a tensegrity which can be distinguished by its twisted form. I think it may be these straight angles that make this tensegrity so popular. Wherever we humans changed the world we used straight angles, and although it may not be a natural form, for us a straight angle is restful and feels normal. I guess one could say that from all tensegrities the icosahedron tensegrity looks the most like a conventional construction.
I made quite a few tensegrities, but the icosahedron was the first, also because mathematics behind it is more easy than for instance the 3 strut prism.

But all this doesn't say anything about nature or biotensegrity of course. If you ask me, I guess nature is full of tensegrities and the icosahedron is the bridge between nature and it's construction methods and us humans with our conventional bricks..

[[Levin, Stephen M.|Levin]]: My understanding of [[Fuller, Richard Buckminster|Fuller]]: there are only three structures that are stable with flexible joints, the [[tetrahedron]], [[octahedron]] and icosahedron. Therefore, biologic structures,( all joints from cell to organism are flexible, held by surface tension, integrins, mucus, ligaments etc.), must be constructed from some combination or permutation of those structures. For reasons previously stated, icosahedrons are the most biologically suited, closest packing, volume for surface area, least energy, etc. Icosahedrons are the most symmetrical of all polyhedrons, it is the most symmetrical system for the subdivision of a spherical surface into modular units and all spherical biologic structures must be icosahedrons to be stable or propped up by the adjacent structure so that, together, they create a stable structure,(fractals?), like bubbles in a foam.

The reasons, (amongst others), the icosahedron-tensegrity would work in biotensegrity is because it close packs and is self-organizing, and the most symmetrical, not because of the parallel struts. It is omnidirectional and has the largest volume for surface area and is the most energy efficient structure with the above characteristics. The icosahedron has already been recognized as the basic structure of carbon 60, viruses, cells, and much more in biology, (there are now well over 2000 scientific articles linking tensegrity and biology). [[3 Struts|3-strut tensegrities]] have a larger surface area for volume, therefore, less energy efficient, and I don't think they close pack. I doubt if they would be self-organizing in nature.

[[Burkhardt, Robert|Burkhardt]]: I can't see the tensegrity icosahedron as a building block for cell structure. But maybe this is just a question of semantics. Bucky did look at it as a building block, for example in [http://www.rwgrayprojects.com/synergetics/s07/p8100.html#784.00|Section 784 of Synergetics I]], but I can't see that work as a model for a cell. For a model of cell walls I would be more comfortable with an approach that talks about a pattern of which the t-icosa (or perhaps della Sala's t-tetrahedron) is the simplest example. Bucky's development of that pattern into more complex single-layer domes or spheres is perhaps a suitable analogy for a cell wall. This approach is what I see in [http://www.rwgrayprojects.com/synergetics/s07/figs/f1701.html|Figure 717.01 of Synergetics I]], but there is a variation here in that the tendons also appear to be attached to the middle of the strut.

[[file:Synergetics_Fig._717.01_Single_and_Double_Bonding_of_Members_in_Tensegrity_Spheres_CROPPED.gif|frame|right|
Detail of Synergetics Fig. 717.01 Single and Double Bonding of Members in Tensegrity Spheres. Copyright © 1997 Estate of R. Buckminster Fuller.]]

Certainly the pattern of the t-icosa has been developed more directly into higher-frequency structures without the tendons being attached to the middle of a strut. This is the approach Kenner develops, and also it is the approach of some of the tensegrity spheres developed by Bucky and his collaborators, although the latter that I see in Dymaxion World of Buckminster Fuller seem to favor the zig-zag approach of della Sala's t-tetrahedron, rather than the diamond approach represented by the t-icosa.

==Unlike close packed spheres, close packing of icosahedra is not well defined==

The notion of close-packing icosahedrons is problematic. [[de Jong, Gerald|de Jong]] notes: I don't feel that this has been described explicitly enough. [[Levin, Stephen M.|Steve]] states that they close pack, like spheres where the middle one surrounded by twelve others nicely shrinks to accommodate, but the description of how one touches the next seems to be overlooked. Two tensegrities join to become a single tensegrity really only when cables meet bars and vice versa.

=Real Icosahedron=
The outline of the six-strut tensegrity as described above does not match the icosahedron as one of the platonic solids because the ratio between the length of struts “s” and the distance between two parallel struts “d” is exactly 2, where for the icosahedron this ratio is equal to approximately 1.618 (that is the “golden” ratio). There has thus been much controversy over the years about the use of this term in tensegrity, with many authors preferring to label it as an expanded octahedron.

[[File:Jessen's icosahedron.jpg|thumb|left]]Borge Jessen described a shape that he referred to as the Orthogonal Icosahedron, where all the faces meet at 90 degrees and match the outline of this 6-strut tensegrity. The card model shown expands and contracts within a similar range to the tensegrity icosahedron. Jessen’s icosahedron is described in:
[https://en.wikipedia.org/wiki/Jessen%27s_icosahedron Wikipedia]
[https://www.youtube.com/watch?v=9-HnJ9n6F20&ab_channel=XYZAidan YouTube]
[https://mathworld.wolfram.com/JessensOrthogonalIcosahedron.html Wolfram Maths]
although a correction to the latter is reported [https://flexiblepolyhedron.wordpress.com/2013/02/09/correcting-the-jessens-orthogonal-icosahedron here]
with more detailed discussion [http://maths.ac-noumea.nc/polyhedr/stuff/shaky_engl-small.pdf here].

Not very well-known are the two tensegrities shown below and invented by [[Pars, Marcelo|Pars]]. The outside of these structures are exact icosahedrons.

[http://www.tensegriteit.nl/afbeelding/tensegrity210.jpg tensegrity210]

[http://www.tensegriteit.nl/afbeelding/tensegrity207.jpg tensegrity207]

The tensegrities can also be seen on [http://www.tensegriteit.nl/e-icosahedron.html Marcelo Pars' icosahedrons]. 3D images of this real icosahedron are shown on [http://www.tensegriteit.nl/e-3dimages.html|Marcelo Pars' 3D images] . Here a small picture of the icosahedron tensegrity without the struts:

[http://www.tensegriteit.nl/afbeelding/vmrlicosahedron03.png vmrlicosahedron03]

[[Category:Portal To Polyhedra]]
[[Category:polyhedron]]

Icosahedron

2026-04-22T17:45:57Z

Unhandyandy:

Read here about the icosahedron, one of the Platonic polyhedra, and the dual of the dodecahedron. Some of the most significant tensegrity structures are argued to be congruent to the icosahedron.

=Overview=

The icosahedron polyhedron has 30 edges, 20 faces and 12 vertexes. A tensegrity that outlines this polyhedron has 6 struts and 24 tensile vectors.

This tensegrity structure is one of the few tensegrities that exhibit mirror symmetry. Burkhardt wrote, its "tendon network would mark out a cuboctahedron if the (non-planar) quadrilaterals in which the struts are nested were changed to squares. The struts are inserted as the diagonals of these squares so each strut is parallel to the strut in the opposite square and so no strut shares a vertex with another strut. This tensegrity is classified as a “diamond” type because each strut is surrounded by a diamond of four.

[[file:icosahedron_by_Burkhardt.png| frame | left | Tensegrity structure of 6 struts and 24 tensile vectors that conforms to the classic icosahedron. From Practical Guide to Tensegrity Design 2nd edition © 2008 by Burkhardt.]]

The icosahedron-like tensegrity structure is argued to be the most significant of all the structures, particularly by researchers in bio-tensegrity of the fascia and muscle system. For example, see the right upper human extremity modelled as a sequence of interconnecting icosahedral tensegrities with compression struts of different lengths, below.

[[file:Right_upper_extremity_icosahedral_model_by_Flemons.png|frame|right|
Right upper extremity modeled as a sequence of interconnecting icosahedral tensegrities with compression struts of different lengths. Model: Graham Scarr as reproduced in Simple Geometry in Complex Organisms 2010. http://www.tensegrityinbiology.co.uk/publications/geometry/.]]

=Comparing the icosahedron to other tensegrities=

Motro published a detailed comparison between the polyhedron and its associated tensegrity structure. “The two geometries can be compared on basis of the ratio between the length of struts “s” and the distance between two parallel struts “d”. For the icosahedron this ratio is equal to approximately 1.618 (that is the “golden” ratio), for the associated tensegrity system it is equal to exactly 2. This resulting tensegrity system can be seen as the expansion of an octahedron, since there are at the end eight triangles of cables (the same as the number of triangular faces for an octahedron), and the three pairs of struts can be understood as the splitting of the three internal diagonals.”

[[file:Comparison_of_icosahedron_and_tensegrity_with_phi_by_Motro.gif|frame|left|
Comparison between the icosahedron polyhedron and its associated tensegrity structure. The blue lengths are the strut length ''s'', and the red lengths are the distance ''d''. phi, the golden ratio. From Structural Morphology Of Tensegrity Systems by Motro.]]

[[file:Comparison_of_icosahedron_tensegrity_and_polyhedron_by_Tibert.gif|frame|right|
Comparison of a tensegrity icosahedron and a polyhedron with which it conforms. Side view, left, and top view, right. The polyhedron is drawn with dashed lines. From Deployable Tensegrity Structures for Space Applications by Tibert.]]
William Brooks Whittier also discusses this tensegrity, calling it a T-6. It has 6 struts and 24 tensile vectors. It outlines an icosahedron with 30 edges, 20 faces, and 12 vertexes.

[[file:6_strut_icosahedron_By_Whittier.gif|frame|left|
Tensegrity structure that outlines an icosahedron. In order from left to right two perspective views and a top view perpendicular to one strut pair. From Kinematic Analysis of Tensegrity Structures By William Brooks Whittier.]]
[[Gómez Jáuregui, Valentín|Gómez Jáuregui]] published a photo of a tensegritoy model he built that conforms with a truncated icosahedron.

[[Motro, René| Motro ]]discusses a tensegrity structure related to the icosahedron. He wrote, “Since it is not possible to design a regular icosahedron with six equal struts, we tried to build one with six struts, one of them being greater than the five others. The basis of this design is a prismatic pentagonal system; a central strut is placed on the vertical symmetry axis. This axis becomes a rotation axis. The lengths of the struts and of the cables are calculated in order to reach an equilibrium state which is characterized by the fact that the twelve nodes occupy the geometrical position of the apices of an icosahedron. The name is chosen by reference to this axis of rotation and to the icosahedron. This system can be classified as a “Z” like tensegrity system according to the classification submitted by Anthony Pugh. There are only two cables and one strut at each node, except for the central strut.

[[file:Spinning_icosahedron_by_Motro.gif|frame|left|
Spinning icosahedron tensegrity structure, perspective and plane view. This can be classified a “Z” tensegrity system in the Pugh system. From Structural Morphology Of Tensegrity Systems by Motro.]]

=Discussion of the importance of the icosahedron=

==Is the icosahedron tensegrity the critical, primitive building block of biological tensegrity?==
[[Pars, Marcelo|Pars]] wrote: The icosahedron-tensegrity has unique characteristics that tensegrities normally never have: The struts are exactly parallel or in straight angle to eachother. This is very rare for a tensegrity which can be distinguished by its twisted form. I think it may be these straight angles that make this tensegrity so popular. Wherever we humans changed the world we used straight angles, and although it may not be a natural form, for us a straight angle is restful and feels normal. I guess one could say that from all tensegrities the icosahedron tensegrity looks the most like a conventional construction.
I made quite a few tensegrities, but the icosahedron was the first, also because mathematics behind it is more easy than for instance the 3 strut prism.

But all this doesn't say anything about nature or biotensegrity of course. If you ask me, I guess nature is full of tensegrities and the icosahedron is the bridge between nature and it's construction methods and us humans with our conventional bricks..

[[Levin, Stephen M.|Levin]]: My understanding of [[Fuller, Richard Buckminster|Fuller]]: there are only three structures that are stable with flexible joints, the [[tetrahedron]], [[octahedron]] and icosahedron. Therefore, biologic structures,( all joints from cell to organism are flexible, held by surface tension, integrins, mucus, ligaments etc.), must be constructed from some combination or permutation of those structures. For reasons previously stated, icosahedrons are the most biologically suited, closest packing, volume for surface area, least energy, etc. Icosahedrons are the most symmetrical of all polyhedrons, it is the most symmetrical system for the subdivision of a spherical surface into modular units and all spherical biologic structures must be icosahedrons to be stable or propped up by the adjacent structure so that, together, they create a stable structure,(fractals?), like bubbles in a foam.

The reasons, (amongst others), the icosahedron-tensegrity would work in biotensegrity is because it close packs and is self-organizing, and the most symmetrical, not because of the parallel struts. It is omnidirectional and has the largest volume for surface area and is the most energy efficient structure with the above characteristics. The icosahedron has already been recognized as the basic structure of carbon 60, viruses, cells, and much more in biology, (there are now well over 2000 scientific articles linking tensegrity and biology). [[3 Struts|3-strut tensegrities]] have a larger surface area for volume, therefore, less energy efficient, and I don't think they close pack. I doubt if they would be self-organizing in nature.

[[Burkhardt, Robert|Burkhardt]]: I can't see the tensegrity icosahedron as a building block for cell structure. But maybe this is just a question of semantics. Bucky did look at it as a building block, for example in [http://www.rwgrayprojects.com/synergetics/s07/p8100.html#784.00|Section 784 of Synergetics I]], but I can't see that work as a model for a cell. For a model of cell walls I would be more comfortable with an approach that talks about a pattern of which the t-icosa (or perhaps della Sala's t-tetrahedron) is the simplest example. Bucky's development of that pattern into more complex single-layer domes or spheres is perhaps a suitable analogy for a cell wall. This approach is what I see in [http://www.rwgrayprojects.com/synergetics/s07/figs/f1701.html|Figure 717.01 of Synergetics I]], but there is a variation here in that the tendons also appear to be attached to the middle of the strut.

[[file:Synergetics_Fig._717.01_Single_and_Double_Bonding_of_Members_in_Tensegrity_Spheres_CROPPED.gif|frame|right|
Detail of Synergetics Fig. 717.01 Single and Double Bonding of Members in Tensegrity Spheres. Copyright © 1997 Estate of R. Buckminster Fuller.]]

Certainly the pattern of the t-icosa has been developed more directly into higher-frequency structures without the tendons being attached to the middle of a strut. This is the approach Kenner develops, and also it is the approach of some of the tensegrity spheres developed by Bucky and his collaborators, although the latter that I see in Dymaxion World of Buckminster Fuller seem to favor the zig-zag approach of della Sala's t-tetrahedron, rather than the diamond approach represented by the t-icosa.

==Unlike close packed spheres, close packing of icosahedra is not well defined==

The notion of close-packing icosahedrons is problematic. [[de Jong, Gerald|de Jong]] notes: I don't feel that this has been described explicitly enough. [[Levin, Stephen M.|Steve]] states that they close pack, like spheres where the middle one surrounded by twelve others nicely shrinks to accommodate, but the description of how one touches the next seems to be overlooked. Two tensegrities join to become a single tensegrity really only when cables meet bars and vice versa.

=Real Icosahedron=
The outline of the six-strut tensegrity as described above does not match the icosahedron as one of the platonic solids because the ratio between the length of struts “s” and the distance between two parallel struts “d” is exactly 2, where for the icosahedron this ratio is equal to approximately 1.618 (that is the “golden” ratio). There has thus been much controversy over the years about the use of this term in tensegrity, with many authors preferring to label it as an expanded octahedron.

[[File:Jessen's icosahedron.jpg|thumb|left]]Borge Jessen described a shape that he referred to as the Orthogonal Icosahedron, where all the faces meet at 90 degrees and match the outline of this 6-strut tensegrity. The card model shown expands and contracts within a similar range to the tensegrity icosahedron. Jessen’s icosahedron is described in:
[https://en.wikipedia.org/wiki/Jessen%27s_icosahedron Wikipedia]
[https://www.youtube.com/watch?v=9-HnJ9n6F20&ab_channel=XYZAidan YouTube]
[https://mathworld.wolfram.com/JessensOrthogonalIcosahedron.html Wolfram Maths]
although a correction to the latter is reported [https://flexiblepolyhedron.wordpress.com/2013/02/09/correcting-the-jessens-orthogonal-icosahedron here]
with more detailed discussion [http://maths.ac-noumea.nc/polyhedr/stuff/shaky_engl-small.pdf here].

Not very well-known are the two tensegrities shown below and invented by [[Pars, Marcelo|Pars]]. The outside of these structures are exact icosahedrons.

[http://www.tensegriteit.nl/afbeelding/tensegrity210.jpg tensegrity210]

[http://www.tensegriteit.nl/afbeelding/tensegrity207.jpg tensegrity207]

The tensegrities can also be seen on [http://www.tensegriteit.nl/e-icosahedron.html Marcelo Pars' icosahedrons]. 3D images of this real icosahedron are shown on [http://www.tensegriteit.nl/e-3dimages.html|Marcelo Pars' 3D images] . Here a small picture of the icosahedron tensegrity without the struts:

[http://www.tensegriteit.nl/afbeelding/vmrlicosahedron03.png vmrlicosahedron03]

[[Category:Portal To Polyhedra]]
[[Category:polyhedron]]

Icosahedron

2026-04-22T17:31:39Z

Unhandyandy: tidied up images displays; deleted dead image link (tensegritoy); changed image links at bottom of page to http links.

Read here about the icosahedron, one of the Platonic polyhedra, and the dual of the dodecahedron. Some of the most significant tensegrity structures are argued to be congruent to the icosahedron.

=Overview=

The icosahedron polyhedron has 30 edges, 20 faces and 12 vertexes. A tensegrity that outlines this polyhedron has 6 struts and 24 tensile vectors.

This tensegrity structure is one of the few tensegrities that exhibit mirror symmetry. Burkhardt wrote, its "tendon network would mark out a cuboctahedron if the (non-planar) quadrilaterals in which the struts are nested were changed to squares. The struts are inserted as the diagonals of these squares so each strut is parallel to the strut in the opposite square and so no strut shares a vertex with another strut. This tensegrity is classified as a “diamond” type because each strut is surrounded by a diamond of four.

[[file:icosahedron_by_Burkhardt.png| frame | left | Tensegrity structure of 6 struts and 24 tensile vectors that conforms to the classic icosahedron. From Practical Guide to Tensegrity Design 2nd edition © 2008 by Burkhardt.]]

The icosahedron-like tensegrity structure is argued to be the most significant of all the structures, particularly by researchers in bio-tensegrity of the fascia and muscle system. For example, see the right upper human extremity modelled as a sequence of interconnecting icosahedral tensegrities with compression struts of different lengths, below.

[[file:Right_upper_extremity_icosahedral_model_by_Flemons.png|frame|right|
Right upper extremity modeled as a sequence of interconnecting icosahedral tensegrities with compression struts of different lengths. Model: Graham Scarr as reproduced in Simple Geometry in Complex Organisms 2010. http://www.tensegrityinbiology.co.uk/publications/geometry/.]]

=Comparing the icosahedron to other tensegrities=

Motro published a detailed comparison between the polyhedron and its associated tensegrity structure. “The two geometries can be compared on basis of the ratio between the length of struts “s” and the distance between two parallel struts “d”. For the icosahedron this ratio is equal to approximately 1.618 (that is the “golden” ratio), for the associated tensegrity system it is equal to exactly 2. This resulting tensegrity system can be seen as the expansion of an octahedron, since there are at the end eight triangles of cables (the same as the number of triangular faces for an octahedron), and the three pairs of struts can be understood as the splitting of the three internal diagonals.”

[[file:Comparison_of_icosahedron_and_tensegrity_with_phi_by_Motro.gif|frame|left|
Comparison between the icosahedron polyhedron and its associated tensegrity structure. “S” is phi, the golden ratio. From Structural Morphology Of Tensegrity Systems by Motro.]]

[[file:Comparison_of_icosahedron_tensegrity_and_polyhedron_by_Tibert.gif|frame|right|
Comparison of a tensegrity icosahedron and a polyhedron with which it conforms. Side view, left, and top view, right. The polyhedron is drawn with dashed lines. From Deployable Tensegrity Structures for Space Applications by Tibert.]]
William Brooks Whittier also discusses this tensegrity, calling it a T-6. It has 6 struts and 24 tensile vectors. It outlines an icosahedron with 30 edges, 20 faces, and 12 vertexes.

[[file:6_strut_icosahedron_By_Whittier.gif|frame|left|
Tensegrity structure that outlines an icosahedron. In order from left to right two perspective views and a top view perpendicular to one strut pair. From Kinematic Analysis of Tensegrity Structures By William Brooks Whittier.]]
[[Gómez Jáuregui, Valentín|Gómez Jáuregui]] published a photo of a tensegritoy model he built that conforms with a truncated icosahedron.

[[Motro, René| Motro ]]discusses a tensegrity structure related to the icosahedron. He wrote, “Since it is not possible to design a regular icosahedron with six equal struts, we tried to build one with six struts, one of them being greater than the five others. The basis of this design is a prismatic pentagonal system; a central strut is placed on the vertical symmetry axis. This axis becomes a rotation axis. The lengths of the struts and of the cables are calculated in order to reach an equilibrium state which is characterized by the fact that the twelve nodes occupy the geometrical position of the apices of an icosahedron. The name is chosen by reference to this axis of rotation and to the icosahedron. This system can be classified as a “Z” like tensegrity system according to the classification submitted by Anthony Pugh. There are only two cables and one strut at each node, except for the central strut.

[[file:Spinning_icosahedron_by_Motro.gif|frame|left|
Spinning icosahedron tensegrity structure, perspective and plane view. This can be classified a “Z” tensegrity system in the Pugh system. From Structural Morphology Of Tensegrity Systems by Motro.]]

=Discussion of the importance of the icosahedron=

==Is the icosahedron tensegrity the critical, primitive building block of biological tensegrity?==
[[Pars, Marcelo|Pars]] wrote: The icosahedron-tensegrity has unique characteristics that tensegrities normally never have: The struts are exactly parallel or in straight angle to eachother. This is very rare for a tensegrity which can be distinguished by its twisted form. I think it may be these straight angles that make this tensegrity so popular. Wherever we humans changed the world we used straight angles, and although it may not be a natural form, for us a straight angle is restful and feels normal. I guess one could say that from all tensegrities the icosahedron tensegrity looks the most like a conventional construction.
I made quite a few tensegrities, but the icosahedron was the first, also because mathematics behind it is more easy than for instance the 3 strut prism.

But all this doesn't say anything about nature or biotensegrity of course. If you ask me, I guess nature is full of tensegrities and the icosahedron is the bridge between nature and it's construction methods and us humans with our conventional bricks..

[[Levin, Stephen M.|Levin]]: My understanding of [[Fuller, Richard Buckminster|Fuller]]: there are only three structures that are stable with flexible joints, the [[tetrahedron]], [[octahedron]] and icosahedron. Therefore, biologic structures,( all joints from cell to organism are flexible, held by surface tension, integrins, mucus, ligaments etc.), must be constructed from some combination or permutation of those structures. For reasons previously stated, icosahedrons are the most biologically suited, closest packing, volume for surface area, least energy, etc. Icosahedrons are the most symmetrical of all polyhedrons, it is the most symmetrical system for the subdivision of a spherical surface into modular units and all spherical biologic structures must be icosahedrons to be stable or propped up by the adjacent structure so that, together, they create a stable structure,(fractals?), like bubbles in a foam.

The reasons, (amongst others), the icosahedron-tensegrity would work in biotensegrity is because it close packs and is self-organizing, and the most symmetrical, not because of the parallel struts. It is omnidirectional and has the largest volume for surface area and is the most energy efficient structure with the above characteristics. The icosahedron has already been recognized as the basic structure of carbon 60, viruses, cells, and much more in biology, (there are now well over 2000 scientific articles linking tensegrity and biology). [[3 Struts|3-strut tensegrities]] have a larger surface area for volume, therefore, less energy efficient, and I don't think they close pack. I doubt if they would be self-organizing in nature.

[[Burkhardt, Robert|Burkhardt]]: I can't see the tensegrity icosahedron as a building block for cell structure. But maybe this is just a question of semantics. Bucky did look at it as a building block, for example in [http://www.rwgrayprojects.com/synergetics/s07/p8100.html#784.00|Section 784 of Synergetics I]], but I can't see that work as a model for a cell. For a model of cell walls I would be more comfortable with an approach that talks about a pattern of which the t-icosa (or perhaps della Sala's t-tetrahedron) is the simplest example. Bucky's development of that pattern into more complex single-layer domes or spheres is perhaps a suitable analogy for a cell wall. This approach is what I see in [http://www.rwgrayprojects.com/synergetics/s07/figs/f1701.html|Figure 717.01 of Synergetics I]], but there is a variation here in that the tendons also appear to be attached to the middle of the strut.

[[file:Synergetics_Fig._717.01_Single_and_Double_Bonding_of_Members_in_Tensegrity_Spheres_CROPPED.gif|frame|right|
Detail of Synergetics Fig. 717.01 Single and Double Bonding of Members in Tensegrity Spheres. Copyright © 1997 Estate of R. Buckminster Fuller.]]

Certainly the pattern of the t-icosa has been developed more directly into higher-frequency structures without the tendons being attached to the middle of a strut. This is the approach Kenner develops, and also it is the approach of some of the tensegrity spheres developed by Bucky and his collaborators, although the latter that I see in Dymaxion World of Buckminster Fuller seem to favor the zig-zag approach of della Sala's t-tetrahedron, rather than the diamond approach represented by the t-icosa.

==Unlike close packed spheres, close packing of icosahedra is not well defined==

The notion of close-packing icosahedrons is problematic. [[de Jong, Gerald|de Jong]] notes: I don't feel that this has been described explicitly enough. [[Levin, Stephen M.|Steve]] states that they close pack, like spheres where the middle one surrounded by twelve others nicely shrinks to accommodate, but the description of how one touches the next seems to be overlooked. Two tensegrities join to become a single tensegrity really only when cables meet bars and vice versa.

=Real Icosahedron=
The outline of the six-strut tensegrity as described above does not match the icosahedron as one of the platonic solids because the ratio between the length of struts “s” and the distance between two parallel struts “d” is exactly 2, where for the icosahedron this ratio is equal to approximately 1.618 (that is the “golden” ratio). There has thus been much controversy over the years about the use of this term in tensegrity, with many authors preferring to label it as an expanded octahedron.

[[File:Jessen's icosahedron.jpg|thumb|left]]Borge Jessen described a shape that he referred to as the Orthogonal Icosahedron, where all the faces meet at 90 degrees and match the outline of this 6-strut tensegrity. The card model shown expands and contracts within a similar range to the tensegrity icosahedron. Jessen’s icosahedron is described in:
[https://en.wikipedia.org/wiki/Jessen%27s_icosahedron Wikipedia]
[https://www.youtube.com/watch?v=9-HnJ9n6F20&ab_channel=XYZAidan YouTube]
[https://mathworld.wolfram.com/JessensOrthogonalIcosahedron.html Wolfram Maths]
although a correction to the latter is reported [https://flexiblepolyhedron.wordpress.com/2013/02/09/correcting-the-jessens-orthogonal-icosahedron here]
with more detailed discussion [http://maths.ac-noumea.nc/polyhedr/stuff/shaky_engl-small.pdf here].

Not very well-known are the two tensegrities shown below and invented by [[Pars, Marcelo|Pars]]. The outside of these structures are exact icosahedrons.

[http://www.tensegriteit.nl/afbeelding/tensegrity210.jpg tensegrity210]

[http://www.tensegriteit.nl/afbeelding/tensegrity207.jpg tensegrity207]

The tensegrities can also be seen on [http://www.tensegriteit.nl/e-icosahedron.html Marcelo Pars' icosahedrons]. 3D images of this real icosahedron are shown on [http://www.tensegriteit.nl/e-3dimages.html|Marcelo Pars' 3D images] . Here a small picture of the icosahedron tensegrity without the struts:

[http://www.tensegriteit.nl/afbeelding/vmrlicosahedron03.png vmrlicosahedron03]

[[Category:Portal To Polyhedra]]
[[Category:polyhedron]]

Icosahedron

2026-04-22T17:01:25Z

Unhandyandy:

Read here about the icosahedron, one of the Platonic polyhedra, and the dual of the dodecahedron. Some of the most significant tensegrity structures are argued to be congruent to the icosahedron.

=Overview=

The icosahedron polyhedron has 30 edges, 20 faces and 12 vertexes. A tensegrity that outlines this polyhedron has 6 struts and 24 tensile vectors.

This tensegrity structure is one of the few tensegrities that exhibit mirror symmetry. Burkhardt wrote, its "tendon network would mark out a cuboctahedron if the (non-planar) quadrilaterals in which the struts are nested were changed to squares. The struts are inserted as the diagonals of these squares so each strut is parallel to the strut in the opposite square and so no strut shares a vertex with another strut. This tensegrity is classified as a “diamond” type because each strut is surrounded by a diamond of four.

[[file:icosahedron_by_Burkhardt.png| frame | left | Tensegrity structure of 6 struts and 24 tensile vectors that conforms to the classic icosahedron. From Practical Guide to Tensegrity Design 2nd edition © 2008 by Burkhardt.]]

The icosahedron-like tensegrity structure is argued to be the most significant of all the structures, particularly by researchers in bio-tensegrity of the fascia and muscle system. For example, see the right upper human extremity modelled as a sequence of interconnecting icosahedral tensegrities with compression struts of different lengths, below.

[[file:Right_upper_extremity_icosahedral_model_by_Flemons.png|frame|right|
Right upper extremity modeled as a sequence of interconnecting icosahedral tensegrities with compression struts of different lengths. Model: Graham Scarr as reproduced in Simple Geometry in Complex Organisms 2010. http://www.tensegrityinbiology.co.uk/publications/geometry/.]]

=Comparing the icosahedron to other tensegrities=

Motro published a detailed comparison between the polyhedron and its associated tensegrity structure. “The two geometries can be compared on basis of the ratio between the length of struts “s” and the distance between two parallel struts “d”. For the icosahedron this ratio is equal to approximately 1.618 (that is the “golden” ratio), for the associated tensegrity system it is equal to exactly 2. This resulting tensegrity system can be seen as the expansion of an octahedron, since there are at the end eight triangles of cables (the same as the number of triangular faces for an octahedron), and the three pairs of struts can be understood as the splitting of the three internal diagonals.”

[[file:Comparison_of_icosahedron_and_tensegrity_with_phi_by_Motro.gif|frame|left|
Comparison between the icosahedron polyhedron and its associated tensegrity structure. “S” is phi, the golden ratio. From Structural Morphology Of Tensegrity Systems by Motro.]]

[[file:Comparison_of_icosahedron_tensegrity_and_polyhedron_by_Tibert.gif|frame|right|
Comparison of a tensegrity icosahedron and a polyhedron with which it conforms. Side view, left, and top view, right. The polyhedron is drawn with dashed lines. From Deployable Tensegrity Structures for Space Applications by Tibert.]]
William Brooks Whittier also discusses this tensegrity, calling it a T-6. It has 6 struts and 24 tensile vectors. It outlines an icosahedron with 30 edges, 20 faces, and 12 vertexes.

[[file:6_strut_icosahedron_By_Whittier.gif|frame|left|
Tensegrity structure that outlines an icosahedron. In order from left to right two perspective views and a top view perpendicular to one strut pair. From Kinematic Analysis of Tensegrity Structures By William Brooks Whittier.]]
[[Gómez Jáuregui, Valentín|Gómez Jáuregui]] published a photo of a tensegritoy model he built that conforms with a truncated icosahedron.

[[file:truncated_icosahedron_tensegritoy_by_Jáuregui.png|frame|right|
Tensegritoy model that conforms with a truncated icosahedron. From Appendix G Tensegrity Models, Tensegrity Structures by Jáuregui.]]

[[Motro, René| Motro ]]discusses a tensegrity structure related to the icosahedron. He wrote, “Since it is not possible to design a regular icosahedron with six equal struts, we tried to build one with six struts, one of them being greater than the five others. The basis of this design is a prismatic pentagonal system; a central strut is placed on the vertical symmetry axis. This axis becomes a rotation axis. The lengths of the struts and of the cables are calculated in order to reach an equilibrium state which is characterized by the fact that the twelve nodes occupy the geometrical position of the apices of an icosahedron. The name is chosen by reference to this axis of rotation and to the icosahedron. This system can be classified as a “Z” like tensegrity system according to the classification submitted by Anthony Pugh. There are only two cables and one strut at each node, except for the central strut.

[[file:Spinning_icosahedron_by_Motro.gif|frame|left|
Spinning icosahedron tensegrity structure, perspective and plane view. This can be classified a “Z” tensegrity system in the Pugh system. From Structural Morphology Of Tensegrity Systems by Motro.]]

=Discussion of the importance of the icosahedron=

==Is the icosahedron tensegrity the critical, primitive building block of biological tensegrity?==
[[Pars, Marcelo|Pars]] wrote: The icosahedron-tensegrity has unique characteristics that tensegrities normally never have: The struts are exactly parallel or in straight angle to eachother. This is very rare for a tensegrity which can be distinguished by its twisted form. I think it may be these straight angles that make this tensegrity so popular. Wherever we humans changed the world we used straight angles, and although it may not be a natural form, for us a straight angle is restful and feels normal. I guess one could say that from all tensegrities the icosahedron tensegrity looks the most like a conventional construction.
I made quite a few tensegrities, but the icosahedron was the first, also because mathematics behind it is more easy than for instance the 3 strut prism.

But all this doesn't say anything about nature or biotensegrity of course. If you ask me, I guess nature is full of tensegrities and the icosahedron is the bridge between nature and it's construction methods and us humans with our conventional bricks..

[[Levin, Stephen M.|Levin]]: My understanding of [[Fuller, Richard Buckminster|Fuller]]: there are only three structures that are stable with flexible joints, the [[tetrahedron]], [[octahedron]] and icosahedron. Therefore, biologic structures,( all joints from cell to organism are flexible, held by surface tension, integrins, mucus, ligaments etc.), must be constructed from some combination or permutation of those structures. For reasons previously stated, icosahedrons are the most biologically suited, closest packing, volume for surface area, least energy, etc. Icosahedrons are the most symmetrical of all polyhedrons, it is the most symmetrical system for the subdivision of a spherical surface into modular units and all spherical biologic structures must be icosahedrons to be stable or propped up by the adjacent structure so that, together, they create a stable structure,(fractals?), like bubbles in a foam.

The reasons, (amongst others), the icosahedron-tensegrity would work in biotensegrity is because it close packs and is self-organizing, and the most symmetrical, not because of the parallel struts. It is omnidirectional and has the largest volume for surface area and is the most energy efficient structure with the above characteristics. The icosahedron has already been recognized as the basic structure of carbon 60, viruses, cells, and much more in biology, (there are now well over 2000 scientific articles linking tensegrity and biology). [[3 Struts|3-strut tensegrities]] have a larger surface area for volume, therefore, less energy efficient, and I don't think they close pack. I doubt if they would be self-organizing in nature.

[[Burkhardt, Robert|Burkhardt]]: I can't see the tensegrity icosahedron as a building block for cell structure. But maybe this is just a question of semantics. Bucky did look at it as a building block, for example in [http://www.rwgrayprojects.com/synergetics/s07/p8100.html#784.00|Section 784 of Synergetics I]], but I can't see that work as a model for a cell. For a model of cell walls I would be more comfortable with an approach that talks about a pattern of which the t-icosa (or perhaps della Sala's t-tetrahedron) is the simplest example. Bucky's development of that pattern into more complex single-layer domes or spheres is perhaps a suitable analogy for a cell wall. This approach is what I see in [http://www.rwgrayprojects.com/synergetics/s07/figs/f1701.html|Figure 717.01 of Synergetics I]], but there is a variation here in that the tendons also appear to be attached to the middle of the strut.

[[file:Synergetics_Fig._717.01_Single_and_Double_Bonding_of_Members_in_Tensegrity_Spheres_CROPPED.gif]]
caption="Detail of Synergetics Fig. 717.01 Single and Double Bonding of Members in Tensegrity Spheres. Copyright © 1997 Estate of R. Buckminster Fuller "

Certainly the pattern of the t-icosa has been developed more directly into higher-frequency structures without the tendons being attached to the middle of a strut. This is the approach Kenner develops, and also it is the approach of some of the tensegrity spheres developed by Bucky and his collaborators, although the latter that I see in Dymaxion World of Buckminster Fuller seem to favor the zig-zag approach of della Sala's t-tetrahedron, rather than the diamond approach represented by the t-icosa.

==Unlike close packed spheres, close packing of icosahedra is not well defined==

The notion of close-packing icosahedrons is problematic. [[de Jong, Gerald|de Jong]] notes: I don't feel that this has been described explicitly enough. [[Levin, Stephen M.|Steve]] states that they close pack, like spheres where the middle one surrounded by twelve others nicely shrinks to accommodate, but the description of how one touches the next seems to be overlooked. Two tensegrities join to become a single tensegrity really only when cables meet bars and vice versa.

=Real Icosahedron=
The outline of the six-strut tensegrity as described above does not match the icosahedron as one of the platonic solids because the ratio between the length of struts “s” and the distance between two parallel struts “d” is exactly 2, where for the icosahedron this ratio is equal to approximately 1.618 (that is the “golden” ratio). There has thus been much controversy over the years about the use of this term in tensegrity, with many authors preferring to label it as an expanded octahedron.

[[File:Jessen's icosahedron.jpg|thumb]]Borge Jessen described a shape that he referred to as the Orthogonal Icosahedron, where all the faces meet at 90 degrees and match the outline of this 6-strut tensegrity. The card model shown expands and contracts within a similar range to the tensegrity icosahedron. Jessen’s icosahedron is described in:
[https://en.wikipedia.org/wiki/Jessen%27s_icosahedron Wikipedia]
[https://www.youtube.com/watch?v=9-HnJ9n6F20&ab_channel=XYZAidan YouTube]
[https://mathworld.wolfram.com/JessensOrthogonalIcosahedron.html Wolfram Maths]
although a correction to the latter is reported [https://flexiblepolyhedron.wordpress.com/2013/02/09/correcting-the-jessens-orthogonal-icosahedron here]
with more detailed discussion [http://maths.ac-noumea.nc/polyhedr/stuff/shaky_engl-small.pdf here].

Not very well-known are the two tensegrities shown below and invented by [[Pars, Marcelo|Pars]]. The outside of these structures are exact icosahedrons.

[[file:http://www.tensegriteit.nl/afbeelding/tensegrity210.jpg]]
Image width="358" height="350"
[[file:http://www.tensegriteit.nl/afbeelding/tensegrity207.jpg]]
The tensegrities can also be seen on [http://www.tensegriteit.nl/e-icosahedron.html Marcelo Pars' icosahedrons]. 3D images of this real icosahedron are shown on [http://www.tensegriteit.nl/e-3dimages.html|Marcelo Pars' 3D images]] . Here a small picture of the icosahedron tensegrity without the struts:

[[file:http://www.tensegriteit.nl/afbeelding/vmrlicosahedron03.png]]
Image width="329" height="211"
[[Category:Portal To Polyhedra]]
[[Category:polyhedron]]

Icosahedron

2026-04-22T16:41:28Z

Unhandyandy: /* Overview */

Read here about the icosahedron, one of the Platonic polyhedra, and the dual of the dodecahedron. Some of the most significant tensegrity structures are argued to be congruent to the icosahedron.

=Overview=

The icosahedron polyhedron has 30 edges, 20 faces and 12 vertexes. A tensegrity that outlines this polyhedron has 6 struts and 24 tensile vectors.

This tensegrity structure is one of the few tensegrities that exhibit mirror symmetry. Burkhardt wrote, its "tendon network would mark out a cuboctahedron if the (non-planar) quadrilaterals in which the struts are nested were changed to squares. The struts are inserted as the diagonals of these squares so each strut is parallel to the strut in the opposite square and so no strut shares a vertex with another strut. This tensegrity is classified as a “diamond” type because each strut is surrounded by a diamond of four.

[[file:icosahedron_by_Burkhardt.png]]
Image width="273" height="220"
caption="Tensegrity structure of 6 struts and 24 tensile vectors that conforms to the classic icosahedron. From Practical Guide to Tensegrity Design 2nd edition © 2008 by Burkhardt "

The icosahedron-like tensegrity structure is argued to be the most significant of all the structures, particularly by researchers in bio-tensegrity of the fascia and muscle system. For example, see the right upper human extremity modelled as a sequence of interconnecting icosahedral tensegrities with compression struts of different lengths, below.

[[file:Right_upper_extremity_icosahedral_model_by_Flemons.png]]
caption="Right upper extremity modeled as a sequence of interconnecting icosahedral tensegrities with compression struts of different lengths. Model: Graham Scarr as reproduced in Simple Geometry in Complex Organisms 2010. http://www.tensegrityinbiology.co.uk/publications/geometry/."

=Comparing the icosahedron to other tensegrities=

Motro published a detailed comparison between the polyhedron and its associated tensegrity structure. “The two geometries can be compared on basis of the ratio between the length of struts “s” and the distance between two parallel struts “d”. For the icosahedron this ratio is equal to approximately 1.618 (that is the “golden” ratio), for the associated tensegrity system it is equal to exactly 2. This resulting tensegrity system can be seen as the expansion of an octahedron, since there are at the end eight triangles of cables (the same as the number of triangular faces for an octahedron), and the three pairs of struts can be understood as the splitting of the three internal diagonals.”

[[file:Comparison_of_icosahedron_and_tensegrity_with_phi_by_Motro.gif]]
caption="Comparison between the icosahedron polyhedron and its associated tensegrity structure. “S” is phi, the golden ratio. From Structural Morphology Of Tensegrity Systems by Motro."

[[file:Comparison_of_icosahedron_tensegrity_and_polyhedron_by_Tibert.gif]]
caption="Comparison of a tensegrity icosahedron and a polyhedron with which it conforms. Side view, left, and top view, right. The polyhedron is drawn with dashed lines. From Deployable Tensegrity Structures for Space Applications by Tibert. "
William Brooks Whittier also discusses this tensegrity, calling it a T-6. It has 6 struts and 24 tensile vectors. It outlines an icosahedron with 30 edges, 20 faces, and 12 vertexes.

[[file:6_strut_icosahedron_By_Whittier.gif]]
caption="Tensegrity structure that outlines an icosahedron. In order from left to right two perspective views and a top view perpendicular to one strut pair. From Kinematic Analysis of Tensegrity Structures By William Brooks Whittier"
[[Gómez Jáuregui, Valentín|Gómez Jáuregui]] published a photo of a tensegritoy model he built that conforms with a truncated icosahedron.

[[file:truncated_icosahedron_tensegritoy_by_Jáuregui.png]]
caption="Tensegritoy model that conforms with a truncated icosahedron. From Appendix G Tensegrity Models, Tensegrity Structures by Jáuregui"

[[Motro, René| Motro ]]discusses a tensegrity structure related to the icosahedron. He wrote, “Since it is not possible to design a regular icosahedron with six equal struts, we tried to build one with six struts, one of them being greater than the five others. The basis of this design is a prismatic pentagonal system; a central strut is placed on the vertical symmetry axis. This axis becomes a rotation axis. The lengths of the struts and of the cables are calculated in order to reach an equilibrium state which is characterized by the fact that the twelve nodes occupy the geometrical position of the apices of an icosahedron. The name is chosen by reference to this axis of rotation and to the icosahedron. This system can be classified as a “Z” like tensegrity system according to the classification submitted by Anthony Pugh. There are only two cables and one strut at each node, except for the central strut.

[[file:Spinning_icosahedron_by_Motro.gif]]
caption="Spinning icosahedron tensegrity structure, perspective and plane view. This can be classified a “Z” tensegrity system in the Pugh system. From Structural Morphology Of Tensegrity Systems by Motro"

=Discussion of the importance of the icosahedron=

==Is the icosahedron tensegrity the critical, primitive building block of biological tensegrity?==
[[Pars, Marcelo|Pars]] wrote: The icosahedron-tensegrity has unique characteristics that tensegrities normally never have: The struts are exactly parallel or in straight angle to eachother. This is very rare for a tensegrity which can be distinguished by its twisted form. I think it may be these straight angles that make this tensegrity so popular. Wherever we humans changed the world we used straight angles, and although it may not be a natural form, for us a straight angle is restful and feels normal. I guess one could say that from all tensegrities the icosahedron tensegrity looks the most like a conventional construction.
I made quite a few tensegrities, but the icosahedron was the first, also because mathematics behind it is more easy than for instance the 3 strut prism.

But all this doesn't say anything about nature or biotensegrity of course. If you ask me, I guess nature is full of tensegrities and the icosahedron is the bridge between nature and it's construction methods and us humans with our conventional bricks..

[[Levin, Stephen M.|Levin]]: My understanding of [[Fuller, Richard Buckminster|Fuller]]: there are only three structures that are stable with flexible joints, the [[tetrahedron]], [[octahedron]] and icosahedron. Therefore, biologic structures,( all joints from cell to organism are flexible, held by surface tension, integrins, mucus, ligaments etc.), must be constructed from some combination or permutation of those structures. For reasons previously stated, icosahedrons are the most biologically suited, closest packing, volume for surface area, least energy, etc. Icosahedrons are the most symmetrical of all polyhedrons, it is the most symmetrical system for the subdivision of a spherical surface into modular units and all spherical biologic structures must be icosahedrons to be stable or propped up by the adjacent structure so that, together, they create a stable structure,(fractals?), like bubbles in a foam.

The reasons, (amongst others), the icosahedron-tensegrity would work in biotensegrity is because it close packs and is self-organizing, and the most symmetrical, not because of the parallel struts. It is omnidirectional and has the largest volume for surface area and is the most energy efficient structure with the above characteristics. The icosahedron has already been recognized as the basic structure of carbon 60, viruses, cells, and much more in biology, (there are now well over 2000 scientific articles linking tensegrity and biology). [[3 Struts|3-strut tensegrities]] have a larger surface area for volume, therefore, less energy efficient, and I don't think they close pack. I doubt if they would be self-organizing in nature.

[[Burkhardt, Robert|Burkhardt]]: I can't see the tensegrity icosahedron as a building block for cell structure. But maybe this is just a question of semantics. Bucky did look at it as a building block, for example in [http://www.rwgrayprojects.com/synergetics/s07/p8100.html#784.00|Section 784 of Synergetics I]], but I can't see that work as a model for a cell. For a model of cell walls I would be more comfortable with an approach that talks about a pattern of which the t-icosa (or perhaps della Sala's t-tetrahedron) is the simplest example. Bucky's development of that pattern into more complex single-layer domes or spheres is perhaps a suitable analogy for a cell wall. This approach is what I see in [http://www.rwgrayprojects.com/synergetics/s07/figs/f1701.html|Figure 717.01 of Synergetics I]], but there is a variation here in that the tendons also appear to be attached to the middle of the strut.

[[file:Synergetics_Fig._717.01_Single_and_Double_Bonding_of_Members_in_Tensegrity_Spheres_CROPPED.gif]]
caption="Detail of Synergetics Fig. 717.01 Single and Double Bonding of Members in Tensegrity Spheres. Copyright © 1997 Estate of R. Buckminster Fuller "

Certainly the pattern of the t-icosa has been developed more directly into higher-frequency structures without the tendons being attached to the middle of a strut. This is the approach Kenner develops, and also it is the approach of some of the tensegrity spheres developed by Bucky and his collaborators, although the latter that I see in Dymaxion World of Buckminster Fuller seem to favor the zig-zag approach of della Sala's t-tetrahedron, rather than the diamond approach represented by the t-icosa.

==Unlike close packed spheres, close packing of icosahedra is not well defined==

The notion of close-packing icosahedrons is problematic. [[de Jong, Gerald|de Jong]] notes: I don't feel that this has been described explicitly enough. [[Levin, Stephen M.|Steve]] states that they close pack, like spheres where the middle one surrounded by twelve others nicely shrinks to accommodate, but the description of how one touches the next seems to be overlooked. Two tensegrities join to become a single tensegrity really only when cables meet bars and vice versa.

=Real Icosahedron=
The outline of the six-strut tensegrity as described above does not match the icosahedron as one of the platonic solids because the ratio between the length of struts “s” and the distance between two parallel struts “d” is exactly 2, where for the icosahedron this ratio is equal to approximately 1.618 (that is the “golden” ratio). There has thus been much controversy over the years about the use of this term in tensegrity, with many authors preferring to label it as an expanded octahedron.

[[File:Jessen's icosahedron.jpg|thumb]]Borge Jessen described a shape that he referred to as the Orthogonal Icosahedron, where all the faces meet at 90 degrees and match the outline of this 6-strut tensegrity. The card model shown expands and contracts within a similar range to the tensegrity icosahedron. Jessen’s icosahedron is described in:
[https://en.wikipedia.org/wiki/Jessen%27s_icosahedron Wikipedia]
[https://www.youtube.com/watch?v=9-HnJ9n6F20&ab_channel=XYZAidan YouTube]
[https://mathworld.wolfram.com/JessensOrthogonalIcosahedron.html Wolfram Maths]
although a correction to the latter is reported [https://flexiblepolyhedron.wordpress.com/2013/02/09/correcting-the-jessens-orthogonal-icosahedron here]
with more detailed discussion [http://maths.ac-noumea.nc/polyhedr/stuff/shaky_engl-small.pdf here].

Not very well-known are the two tensegrities shown below and invented by [[Pars, Marcelo|Pars]]. The outside of these structures are exact icosahedrons.

[[file:http://www.tensegriteit.nl/afbeelding/tensegrity210.jpg]]
Image width="358" height="350"
[[file:http://www.tensegriteit.nl/afbeelding/tensegrity207.jpg]]
The tensegrities can also be seen on [http://www.tensegriteit.nl/e-icosahedron.html Marcelo Pars' icosahedrons]. 3D images of this real icosahedron are shown on [http://www.tensegriteit.nl/e-3dimages.html|Marcelo Pars' 3D images]] . Here a small picture of the icosahedron tensegrity without the struts:

[[file:http://www.tensegriteit.nl/afbeelding/vmrlicosahedron03.png]]
Image width="329" height="211"
[[Category:Portal To Polyhedra]]
[[Category:polyhedron]]