Tetrahedron


Read here about the simplest poyhedron, the tetrahedron, a basic building block of the real world, hence an important concept in tensegrity structures. The tetrahedron is often deployed or described by tensegrity structures. See also octahedron, icosahedron, and the other polyhedra.

Overview

A tetrahedron (plural: tetrahedra) in classic geometry is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids. The tetrahedron is the only convex polyhedron that has four faces.The tetrahedron is the three-dimensional case of the more general concept of a simplex.

Truncated Tetrahedron


The truncated tetrahedron has 18 edges, 8 faces and 12 vertexes. A tensegrity that outlines this polyhedron has 6 struts and 18 tensile vectors. It was first exhibited by Francesco della Sala at the University of Michigan in 1952, according to Burkhardt.

Motro calls such structures tensypolyhedra. He rendered the structure with 6 struts and 18 tensile vectors.

truncated_tetrahedron_by_Motro.png
Tensegrity structures that conform with a truncated tetrahedron. From Structural Morphology Of Tensegrity Systems by Motro

Tibert compares in the drawing below an idealized polyhedron drawn with dashed lines, and a tensegrity that outlines the polyhedron, drawn with black heavy struts and light black tension members. He renders a truncated tetrahedron.

truncated_tetrahedron_by_Tibert.gif
Comparison of truncated tensegrity tetrahedron 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.

Burkhardt classifies this model as a “zig-zag” tensegrity because each strut is supported by two other struts tied into a zig-zag of three tendons spanning the strut. The 6 strut tensegrity structure that conforms with the truncated tetrahedron is the zig-zag counterpart of the 6 strut icosahedron, as both structures have 6 struts. But this truncated tetrahedron-like model has four tendon triangles, whereas the icosahedron-like model has eight.

truncated_tetrahedron_by_Burkhardt.png
Tensegrity structure of 6 struts and 18 tensile vectors that conforms with the truncated tetrahedron. From Practical Guide to Tensegrity Design 2nd edition © 2008 by Burkhardt


Tensegritoy_truncated_tetrahedron_by_Jáuregui.png
Tensegritoy model that conforms with the truncated tetrahedron. From Appendix G Tensegrity Models, Tensegrity Structures by Jáuregui


C. R. Calladine uses this tensegrity as the basis for his analysis of tensegrity structures in terms of Maxwell’s stiffness rule.

truncated_tetrahedron_by_Calladine.gif
The tensegrity structure analyzed by Calladine. It outlines a truncated tetrahedron. From Buckminster Fuller’s Tensegrity Structures And Clerk Maxwell’s Rules For The Construction Of Stiff Frames by Calladine


Image Gallery of Tetrahedron-like Tensegrity Structures


Pars shared the image below:

24_strut_tetrahedron_Marcelo_Pars_tns098.jpg
4 strut tetrahedron. Each strut has 5 redundant neighbors, for a total of 24 struts. Construction and photo by Marcelo Pars.


angelo shared the image below, showing a 4-struts, "real" tetrahedron (4 vertices, 6 edges, 4 equal triangular faces etc.- link):
external image bild14-small.JPG
this is a first attempt to generate tensegrities shaping the platonic solids - see a cube from the same author

Below Marcelo Pars version of a real tetrahedron. Probably the most simple tetrahedron possible, with only 4 struts and ten strings.
external image tensegrity217.jpg
The tetrahedron is also available on Marcelo Pars' 3D images
Another variation of this perfect tetrahedron is a tensegrity with two bowed struts (each strut is half a circle):
external image tensegrity218.jpg
See also Pars' tetrahedrons

Online 3d Application Modeling The Tensegrity Tetrahedron


The tetrahedron, both ordinary and "Snelson" version, is available in the Xozzox online 3D simulation java application. In the object selection box on left, select "TetrahedronSnelson" or "Tetrahedron."

Links and References


Marcelo Pars' 3D tensegrities

Portal to Polyhedra
A series on polyhedra and associated tensegrities
  1. Platonic: Cube, Dodecahedron, Icosahedron, Octahedron, Tetrahedron
  2. Archimedan: Cuboctahedron, Jitterbug, Rhombic Dodecahedron, Stella Octangula, Tricontahedron
  3. Form: Prism, Torus
  4. Concepts: Naming
Access by no. of struts: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 21, 30, 60, 90, 270, 540; Procedures: 3, 30