92 Prism Tensegrity

Read here about tensegrity structures composed of 540 struts, part of a series of pages organized by strut count.

92 Prism Tensegrity

Photographs dated to April 1979 show R. Buckminster Fuller holding a tensegrity sphere composed of 92 Prisms. Its structure, as analyzed by Taffgoch and Adrian Rossiter is summarized below.

RBF with 92 prism tensegrity, April 1979.

RBF with 92 prism tensegrity, April 1979.

Tensegrity Prism 960 struts by Taffgoch.

Tensegrity Prism Animated Overlay by Taffgoch.

Tensegrity Prism inner tendons icosahedron shown by TaffGoch.

Tensegrity Prism one highlighted by Taffgoch.

Tensegrity Prism outer and inner tendons by TaffGoch.

Tensegrity Prism outer tendons icosa by TaffGoch.

The sphere is a tensegrity sphere, composed of many adjacent prism tensegrties. See "prism" tensegrities: http://tensegritywiki.com/Prism

Mexican method 6V sphere, composed of 92 prisms. 540 struts total 12 pentagonal prisms. = 60 struts. 80 hexagonal prisms = 480 struts

Assuming uniform struts (all the same length), the pentagonal prisms will be 20% higher if they had the same approximate edge length as the hexagons. This longer edge should be visible in the photograph, and may result in a slight twist.

The tendon nets can be comprehended by outer, inner and interstitial Outer purple Inner blue Interstitial orange

The tendon nets could be continuous. "An undirected graph has an Eulerian cycle if and only if every vertex has even degree, and all of its vertices with nonzero degree belong to a single connected component."

Each prism has a winding direction.

To physically construct this tensegrity: Since each prism is rigid, each can be assembled separately, then connected to adjacent prisms, by the corners (not the edges.)

To construct this tensegrity in Rossiter's Antirpsim dome_layer program:

// tens_strut_string.inc

1. include "woods.inc"
2. include "metals.inc"

1. macro disp_edge(edge, col)

  #if(!v_equal(verts[edges[edge][0]], verts[edges[edge][1]]) )
     #if (col.x=0)
        cylinder{verts[edges[edge][0]] verts[edges[edge][1]] edge_sz
                   texture{ T_Wood3 }
                }
     #end
     #if (col.x=1)
        cylinder{verts[edges[edge][0]] verts[edges[edge][1]] edge_sz/3
                   texture{pigment{colour <0.8, 0.8, 0.6>}}
                }
     #end
  #end

1. end

I have attached images showing the result of applying it to a spherical 3F icosahedron dual (the commands will be valid for the next snapshot of Antiprism, 23.99+07 or greater). One is the basic model viewed in Antiview

  dome_layer -t prism geo_3_d | antiview -v 0.015

The other is a the model processed by off2pov, with a custom include file, and rendered in POV-Ray

  off_trans -y D5 geo_3_d | dome_layer -i -t prism | off_color -e M -m map_0:1:1:1 | off2pov -v 0.015 -i tens_strut_string.inc > pri_tens_geo_3_d.pov

[off_trans -y lines up the model on a 5-fold axis. dome_layer -i uses index numbers for colouring. The off_color command makes the strut colour 0 and the strings colour 1. off2pov converts it to POV-Ray format, and the custom include file tens_strut_string.inc (included below) draws the elements to make the model look like a tensegrity.]

On my system, I rendered this with

  povray declare=AspectRatio=1 +a +p +H2000 +W2000 pri_tens_geo_3_d.pov

A similar 4V is proposed by TaffGoch. This Mexican method 4V sphere, composed of 42 prisms. 240 struts total 12 pentagonal prisms. = 60 struts. 30 hexagonal prisms = 480 struts