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Four golden ratio tetrahedra


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Phi scaled tetrahedra that face bond to make numerous golden quasicrystalline structures and the golden ratio spiral.

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Four golden ratio tetrahedra

  1. 1. Golden Ratio Tetrahedra Nets and building instructions Grayham Forscutt
  2. 2. Prime net is a square with edges marked in phi ratio Left: Square with edge’s marked in Golden Ratio. Right: Flip the small triangle to make the E tetra. Centre top: folding. Centre: Examples of packing the E tetra.
  3. 3. Rhombic Triacontahedron frame with E tetra We will focus on constructing the four golden tetra, EUVW, and show how they combine to create larger EUVW’s.
  4. 4. E Tetrahedron Phi tetrahedron called ‘E’ tetra after Bucky Fuller Below: smaller by phi E’s pack to make larger E tetrahedron.
  5. 5. E Tetra can also be made with V & W’s Three further phi tetra: ‘UVW’ discovered by David Koski. These pack with ‘E’s to form higher frequency ‘E’s.
  6. 6. E tetra can be any size Every phi tetra can pack with smaller & smaller phi tetra. Infinite compression, maximum store, infinite energy.
  7. 7. Net for UVW tetrahedra By sub dividing the edges of the square further, using phi ratio lengths, the two triangles used to make the UVW’s can be plotted.
  8. 8. Colour Coding The UVW tetra are colour coded. By following the colours shown in the nets each tetra & its mirror can be constructed. U = red & green V = violet & yellow W = blue & orange A little trial and error will give the needed arrangements for all six tetrahedra.
  9. 9. U tetrahedron There is only one triangle needed to make the U tetra, there are two versions. Larger U tetra can be made by combining UVW’s, colour coding indicates the tetra involved and how they pack together.
  10. 10. Every division along the edge length of the larger U is phi ratio Graphics using the vZome software.
  11. 11. Nets for the W tetrahedron Close study of these nets reveals two different triangles, you can see the lengths of these triangles are different, a mark is drawn on two edges to indicate they are equal in length, this shows how to arrange the four triangles that make the W.
  12. 12. Larger W’s Arranging the UVW’s in this way will construct a larger W. It can be fun trying to discover how they are packed. Above: Three phi ratio sizes of W’s are shown, each made from UVW’s.
  13. 13. V tetrahedron Nets showing how the two triangles attach to make the V tetra.
  14. 14. Larger V made from UVW’s The same procedure is taken to make larger V’s, colour coding reveals how to arrange the smaller UVW’s.
  15. 15. V tetra packing There is enough here to make the larger V. Trial and error is to be expected.
  16. 16. Original UVW models Making these gave enough information to make the coloured versions.
  17. 17. EUVW combinations
  18. 18. Acute Phi Rhombohedron We will explore how EUVW’s pack to make rhombic structures
  19. 19. Golden Ratio Spiral We will also explore how to pack the UVW’s to make a volumetric phi spiral.