Golden Ratio

Tetrahedra

Nets and building instructionsGrayham Forscutt

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.

Rhombic Triacontahedron frame with E tetraWe will focus on constructing the four golden tetra, EUVW, and show how they combine to create larger EUVW’s.

E TetrahedronPhi tetrahedron called ‘E’ tetra after Bucky Fuller Below: smaller by phi E’s pack to make larger E tetrahedron.

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.

E tetra can be any size

Every phi tetra can pack with smaller & smaller phi tetra. Infinite compression, maximum store, infinite energy.

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.

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 & yellowW = blue & orange

A little trial and error will give the needed arrangements for all six tetrahedra.

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.

Every division along the edge length of the larger U is phi ratio

Graphics using the vZome software.

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.

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.

V tetrahedron

Nets showing how the two triangles attach to make the V tetra.

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.

V tetra packing

There is enough here to make the larger V.Trial and error is to be expected.

Original UVW modelsMaking these gave enough information to make the coloured versions.

EUVW combinations

Acute Phi Rhombohedron

We will explore how EUVW’s pack to make rhombic structures

Golden Ratio Spiral

We will also explore how to pack the UVW’s to make a volumetric phi spiral.