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Nesting Structures

  How to Build Physical Models
  Nesting Dolls
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Nesting dolls are a toy where a doll is opened up and a slightly smaller doll is inside. You open that one and there is a still smaller doll inside it; on and on. This is a way to understand how basic geometry is one synergetic whole system where geometrical systems can fit exactly inside another in a series.

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Image Buckminster Fuller

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The inspiration for my organizing tool of thinking of structures as containers came from this illustration in Fuller's Synergetic Geometry. It was difficult to read which systems were where so I set out the build physical models of them to understand how their synergy worked.

  Dodecahedron Inside Icosahedron
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The icosahedron's edge is multiplied by 3GR to get the edge of the dodecahedron. The mass production grids can be built with this information. (See Guide to Basic Tool Kit)

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The clear material is called DuraLar which can be bought to most arts supply stores. The main reason I use the term geometrical system instead of polyhedron is system recognizes that the material a structure is made of has a thickness. When you do a series of systems one inside the other you have to compensate for the thickness by always measuring the outside of the system you are going to contain. Generally 2 mm (millimeter) is enough to deal with the thickness and any tape or glue.

In the series above, each system is broken down into its modular pieces so you can build the octa, tetra, cube and dodecahedron at their actual size. Only the icosahedron needs to swell to compensate for the thicknesses. The actual measurement changes from 9.72326 cm to 9.9 cm. Adjust your mass production grid where/when numbers.

  Dodecahedron's Modular Unit Surrounding the Cube
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The dodeca-cap is made from 2 pentagon cut along its long axis. To interconnect a cube and dodeca have the cube's edge be the same as the long axis of the pentagon.

  Cube Inside Dodecahedron
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  Tetrahedron Inside Cube
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To fit a tetrahedron exactly inside a cube have the long axis of the cube's square face equal to the edge of the tetra. The numerical constant is √2.

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  The Interconnectedness the Cube, Octahedron and Tetrahedron
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In this example, the volume difference between the tetra and the cube is an 4-sided asymmetrical tetrahedron. Each is 1/8 of the volume of an octahedron of the same edge as the tetra.

Two tetras can share a common center and 'spin' in opposite directions to interconnect with the 8 outside corners of a cube. Fuller called this arrangement a     dou-tet. The volume difference between the dou-tet and the cube are another 4-sided asymmetrical tetra that is 1/4 volume of an octa the is half-size of the tetras. The interconnectedness, synergy, of systems is the rule not the exception.

  Octahedron Inside Tetrahedron
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There will always be a half-sized octahedron inside any equilateral triangle. A simple constant of multiplying/dividing by 2.

 Icosahedron Inside Octahedron
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There is a natural tendency at the end of a building pattern to take an 'and/ah' pause and begin again on a different scale. This is a simple fractal pattern. The numerical constant between the two triangles =           .

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  The Interconnectedness of the Octahedron and Icosahedron
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Fuller called the modular volume difference between the octa and the icosa inside the 'S' module. There is also an octa that fits exactly inside our original icosa.

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Octa Inside Icosa
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What I call the Periodic Table of Structural Interconnectedness is the examination of the 20 possible combinations of one system inside another among the Five Platonic Solids. Each is defined as the outside container with the modular units explained. This synergy is true for all of basic geometry.

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