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Introduction to Measurement
Limiting the Scope of the Work
The Five Platonic Solids

In 2023 it will be 50 years that I have been working on these ideas. Over the years I have amassed a large body of work. In the past when I have tried to present my ideas and techniques the effort would collapse under the weight of the large amount of details I would try 

to present.

Five years ago, I begin anew. I spent a great deal of time reorganizing my archives and securing them. From that effort I saw a new critical path through the material. What is unique and practical about Newtools and naturalmodular could be expressed with a minimum of systems.


More than 2,000 years ago, the ancient Greek determined the Five Platonic Solids were most fundamental because of the 'regularity' of their outside appearance. They discovered only these five have these 3 regular characteristics.

1) They are constructed using just one shape

2)The edges of each shape are equal

3) The outside corners of each system are made with the same combination of shapes

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  The Vector Equilibrium (VE)

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Fuller called this 14-sided system a vector equilibrium (VE). The ancient Greeks called it the cuboctahedron. It has 8-60˚triangles and 6 squares. Newtools uses it as a 'push-off-start' structure to explore the interconnectedness of useful information in basic geometry. In the mapping tool box the VE will serve as the all-purpose outside border of any map. It has two characteristics that warrant this beginning position.

One, is that it is as close as we can come in a fundamental geometrical system to achieving a 'balanced state'. The lines of communication between its center and outside corners are equal to the lines of communication between the outside corners themselves (outside edges).

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Two, its structure grows out of the lines of communication between the centers of same-sized spheres that are packed together in a MEM (Most Effective Minimum) arrangement. If you have a center sphere and ask the question, "What is the minimum number of same-sized spheres to construct a continuous first layer around it?", the answer will always be 12 around the original 1. These 13 spheres are one of the most fundamental structural and numerical constants found in Nature. 

  How To Measure Structural Interconnectedness

Four “Not-Us” Numerical Constants



GR = Golden Ratio



√2 = 1.41421356237095...


Most people avoid any long string of decimals like the value for √2. If you are interested in exploring fundamental structure on your own they are a very practical tool. Because of the power of the calculators on our devices, the only skill needed is the ability to exactly copy a long series of numbers.


√3 = 1.732050807568877...


GR = Golden Ratio = 0.618033988749895...


Traditionally, GR has been a ratio between 1 and 2 expressed as 1.618... Newtools expresses it between 0 and 1 for easier calculation with our BB measuring system. Both are the same ratio.

What is important about this number is the .6180339... portion of the number which defines the Fibonacci series. It is just a convention to define it between 1 and 2.

GR is a new symbol for the number, the Greek letter used before is harder to use.

One of the only formulas in Newtools is how to calculate GR.

GR = √5 -1 divided by 2

GR = 2.23606797749979 – 1 / 2 (/ = divide by)

GR = 0.618033988749895...


√3.618... = √3.618033988749895... = 1.902113032590307...

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This number builds many shapes and systems in the Bubble family (Five-Fold Symmetries). In a pentagon, if you multiply √3.618...with half of the long axis of the pentagon you will have the altitude of the shape. The divisions of the altitude are made by multiplying by GR. To the best of my knowledge, this is new information not known before. 


Exactitude as a Practical Tool

Exploring fundamental structure on your own uses the process of trial and error to find hidden building patterns embedded in a system you are studying. Using the full decimal reach of the calculators on our devices helps confirm that two numbers are the same value. If your calculations work out to one-millionth of a millimeter you can say they are exact.


Metric Measuring System

Newtools only uses the metric system. Multiplying and dividing decimals is so much easier than fractions. Building scale models of real life designs are easy with centimeters and meters as your scales.

Six "Not-Us" Structural Constants


60˚ Triangle



Sphere Diamond

Bubble Diamond


The circle is the first word in the 'language of the spheres' (sphere-speak). This primordial (at the beginning) language is how structure's integral parts learned to communicate with each other. The first circle was the equator of the first sphere which had the ability to reproduce exact copies of itself. It learned there was a most effective way for same-sized spheres to pack together. The first center sent a line of communication to another center and networks were born. 'Closest-packed' wanted every center as close as possible to the other centers and urged all spheres to attempt to be in contact with as many other spheres as possible. From this beginning the interconnectedness of structure was born.

We can build models of this origin myth. If you closest-pack spheres around a center sphere on a table, it takes the form of the 6-sided hexagon. The hexagon and the circle speak the same language. The self-organization of circles can produce mass production grids of the 60˚triangle or four circles can share a common center and rotate in 60˚orientation with each other to form a sphere. This sphere defines the 14-sided VE.

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  Self-Organization of Spheres Produces a Mass Production Grid

A Self-Organizing Pattern as Four Basic Abilities





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Draw a line across a sheet of paper horizontally. Choose a radius for a drawing compass and draw a circle near the left side of the paper. Where the circle intersects the line are locations-in-common. On this location draw another circle and repeat this to the other side.

Each shape has a different code of how the locations-in-common communicate with each other. Here, the lines define a 60˚triangle, diamond and hexagon. Self-stabilization begins with the first triangles (triangles are the only self-supporting shape). Self-correction is achieved by aligning the locations-in-common. If your original drawing is off, these alignments will self-correct. Self-regeneration happens from the repetition of the patterns.

Once you complete this first cycle of connection two self-innovations happen. One, you have defined the centers of each triangle and two, each edge is divided in half. Both of these innovations are very helpful in building models of modular building patterns.

There seems to be a 'half-force' that exist in simple design in Nature. Connecting the midpoint horizontally begins another cycle of self-organizing; the exact same pattern at half-size. This scale helps with the self-correcting abilities, many more locations-in-common to align. The circle is the push-off start for the other 5 "Not-Us" structural constants.

  The Other Five "Not-Us" Shapes
   The Color Coordination of Shape



Sphere Diamond

Bubble Diamond 


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The reason to set a standardized color coordination for the shapes is convenience. At a glance it is easy to determine which shapes were used to construct these models. Each color represents four bits of useful information:

A Shape

"Not-Us" number or numbers

Angles inside the shape

Structural/numerical constants of the 90˚triangle embedded in each shape

In the images, all measurements will be relative to the base of the triangle, X.



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Sphere Diamond

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  Bubble Diamond

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A diamond is a shape with four equal sides and an internal order of a long and short axis. The Sphere and Bubble diamonds follow the same pattern in their 90˚triangles. The only difference is the pair of numbers,(√2, √3 Sphere family) and (GR, √3.618...Bubble family).


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