Among a dozen of the earliest articles about this project, this page was within Wikipedia (March-May 2012). It is reconstructed and opened for editing and development.
Base 2 exponential notation: Proposed, produced, and then rejected by Wikipedia as “Original Research.”
Please note: We started this project in December 2011 and it now continues as a “work in progress.” This project is a result of substituting for a nephew’s high school geometry classes. At one time the words on the right of some sections below, “[edit the original]” went to the article within Wikipedia. Today, they go to an invitation to join the editing team! At some time in the “not too distant future, it just might be re-posted on Wikipedia and then, this page could be edited by all others. My hope in pulling all those pieces and history together is to open the discussion to a wider audience so if parts of it are wrong or could be improved, you can do it or advise me so I can update this article below and the one on Wikipedia.
Base-2 exponential notation is based on the power of two. It should not be confused with a base-2 number system – the foundation of most computers and computing. Exponential notation is used within computer programming, however, its use in other applications to order data and information has wide implications within education. Here the form and function of space and time — measurement — operates in the range between the Planck length and the edges of the observable universe.
Base 10 scientific notation is widely studied and used to depict the universe in colorful ways.      Base-2 scientific notation is more granular and relational. It is an ordering system that de facto can be used within any academic discipline. A didactic example is given within the substantial work that has been done in mathematics, particularly geometry.
Base-2 scientific notation in geometry uses a nested hierarchy of objects, particularly space filling polyhedron and other basic structures that create polyhedral clusters and apply combinatorial geometries.
The simplest of the platonic solids, the tetrahedron, is also a simple starting point. Take as a given that the initial measurement of each edge is just one meter. Starting at the human scale, that object is both divided and multiplied by 2. If one starts at the Planck length, it would always be multiplied by 2. If one were to start at the edges of the observable universe, the result would always be divided by 2.
The limits of base-2 scientific notation
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There are limits. Going within, the limit of the smallest division is the Planck length. It is reached in 115 notations by dividing by 2. Going out through multiplication, the limit is to the edges of the observable universe. It is reached in 91 notations multiplying by 2. The result is similar to the orders of magnitude using base-10 scientific notation.
With each successive division and multiplication, base-2 scientific notation within geometry readily expands to include the other four basic platonic solids, then the Archimedean and Catalan solids, and other regular polyhedron. Cambridge University maintains a database of some of the clusters and cluster structures.
Base-2 scientific notation in geometry involves every form and application of geometry and geometric structures. Arthur Loeb (Space Structures, Their Harmony and Counterpoint ) analyzes Dirichelt Domains (Voronoi diagram) in such a way that space-filling polyhedra can be distorted (non-symmetrical) without changing the essential nature of the relations within structure (Chapters 16 & 17).
There is no necessary and conceptual limitation of the diversity of embedded or nested objects .
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Geometers throughout time — people such as Pythagoras, Euclid, Euler, Gauss, Buckminster Fuller, Robert Williams, Károly Bezdek, John Horton Conway, and thousands of others have contributed to this knowledge of geometric diversity. These manifestations of structure are well-documented within many notations (see Buckyballs and Carbon Nanotubes, using electron microscopy). The Frank-Kasper phases including the Weaire-Phelan polyhedral structure have even contributed to architectural design within the human scale, i.e. Beijing National Aquatics Centre.
Constants & Universals
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There are constants, inheritance (in the legal sense as well as that used within object-oriented programming) and extensibility between notations. Each notation has its own rule sets. Taken as a whole, from the smallest to the largest, this polyhedral cluster has been described as dodecahedral by astrophysicist Jean-Pierre Luminet at the Observatoire de Paris in France.
Polyhedral combinatorics is a subgroup of base-2 scientific notation in geometry.
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In 90 steps of multiplying, one reaches the edges of the observable universe, the largest possible representational geometric number. In 112 steps of dividing, one enters the area of Planck’s constant, the smallest possible representational geometric number. In 202 notations every scientific discipline is necessarily related between notations. Every act of dividing and multiplying involves the formulations and relations of nested objects, embedded objects and space filling. All structures are necessarily related. Every aspect of the academic inquiry from the smallest scale, to the human scale, to the large scale is defined within one of these 206 notations. Both calotte model of space filling and the pleisohedron of space filling are used and continuity, symmetry, and harmony are taken as given to define order, relations, and dynamics respectively.
Geometries within base-2 scientific notations have been applied to virtually every academic discipline from game theory, computer programming, metallurgy, psychology, econometric theory, linguistics  and, of course, cosmological modeling.
- Base 10 Orders of Magnitude
- Space filling polyhedron
- Combinatorial geometries
- Carbon Nanotube
References and External Links
- ^ Loeb, Arthur (1976). Space Structures – Their harmony and counterpoint. Reading, Massachusetts: Addison-Wesley. pp. 169. ISBN 0-201-04651-2.
- ^ Thomson, D’Arcy (1971). On Growth and Form. London: Cambridge University Press. pp. 119ff. ISBN 0 521 09390.
- ^ Frank, F. C.; Kasper, J. S. (1958). “Complex alloy structures regarded as sphere packings. I. Definitions and basic principles”. Acta Crystall. 11. Frank, F. C.; Kasper, J. S. (1959). “Complex alloy structures regarded as sphere packings. II. Analysis and classification of representative structures”. Acta Crystall. 12
- ^ Smith, Warren D. (2003). “Pythagorean triples, rational angles, and space-filling simplices”. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.124.6579&rep=rep1&type=pdf .
- ^ Gärdenfors, Peter (2000). Conceptual Spaces: The Geometry of Thought. MIT Press/Bradford Books. ISBN 9780585228372
- Kees Boeke (Holland), Cosmic View, The Universe in 40 Jumps. In 1957 Nobel laureate in physics, Arthur Compton, wrote the introduction for this work. In 1968 Charles Eames and his wife, Ray, produced a documentary, Powers of Ten based on that book. MIT physics professor, Phil Morrison, narrated the movie. See also, Phil and Phyllis Morrison, Powers of Ten: A Book About the Relative Size of Things in the Universe and the Effect of Adding another Zero (1982)
- Bergman clusters, http://met.iisc.ernet.in/~lord/webfiles/Coordinates.pdf, Indian Institute of Science, Bangalore, India, Department of Materials Engineering
- Jonathan P. K. Doyle, Cluster structures, http://doye.chem.ox.ac.uk/research/cluster_structure.html See J. Chem. Phys., 119, 1136-1147 (2003)
- Econometric modelling http://www.spacefillingdesigns.nl
- Howard Eves, page 131, 1990. Foundations and Fundamental Concepts of Mathematics. 3rd. ed. Boston: PWS-Kent. [Reprint: 1997. Dover Publications.]
- Frank-Kasper coordination polyhedra http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.188.2092&rep=rep1&type=pdf
- Polyhedral Clusters, Indian Institute of Science, Bangalore, India, Department of Materials Engineering, http://met.iisc.ernet.in/~lord/webfiles/clusters/polyclusters.pdf
- Qisheng Lin and John D. Corbett, “New Building Blocks…” “According to higher dimensional projection methods, a series of cubic ACs (approximant crystals) exist with orders (q/p) denoted by any two consecutive Fibonacci numbers, i.e., q/p = 1/1, 2/1, 3/2, 5/3 … F n+1/F n (1).” http://www.pnas.org/content/103/37/13589.full
- Jean-Pierre Luminet et al. 2003 Nature 425 593 http://physicsworld.com/cws/article/news/18368
- Weisstein, Eric W., “Space-filling polyhedron” from MathWorld” http://mathworld.wolfram.com/Space-FillingPolyhedron.html