The original Wikipedia article as written in March 2012

The Planck length, base-2 exponential notation, and nesting geometries

Introduction:  This article for Wikipedia was written in March 2012.  It was publicly posted within Wikipedia for a few weeks in April; but on May 2, 2012, it was deleted as original research.” Though there are many referenced scholarly journals, there were no scholarly articles from published academic journals regarding the integration of base-2 exponential notation, nested geometries, and the Planck Length. Wikipedia requires such attributions. It is an encyclopedic reference and the primary references for each article protect the integrity and quality of their published articles. So now, we are attempting to prepare these pages to be read by scholars as well as students.

To date, none of these pages have been formally engaged by a senior editor. Some of this writing has been influenced by students, teachers, other interested thinkers, and by faculty within universities and institutes; however, I (Bruce Camber) take full responsibility for all the mistakes of any kind.  Please let me know when you find one.

Some of the links (to Wikipedia articles and others) have been added.  There have been small edits, yet essentially this is the article that had been submitted, initially posted, and eventually deleted by Wikipedia. Also, to go to the page of calculations, Notations 1-to-203, please click here. To go to a general overview, click here, and here to go to more recent overviews. This work has roots with a display project in 1979 at MIT  with 77 leading living scholars.


On measuring the universe using the Powers-of-Two, Exponentiation, and the Planck Length

Base-2 exponential notation (abbreviated here as “B2”) uses the powers-of-two, exponentiation, and the Planck length to provide a simple, granular, ordering system for information. Also, the process of dividing and multiplying by two is the basis for key functions in science, particularly biological systems (cellular division) and chemical bonding, i.e. bond strength. Although base-2 is more granular than dividing or multiplying by ten, base-ten scientific notation has gotten all the attention.

Base-ten scientific notation (B10).  Within the study of orders of magnitude, base-ten scientific notation, is a simple study.  In 1957  Kees Boeke, a Dutch high-school educator, published Cosmic View.

A Nobel laureate in physics, Arthur Compton, wrote the introduction for this work. By 1968 Charles Eames and his wife, Ray, produced a documentary, Powers of Ten based on that book. MIT physics professor, Philip Morrison, narrated the movie and with his wife, Phyllis, they wrote a book, Powers of Ten: A Book About the Relative Size of Things in the Universe and the Effect of Adding another Zero (1982).

NASA and Caltech maintain a website that keeps Boeke’s original work alive and now people have expanded and corrected Boeke’s work.

There is the on-going work of the National High Magnetic Field Laboratory at Florida State University; they give Boeke credit for inspiring their effort called “Secret Worlds: The Universe Within.”

Just fourteen-years old at the time they initiated their online work, genetic twins Cary and Michael Huang developed a most colorful online presentation that opens the study of scientific notation to a young audience. The concepts were widely popularized with the 1996 production of Cosmic Voyage by the Smithsonian National Air and Space Museum for their 150th anniversary (the 20th for the museum). With IMAX distribution and Morgan Freeman as the narrator, many more people are experiencing the nature of scientific notation.

Yet, the work within base-ten scientific notation has not had consistent limits. Most of this work starts at the human scale and goes inside the small-scale universe and stops well-short of the Planck length. Going out to the large-scale universe, the limit was generally-accepted measurement of the observable universe at that time.

Base-2 Exponential Notation (B2), though analogous to base-ten scientific notation, starts at the Planck length and is based on multiples of the Planck Length. Each notation is a doubling of the prior notation. Here the word, notation,  is also referred to as doublings, groups, layers, sets and steps. Though the edges of the observable universe will continue to be studied, scored, and debated, within the B2 system that measurement will always be a ratio of the Planck length. The power-of-2, instead of power-of-ten, provides a very different key to explore a fully-integrated universe in 201+ necessarily inter-related notations.

Use in computer science and throughout academia

See other bases for scientific notation (within Wikipedia).

1234 = 123.4×101 = 12.34×102  = 1.234×103   =  210 + 210

The powers of two are basic within  exponentiation, orders of magnitude,  set theory, and simple math. This activity should not be confused with the base-2 number system – the foundation of most computers and computing.  Though exponential notation is used within computer programming,  its use in other applications to order data and information has wide implications within education.The term, Base-2 exponential notation is also used to describe the number obtained at each step in an algorithm designed to clarify the form and function of space and time — measurement — operates in the range between the Planck length and the edges of the observable universe.

B2 has applications throughout education.

Geometrical visualization

Consistent across every notation is (1) the Planck length, (2) its inherent mathematics (doubling each result across the 202.34 notations) and (3) basic geometries that demonstrate encapsulation, nested hierarchies of objects, space-filling polyhedra (Wolfram), honeycomb geometries (Wikipedia) and other basic structures that create polyhedral clusters (opens a PDF from Indian Institute of Science in Bangalore). It also opens the door to the work within combinatorial geometries.

These are the inherent simple visuals of base-2 exponentiation.

A simple starting point is to take the tetrahedron within the platonic solids and take as a given that the initial measurement of each edge is just one meter. This is the human scale. If each edge is divided by two and the dots are connected, a tetrahedron that is half the size of the original is in each corner and an octahedron is in the middle. If each edge of the octahedron is divided by two, and the dots are connected, an octahedron that is half the size of the original is observed in each of the six corners and a tetrahedron in each of the eight faces. In a similar fashion those two platonic solids can be multiplied by two. These nested objects have been observed and documented by many geometers including Buckminster Fuller, Robert Williams, Károly Bezdek, and John Horton Conway.

Taking just the tetrahedron and octahedron, base-two exponential notation can be visualized. With just these two objects, each could be divided and multiplied thousands of times to fill space, theoretically without limit. Yet, in the real world there are necessary limits. The Planck length is the limit in the small-scale universe. The edge of the observable universe is the limit in the large-scale universe.

Counting Notations

In this context, the numerical output of any given step or doubling is called a notation and  each instance is represented as a multiple of the Planck length.

Starting at the smallest unit of measurement, the Planck length (1.616199(97)x10-35m), multiply it by 2; each notation is progressively larger. In 116 notations, the size is 1.3426864 meters. From here to the edge of the observable universe (1.6×1021 m) is  approximately 86+ additional notations. The total, 202.34 notations, is a number calculated for us by a NASA physicist using data from the Baryon Oscillation Spectroscopic Survey (BOSS). A figure of 206 notations was given to us by the chief scientist of an astrophysical observatory. The total number of notations will be studied more carefully. Compared to the orders of magnitude using base-ten scientific notation, the first guesses had as few as 40 notations while others more recently have calculated as many as 56. The actual number is between 61 and 62.


With each successive division and multiplication, base-2 scientific notation using simple geometries and math can encompass and use the other platonic solids to visualize complexity within each notation.

The Archimedean and Catalan solids, and other regular polyhedron are readily encapsulated simply by the number of available points at each notation. Cambridge University maintains a database of some of the clusters and cluster structures.

Base-2 exponential notation using simple geometry and simple math opens the door to study every form and application of geometry and geometric structures. In his book, Space Structures, Their Harmony and Counterpoint,[1] Arthur Loeb 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).

The calotte model of space filling  will also be introduced.

Because each notation encapsulates part of an academic discipline, there is no necessary and conceptual limitation of the diversity of embedded or nested objects.[2]


Geometers throughout time have contributed to this knowledge of geometric diversity within a particular notation. From Pythagoras, Euclid, Euler, Gauss, and to hundreds of thousands living today, the documentation of these structures within notations is extensive. Buckyballs and Carbon Nanotubes (using electron microscopy) use the same platonic solids as the Frank–Kasper phases[3]. The Weaire–Phelan polyhedral structure has even been used within the human scale for architectural modelling and design, i.e. see the Beijing National Aquatics Centre in China, as well as within chemistry and mineralogy. Each notation has its own rule sets.[4] Some geometers have taken the universe as a whole, from the smallest to the largest, and have described this polyhedral cluster as dodecahedral first in Nature magazine and then in PhysicsWorld (by astrophysicist Jean-Pierre Luminet at the Observatoire de Paris and his group of co-authors.

Constants and universals

There are constants, inheritance (in the legal sense as well as that used within object-oriented programming) and extensibility between notations which has become a formal area of study, Polyhedral combinatorics.

Every notation has a Planck length in common.

Every scientific discipline is understood to be classifiable within one or more of these 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 notations.

Geometries within the 202.34 base-2 exponential notations have been applied to virtually every academic discipline from game theory, computer programming, metallurgy, physics, psychology, econometric theory, linguistics [5] and, of course, cosmological modeling.

See also


  •     Kees Boeke, Cosmic View, The Universe in 40 Jumps, 1957
  •     An Amazing, Space Filling, Non-regular Tetrahedron Joyce Frost and Peg Cagle, Park City Mathematics Institute, Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540
  •     Aspects of Form, editor, Lancelot Law Whyte, Bloomington, Indiana, 4th Printing, 1971
  •     Foundations and Fundamental Concepts of Mathematics, Howard Eves, Boston: PWS-Kent. Reprint: 1997. Dover, 1990
  •     Jonathan Doye’s Research Group at
  •     Magic Numbers in Polygonal and Polyhedral Clusters, Boon K. Teo and N. J. A. Sloane, Inorg. Chem. 1985, 24, 4545–4558
  •     Pythagorean triples, rational angles, and space-filling simplices PDF, WD Smith – 2003
  •     Quasicrystals, Steffen Weber, JCrystalSoft, 2012
  •     Space Filling Polyhedron
  •     Space Structures, Arthur Loeb, Addison–Wesley, Reading 1976
  •     Structure in Nature is a Strategy for Design, Peter Pearce, MIT press (1978)
  •     Synergetics I & II, Buckminster Fuller,
  •     Tilings & Patterns, Branko Grunbaum, 1980


  1. Loeb, Arthur (1976). Space Structures – Their harmony and counterpoint. Reading, Massachusetts: Addison-Wesley. pp. 169. ISBN 0-201-04651-2.
  2. Thomson, D’Arcy (1971). On Growth and Form. London: Cambridge University Press. pp. 119ff. ISBN 0 521 09390.
  3.  Frank, F. C.; J. S. Kasper (July 1959). “Complex alloy structures regarded as sphere packings”. Acta Crystallographica 12, Part 7 (research papers): 483-499. doi:10.1107/S0365110X59001499.
  4. Smith, Warren D. (2003). “Pythagorean triples, rational angles, and space-filling simplices”. [1].
  5.  Gärdenfors, Peter (2000). Conceptual Spaces: The Geometry of Thought. MIT Press/Bradford Books. ISBN 9780585228372.

External links

Categories: Exponentiation, Base-2, Powers of Two, order of magnitude


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