What is the Planck Length?

Follow the progression of the Planck Length to the Observable Universe in just 201+ notations within this horizontally-scrolled, working chart of the universe.  We invite you to help us interpret what it means!


“In physics, the Planck length, denoted <span P, is a unit of lengt, equal to 1.616199(97)×10−35 metres. It is a base unit in the system of Planck units, developed by physicist Max Planck. The Planck length can be defined from three fundamental physical constants: the speed of light in a vacuum, the Planck constant, and the gravitational constant.”


“The Planck length is the scale at which classical ideas about gravity and space-time cease to be valid, and quantum effects dominate. This is the quantum of length, the smallest measurement of length with any meaning. And roughly equal to 1.6 x 10-35 m or about 10-20 times the size of a proton.”


“All of the quantities that have “Planck” attached to their name can ultimately be understood from the concept of the “Planck mass.” The Planck mass, roughly speaking, is the mass a point particle would need to have for its classical Schwarzschild radius (the size of its event horizon, if you like) to be the same size as its quantum-mechanical Compton wavelength (or the spread of its wave-function, if you like). That mass is 1019 GeV/c2, or about 10-8 kilograms.”

“The significance of this mass is that it is the energy scale at which the quantum properties of the object (remember, this is a point particle!) are as important as the general relativity properties of the object. Therefore it is likely to be the mass scale at which quantum gravity effects start to matter. Turning this into a mass is as simple as using the formula for the Compton wavelength given in the above link and plugging in the Planck mass. Thus, the Planck length is the typical quantum size of a particle with a mass equal to the Planck mass. As you point out, the Planck time is then just the Planck length divided by the speed of light.”

“Since the Planck length has this special property of being the length scale where we can’t ignore quantum gravity effects, it is typically taken to be the size of a fundamental string, in string theory. Alternatively, if we consider more general theories of quantum gravity, one might speculate that it is the typical size of the “fuzziness” of spacetime. It’s a length scale (or energy scale) we are unlikely to probe in any future experiments so we tend to interpret it as the length scale at which classical general relativity (GR) “breaks down” — i.e. at which classical GR fails to provide an accurate description of nature. This is very similar to the way that the speed of light is considered the velocity scale at which Newtonian mechanics “breaks down” and special relativity is called for.”
Answered by: Brent Nelson, Ph.D., Research Fellow, University of Michigan

A video

with Alex Filippenko, Clifford Johnson, Max Tegmark and Sean Carroll

The Chart

Follow the progression of the Planck Length to the Observable Universe in just 201+ notations within this horizontally-scrolled, working chart of the universe.  We invite you to help us interpret what it means!


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 http://physchem.ox.ac.uk/~doye/
  •     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 http://mathworld.wolfram.com/Space-FillingPolyhedron.html
  •     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 http://www.washington.edu/research/pathbreakers/1980d.html


  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

An exploration of 101 steps from the smallest measurement, the Planck length, to the human scale, and then 101 more steps out to somewhere near the edge of the observable universe.

Editor’s Note: The very first posting in January 2012 about our work within geometry and base-2 exponential notation (doublings) was within our Small Business School website by Bruce Camber and Hattie Bryant. That site had been live on the web since December 1994.

Here is the very first time we would see the entire universe in just over 201+ steps, all necessarily-related notations.

Perhaps this work could be called, “From praxis-to-theoria.” This working project is dubbed, a Big Board for our little universe. This page is part of a high school geometry class project to use base-2 exponential notation (praxis) whereby the entire universe, from the smallest measurement (Planck length) to the largest (the Observable Universe), is represented in 201+ steps. This project started as a result of studying nested platonic solids. So from the very first notation,  every point is seen as a vertex for constructions.  From a point to a  line to a triangle, then a tetrahedron, octahedron, icosahedron, cube and dodecahedron, form-and-function builds upon itself and within itself. The board’s many blank lines will be filled with facts or conjectures (ideas and concepts, also known as theoria). Eventually real data will be added. The original was created in just a week (December 12-19, 2011). An article about it was posted online in January 2012. It was then updated to include Version of the board, posted  on Saturday, September 15, 2012, however, it is still being updated and will be for a long time to come.  Each notation is to be linked to some of the best research scholars within a discipline that studies things within the range of lengths with each notation.

Big Board-little universeSo, a warm welcome to you… this page provides access to a work-in-progress. Friends and family were the first to be invited to begin a critical review. Now, friends of friends are also being invited! The hope is that the project will be validated in its scope and logic. If the logic and scope are invalidated, the results of that process will be fully reported and analyzed. Is the Planck length the right place to start?  Can a dimensionful number be multiplied by 2?  What are the constants?  Why are universals universal?  To open these questions to discussion, more high school students will be invited to think about this model as a relatively simple way to organize information. College students, graduate students, doctoral candidates, and post-docs will be invited to consider how base-2 exponential notation —  praxis — can become the basis for theoria. Everyone is invited to consider if and how these concepts might be integrated within their own.

Here are links to key working pages for the big board.
• Our first Big Board and today’s Big Board-little universe Chart
Today’s overview of some of the key ideas
First article about the unfolding of the key ideas
An article posted-then rejected by Wikipedia editors

Summary description of this page:   An introduction to collaborative research of an indexed  model of the universe using base-2 exponential notation. Because we start at the Planck Length and go to the Oobservable Universe, these notations are called Planck Notations (PN).

The small-scale universe: PN1 to PN67
The human-scale universe: PN67 to PN135
The large-scale universe: PN135 to PN202+

The back story:  This project began within a high school geometry class in the metro New Orleans area.

202 base-2 notations from the Planck scale render a highly-integrated model of the universe

The Big Board–Little Universe Project USA

Last update: 8 August 2017

IntroductionBig Board-little universe
In December 2011 two teachers and about 80 high school students rather naïvely began to explore a geometric progression that first went down in size to the Planck Length then reversed to go back up all the way to the Observable Universe (most links open a tab or window and go to an in-depth Wikipedia page).

The first chart to be developed, pictured on the left, measures 60×11 inches. It is a view of the entire universe and has just over 200 base-2 exponential notations (dividing or multiplying by 2, over and over again). Thinking that this simple math was already part of academic work, they began asking friends and family, “What is right or wrong within our logic for this model?” A two-year search did not uncover any references to base-2 and the Planck Length.* In that time, asking around locally and then globally, many people were puzzled and asked, “Why haven’t we seen a base-2 scale of the universe before now?”

An Integrated Universe View
Dubbed Big Board – little universe, this project started as a curiosity; today, it is an on-going study to analyze and develop the logic and potential links from their simple mathematics to all the current mathematics that define the universe, all its parts, everything from everywhere, and from the beginning of time to this very moment in time. Their hope is that this simple logic has simple links to real realities. Their standing invitation is, Open To Everyone, to help. This chart follows the progressions from the smallest to the largest possible measurement of a length. Subsequent charts engage the other Planck base units. With more questions than answers, this group is trying to grasp the logic flows in light of current academic-scientific research. Progress is slow.


Yes, on December 19, 2011 the geometry classes in a New Orleans high school were introduced to the chart on the left (Planck Length to the Observable Universe). In December 2014 they began to track Planck Time to the Age of the Universe. When they added the other the Planck base units to each maximum value, it seemed to call out for a horizontally-scrolled chart to follow each line of data more easily. Natural inflation becomes self-evident. And, that opened the way to question the big bang theory, especially the first four epochs — the Planck Epoch, the Grand Unification Epoch, the Inflationary Epoch, and Electroweak Epoch. In their search for answers about this model, questions abound.

This first chart is very early work.
Click on it, then click on it again to enlarge it

What’s next?
They ask, “Where are the informed critics to tell us where we are going wrong?” One rather brilliant, young physicist told them that the concept for this project is idiosyncratic. They quickly learned how right he was. Nobel laureates and scholars of the highest caliber were asked, “What is wrong with our picture? Where is our fallacy of misplaced concreteness?” The group is slowly analyzing the logic and developing their thoughts as web postings with the hope that somebody will say, “That’s wrong” and be able to tell them in what ways they have failed logically and mathematically.

The first 36 of 200+ notations of the horizontally-scrolled chartIf not wrong, the extension of their basic logic could begin to yield rather far-reaching results. For example, the Big Bang theory could get a special addendum, the first 67 notations. That would make it simple, symmetric (entirely relational), predictive, and totally other. The entire universe could get an infrastructure of geometries whereby many issues in physics, chemistry and biology could be redressed. The finite-infinite relation is opened for new inquiries. In this model of the universe, time-and-space are derivative of two quantitative qualities of infinity: continuity-and-symmetry. As a result, these derivative relations begin to have an inherent qualitative or value structure. If so, ethics and the studies of the Mind (the discernment of qualities) just might, for the first time in history, become part of a scientific-mathematical continuum. A trifurcated definition of the individual may emerge whereby people are simultaneously within the small scale, human scale, and large scale universe. Embracing a different sense of the nature of space and time by which both are localized by notation is surely enough; yet there will always be more. There are many working postings that have been written since their first chart; all of it needs constant updating. Many can be found through the top navigation bar option, INDEX.

Notes, lesson plans and posts (and all new posts) are being consolidated and linked from this homepage. Now called, The Big Board – little universe Project, it is a Science-Technology-Engineering-Mathematics (STEM) application. Secondary schools from around the world are being invited to join the exploration. Daily work on the topic is being researched, developed, and communicated through a sister website, http://81018.com.

The earliest postings and blogs were done by Bruce Camber within a section of his website — SmallBusinessSchool.org. That site supports a television series, Small Business School, that he and his wife, Hattie Bryant, started. It aired for 50 seasons on most PBS-TV stations throughout the USA and on thousands around the world via the Voice of America-TV affiliates.

Articles and blogs have been posted on WordPress, LinkedIn, Blogger, and Facebook (often those links open in new windows). An April 2012 article, formatted for and displayed within Wikipedia for a few weeks, was deleted on May 2, 2012 as “original research” by highly-specialized Wikipedia editors. Only then did this little group of teachers and students finally begin to believe that base-2 notation had not already been applied to the Planck base units. And, as they have grown in their analyses, it has become increasingly clear that this area of simple math and simple logic is a relatively new exploration and that notations 1-to-67 may be a key to unlock a new understanding of the nature of physical reality.

The challenge is to study the logic flow within their many charts, all based on the Planck base units, both up and down and across, to build on the question, “Is this logic simple and consistent? What does it imply about the nature of the universe?”

So, even now, there is much more to come. At the end of the year, 2015, a Lettermanesque Top Ten was added. In January 2016 an article, Constructing the Universe from Scratch, emerged. In April 2016 the horizontally-scrolled chart provided a better sense of the flow and of phase transitions. Still a “rough draft” this project has a long way to go! Bruce Camber says, “You are most welcome to add your comments, questions, ideas and insights!


* Footnote: In 1957 Kees Boeke did a very limited base-10 progression of just 40 steps. It became quite popular. In July 2014, physicists, Gerard ‘t Hooft and Stefan Vandoren wrote a scholarly update using base-10. Notwithstanding, base-2 is 3.3333+ times more granular than base-10 plus it mimics cellular reproduction and other naturally bifurcating processes in mathematics, physics, chemistry, biology, topology, botany, architecture, cellular automaton and information theory; it has a geometry; it has the Planck base units, and, it has a simple logic and so much more.


If you would like to contribute content to this site, please contact Bruce Camber
at camber – (at) – bblu.org or click here for more contact information. Thank you.