Wilson, Edwin O.

E.O. (Edward Osborne) Wilson
Harvard University

Book: Consilience: The Unity of Knowledge  
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First email: Thu, Jul 21, 2016 at 8:14 AM

https://www.ted.com/talks/e_o_wilson_on_saving_life_on_earth

My dear Prof. Dr. E.O. Wilson:

In December 2011 a group of high school people went inside the tetrahedron, dividing by 2, and found the half-sized tetras in the four corners and an octahedron in the middle.  We went inside that octahedron, dividing by 2, found the half-sized octas in each of the six corners and eight tetras in each face, all sharing a common center point. We kept going within all 19 objects.  Within just a few steps we found our nematode friends. In another few steps the prochlorococc greeted us, “Set em up baby…”

In just 45 steps within we were zipping by the fermions and protons and just kept going!  In the next 67 steps, you wouldn’t believe what we saw! We were at the door of a singularity that Max Planck gave us and all those secret codes, but it took 100 years and Frank Wilczek to begin to interpret them (2001, Physics Today, Scaling Mt. Planck I-III).

Just over 112 notations.  What was that?

It didn’t take too long before we got the bright idea, “Let’s multiply by 2.” What an epiphany! In less than 90 steps we were out to the Age of the Universe and the Observable Universe. Looking at ourselves, we were lost within all this new information, so we decided to turn to the experts. Huh? We found Kees Boeke’s base-10 work from 1957 but he only had 40 quick jumps (Cosmic View) and missed so much of life!  We found Stephen Hawking but he was in tight with big bang theory. Where are our experts?

What? Huh?  Our knowledge of the universe is so incomplete, our sense of the universal is so limited, our understanding of the constants is so elementary, we are flying blind.

The Encyclopedia of Life truly needs a wonderfully integrative, expansive container so it doesn’t get walled in!  Of course, its website opens it to our world.  Let’s open it to the universe.  Yes, a wall-less container where ideas and creativity can explode old boundary conditions and creatively new parameter sets emerge.

Now we are amateurs, but we really feel that biology and the search for life must begin with that initial creation, the first moment, when there was a profound integration, and come through it all right to the 200th notation to our present day.  Let’s encapsulate the universe so we can truly address the “… transcendent qualities in the human consciousness, and sense of human need” (from your Ted Talk).

Are we crazy?  Of course, we are, but hopefully delightfully so! Thanks.

Most sincerely,

Bruce
*****************
Bruce Camber

http://bblu.org

https://bbludata.wordpress.com/2016/05/25/timeline/
https://bbludata.wordpress.com/1-204/

PS.  I grew up not far from the Peabody and all the glass flowers. My father was an HVAC machinist for the Mark I while my mother had been a nanny for Shady Hill characters.

In 2002, Wilczek reflects, “It therefore comes to seem that Planck’s magic mountain, born in fantasy and numerology, may well correspond to physical reality.” (PDF)

“Can’t you see, we are in a dialogue with the universe?” asks Charles Jencks.

Dark Matter, Cosmology and the Large-Scale Universe

First draft. In process. Your feedback is key.  Last update: January 19, 2016

Yes, today is Thursday, November 26, 2015. Paris is healing. Al-Raqqah is burning. And, without question, we all need a new model of who we are and why we are. There is simply too much darkness within this world and within our views of the universe.

Many people are aware that we have been on a speculative journey 1 that began in 2011 in a high school. Our journey is based on an idiosyncratic way of looking at our universe.2 Everything, everywhere, throughout all time is contained within just over 201 base-2 exponential notations from the Planck Base Units out to their largest possible measurements, always “this given moment”  or the Now.  Although our charts are simple, logical, and comprehensive,  and all the numbers show us an exquisitely interconnected universe, they challenge some of our commonsense worldviews.

It is not easy to change one’s conceptual frame of reference. Oftentimes I feel a little slow because it becomes so difficult to put one-and-one and two-and-two together. But, we are motivated to break through the bottleneck of questions that have remained unanswered over our lifetime and throughout the centuries. We are pushed to ask, “What are we missing?” Taunting myself, I say, “Go over this one more time; you really are missing something here.”

Dark Matter and Dark Energy are in that camp for me. Yet, recently a faint light has come on. A new book by physicist, Lisa Randall, had become a subject of a discussion over lunch.  The subjects of cosmology have always been way out there for me. Since childhood, especially on the occasion of a dark, clear Maine night, I have had a visceral reaction to it all. Spooked, jammed and suffocated by the vastness of the seemingly infinite universe, truly impossible to take it in, I learned to avoid cosmology and astronomy.

Dark Matter 3, 4 (links that go to Wikipedia open new tabs or windows).  I had begun thinking that Dark Matter was a misnomer. Within the Big Board – little universe model, it is not dark per se, especially if part of the first 60 notations where there is an absence of light. Within our isotropic and homogeneous universe, space-time-charge-mass-and-temperature have a tightly-woven fabric, tiled and tessellated defined by base-2 exponential notation and the Planck Base Units.

Though associated with analyses of structures on the order of galactic scale, Dark Matter may best be seen within the context of all 201+ base-2 notations, yet especially the first 60 or so notations.

Everything, everywhere, throughout all time.  That is the simple mathematical model we call the Big Board-little universe.  We can see that space and time are contained.  We can see a very simple beginning  and we see-experience-feel the current time.  We have our seeing-experiencing-feeling moments within the current time which is within the 201st notation.  We have the physical definition of parts, for now I call these our physical archetypes  between the 67th and 116th notations within the Human-Scale Universe.  Our most essential being, the archetypes of our archetypes, are within the small-scale universe. Everything-everywhere-throughout-time has some kind of presence with the small-scale, human-scale, and large-scale universe. 

There will be many challenges ahead to texture, define and defend such statements especially given that Dark Matter now constitutes only as much as 26% of the universe.5   Out of our naïveté, it just may be that our simple model, this Big Board – little universe,  gives Dark Matter a space and time, mass and charge, and a reason for being, and for being where it is.Gold & Diamonds

I believe our little chart is like a gold and diamond mine because the key figures of the universe are all there just waiting to be examined.  These figures have been given a cursory review.  It is now time for the professionals to go in and take a hard look.  I believe we can find configurations within those first 60 notations that can give rise to the observations and research throughout the entire history of cosmology and astronomy.

Most speculative:  The small-scale universe is the human-scale universe is the large-scale-universe. It is discrete, quantized and derivative of continuities and symmetries and even an overarching harmony of the universe. Within this speculative journey we are defined by every notation. Our particularity and humanity come out within the 101 to 116th of the Planck Length but we are fully at the 201+ notation within “the moment” or “the now.”

A special challenge will be to  come up with a better name than Dark Matter.  Although the inherent continuity equation for those 201+ notations has classic characteristics generally used to describe God, here we will rely strictly on the simple logic and mathematics, the Planck Base Units, and basic geometries to create the parameters for our naming convention.

Dark Energy. Though this conceptual frame of reference is related to the expansion of the universe, and dark energy weighs in at 68.3% of the total energy in today’s observable universe, we would postulate that it, too, is related through the Planck Base Units and the systemic expansions within the first 60 steps of the Big Board – little universe model. Light (a light second) does not appear until the 143 notation; that requires another 80 or so steps or notations or doublings from those first 60.

Projections. It has been almost four years since we began the discussions about our charts of the Big Board – little universe.  It was only three years ago that we were encouraged to continue our study of the Planck Base Units by Prof. Dr. Frank Wilczek6,7 of MIT (a world-renown physicist and champion of Planck’s work). In December 2014 Planck Time was added to the chart.  Then, in February 2015 the other Planck base units were included.

Obviously, for me, concepts need time to incubate.

First, as uneven as we are, I hope we are getting closer to justifying to the academic communities to come in to take a look at the inherent structures of the first 60 notations. To begin examining how those first 60 notations came to be and why they have been ignored, we have begun to focus on all the research and all the data from cubic-close packing (ccp), face-centered cubic (fcc), hexagonal close-packed (hcp), root system (A3) and Lie theory. It seems that all these applications and functions apply to the first 60 notations. And, inside those first 60 steps come all the dimensionless constants and soon thereafter, all the physical constants, and soon thereafter, the fermions and every expression of physicality.

That’s a rough outline within which our research of the first 60 notations will proceed. Now, where’s Lisa Randall 8 this morning and what has she been saying about dinosaurs?

Thank you.

***

Endnotes:
1  Speculative journey, dubbed the Big Board – little universe, began in December 2011 in our high school geometry classes.
2  Idiosyncratic way of looking at our universe is actually based on simplicity itself. In December 2014, Planck Time was added alongside the Planck Length. That was filled with pleasant surprises. Then in February 2015, the Planck Mass, Planck Charge, and Planck Temperature were added.
3,4,5  Dark Matter is the subject of many scholarly articles.  The Wikipedia entries, however, are straightforward, easy reading:
https://en.wikipedia.org/wiki/Dark_matter
https://en.wikipedia.org/wiki/Lambda-CDM_model
https://en.wikipedia.org/wiki/Planck_%28spacecraft%29#2015_data_release
6,7  Prof. Dr. Frank WilczekThroughout December 2011 and then throughout the following year, I asked literally hundreds of people, “Can you responsibly multiply the Planck Length by 2?” In December 2012, we got a response, an answer from Prof. Dr. Frank Wilczek,  “One can certainly divide (or multiply) by two repeatedly, and generate different numbers or scales, just as one can with powers of ten.”
https://bblu.org/2012/09/03/analysis/#Wilczek
https://bblu.org/2015/07/24/simple/#CMO
8 Many years ago, back when we were doing our weekly television productions, I called and talked with Lisa Randall about doing a PBS-special about her book, Warped Passages. She had been inundated with such proposals. Subsequently, in 2012, I sent a note to ask about the cogency of our base-2 model. We had been told by an astute-but-young theoretical physicist (who asked to remain anonymous) that our work was idiosyncratic, but he wasn’t quite ready to say that it was not even wrong. We’ve asked some of the finest scientists from around the world and the brave among them in some way said, “Carry on (…but please don’t quote me).”

Lisa’s new book, Dark Matter and the Dinosaurs: The Astounding Interconnectedness of the Universe, reaches the same general conclusion as we do about the connectedness of the universe, yet I am rather confident that she would distance herself from our work and the reasons for our conclusion!

*************

A few of the articles that have been written over the years.  These are all being updated and consolidated here.

Extremely-Small and Extremely-Large Numbers

Let us start with the two key numbers:
1. The Planck Length: 1.61619926×10-35 meters which is 0.0000000000000000000000000000000000161619926 meters

2. The Observable Universe: 8.79829142×1026meters or 879,829,142,000,000,000,000,000,000 meters

There are many numbers in between the two. Each “0” represents a major base-10 transformation; and within each base-10, there are three or four base-2 notations. Though some say that the Planck Length is a special type of singularity, it has a specific length. Yet, that length is so small, for about 100 years, it was virtually ignored by the entire scientific community. Perhaps a better way of looking at the Planck Length is through the lenses of geometry. If we make it one of Alfred North Whitehead’s point-free vertices of a specific length, each time we multiply by two we grow the size as well as the number of vertices.

The Numbers of Vertices at Key Notations Between 1 and 65. When you assume that the Planck Length is a vertex, unusual concepts flow. First, consider the generation of vertices just by multiplying by 2, then each result by two, over and over again. By the tenth doubling there are 1024 vertices. By the 20th doubling, over a million more are added. On the 30th, another billion+ are added. Then, comes another trillion+ at the 40th, a quadrillion+ at the 50th notation and a quintillion+ at the 60th. At the 61st there are another 2+ quintillion vertices added. These vast arrays and systems of vertices cannot be observed.

This is the domain of postulations and hypostatizations. Consider this concept: going within from about the 65th notation, the domains begin to be shared. More and more is shared by everything as the Planck Length approaches. Each notation organizes uniquely, yet within groups. And these natural groupings reflect all the diversity within all the notations 65 and higher. It seems that the mathematics of cellular automaton may figure into the first 20 or 30 notations. We start with the most basic Forms, then Structures, which become the pre-structure for Substances, archetypes for Qualities, then Relations, then the Mind. We turn to systems theory, group theory, and set theory to discern the order of things.

Perhaps there are five hot spots for immediate research:
* Notations 1-20 and the foundations of cellular automaton and fractal geometries by using the functions created by more than one million vertices
* Notations 50-60 and the foundations of the Mind, logic, psychology, memory, thought, epistemology and learning with over 500 trillion vertices at the 59th notation and then another quintillion+ vertices within the 60th notation.
* Notations 60-80, the emergence of the particles and atoms and the most basic structures of all physical matter
* Notations 100-103, the emergence of the human life and most all life as we know it
* Notations 135-138, the transition to the Large-Scale Universe with the possibilities of uncovering pathways to the Einstein-Rosen bridges and tunnels also known as wormholes.
Key references for more: The numbers

Facts & Guesses. The Facts are what is measurable and what fits within each domain. The Guesses are about what goes on with those domains (aka steps, notations, layers or doublings) especially those that remain blank. Is there a pattern, especially a cyclic pattern that manifests in another notation? We followed Max Planck where he took the constants of nature, starting with the speed of light to calculate the smallest number. We took the age of the universe, with some help from scientists, to learn the largest calculation of a length, the Observable Universe. Making sense of these numbers is another story. So, over the forthcoming weeks, months and years, we will be looking even deeper. Would you help us now and take the little survey?

More notes about the how these charts came to be:
1Three downloads authored by Prof. Dr. Frank Wilczek: Scaling Mt. Planck (from Columbia University), C. Alden Mead’s letter and Wilczek’s response in Physics Today, and Wilczek’s August 2013 Lecture notes on units and magnitude (If you like this paper, also read this one).

The simple conceptual starting points
An article (unpublished) to attempt to analyze this simple model. There are pictures of a tetrahedron and octahedron.
A background story: It started in a high school geometry class on December 19, 2011.
The sequel: Almost two years later, a student stimulates the creation of this little tour.

Wikipedia on the Planck length
Wikipedia on the Observable Universe

Take it as a given that it is also a vertex. By the second doubling, there are four vertices, just enough for a tetrahedron. By the tenth doubling there are 1024 vertices. The number doubles each notation. By the 20th doubling, over a million more are added. On the 30th, another billion+ are added. Then, comes a trillion+ at the 40th, a quadrillion+ at the 50th notation and a quintillion+ at the 60th. At the 61st there are another 2+ quintillion vertices. What does it mean?

The simplest geometries yield a deep-seated order and symmetries throughout the universe. Those same simple geometries also appear to provide the basis for asymmetry and the foundations of quantum fluctuations and perhaps even human will.

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.

Diversity

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]

History

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

Bibliography

  •     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

References

  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