Category Archives: Base-2 exponential notation
Email: Stephen Wolfram, creator of Mathematica
Date: Sat, 4 Jan 2014 Subject: Cellular automaton References: Key questions: Dear Stephen: Thank you for your 2003 H. Paul Rockwood Memorial lecture on cellular automata. I just finished watching the YouTube version of it and I have learned substantially and I have been challenged. It was all quite brilliant. In light of your work, I need to examine further several simple facts: 1. The universe is mathematically very small. Using base-2 exponential notation from the Planck Length to the Observable Universe, there are just 202.34 (NASA, Kolecki) to 205.11 (Paris, Luminet) notations, steps or doublings. This work (the mathematics) actually began in a high school geometry class when we started with a tetrahedron and divided the edges by 2 finding the octahedron in the middle and four tetrahedrons in each corner. Then dividing the octahedron we found the eight tetrahedrons in each face and the six octahedron in each corner. We kept going within until we found the Planck Length. It was easy to decide to multiply by 2 out to the Observable Universe. Then it was easy to standardize the measurements we just multiplied the Planck Length by 2 until we were in the range of the Observable Universe. 2. The small scale universe is an amazingly complex place. We had to assume the Planck Length is a singularity of one vertex and then we followed the expansion of vertices along each notation. By the 60th notation, of course, there are over a quintillion vertices and at 61st notation another 3 quintillion vertices are added. Yet, it all must start most simply and here the principles of computational equivalence has its a great possible impact. We are right now researching to see how and if AN Whitehead’s point-free geometries could also have applicability. 3. This little universe is readily tiled by the simplest structures. 4. Yet, the universe is delightfully imperfect. In 1959, Frank/Kaspers discerned the 7.38 degree gap with a simple construction of five tetrahedrons looking a lot like the Chrysler logo. The icosahedron can be constructed with two facing pentastars and a band of ten tetrahedrons between them. When made up of 20 tetrahedrons, the icosahedron is more than irregular, it is quite squishy. We call it quantum geometry in our high school. It is the opening to, or the beginning of, randomness. 5. The Planck Length as the next big thing. The behavior may not be so complicated on the surface, but far more complicated just below it. Computers generate rules and these simple rules might be what nature is using below the thresholds of measuring devices. I could go on, but let’s see if these statements are at all helpful. Our work is just two years old yet relies on several assumptions that have been rattling around for 40 years. I’ll insert from references below. Many thanks again for your cellular automaton lecture. Warmly, Bruce Camber First principles: |
Tour #2 Step 3: Extremely-Small and Extremely-Large Numbers
Let us start with the two key numbers: 2. The Observable Universe: 8.79829142×10^{26}meters 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: 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? ________________________________________________________________________________________ Notes about Look-and-feel and Navigation: If a little thumbnail of any picture is displayed, simply refresh your browser and the full-size version hopefully will paste in. Also, if any of the letters from right column, particularly the Archives and Meta listings, are bleeding through the image of the Universe Table, please open your window larger (possibly to full screen). Usually if you click on the last sentence in each description you will go to the next page. More notes about the how these charts came to be: The simple conceptual starting points Wikipedia on the Planck length |
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.
TOUR #1. NEEDS EDITING. BENEFITS STATEMENT
REGARDING PRODUCTIVITY, INSIGHT AND OPTIMISM.
The universe is mathematically very small:
http://doublings.wordpress.com/2013/07/09/1/
TOUR #2. The very small scale universe is amazingly complex.
Assuming the Planck Length is a singularity of one vertex, consider
the expansion of vertices. By the 60th notation, of course, there are
over a quintillion vertices and at 61st notation well over 2 quintillion more
vertices. Yet, it must start most simply and here we believe the work
within cellular automaton and the principles of computational equivalence
could have a great impact. It’s mathematics of the most simple. We also
believe A.N. Whithead’s point-free geometries should have applicability.
Key references for more: http://doublings.wordpress.com/2013/04/17/60/
TOUR #3. This little universe is readily tiled by the simplest structure.
The universe can be simply and readily tiled with the four hexagonal plates
within the octahedron and by the tetrahedral-octahedral-tetrahedral chains.
Key references for more: http://bigboardlittleuniverse.wordpress.com/2013/03/29/first/
TOUR #4. And, the universe is delightfully imperfect.
In 1959, Frank/Kaspers discerned the 7.38 degree gap with a simple
construction of five tetrahedrons (seven vertices) looking a lot like the Chrysler
logo. As I said in the restaurant, we have several icosahedron models with its
20 tetrahedrons and call squishy geometry. We also call it quantum geometry
(just in our high school) and we guess, “Perhaps here is the opening to randomness.”
Key references for more: YET TO BE WRITTEN
Future tours: The Planck Length as the next big thing.
Within computational automata we might just find the early rules
that generate the infrastructures for things. Given your fermions and proton
do not show up until the 66th notation or doubling, what are we to do with those
first 65?
Start at the Planck Length, use base-2 exponential notation and the five Platonic Solids, and go to the Observable Universe in less than 206 necessarily-related steps
- Introduction. The Big Board-little universe was first used in class on December 19, 2011. This Universe Table (a full-sized version is just below) began a year later for those same high school geometry classes. The big board measures 62″x10″ and it is just too big and awkward for desktop work. The table is designed to be printed on 8.5″x11″ paper and displayed on a smartphone. Now, the origin of these two charts can be found by looking inside a tetrahedron and an octahedron. Half-sizes of both objects can perfectly fill each other. The question was asked, “How far can we go inside each by dividing in half over and over again?” The simple answer is, “Not very far.” Within 50 steps we were at the size of a proton. Within another 65 steps we were at the Planck Length. This particular table was created by multiplying the smallest possible length -the Planck Length- by 2 until we got to the edges of the Observable Universe. Distinguished scientists have helped us. One scientist calculated as few as 202.34 notations (doublings or steps) from the smallest to the largest. Another calculated 205.11.
- We use both figures. Within over 202 notations, here is a simple view of the universe, highly-ordered and structurally integrated from the smallest to the largest measurements of a length, all within a geometrically homogeneous group.
- Examples. Initially the examples came mostly from the Paul Falstad’s Scale. Then over the following year other objects of special interest to students were selected. Each should be within the range of the notation defined as no less than 50% smaller or no more than 50% larger than that particular Planck multiple. As we go forward, every entry will have footnotes and links for further explanation and exploration.
- Exhausting. Today’s information glut is so chaotic and overwhelming, it seems that it actually depresses creative thinking. So, our hope in sharing this simple little table is that students will feel empowered to search for new insights to understand this universe more deeply and as we do, to instill some optimism about our common future.
- Notice the first 66 notations. In light of the Planck Length doublings, this is rather new ground. Here, there are very few facts and many, many guesses. And, we believe that there are many more surprises to be found throughout this chart. One might readily conclude that all the surprises are new domains for research and study.
Small Scale Speculations Ideas |
Concepts and Parameters |
Boundaries |
Trans- |
Numbers and Number Theory |
Forms Order Relation Dynamics |
Functions Continuity Symmetry Harmony |
Large Scale |
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1 Planck Length ( ℓP ) |
Transition: Small-to-Human Scale |
1. Display area: Every number/word hyperlinked – quick results display here 2. Options: Open full screen, new tab or window to the research of the experts 3. Also: Related videos-images and online collaborations with up to nine visitors 4. Key Links: http://Universe-View.org and http://BigBoardLittleUniverse.org |
Transition: Human-to-Large Scale |
205+ Observable Universe |
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2- 10 Forms^{1} Vertices:1024 |
77 Research ℓP:2.44×10^{-12}m |
78 X-ray Wavelength |
95 Range: Visible Light |
96 Bacteria Red Light |
113 Hand-size^{H} 16.78^{+}cm |
114 Textbook^{T} 12.8^{+}inches |
131 Marathon 27^{+}miles |
132 54+ miles 87.99^{+}km |
204+ Observable Universe |
11-20 Structure-Ousia V: 1^{+}million |
76 Gamma Wavelength |
79 Huang Scale |
94 Nanoparticles 100-10000^{+}nm |
97 Blood cell^{R} 2.4^{+}microns(µm) |
112 Finger-size 3.3″(inches) |
115 Things 67.134^{±}cm |
130 Race 21.998^{+}km |
133 Drive 108^{+}miles |
202-203+ Observable Universe |
21-30 Substances V:1^{+} billion |
75 Falstad Scale |
80 Periodic Table |
93 Gold Leaf^{G} 160.06^{±}nm |
98 Capillary 5.12^{+}microns |
111 Spoonful 4.19^{+}cm |
116 A child 52.86^{±}in |
129 Distances: 6.834^{+}miles |
134 Gravity-free 351.97^{+}km |
198-201 Superclusters 6.1-54^{+}yottometers |
31-40 Qualities V:1^{+} trillion |
74 Research 1.52^{+}x10^{-13}m |
81 Hydrogen^{H} 31^{±}pm |
92 Nanowires 80.03^{±}nm |
99 Cells 10.24^{±}microns |
110 Makeup^{M} .82^{±}inches |
117 A bed 105.72^{±}inches |
128 Village 3.41^{±}miles |
135 Distance 437.41^{±}miles |
191-197 Virgo Supercluster^{3} |
41-50 Relations V:1+ quadrillion |
73 Research: Tunneling^{4} |
82 Hydrogen^{H} 78^{+} pm |
91 Little chips^{lc} 40.01^{+}nm |
100 Sperm 20.48^{+}microns |
109 Lipstick^{L} 1.04^{+}centimeters |
118 Bedroom 5.37^{+}meters |
127 Walk 1.7^{+}miles |
136 Fly 874^{+}miles |
181-190 Galactic Group^{6} |
51-60 Systems The Mind^{M} |
72 Nucleus^{N} 7.63^{+}x10^{-14}m |
83 Carbon^{C} 70^{±}pm^{2} |
90 Viruses 20.007^{+}nm |
101 HAIR 40^{+}microns |
108 Diamond^{D} 5.2^{+}mm^{M} |
119 Home 35.24^{+}feet |
126 Downtown 1.37^{+}km |
137 Rivers 2815.81^{+}km |
171-180 Milky Way |
61-65 Elementary Particles |
71 Gold^{AU Nucleus}^{ } |
84 WATER^{W} 3.12^{+}x10^{-10}m |
89 Cell Wall 10^{+}nm |
102 Paper 81.95^{+}microns |
107 Ants 2.62^{+}mm |
120 Property 21.48^{+}m |
125 Superdome 687.45^{+}m |
138 USA-to-UK 3500^{+}miles |
161-170 Solar^{S} Interstellar |
65-67 Neutron Proton-Fermion |
70 Aluminum^{Al} 1.90^{+}x10^{-14}m |
85 DNA^{D} 6.25^{+}x10^{-10}m |
88 Insulin 5.00^{+}x10^{-9}m |
103 Egg^{E} .16^{+}millimeters |
106 Sand 1.31^{+}mm |
121 Yacht 142^{+}feet |
124 Skyscraper 343.7^{+}meter^{+} |
139 Earth^{E} 11,263^{+}km |
151-160 Solar System^{S} |
68 Helium^{He} 4.77^{+}x10^{-15} m |
69 Electron 9.54^{+}x10^{-15}m |
86 Buckyballs 1.25^{+}nm |
87 Ribosomes 2.50^{+}nm |
104 >^{.}< Period .32^{+}mm |
105 Bacterium .65^{+}mm |
122 Sequoia 85^{+}meters |
123 Tall Building 171.86^{+}m |
140 GPS Satellite 22526^{+}km |
141-150 Earth Systems |
THIS IS A VERY ROUGH DRAFT AND A WORK IN PROGRESS: Links and references to the work of scholars within each notation, 1-to-205, will be added such that every entry on this chart will be an active link. On-going Investigations: Distance between bonded hydrogen atoms, diameter of a carbon atom, and a better definition of the ranges within the small scale and large scale universe, i.e. 191-197: 48 zettameters-to-4.7± yottometers and the Local Galactic Group Tunneling — a wild-and-crazy speculation: There may be parallel constructs whereby tunneling is part of the transformation More than things, as in protons and fermions, could the results of cellular automaton be understood as Plato’s forms (perhaps notations 3-to-20) and Aristotle’s ousia (perhaps doublings 20-to-30)? Assuming the Planck Length to be a vertex, and assigning the area over to pure geometries, do we have the basis for form, structure, and the architecture for substances? Then, could it be that this architecture gives rise to an architecture for qualities (notations 30-to-40)? And, as we progress in the evolution of complexity, could it be that in this emergence, there is now an architecture for relations (notations 40-to-50)? If we assume an architecture for relations, could the next be an architecture for systems (notations 50-to-60) and this actually becomes the domain of the Mind? It is certainly a different kind of ontology given it all begins with cellular automaton and base-2 notation to provide a coherent architecture (with the built-in imperfections of the five-tetrahedral cluster also known as a pentastar). More: https://utable.wordpress.com/2014/01/20/osi/ |
#2 The Planck Length is the smallest; the largest length is the Observable Universe.
This may well be the first time you will have seen the entire universe notated on a chart, all numerically and geometrically ordered, in a very granular relation. It is a simple model that opens many questions. The Planck Length is not a point because it has length. Points do not have length. And, if you were to multiply that length over and over again by 2, anywhere from 202.34 times to 206 times, you will be out to the largest possible length, the Observable Universe. It is hard to believe, yet the simple math tells the story. This project uses that range, 202.34 to 205.11 because even though math can be exacting, our knowledge of the age of the universe is not. |
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By the way, these notations are already helping students to understand the relations between disciplines. There is a sense of real order where there was chaos. _______________________________________________________________________________________________________________________ Notes about Look-and-feel and Navigation: If a little thumbnail of any picture is displayed, simply refresh your browser and the full-size version should paste in. Also, if any of the letters from right column, the Archives and Meta are bleeding through the image of the Universe Table, please open your window larger (possibly to full screen). Usually if you click on the last sentence in each description you will go to the next page. More notes about the how these charts came to be: Wikipedia on the Planck length Key Surprises: First, just the very small number of notations was a surprise. When we started at one meter and got down to the size of a proton in about 50 steps (at that time, we were dividing each notation in half), it seemed hard to believe. Then, when we got in the range of the Planck Length in another 66 steps, it was puzzling. When we went to the large scale and found only another 90 or so notations, we were flummoxed. The universe divided in thirds. The old logic of the Small Scale, the Human Scale, and the Large Scale universe is in play, yet here the start points are quite different than the historical use of those terms. Here the very logical range for each is defined by simple math. What does it mean to look at the totality of the universe and it in thirds. The simple geometries. If the Planck Length is defined as a single vertex at notation 1, by step 60, it becomes a major structural enterprise. The proton-fermion-electron appear to manifest at step 66. This small scale universe of geometry and numbers is unexplored. Perhaps it is the first step for a science and basic logic for perfected states within this continuum. |
#4 Just fifteen steps up the scale to step 116, we find the child within.
Human life starts right within that midrange, at Step 103, but we grow up and up into step 116, and just a few into step 117. The Planck Length has now been multiplied by 2, and each result by 2, over and over again, 116 times. This length is rendered here in inches (52.86) and meters (1.342864). Within this notation there are all kinds of very familiar things. We selected children to represent the category. |
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Think about the birth cycle. Growth in nine months usually renders an infant mostly within Step 113 and usually through a birth canal. Most of us hardly give it a thought. Yet there are parallel constructs throughout nature. By approaching the entire universe as an ordered set, a group that has well-defined subsets, we can re-examine our presuppositions one more time; and in a very new way, look for new insights to old enduring questions. Key idea: Within this notation there are all kinds of very familiar things. We selected children to represent the category so you might suspend your judgment and become like a child. Engage the entire universe like you have never done before today. This discovery process will challenge your imagination and may press in on your belief systems. Please remind yourself that all these numbers are the result of simple math — multiplying by 2. It is also the result of following embedded geometries within two of the most simple, pure objects known to our academic community. We propose, and you will soon see, how these geometries pervade everything. But first, our mea culpa. _______________________________________________________________________________________________________________________ Notes about Look-and-feel and Navigation: If a little thumbnail of any picture is displayed, simply refresh your browser and the full-size version should paste in. Also, if any of the letters from right column, the Archives and Meta are bleeding through the image of the Universe Table, please open your window larger (possibly to full screen). Usually if you click on the last sentence in each description you will go to the next page. Sizes: The range for this notation goes from about 36 inches to 79 inches (6′ 7″ feet). So much of our life is within this notation. Each notation ranges from just under 50% larger than the Planck Length multiple (115) in the smaller notation (2.20 feet or 67.134 cm) to just under 50% smaller than the Planck Length multiple (116) in the larger notation (105.72 inches). Boundary conditions between notations are a key part of this study. |
#5 Our caveat: We are still at the very beginning of this study.
There are mistakes, but we hope our intention is clear. We move to Step 97. To represent this step we chose the diameter of a single cell of blood. It’s been said, “Blood is thicker than water” and it is. The study of blood is called hematology. The study of the genetic transmission of characteristics from parent to offspring is called heredity. . |
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Questions abound: What is blood? What within it makes it so very important to sustain and transfer human life? Why does blood manifest within this range (notation, step or layer)? Yes, though blood is four times smaller than a human hair, a water molecule is even smaller. Key Idea: Each notation is important. Each has its own geometry and parameters, boundary conditions and transformations. There is a lot to study. We are inching along at the beginning of this study within this construct of the universe. Though this scale of the universe may not prove to be scientifically useful, heuristically it has already made a mark. So, let us move on toward the Planck Length by next going to a water molecule. _______________________________________________________________________________________________________________________ Notes about Look-and-feel and Navigation: If a little thumbnail of any picture is displayed, simply refresh your browser and the full-size version should paste in. Also, if any of the letters from right column, the Archives and Meta are bleeding through the image of the Universe Table, please open your window larger (possibly to full screen). Usually if you click on the last sentence in each description you will go to the next page. |