Please note: It appears that China has disabled the links to servers coming through the USA. We’ll will try to re-route some of those links.
Third email: Wednesday, May 28, 2020
Dear Prof. Dr. Chuanming Zong:
First, let me congratulate you on your new location. Wonderful. It appears that you are still within 100 miles of Beijing. That’s excellent.
I am still quoting you after all these years (see above). Because the citations were getting so numerous, I created references page for you and Prof. J. Lagarias. My page for you: https://81018.com/2020/05/28/zong/
In these days and times, my most important conclusion is here about all our work, collectively and individually: https://81018.com/duped/#R3-2 Of course, if you would like anything changed, deleted, or added, I will be glad to accommodate your request. Thank you.
Second email: Wednesday, January 8, 2014
Your paper is sensational. It is exactly what I needed to be assured that Frank-Kaspers and many others were not leading us astray.
Your mathematics and analysis are spot on.
Let me share my reasons for my enthusiasm below this note to you. Thanks.
PS. Your work helps us with #2 and #4 below:
1. The universe is mathematically very small. Using base-2 exponential notation from the Planck Length to the Observable Universe, there are somewhere over 202.34 and under 205.11 notations, steps or doublings. NASA’s Joe Kolecki helped us with the first calculation and JP Luminet (Paris Observatory) with the second. Our work began in our high school geometry classes 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 tetrahedron in each face and the six octahedron in each corner. We kept going inside until we found the Planck Length. We then multiplied by 2 out to the Observable Universe. Then it was easy to standardize the measurements by just multiplying the Planck Length by 2. In 202 notations we go from the smallest to the largest possible measurements of a length.
2. The very small scale universe is an amazingly complex place. Assuming the Planck Length is a singularity of one vertex, we also noted the expansion of vertices. By the 60th notation, of course, there are over a quintillion vertices and at 61st notation well over 3 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. The mathematics of the most simple is being done. We also believe A.N. Whitehead’s point-free geometries should have applicability. 3. This little universe is readily tiled by the simplest structures. The universe can be simply and readily tiled with the four hexagonal plates within the octahedron and by the tetrahedral-octahedral-tetrahedral chains.
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. We have several icosahedron models with its 20 tetrahedrons and call squishy geometry. We also call it quantum geometry (in our high school). Perhaps here is the opening to randomness.
5. The Planck Length as the next big thing. Within computational automata we might just find the early rules that generate the infrastructures for things. The fermion and proton do not show up until the 66th notation or doubling.
I could go on, but let’s see if these statements are interesting to you in any sense of the word. -BEC
Congratulations on all that you doing to preserve the best among our scholars.
I have a deep and abiding affection for Max Planck and his base units.
Is it your office where the Vetter printing of Max Planck hangs today? Does your museum/library gift shop sell a replica?
So entirely delightful. That it was painted in my birth year, 1947, is sweet. That it is an artistic image, probably the last prior to his death, gives it even more vibrancy.
Though Freeman Dyson and I go back to work in 1979, I remember writing to you in and around 1970! I am getting too forgetful and these postings are a way for me to check on my most recent communications with a person. If you would like anything changed, updated or deleted on this page, just let me know and I will be as expeditious as possible. Thanks.
I just came upon the work of Nassim Haramein where I learned about his Unified Field And Sacred Geometry that you were attempting to write a scaling law essentially to encapsulate the universe. How is that going?
Might the 202 doublings of the Planck base units to the Age of the Universe
and the size of the universe be a simple solution? Perhaps too simple, at least
it is a start: https://81018.com/chart
First email: Sunday, July 17, 2016, 9:05 AM
We corresponded back in the ’70s. Noyes, Bastin, Bohm were all mutual friends. My interim story is too arduous, perhaps for another time. You have been prodigious. What a vitae!
In 2011 in a high school I had the five geometry classes go inside the tetrahedron and octahedron. Dividing by 2, this perfect, interior tessellation brought them face-to-face with the proton in just 40 steps, and then face-to-face with the Planck base units in another 67 steps. A sweet journey it was. In our next time together, we multiplied by 2 and in about 90 steps we were out to the edge of the universe, well beyond Kees Boeke.* 3.33 times more granular with imputed geometries and the Planck base units, what was not to love about it?
The project got away from us and has it own life: /https://81018.com/chart
That link goes to a rather large, horizontally-scrolled file.
Is it all wet? …too idiosyncratic? …too simple?
I thought you would find it of some interest, if just as a novelty. A penny for your thoughts? Thanks.
New Orleans http://81018.com
*Of course, Kees Boeke’s base-10 is great fun, but it doesn’t mimic
cellular reproduction and bifurcation theory, nor does it engage
cellular automaton, or the automorphic forms of the Langlands
We took the Planck base units of Length, Time, Mass and Charge and
applied base-2 exponentiation. In 202 doublings, the chart is out to the
Age of the Universe. It seems straight forward, however, the results are
First, it is a simple, logical, mathematical map of the universe.
I am not sure… are there any others?
By studying the numbers associated with each doubling, we see
that most of the 202 doublings are about the early universe.
Notation 143 contains the first second.
Notation 197 contains the beginning of large structure-formation.
All the numbers are here: https://81018.com/chart/
Though entirely idiosyncratic, I think there is something here.
1. Of course, the Planck Length doubling at one second, divided
by the Planck Time doubling at one second is very close to the value
of the speed of light in a vacuum. It is consistent with Planck’s initial
equation for Planck Time. Yet, it is naturally also consistent within
each of the other 202 notations.
3. All notations appear to be forever active and necessarily
interdependent and appears to define the Now for Richard Muller
and Carlo Rovelli.
We started this project in a high school geometry class in 2011
( https://81018.com/home/ ) with just the Planck Length. We
did not introduce Planck Time until 2014 and Planck Mass and
Planck Charge until 2015. So, really we have just begun to study
and attempt to understand these numbers in light of current theories
within cosmology and physics. It is entirely provocative!
Even though it is idiosyncratic, is there any hope for it?
The current homepage is my latest attempt to spotlight key ideas
and problems. Thank you.
Yes, you are back up on our homepage today: https://81018.com/believed/ Essentially, inspired by Murray Gell-Mann, I thought you would want to know.
In light of the Ellis Physics on Edge harangue, virtually touching everyone who has been a leading thinker in the past 20 years, right to the final paragraph with Dawid-Rovelli, I repeat the John Wheeler 1986 statement within his article, “How Come the Quantum?” where he says, “Behind it all is surely an idea so simple, so beautiful, that when we grasp it — in a decade, a century, or a millennium — we will all say to each other, how could it have been otherwise?”
These simple numbers may bear him out: https://81018.com/chart/ Simple processes, like Euler’s equations and base-2 notation give us an entire range of unexplored numbers from the second notation to at least the 64th notation. That’s a science unto itself. Pure math, perhaps the string theorists could finally claim a home.
It is easy to write off simplicity, yet someday these numbers will be explored by the likes of somebody as informed as you are. Thank you.
Second email: Sunday, 27 January 2019
Perhaps the earlier email (below) was buried. Perhaps this base-2 model is just too absurd to acknowledge. I am just a simple guy following simple logic.
1. The Planck base units of length, time, mass and charge describe a real reality.
2. The conceptual door to this infinitesimal universe is where all four Planck base units concresce (grow together, yet individuate) to create a stream of infinitesimal spheres. Though physical, length-time are well below thresholds of measurement, the progression of mass-charge units can be studied. These four units are, in some manner of speaking, the Janus-face of each other and of light.
Since December 2011, I have been carrying on in this light, slowly, intentionally, but naively. I wish somebody of your stature and command of all the academic fields involved would take a moment and put a stop to this effort if it is sheer poppycock. Thank you ever so much. Sincerely,
Our focus has been on the Planck scale. We believe there it has more to contribute than meets the eye.
Between the Planck scale and “CERN-scale of measurements,” there are 67 doublings (or notations or causal sets) of the Planck base units.
[If assume the very first instant of the universe is that which is defined by Planck Time and Planck Length and Planck Mass and Planck Charge and that there is a natural base-2 expansion, the 202nd notation includes the current day and time].
Perhaps it might be better to start at the first doubling and to observe the logical possibilities.
Might other factors like Causal Dynamical Triangulations (CDT), Regge calculus, fractal structure, 2-D spacetime, and the flavors of the simplex be included in an appropriate build-out and within an appropriate doubling?
I think that the emergence at the Planck base units, the simplest planckspheres may well account for what we know as dark matter and dark energy. I believe it’s a relatively simple calculation.
Among all the people to whom I write, I suspect you can debunk this concept most quickly; or, you may be surprised at its simplicity and possibility. Of course, the derivative, discrete nature of space-time is necessary and I think we would do well to redefine the infinite with mathematical terminology and anticipate a finite-infinite transformation possibly further defining the renormalization process..
Follow-up (from CDT and Cosmology): • Non-renormalizability of perturbative quantum gravity • FLRW paradigm: Physics Beyond the Standard Models of Particles, Cosmology and Astrophysic. • Imposing homogeneity and isotropy on spatial slices of constant time t. • So-called “backreaction” effect of inhomogeneities on smaller scales on the dynamics of the universe on larger scales • “…an explicit realization of a non-perturbative, Planckian quantum dynamics…”
More Work to do (further research): • Path integrals and Gaussian fixed point. See Assaf Shomer’s on page 7: “The derivation of the path integral formula in quantum mechanics of a massive particle involves chopping up the quantum evolution into very short time intervals and inserting complete sets of states between them.” • Doplicher S, Fredenhagen K, Roberts JE (1995) The quantum structure of spacetime at the planck scale and quantum fields. Communications in Mathematical Physics 172(1):187–220 • Scale invariance and conformal symmetries
This website is a study of a model of the universe based on an application of base-2 exponentiation (multiplying by 2 or doublings) that eventually encapsulates-and-relates everything, everywhere, for all time in just 202 notations. This model of the universe starts with Planck Length and Planck Time, the smallest possible measurements, and goes to the largest. It starts with the very first moment in time and goes to this very moment in time.
There has never been a mathematical model of the universe quite like it.
To get an intuitive sense of this model is difficult. Four sacred cows of science need to be reexamined; these are best summarized as continuity, symmetry, harmony and the finite-infinite relation. Continuity applies to the nature of time. Symmetries apply to the structure of space, focusing here particularly on the very small-scale. Harmony is a focus on the dynamics of perfection and imperfection whereby chaos, indeterminacy, creativity, free will and fluctuations all emerge. And, all three are the face of the finite-infinite relation. 
The dynamic image at the top of this article opens this analysis. The first sphere is the first instant of space and time with a very specific mass and charge. These numbers were all calculated by Max Planck in 1899 and have been studied in earnest since 2001 when Frank Wilczek wrote a three-part series, “Scaling Mt. Planck” for Physics Today.  When he received his Nobel prize in 2004, these three articles took on the patina of authority.
The next step for scholarship was obvious, but everybody seems to be ignoring it.
The first moment of time is derivative, finite and discrete. Newton’s absolute space and time are sidelined to introduce a new scale of the universe that begins with Planck Length and Planck Time. The focus of that image, including the ellipsis (36 displayed), is “the first emergence.” 
Seemingly out of nothing – no space, no time – it all starts with just one sphere, defined by the Planck base units and many dimensionless constants, that is followed by another sphere, then another and another…. For now, we’ll call these spheres, planckspheres. If these spheres could be observed — obviously much-much-too-fast-and-too-small to measure — perhaps this process might be described as a line or a string coming out of nowhere, literally defining space and time as it emerges. This is the beginning of time, and this first moment is still emerging, today, at this moment…. It is still creating space/time and the dynamics that are mass/energy (or charge). It is assumed that all four Planck units are inextricably interwoven throughout the 202 doublings  (or notations) that bring us to this very moment within this day.
In that light, our first principle is that our Universe begins and is sustained by the dynamics that are defined within Planck Length/Planck Time and Planck Mass/PlanckCharge. Given these are inextricably interwoven, one of our challenges is to loosen, then disentangle all the knots.
Be assured, this is not your daddy’s or your mommy’s sense of time. Here it is a rate of encoding and imprinting on a universe that has no past. It has no future. It is only right now. This instant. Everywhere, everything shares this same moment and this same infrastructure.  This first notation is always the same, yet it is always unique just like pi. Impenetrable, there is nothing smaller; and these spheres penetrate and sustain all things.
So, another principle is that time is not a measurement of duration but of processing speed.
Here is the operational nexus between the finite and the infinite. Here is the beginning of an integrated, mathematical model of the universe and a quiet expansion with a most-natural inflation. Here is our little universe displaying its deep-seated order; yet very quickly, it begins to reveal how disorder, chaos, uniqueness, and creativity emerge.  Those geometries are well-known and the dynamics within each manifestation are now being explored and will be discussed in subsequent homepages.
Doublings. In the second notation the most basic projective geometry begins to emerge and structure begins building on basic structures  that creates a logical continuum from the infinitesimally small scale structures right up to the 67th doubling where now things can, in some sense of the word, be measured by accelerators like CERN in Geneva, Switzerland.
One of the key purposes of this site is to chart a map that takes us right down into these assumed infinitesimal structures.
Academics and scholars have not adopted this model. Questions should be asked first, about the jump or “quantum leap” from the CERN-scale to the Planck scale.
To date, there appears to be no other attempt to define this exquisitely small space using a simple application of base-2 notation and a profound respect for the Planck base units. In this study Planck temperature is derivative of mass/charge; so to approach the Planck temperature value, it has been reverse ordered. Just for convenience, it is now started just one notation above the 202nd notation. The logic supporting such a positioning is still being formulated.
Can the deep nature of that “quantum leap” be calculated today using just the four Planck base units and doublings? Could the first doubling from Notation 1 to Notation 2 be the foundation for all doublings? There are many different types and applications of doublings that have already been well-researched and defined. To learn a little about each, on one page within this site, these key types of doubling will be studied and further researched in light of the continuity equations from the first doubling to the 202nd doubling.  Hopefully period-doubling bifurcation, cellular division, double field theory and gauge-symmetry for T-duality-and-doubled geometry, and multiscale modeling and simulations will inform us.
What other kinds of doublings should be considered? The 64 doublings from the Planck scale to just under the CERN-scale (at the 67th doubling) have been well-enumerated through the study of geometric expansions, especially as outlined by the Wheat & Chessboard story. It begs the question, “Is there a logical progression by which numbers and geometries progress?” Does every kind of mathematics, geometry and logic build upon each other? 
Given recent scholarship within the studies of prime numbers, the question should also be asked, “What is the role of prime numbers in this expansion?” There are 45 prime numbers between notation 1 and 202; there are 19 primes from 1 and 67. Could each notation that is a prime open a path for more complex mathematics? That question is being pursued within the development of the following pages: https://81018.com/1-202https://81018.com/a0https://81018.com/a1https://81018.com/a2 …
So, even as we study these possibilities, a simple stacking renders our first doubling and an infrastructure for all subsequent doublings. Our centerfold image at the top captures the dynamics of doublings. Cubic-close packing, both face-centered cubic (fcc) and hexagonal close-packed (hcp), has a rich history beginning in-and-around the 1570s starting with the problem of stacking cannonballs on the deck of a ship. Today there are purely mathematical packing challenges as well as applications of atomic and crystallographic stacking and packing. By starting with planckspheres, this most-simple doubling application becomes discernible as the second, third, and fourth doublings are assumed. Further, subsequent doublings are assumed right on up to 202nd doubling and the current time. Yet, something unusual is captured within the 67th doubling, we begin to measure it. That length opens the possibilities of particle physics revealing the potential science of the first 64 steps. It begs the question; is this a logical continuum from the infinitesimally small scale structures up to those being measured by accelerators like CERN?
These planckspheres, a key element of the finite-infinite bridge, are defined by pi, the Planck base units, dimensionless constants and simple logic. Every finite-infinite discussion-and-debate should be re-examined. Though tedious, it must be re-engaged. There are too many fine scholars who are being torn up and their logic being shredded to not engage every idea that has been posited throughout human history. All of that is within the 202nd notation. The 197th notation takes us up to 343+ million years. Our first 196 notations open a deep study of the earliest cosmological epochs.
Of course, the question must also be asked, “Is this model overly simplistic and naive?” Yet, even if so, could this model of the early universe be closer to the truth than the big bang theory? I believe it is. Thank you. -BEC
 The three faces of the finite-infinite relation extend our earlier discussions about David Hilbert’s understanding of infinity and Max Tegmark’s disdain for the word. Continuity-symmetry-harmony are the mathematical-scientific faces of infinity and each face is captured by the dynamics of pi and the emergence of lattice, tetrahedrons and octahedrons, and eventually complex structure.
 Frank Wilczek wrote his three-part series, “Scaling Mt. Planck” for Physics Today, yet has not acknowledged that Planck base units are the best conceptual orientation to start constructing this universe. As a result of this analysis, we will ask him, “Why not?” To our knowledge, the writings within this website are the first to lift up Max Planck’s base units as the starting point for the universe. We are anxious to discover and understand any articles that analyze their place, power and conceptual richness.
 “The first emergence” is a steady stream of planckspheres being uniquely created just like they were in the very first moment. Every notation has a unique function and every notation is evolving at the same time it continues to do what it has done. Here, perhaps are Neil Turok’s perpetual starts of the universe. Here everything, everywhere is built up and emergent from this fabric of the universe, called an aether and/or dark matter and dark energy, that gives this universe its isotropy and homogeneity.
 The Planck base units and dimensionless constants are inextricably interwoven throughout the 202 doublings (or notations) are based on the “Plancksphere” and that analysis is just beginning. I googled the word, for example, on June 25, 2018 with those quotes and there are just nine results. Using two words, “Planck sphere” with the quotes, there are 320 results. And without the quotes there are just over four million combinations that come close. Such results suggest that this is a new or emergent science.
Within our dynamic image about sphere stacking, the tetrahedral-octahedral architecture begins to emerge. Here, the possibilities for getting things inextricably woven together become staggering. By the tenth doubling there are 134,217,728 scaling vertices with which to work. By the 20th notation it catapults to 1.4411519×1017 — there are no limits to the entanglement of strings and knots and yet-to-be-fathomed geometries to create. By the 64th notation those scaling vertices have jumped up to 6.2771017×1057 and the first particle has yet to emerge!
Unless this simple logic is mistaken, there is altogether too much potential to ignore these possibilities and this orientation any longer.
 Everywhere, everything for all time shares this same moment and this same infrastructure. There are many books and articles about the nature of time. Within this study, most have fallen short. Einstein and Planck opened the door to re-analyze Newton’s earlier conceptual frame of absolute space and time, yet nothing more compelling emerged. Newton continues to define our commonsense logic, but should it? If it is established that period-doubling bifurcation, cellular division, double field theory and doubled geometry, and multiscale modeling all share the same common denominator that starts at the Planck base units, absolute space and time can be placed on the historical shelf as a footnote within the imprinting on the sentience of this universe. More work needed…
 Structures begin building on basic structure. There are two areas where our analysis is focused. The first is on a notation-by-notation analysis but progress is slow because there is so much mathematical logic to be learned. The other is basic geometries, quantum geometries and dynamic geometries. Here, too, there is so much to learn and, of course, more to come…
 There appears to be a rigorous academic study of the logical construction of concepts, geometries, and equations. Mathematics and geometries do build upon each other! These studies will become our studies and as quickly as possible, each will be integrated within our map of the universe. Yes, there is more to come…
“In philosophy, systems theory, science, and art, emergence occurs when “the whole is greater than the sum of the parts,” meaning the whole has properties its parts do not have. These properties come about because of interactions among the parts.” -Wikipedia
* This page was started on June 21, 2018 in South San Francisco while on our tour of America. On many occasions Hattie and I been challenged to look at the world and ourselves in new ways. Along our route, we’ve spent time engaging with people:
At John Hendrick’s retreat, Gateway Canyon Ranch, an hour south of Grand Junction, Colorado, we discovered his CuriosityStream retreat center. That got me thinking.
We were in the highly-overpriced Yellowstone Hotel in Wyoming where the National Park Service is attempting to create a Disney-like experience, highly-controlled-and-organized wilderness. That compressed conflict got me thinking.
On our drive to Bend, Oregon, I discovered the Simplot Don plant near Pocatello, Idaho and learned about fertilizers and growth and even that challenged my thinking.
In Bend, while visiting with friends, I was challenged again to understand why there is such disparity, both political and economic, within our world.
On to Portland, the land of inclusivity, two different sets of friends challenged me to see the world through their eyes. There is so much to process and process it we will until each becomes a homepage.
This homepage was simply to clarify the last three homepages:
As you might imagine, the first second is between notations 142 and 143. It is a natural inflation (aka Euler’s identity) and, to date, nobody has provided a rational reason to stop studying this progression. I am most fascinated with the potentials within the first 60+ notations. I do not believe these have ever been carefully examined by mathematicians.
I sent an introductory note to you on 8 April 2018!
Good work! Nice thoughtful reflections. And, jsomers.net is nicely done. You’re a mensch!**
I sent this note through Twitter, but whoever sees a tweet?
So, just for the record: @jsomers@TheAtlantic Excellent. At first I thought it was going
to be a Wolfram promo piece,
but that quickly changed!
Thanks for the introduction to Jupyter.
Thanks for the thoughtful reflections.
What ArXiv entry has the most signatures? http://81018.com – An integrated universe view: visuals?
I am one of those idiosyncratic fellows who has been
chasing the EPR paradox since 1971. Actually met JS Bell
at CERN through MIT’s Vicki Weisskopf… so many stories.
In 2011 helping a nephew, I finally found a mathematical container
for the universe: https://81018.com/home It’s been six plus
years attempting to exegete that initial work.
Here’s a chart/map of the universe that is
laughable for its simplicity and grandiosity! https://81018.com/chart
Purpose of this group. To develop a special graciousness and openness about life and beliefs, this group is for the people who want to get along with believers and non-believers. Can we find a deeper truth that holds up some new insights that will embrace both sides of an equation by focusing on the nature of the relation. We’ll engage the edges of scientific research and its implications for our current theories about the universe and its origins, and about who we are and the meaning and value of life.
Overview: One of the best sources for a study of the relation between the finite and infinite are the sacred texts within our religious traditions. Many of us who grew up in the Christian tradition and like typical college students, we tend to reject the old to begin to self-actualize. Then we begin studying at the limits and boundaries of our knowledge and the challenges can become so daunting, they get left in those undergraduate and graduate classrooms. Personally I went out as far as I could on the edges of physics, working with Bob Cohen, the chairman of Boston University’s physics department, and then with physicists from around the world. I began discovering there are conceptual overlaps between all the departments within the university. The most extreme appeared to be those who were religious and those who demonized all religions.
What can be more different that the texts within The Bible, both Old Testament and the New Testament, and most texts within the scientific community, that is between Genesis 1 and John 1 and Stephen Hawking’s big bang theory. These sessions are designed to examine concepts within the sciences, philosophy, ethics, and mathematics to see how and where these overlap with concepts about eternity, infinity, light, and love. We start with an integrated view of the universe, and that begins to inform our understanding of the infinite and infinity.
Structure: This small group will have just nine gatherings, no longer than 90 minutes each. A goal for these sessions is to chart a way to empower people to create such a small group study. • The first two sessions. We will re-examine cosmological models of the universe (2 weeks). • The 3rd and 4th sessions: We will explore various ways of approaching an understanding of the finite-infinite relation. • The 5th and 6th sessions: We will explore a rather different understanding of light, a light that permeates and defines every notation and all of space and time. • The 7th and 8th sessions. We will explore the challenges to our understanding of basic concepts like space and time. Both become finite and transaction oriented. • The final session: Beyond the summaries, we will be searching for answers to the question, “What do we do now?”
Simple and small goals: The first goal is to open the door to a very simple orientation to science and faith that (1) works with science and mathematics and (2) allows for, and possibly informs, religious beliefs. The next goal is to explore the entry points between the finite and infinite. Another goal is to explore the physics, philosophy, and psychology of light. If we have even limited success, we’ll all begin to shrink space and time and open up an intimacy with the universe.
1 The first link goes to our correspondence (mostly emails) to Nima Arkani-Hamed. He has also received a few tweets as well.
2 Spacetime is doomed is a YouTube video. If you go to 5:45 minutes of his 1 hour 22 minute lecture, you will hear that indeed, spacetime is doomed. The entire lecture is his focus on this concept.
3 Pace-Man is our affectionate name for Nima, from his pacing back and forth when he gives a lecture.
4 The amplituhedron has a special place within our work. It was one of many basic structures and by no means is the most basic. The transformations between circles and spheres to lines and tetrahedra and then octahedra are.
A large image, it may load a little slowly. Click on the image (or here) to go to the page where seven reasons are given why this image is fake news. Here is a real Forbes article about the same phenomenon from two years ago! (new window) *** This page is part of our analysis […]