This chart has been used in several blogs. The first time it was used was in June 2014 when the question was asked, “Is there order in the universe?” Click on the chart itself (or here) to go to an another blog, *Where is the Good in Science, Business, and Religion?*

# Monthly Archives: September 2014

# Where is the Good in Science, Business, and Religion?

**June 5, 2014**: This chart was first used in a blog, *Is There Order In The Universe?*

**All three of these major domains of human activity — Science, Business & Religion — are fraught with travail and have been blemished with the worst of human behavior**. Notwithstanding, there is a deep ethical bias within science which is also an essential infrastructure of business, and it is the heart of good religion.

Though this is the first time this ethical bias has been placed on a chart, it had became so obvious, so entirely self-evident, it had become a truism for me.

This circular color chart opens the door on the story. The chart seems to capture all the energies — both negative and positive — within our finite universe, including our finite world, and our finite life. Though using a Cartesian coordinate system as its container, here the x-axis (horizontal axis) is the totality of time. The vertical y-axis becomes the totality of space. The thrusts — energy and purpose — are the most basic forms/functions of life. Above the x-axis are all the constants and universals that define who we are, our life, the arts, sciences, business and religion. Below that x-axis — domains that involve so much of human activity — is the antithesis of Order/Continuity, Relations/Symmetry, and Dynamics/Harmony.

The antithesis of these form/functions create a de facto ethical platform by which we can begin to judge ourselves, our businesses, our religion (including atheism), and our political and social organizations.

Of course, this is a first pass at a complex subject addressed by a relatively simple person trying to make sense of it all. There will be many more updates to come.

You might find an earlier article about the constants and universals to be helpful. In many ways, it anticipates this chart but was prior to the chart’s development. Title: *Belief systems https://bigboardlittleuniverse.wordpress.com/2013/03/21/belief/*

Again, the first use of the chart was within the article, *Is There Order In The Universe?*

The Tiny URL for this post: http://tinyurl.com/GoodSBR

# Did a Quiet Expansion precede the Big Bang?

If you think about it, most of the world’s people have never heard of the Big Bang theory (Reference 1) (the cosmological model, not the TV series). Of those who know something about it, a few of us are somewhat dubious, “How can the entire physical universe have originated from a single point about 13.8 billion years ago?” It seems incomplete, like there are major missing parts of the story.

To open a dialogue about this pivotal scientific theory is the reason for this article. And, if we are successful, all of us will have re-engaged our ninth grade geometry classes and we will begin to ask a series of “what if” questions about the origins of this unvierse.

Some of you are aware of the work of our high school geometry classes (Reference 2) and our developing models within the Big Board-little universe (Reference 3). Possibly you even know a little about the 205.1 base-2 exponential notations from the Planck Length to the Observable Universe. It is a study that informally began on December 19, 2011, so most of us have only begun to explore the inner workings of each of the 205.1 notations. Because we believe all things start most simply, the first 60+ notations are potential keys for understanding. Also, these notations *per se *have not yet been studied by our academic communities (Reference 4). The best guess at this time is that the range of our elementary or fundamental particles begins within the 64th and 65th notations.

The simple mathematics (Reference 5) and the simple geometries are taken as a given; the interpretation of the chart is wide open..

This little article is an attempt to enlist you and some of our best academic thinkers, especially the best geometers among them, to look at those first 60+ notations. What kinds of what-if questions could we ask? Can we speculate about how geometries could grow from that singularity to a bewildering complex infrastructure within and throughout those first 60+ domains, doublings, layers, notations, and/or steps? What if in these very first steps, well prior to the Big Bang, there is a complexification of geometries?? Might we might call it a quiet expansion? Though we have always been open to suggestions, questions and criticisms, we are now also asking for your insight and help.

Special versions of both models are being prepared. Within the Big Board-little universe, those first 60+ notations will be highlighted. Also, a special version of the Universe Table will be created to emphasize every notation from 1 to 65. And, with this article, we will initiate a base-2 exponential notation of time from the Planck Time. Some of that work has been recently done within the 2014 book, *Time in Powers of Ten, *Natural Phenomena and Their Timescales, by Gerard ‘t Hooft and Stefan Vandoren of Utrecht University (Reference 6). Plus, there are an abundance of very informative articles from all the research since the concept was proposed in 1927 by Georges Lemaitre.

Steven Weinberg, the author of The First Three Minutes (Reference 7), begins his journey through the origin of the universe at the 1/100th of a second mark. Our hypothesis is that we can mathematically go back to a much, much smaller duration. We believe that we should start at the Planck Time and multiply it by 2. And, just as the fermion within notation 66 would be the size of a small galaxy compared to the Planck Length, we anticipate that 1/100 of a second will be no less than seven days, maybe ten years, maybe a millennium when compared to the Planck Time (Reference 8).

We’ll see what the numbers tell us.

A lot of pre-structuring of the universe could be quietly happening within such a duration (1/100th of a second). Using our most metaphorical, speculative thinking, one could imagine that the actual event within those first sixty notations was a gentle, symphonic unfolding, fully homogeneous and isotropic.* Although we should embrace all the key elements of today’s theory, we should also be constantly asking, “What kinds of geometries would be required within each of the first 60 notations to render these effects?”

Perhaps the universe and our future belong to the geometers.

So, this article is to empower all of us to find the best geometers around the world to engage the Big Board-little universe model within what we call “the really-real small scale universe.” Of course, some of the work has already been done within the study of spheres, tilings, and combinatorial geometries.

If you would like to join us, drop me a quick note (camber-at-81018.com). Thanks.

###

**References**:

*3* A WordPress blog page. This image of the Big Board-little universe is Version 2.0001.

*7* This article about the book, *Time in Powers of Ten* by Gerard t’Hooft and Stefan Vandoren, is the most comprehensive that I could find at this time. If you happened to find a better review, please advise us.

*8* An online version of the entire book, *The First Three Minutes* by Steven Weinberg.There are many reviews, yet this one provides a little counterweight. Weinberg also wrote the forward to *Time in Powers of Ten*. Gerard t’Hooft (1997) and Steven Weinberg (1979) are Nobel laureates.

*9* A WordPress article about very small and very big numbers. There is our initial discussion about the first 65 notations. The range with which we will be working is from the Big Bang at about 10[90] seconds (13.78+ billion years) to Planck time, 10[-44] seconds.

**Notes**:

The TinyURL for this article: http://tinyurl.com/QuietExpansion

*homogeneous: Having the same property in one region as in every other region

isotropic: Having the same property in all directions.

# Just what’s happening here?

For over 100 years, the Planck Length was virtually ignored. That length was so small, it seemed meaningless.¹ Nothing and nobody could measure it. It was just a ratio of known constants. Yet, it created a conceptual limit of a length which gave a New Orleans high school geometry class a goal or a boundary beyond which they did not have to go. Recent measurements from the Hubble telescope provided the upper limit so this class could define the number of base-2 exponential notations from the smallest measurement of a length to the Observable Universe, the largest.

Within that continuum everything can be placed in a mathematical and geometric order. Everything. That is, everything in *the known universe*. The most remarkable discovery was that it took no more than 205.1 base-2 exponential notations. It would be our very first view of an ordered universe. And, it readily absorbed all of our worldviews.

That was December 19, 2011. Formally dubbed, “The Big Board – little universe,” we then asked, “What does it mean? How do we use it?” When we engaged the experts, they appeared a bit puzzled and seemed to be asking, “Why haven’t we seen this chart before?” Those who knew Kees Boeke’s 1957 book, *Cosmic Vision*, asked, “How is it different from Boeke’s work using base-10 exponential notation?” That was a challenge. Our best answers to date – it’s more granular, it mimics chemical bonding and cellular reproduction; it’s based on cascading, embedded, and combinatorial geometries – were not good enough. In April 2012 even the Wikipedia experts (Steven Johnson, MIT) protested. He classified our analysis as “original research” and within a very short time our Wikipedia article was taken down. Others called it idiosyncratic (John Baez, UC-Riverside), but they did not tell us what was wrong with our analysis.

“Let’s just make as many observations as we can to see what can we learn?” A NASA senior scientist and a French astrophysicist helped us with our calculations. Their results gave us a range; the low was 202.34 notations and the high, 205.11. We could identify many things between the 66^{th} notation and the 199^{th} notation. But, there were blanks everywhere so we got busy speculating about them. The biggest group of empty notations was from 2 to about 65. We asked, “Conceptually, what could be there?” Max Planck may have given us a clue when in 1944, in a speech in Florence, Italy; he said, “All matter originates and exists only by virtue of a force which brings the particle of an atom to vibration and holds this most minute solar system of the atom together. We must assume behind this force the existence of a conscious and intelligent mind. This mind is the matrix of all matter.” (The Nature of Matter, Archiv zur Geschichte der Max-Planck-Gesellschaft, Abt. Va, Rep. 11 Planck, Nr. 1797, 1944) Matrix is a good word. Throughout history others have described it as the aether, continuum, firmament, grid, hypostases, plenum and vinculum.

We made two columns and within the top notations, 100-to-103, we found humanity. That seemed politically incorrect until we discovered the cosmological principle that the universe is isotropic and homogeneous. So, if it is true for us, it would also have to be true for “everybody” anywhere in the universe.

This is high school. We had been following embedded geometries, particularly the tetrahedron and octahedron. We observed a tetrahedral-octahedral-tetrahedral chain. In no more than 206 layers everything in the universe is bound together. We learned about tilings and could see that the four hexagonal plates we discovered within the octahedron also created tiles in every possible direction.

“What is this all about? Just what’s happening here?”

We knew we were imposing a certain continuity and order with our mathematics (base-2 exponential notation), and we were also conveying certain simple symmetries and relations with our geometries. That wrapped our work within a conceptual framework that was quite the opposite of the chaotic world of quantum mechanics. Our picture of the known universe was increasingly intimate and warm; it was highly-ordered and had immediate value. And the more we looked at it, the more it seemed that all of science and life had an inherent valuation structure. Here numbers became the container for time, and geometries the container for space. How each was derived became our penultimate challenge. Ostensibly we had backed into a model of the universe and somehow we began to believe that if we could stick with it long enough, it just might ultimately give us some answers to the age-old question, “What is life?”

We had strayed quite far from those tedious chapters in our high school geometry textbook. Yet, we also quickly discovered how little we knew about basic structure when we attempted to guess about the transitions from one notation to the next. We asked, “How can we get from the most-simply defined structure, a sphere, to a sphere with a tetrahedron within it?” We needed more perspective.

“Who is doing this kind of work?” We began our very initial study of the Langlands Program and amplituhedrons. Then, we walked back through history, all the way to the ancient Greeks and we found strange and curious things all along the way. There were the circles of Metatron that seemed to generate the five platonic solids. “How does that work? Are there experts who use it? How?” We still do not have a clue. All the discussions about infinitesimals seemed to come to a crescendo with the twenty-year, rancorous debate between Thomas Hobbes and John Wallis. It was here that we began to understand how geometry lost ground to calculus and algorithms.

The Big Board-little universe was awkward to use. It was five feet tall and a foot wide. Using the Periodic Table as a model, an 8½-by-11 chart was created and quickly dubbed, The Universe Table. It would be our Universe View into which we could hopefully incorporate any worldview. It was an excellent ordering and valuation system.

Though the Planck Length became a natural unit of measurement, a limit based on known universal constants, it wasn’t until Frank Wilczek of MIT opened the discussion did things really begin to change. In an obscure 1965 paper by C. Aldon Mead, his use of the Planck Length was pivotal. In 2001 Wilczek’s analysis of Mead’s work and their ensuing dialogue was published in *Physics Today. *Wilczek, well on his way to obtaining a Nobel Prize, then began writing several provocative articles, *Scaling Mt. Planck. * Even his books were helpful. In January 2013 he personally encouraged us on our journey.

In 1899 Max Planck began his quest to define natural units. At that time he took some of the constants of science and he started figuring out natural limits based on them. There are now hundreds that have been defined. Each is a ratio and each can be related to our little chart and big board. The very nature of a ratio seems to be a special clue. It holds a dynamic tension and suggests that the relation is primary and all else is derivative.

We have a lot of work in front of us! And, we are up for the challenge.

Who would disagree with the observation that our world has deep and seemingly unsolvable problems? The human future has become so problematical and complex, proposals for redirecting human energies toward basic, realizable, and global values appear simplistic. Nevertheless, the need for such a vision is obvious. Rational people know that there is something profoundly missing. So, what is it? Is it ethics, morality, common sense, patience, virtues like charity, hope and love? We have hundreds of thousands of books, organizations and thoughtful people who extol all of these and more. The lists are robust. The work is compelling, but obviously none of it is quite compelling enough.

First, it has to be simple. Our chart is simple.

Second, it has to open up to enormous complexities. Using simple math, by the tenth notation there are 1024 vertices. We dubbed it the *Forms* or *Eidos* after Plato. The 20th notation would add a million vertices; we called it *Structure*. The 30th adds a billion new vertices. We ask, “Why not *Substances*?” The 40th adds a trillion so we think *Qualities*. The 50th adds a quadrillion vertices. We speculate *Relations*. By the 60^{th} notation there are no less than a total of 2 quintillion vertices with which to create complexity. We speculate *Systems* and within *Systems* there could be *The Mind*. As if a quintillion vertices is not enough, the great physicist, Freeman Dyson, advises us that really we should be multiplying by 8, not by 2, so potential complexity could be exponentially greater.

Three, it should be elegant. There is nothing more elegant than complex symmetries interacting dynamically that create special harmonies. We can feel it. And, we believe the Langlands program and amplituhedrons will help us to further open that discussion.

*What is life? *Let us see if we can answer very basic questions about the essence of life for a sixth grade advanced-placement science class and for very-average, high-school students. These are our students. The dialogue is real. The container for these questions and answers is base-2 exponential notation from the Planck Length to the Observable Universe. To the best of our knowledge, December 19, 2011 was the first time base 2 exponential notation was used in a classroom as the parameter set to define the universe. Though our study at that time was geometry, this work was then generalized to all the scientific disciplines, and more recently it was generalized to business and religion. So, as of today, readers will see, and possibly learn, the following:

1. See the totality of the finite, highly-ordered, profoundly inter-related, very-small universe where humanity is quite literally back in the middle of it all.

2. Engage in speculations about the Infinite and infinity whereby the Creative and the Good take a prominent place within the universal constructs of Science.

3. Extend the scale of the universe by redefining the Small Scale and engaging in speculations about the deep symmetries of nature, giving the Mind its key role within Systems, and demonstrating the very nature of *homogeneity and isotropy.*

4. Adopt an integrated universe view based on Planck Length and Planck Time such that Science, Technology, Engineering and Mathematics are demythologized, new domains for research are opened, and philosophies and religions are empowered to be remythologized within the constraints of universals and constants.

People ask, “Aren’t you getting ahead of yourself? Isn’t this a bit ambitious?” The concepts of space and time raise age-old questions about who we are, where we have come from, and where we are going. With our little formulation, still in its infancy, we are being challenged to see life more fully and more deeply. And so we reply, “What’s wrong with that?”

###

**Footnotes still to be added**.

^{1} http://www.phys.unsw.edu.au/einsteinlight/jw/module6_Planck.htm Physics professor, Joe Wolfe (Australia), says, “Nothing fundamentally changes at the Planck scale, and there’s nothing special about the physics there, it’s just that there’s no point trying to deal with things that small. Part of why nobody bothers is that the smallest particle, the electron, is about 1020 times larger (that’s the difference between a single hair and a large galaxy).

^{2}

^{3}

^{4}

# Finite or Infinite? Is That The Question?

We have been studying simple math and simple geometries from the smallest possible measurement of a length to the largest (Reference #2). It appeared to some of the students, based on this work, that the universe is obviously finite. They have been told that intellectually and historically, it is an open question. For them, “Make a choice and see where it takes you.”

These students also conclude, “*Only God is Infinite*. All things within space and time are finite.” (Reference #3) When asked about all the universals-and-constants and space-and-time, the concurrence is that these are the access paths, interconnections and transformations between the Finite and the Infinite.

For the best of these students, asking the question, “What is the Infinite?” is like asking the question, “Who is God?” And, they have answers.

Of course, as a result of a little coaching, they say, “First, God is Perfect.” When asked, “What is perfection?” they echo their coach: “Perfection is order-continuity, relations-symmetry and dynamics-harmony, all rolled into one.” (Reference #4) That amounts to an understanding of the Infinite without importing all the related history and revelation from the various faith statements with our very short history throughout our little world. The Finite is another story. We turn to Euclid for inspiration to provide the academic and religious communities with our simple observations and assumptions.

Hardly postulates and axioms, our statements are a *praxis in-search-of theoria*:

**Interior geometries**. There are basic geometries within basic geometries. Two of the most simple platonic solids are the tetrahedron and octahedron. The simple interior structures can become exceedingly complex. (Reference #5)**A domain that remains largely unexplored**. We assume that among its many possible descriptions, the Planck Length could be taken to be a vertex. If so, there are over a million-trillion vertices within just the 60th base-2 exponential notation and then another two million-trillion with the 61st notation. (Reference #6) There are over 60 virtually unexplored domains from the Planck Length to particle physics.**A range of notations**. There are around 205.11 base-2 exponential notations (doublings, domains, layers or steps) within the Known Universe. (Reference #7)**There are imperfect geometries.**These appear to begin within the five tetrahedral “pentastar’ cluster and these are extended within the icosahedron and the Pentakis dodecahedron. It appears that this imperfection was first recorded within history in 1958 (Frank & Kaspers).**The universe can be perfectly tiled.**The simplest 3D tiling is with tetrahedral-octahedral-tetrahedral chains and clusters within those 205.11 layers. Also, within the octahedron are four hexagonal plates at 60 degree angles around the center point. These each extend as planes with abutting octahedrons for 2D tiling. This is the simple beginning of possible ways to tile the universe. For more, we turn to Penrose, Conway, and many others.**In theory, the Planck Length is indivisible**. It is, however, a specific length. If that conclusion is taken as a given, the Planck Length could be considered the indivisible unit in the historic Theory of Indivisibles and the start of the Finite and the transformation between the Finite and Infinite.

If these statements are taken as a given, then what kind of universe and what kind of science do we have? Should we re-examine the use of infinity throughout the ages going back to the ancient Greeks? Should we reconsider the theory of indivisibles? And, perhaps we should even reconsider the very nature of the Big Bang and its theory.

Of course, that is our agenda (Reference #8), our current focus for the immediate future.

**References**:

1. An associated article within LinkedIn titled,* Order in the Universe*

2. One of the earliest reflections on this work: https://bigboardlittleuniverse.wordpress.com/

3. Initially written in November 2012, this article was first posted on March 2013 within WordPress: https://bigboardlittleuniverse.wordpress.com/2013/03/21/belief/

4. In light of those constants, universals and the finite-infinite relation, the nature of perfection seems to follow: http://smallbusinessschool.org/page1695.html

5. Examining basic structure in basic ways: http://smallbusinessschool.org/page2546.html#TetraInside

6. A new domain to explore with exponential numbers of vertices: http://doublings.wordpress.com/2013/04/17/60/

8. An analysis of 15 key points: http://doublings.wordpress.com/2014/05/08/15/

The TinyURL for this page is: http://tinyurl.com/finitePlanck

# Where is the Good in Science, Business, and Religion?

**Please note**: The first use of this chart was on June 5, 2014 in a blog on LinkedIn, *Is There Order In The Universe?*

**All three of these major domains of human activity — Science, Business & Religion — are fraught with travail and have been blemished with the worst of human behavior**. Notwithstanding, there is a deep ethical bias within science which is also an essential infrastructure of business, and it is the heart of good religion.

This ethical bias is so obvious, entirely self-evident, it has become a truism for many.

The chart begins to tell that story. This circular color chart is our finite universe, including our finite world, and our finite life. Though using the Cartesian coordinate system as its analogue, here the x-axis (horizontal axis) is also time. The vertical y-axis becomes the totality of space. The thrusts — energy and purpose — are the most basic forms/functions of life. Above the x-axis are all the constants and universals that define who we are, our life, the arts, sciences, business and religion. Below that x-axis — domains that involve so much of human activity — is the antithesis of Order/Continuity, Relations/Symmetry, and Dynamics/Harmony.

The antithesis of these form/functions create a de facto ethical platform by which we can begin to judge ourselves, our businesses, our religion (including atheism), and our political and social organizations.

There is an earlier discussion about these constants and universals that anticipates this chart but was prior to its development. It is simply titled, *Belief systems https://bigboardlittleuniverse.wordpress.com/2013/03/21/belief/*

Again, the first use of the chart was within the article, *Is There Order In The Universe?*

The Tiny URL for this post: http://tinyurl.com/GoodSBR

# Did a Quiet Expansion precede the Big Bang?

If you think about it, most of the world’s people have never heard of the Big Bang theory (Reference 1) (the cosmological model, not the TV series). Of those who know something about it, a few of us are somewhat dubious, “How can the entire physical universe have originated from a single point about 13.8 billion years ago?” It seems incomplete, like there are major missing parts of the story.

To open a dialogue about this pivotal scientific theory is the reason for this article. And, if we are successful, all of us will have re-engaged our ninth grade geometry classes and we will begin to ask a series of “what if” questions about the origins of this unvierse.

Some of you are aware of the work of our high school geometry classes (Reference 2) and our developing models within the Big Board-little universe (Reference 3). Possibly you even know a little about the 205.1 base-2 exponential notations from the Planck Length to the Observable Universe. It is a study that informally began on December 19, 2011, so most of us do not yet know too much about the inner workings of each of the 205.1 notations. Because we believe all things start most simply, the first 60+ notations are potential keys for understanding. Also, these notations *per se* have not yet been studied by our academic communities (Reference 4). The best guess now is that range of our elementary or fundamental particles begins within the 64th and 65th notations.

The simple mathematics (Reference 5) and the simple geometries are a given; the interpretation is wide open.

This little article is an attempt to enlist you and some of our best academic thinkers, especially the best geometers among them, to look at those first 60+ notations. What kinds of *what-if *questions could we ask? Can we speculate about how geometries could grow from that singularity to a bewildering complex infrastructure within and throughout those first 60+ domains, doublings, layers, notations, and/or steps? What if in these very first steps, well prior to the Big Bang, there is a complexification of geometries? Might we might call it a *quiet expansion?* Though we have always been open to suggestions, questions and criticisms, we are now also asking for your insight and help.

Special versions of both models are being prepared. Within the Big Board-little universe, those first 60+ notations will be highlighted. Also, a special version of the Universe Table will be created to emphasize every notation from 1 to 65. And, with this article, we will initiate a base-2 exponential notation of time from the Planck Time. Some of that work has been recently done within the 2014 book, *Time in Powers of Ten, *Natural Phenomena and Their Timescales, by Gerard ‘t Hooft and Stefan Vandoren of Utrecht University (Reference 6). Plus, there are an abundance of very informative articles from all the research since the concept was proposed in 1927 by Georges Lemaitre.

Steven Weinberg, the author of *The First Three Minutes *(Reference 7), begins his journey through the origin of the universe at the 1/100th of a second mark. Our hypothesis is that we can mathematically go back to a much, much smaller duration. We believe that we should start at the Planck Time and multiply it by 2. And, just as the fermion within notation 66 would be the size of a small galaxy compared to the Planck Length, we anticipate that 1/100 of a second will be no less than seven days, maybe ten years, maybe a millennium when compared to the Planck Time (Reference 8).

We’ll see what the numbers tell us.

A lot of pre-structuring of the universe could be quietly happening within such a duration (1/100th of a second). Using our most metaphorical, speculative thinking, one could imagine that the actual event within those first sixty notations was a gentle, symphonic unfolding, fully homogeneous and isotropic.* Although we should embrace all the key elements of today’s theory, we should also be constantly asking, “What kinds of geometries would be required within each of the first 60 notations to render these effects?”

Perhaps the universe and our future belong to the geometers.

So, this article is to empower all of us to find the best geometers around the world to engage the Big Board-little universe model within what we call “the really-real small scale universe.” Of course, some of the work has already been done within the study of spheres, tilings, and combinatorial geometries.

If you would like to join us, drop me a quick note (camber-at-81018.com). Thanks.

#

**References**:

*3* A WordPress blog page. This image of the Big Board-little universe is Version 2.0001.

*7* This article about the book, *Time in Powers of Ten* by Gerard t’Hooft and Stefan Vandoren, is the most comprehensive that I could find at this time. If you happened to find a better review, please advise us.

*8* An online version of the entire book, *The First Three Minutes* by Steven Weinberg. There are many reviews, yet this one provides a little counterweight. Weinberg also wrote the forward to *Time in Powers of Ten*. Gerard t’Hooft (1997) and Steven Weinberg (1979) are Nobel laureates.

*9* A WordPress article about very small and very big numbers. There is our initial discussion about the first 65 notations. The range with which we will be working is from the Big Bang at about 10[90] seconds (13.78+ billion years) to Planck time, 10[-44] seconds.

**Notes**:

The TinyURL for this article: http://tinyurl.com/QuietExpansion

**homogeneous**Having the same property in one region as in every other region

**isotropic**Having the same property in all directions.