This article is really by a rather large group of high school students with some help from Bruce Camber who helps our in their geometry classes on occasion. We were studying symmetries and Plato’s solids when we rather circuitously developed a model of the universe from the smallest-to-the-largest possible measurements of a length. We had divided each edge of a tetrahedron in half and connected the new vertices. We kept on dividing by 2 until we were at the Planck length. Then we multiplied by 2 until we were at the Observable Universe. We wondered why we had not seen this particular scale of the universe anywhere on the web (of course, Kees Boeke’s 1957 work on base-10 is well-known, but we had not learned about it yet). So, we wondered, “Could this be an outline to begin doing the smallest-and-the-biggest, yet possibly-the-most-simple, scientific experiments?” We cautiously thought, “Maybe it is.” Yes, this journey began by looking inside our clear plastic models of the tetrahedron and octahedron. We wanted to see what was perfectly inside. That was easy. We then asked, “How many steps within would it take to get to the Planck Length?” The Planck Length was a topic in our physics classes. When we found just 50 steps to the diameter of the proton and another 68 steps to the vicinity of the Planck Length, we were surprised. We were expecting many, many more steps. Within a couple of days, we began multiplying each edge by 2. Our goal was to get out into the range of the Observable Universe. It appeared that we were getting into the general vicinity in just 91 steps. What? …so few steps? We were dumbfounded but undeterred. To make our emerging chart of the universe consistent, we decided to start at the Planck Length and just multiply by 2 and assume those simple geometries as a given. But, we weren’t sure we could multiply the Planck Length by 2, so we turned to experts on the Planck Length. One of them said our work was idiosyncratic and would go no further. Another cautioned us to work within group theory but encouraged our work. Frank Wilczek, one of the world’s experts on the Planck Length and an MIT Nobel Laureate, also encouraged our explorations and use of the Planck Length. Further checking the web to see if we were making stupid logic errors, we quickly found the base-10 work of Kees Boeke, a high school teacher from Holland. Back in 1957 he published a little picture book, Cosmic Vision, that became quite popular within academic circles. Many people have subsequently refined his work. In the ’60s Charles & Ray Eames made a movie about it and Phil & Phyllis Morrison of MIT did an expanded picture book. And, then in 1997 the Smithsonian’s IMAX Theater did an expanded version of the movie with Morgan Freeman. More recently Cary Huang did a lovely online, Flash-based, version dubbed, The Scale of the Universe. All very good, but for us it just doesn’t go far enough in either direction. Certainly it is not granular enough, and it has no inherent geometries. It amounts to what one might call, “exponential notation light”. As Phil Morrison once said, “Just add zeros!” “It’s exponential notation for dummies.” To which our students add, “It’s exponential notation for Dummies.” The Planck Length was first defined in 1899 by Max Planck. Though little used, it is generally accepted to be the smallest meaningful measurement of a length; and because it is based on three constants including the speed of light in a vacuum and the gravitational constant, most scholars believe Planck was right. Though the Observable Universe is a difficult figure to discern, there have been many published guesses over the years. But to give us some confidence that we were on the right track, we turned to a NASA scientist and a French astrophysicist. In this early search, we could not find anybody using base-2 exponential notation to extrapolate a view of the universe. Although our intention was not to create a so-called perfect sequence or scale of lengths from smallest to the largest, we backed into using base-2 exponential notation because it was simple and we were studying embedded-and-nested geometries. So, we could readily number and spatialize each sequence of the scale. It seemed like a logical way to create ordered sets, groups and real relations between everything. Yet, we were quickly approaching our limits of knowledge. Questions abound. What happens in the first 66 notations? Could this be an area for pure math and geometries? Might the first 20 or 30 notations be a domain for cellular automaton? Could the Mind be located somewhere just before the proton? What happens between each doubling? What are the boundary conditions? What is the relation between number and space? Currently we do not know any answers to those questions. Although somebody else might know, we have begun trying to discern a path to discover answers. We began to apply what we knew about the scientific method. How will we identify the types and functions of the constants and variables all along the scale? How will we define the independent and dependent variables? Are there control and moderating variables? What would we use for a control group? Can we begin to make a few hypotheses to test? We are still crawling, trying to find our walking legs. We often find ourselves in a fog of unknowns, but we did come up with a few ideas to create an experiment. Could this be a simple experiment? First, our math gave us somewhere between 202.34^{1} and 205.11 notations,^{2} or doublings, layers, or steps between the smallest and largest measurements of a length. Our first hypothesis is that there will be significant transitions points along the scale. A most simple projection would be to look for significance within the midpoint (dividing in the entire scale by 2, 1:2). We also hypothesize that there will be something significant at the thirds (dividing the entire scale by 3, 1:3), as well as by fourths, fifths, sixth, sevenths, eights, ninths and tenths. We are currently on that path of discovery. It was easy and straightforward to find a range that we could call the midpoint. It was also easy to discover that these notations are where life begins. Yes, the sperm and egg are within that range. Life appears to be the concrescence of the two extremes. That was a very simple experiment and our hypothesis seems to be validated. It is a significant transition. Next, by dividing by 3, we have begun looking at notations 67 to 69 and 134 to 137. Being relatively inexperienced at understanding such small measurements, we made many initial mistakes. Today, we believe at notation 65 we are close to the size of a proton, (r_{p} = 0.84184(67) fm).^{3} Here is the basic building block of the visible universe between notation 65 at 5.96272544×10^{-16} meters and notation 66 at 1.19254509×10^{-15} meters. Though this scale amounts to a view of the known universe, on a serious note, we will call it a Universe View. On a whimsical note, we call it the Big Board-little universe. It feels unique. We have not seen it within our studies. It gives us perspective. It puts everything in a nice grid. It appears to open new areas to study. Another simple hypothesis emerged. We thought, “People with a universe view would be more optimistic, more insightful, and more productive than people who rely on their own views or even a worldview.” It became a project for the National Science Fair.^{4} We developed a ten-step tour of this Universe View, then asked people if they felt more optimistic, insightful, and productive as a result. Our results were not conclusive, however, we have not given up on developing a cleaner methodology and a better control group. We are at the beginning of this study and we have a lifetime of work ahead of us!^{5} —————————————————————— 1. It is a helpful ordering tool. It puts everything in a necessary sequence and a place on a somewhat granular scale. 2. It has inherent geometries. Seen or unseen, we believe these geometries are inherent throughout the known universe and we believe there should be testable hypotheses that we can discern. Perhaps, we are talking about Alfred North Whitehead’s point-free geometries. Our life is filled with things we do not see, but assume there is a structure for it. It could be a memory of times gone by, a creative insight of a possible solution to a problem, or a recurring dream of things to come. 3. Geometry tiles the known universe, first with the four hexagonal plates within the octahedron and then with tetrahedral-octahedral chains and clusters. This remains to be studied, but it certainly suggests that from the Planck Length to the Observable Universe in just under 206 notations, steps, layers, doublings, or sections, everything is necessarily related to everything. The geometer John Conway (Princeton emeritus) with Yang Jiao and Salvatore Torquato recently expounded on this basic configuration.^{6} 4. The known universe has deep-seated symmetries that create a diversity of relations. BUT, we know because of Max Planck’s work to define quantum mechanics, there are also asymmetries, indeterminacy and chaos. We readily find the geometric imperfection and logical opening, first from within the work of Frank-Kaspers and now within the work of many others (just below).^{7} 5. A speculative summary. Historically space and time have been the container within which everything takes place. As much as Einstein tried to dislodge Sir Isaac’s commonsense worldview, nothing has really taken its place. We started with simple geometries and mapped the known universe including a very, very small universe that is ignored by most. In our work, number and geometry pervade everything. We can see the Frank Wilczek’s Grid emerging from the Planck Length; there is order and continuity within a deep substructure. As we move up through the notations, we can see and feel relations and symmetries building. Yet, we can also feel and see asymmetries and discontinuities taking shape. I believe ultimately we will see how space becomes derivative of geometry and time derivative of number. We just might be able to move on from Newton’s commonsense and develop a simple-but-compelling universe view, whereby order, relations, and dynamics are known through continuities and discontinuities, symmetries and asymmetries, and harmony and discordance. ^{1} NASA physisicist, Joe Kolecki was the first to help us with the calculation of 202.34 notations ^{3} Davide Castelvecchi, “What Do You Mean, The Universe Is Flat?, Part 1,” Scientific American, July 25, 2011 ^{4}National Science Fair project by Bryce Estes ^{5}More background by Bruce Camber ^{6} “New family of tilings of three-dimensional Euclidean space by tetrahedra and octahedra” by John H. Conway, Yang Jiao, and Salvatore Torquato, PNAS, 2011 To download a PDF of the entire paper) |