**Note**: This research paper is done in conjunction with *Walk the Planck*, a project for the National Science Fair which involve online surveys and tours of the *Big Board-little universe* and the *Universe Table*. Based on the school fair and the constructive comments of the judges, shorter and more frequent surveys are being developed.

**Research**

Physicists use the Planck length to put things that are insanely small into perspective. By the time most scientists get anywhere near the Planck length, they believe it stops making much sense to talk about the difference between two points in any measurable situation. Historically, because of the uncertainty principle, there is no useful, or physically relevant, difference between the positions of things separated by such distances. It seems that most scientists today who think about the Planck Length believe nothing fundamentally changes at the Planck scale, and there is nothing special about the physics there. They conclude that there is no point trying to deal with things that small. Prof. Dr. Joe Wolfe says, “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).”^{1}

Yet, that all could change.

**Planck’s Length**

According to Special Relativity’s time dilation and length contraction, two observers with relative motion are condemned to eternal disagreement about times and lengths. (http://galileoandeinstein.physics.virginia.edu/lectures/spedlite.html) But is this true? They will disagree over the times measured in seconds, and lengths measured in meters, (or whatever units) as well as disagreeing over clock ticks, atomic vibrations, and light wavelengths. And what if there are absolute times and lengths, determined by the nature of the universe, and what if these quantities could be measured in different frames of reference? The thinkers have to agree on those having the same value, if they agree on the same laws of physics, would they not? If there is an absolute length and an absolute time, might there be some other approach within a consistent framework with known parameters?

The Planck length LP is defined by taking the constants of nature and combining them in such a way that their units combine to give a length. Planck’s constant, h, has units of Joule seconds. Physicists usually work with h/2π. Cavendish’s constant G (the constant of gravitation) has units of N.m2kg-2. And the speed of light c has units of m.s-1. The combination that works is: LP = (hG/2πc3)1/2 (Planck Length is equivalent to [joule seconds multiplied by the constant of gravitation] divided by [two, multiplied by pi multiplied by the speed of light cubed], all to the one half power). (http://www.askamathematician.com/2013/05/q-what-is-the-planck-length-what-is-its-relevance)

The Planck length is 1.6 x 10-35 meters (0.000000000000000000000000000000000016 meters). Compare it with the size of an atom, for example a carbon atom, which is already about 100,000 times smaller than anything able to be seen with the naked eye (an atom size is about 0.0000000001 meters). Suppose that the diameter of an atom in Planck lengths was measured, and that you counted off one Planck length per second. To measure the atomic diameter in Planck lengths would take you 10,000,000 times the age of the universe. Divide the Planck length by the speed of light and the result is a really small unit of time, the Planck time, tP, which is: tP = (hG/2πc5)^{1/2} (Planck Time is equivalent to [joule seconds multiplied by the constant of gravitation] divided by[two, multiplied by pi multiplied by the speed of light to the fifth power] all to the one half power).( http://fractalfoundation.org/resources/what-is-chaos-theory) The Planck time is 5.4 x 10^{-44} seconds. There is also a Planck mass, which is (ch/2πG)1/2 = 22 μg. This doesn’t sound like much, until one thinks of a fundamental particle with that mass, or until it is converted it into energy by multiplying by c2 to get 2.0 109 joules or 1.2 x 1028 eV.

The Planck length and time are very small, but they are results of the values that appear naturally in physical laws. So according to the principle of Special Relativity, it seems that different thinkers should “observe” them to be the same. So what about time dilation and length contraction? If these lengths and times are observable as physical lengths and intervals in moving frames, it appears that Special Relativity needs to be modified to include them or a new language for the very small scale universe needs to be created. One theory that does so is called Doubly Special Relativity, suggested in 2002 by Giovanni Amelino-Camelia.

These effects are immensely beyond current experimental technology. Particle accelerators are described by the energies that can be produce by them, and the latest generation produces energies of terra electric volts, or 1012 electric volts. The Planck energy is 1.2 x 1031 electric volts. Scientists are short by 1019.

Further, it is not clear what it would mean to measure these lengths and times in or from different frames of reference. On the Planck scale, time and space no longer have their ordinary, macroscopic meaning and so naïve applications of relativity are most likely inappropriate.

**Quantum mechanics, gravity and relativity**

So, where do these quantities come from? The speed of light c is the natural unit that relates time and space. G is the constant of gravity, and h is the constant of quantum mechanics. The Planck scale defines the meeting point of gravity, quantum mechanics, time, and space. Not much is known about this interaction, because gravity is feeble and its influence on things as small as quantum systems is minute.

Special Relativity and quantum mechanics work well together. Relativistic quantum electrodynamics is a very accurate theory. Richard Feynman once described how accurate it was by saying: “If you asked me how far it was to the moon and I said, ‘Do you mean from my head or from my feet?’” (http://www.phys.unsw.edu.au/einsteinlight/jw/module6_Planck.htm)

Quantum mechanics and gravity (Newton’s theory of gravity or Einstein’s theory of General Relativity) do not fit so neatly together. The problem can be put in several different ways. From the discussion of virtual particles (why there would be no chemistry without relativity), we saw that virtual particles could be larger if their lifetime and range were smaller.

Both Newton’s and Einstein’s gravity predicts that enough mass in a small enough space can produce a black hole: a region with a gravitational field so strong that its escape velocity is c. When these two ideas are put together, there is a scale small enough for virtual black holes to exist. This is the Planck scale. On this scale, all of the singular behavior associated with black holes asserts itself. Space and time as continuous entities cease to have meanings when discussing distances of 10^{-35} meters and times of 10^{-44} seconds. So relativity, a theory of space and time based on a continuum, must run into serious difficulties, which is perhaps not surprising due to the fact that the Planck scale is a very long extrapolation from current knowledge.

On this topic, the little collected direct knowledge shows that there are few hints to guide the development of theories, and even fewer constraints upon those theories. There are several different families of theories that aim to produce a consistent theory of quantum gravity. Usually they include a larger number of spatial dimensions, not all of which are macroscopic. At the moment, these theories are speculative. Perhaps one of them will turn out to be a good, useful theory, and the others will fall. At the moment, we cannot put them to the test.

** Brief History of Max Planck** (1858-1947)

The German scientist, Max Planck, was born in the city of Kiel, and was the son of a Professor of Constitutional Law. He received his doctorate of philosophy in 1879. He held the title of Privatdozent (Private lecturer) in Munich from 1880 to 1885, and then he was the Associate Professor of Theoretical Physics at Kiel until 1889. He then succeeded Kirchhoff as Professor at Berlin University, where he stayed until his retirement over thirty-five years later. After, he became President of the Kaiser Wilhelm Society for the Promotion of Science, which he held until 1937. Also, he was an appointed member of The Prussian Academy of Sciences (1894) and Permanent Secretary (1912).

He began his studies in the physics field of thermodynamics. During this time, the problems of radiation processes peaked his attention and he showed that these were electromagnetic in nature. From these studies he was led to the problem of the distribution of energy in the spectrum of full radiation. Experimental observations on the wavelength distribution of the energy emitted by a black body as a function of temperature were at variance with the predictions of classical physics. Planck was able to publish a paper explaining the relationship between the energy and the frequency of radiation. This was based on the revolutionary idea that the energy emitted by a resonator could take on only discrete values, or quanta. The energy for a resonator of frequency *v* is h*v* where h is a universal constant, which is now called Planck’s constant.

This was Planck’s most important feat. This feat marked a turning point in physics. The importance of the discovery, with its effect on classical physics, was not appreciated at first. The evidence for its validity gradually became overwhelming as it was accounted for many disagreements between observed phenomena and classical theory. Among these applications and developments, Einstein’s explanation of the photoelectric effect may have been mentioned.

**Very Brief History of Albert Einstein** (1879-1955)

Einstein was also German and was born in the year 1879. As a child, he began wondering about the mysteries of science. After moving to Italy, and then Switzerland, he graduated high school in the year of 1896.

In 1905, while working in Bern, Switzerland, Einstein had what came to be known as his “Annus Mirabilis” (miracle year). It was during this time that he obtained his doctorate degree and published four of his most influential research papers, including the Special Theory of Relativity. In that, the now world famous equation “e = mc^{2} unlocked mysteries of the Universe.

Ten years later, Einstein completed his General Theory of Relativity and in 1921 he was awarded the Nobel Prize in Physics. It also launched him to new popularity heights in the public eye, and his name became synonymous with “genius” worldwide. Einstein immigrated to the United States in the autumn of 1933 and lived in Princeton, New Jersey where he became a professor at the Institute for Advanced Study.

The applications of Einstein’s theories include the development of the television, remote control devices, automatic door openers, lasers, and DVD-players. Recognized as TIME magazine’s “Person of the Century” in 1999, Einstein’s intellect, his strong passion for social justice, and dedication to pacifism, left the world with infinite knowledge and pioneering moral leadership.

**Introducing the Big Board**

In December of 2011, Bruce Camber introduced the sophomore geometry class to the platonic solids. He had objects nested inside clear plastic models. Camber came up with a board with 202-206 steps from the Planck Length to the edge of the observable universe. (For more go to: http://bigboardlittleuniverse.wordpress.com/2013/03/18/ The background story for the big-board for our little universe.) If inside a tetrahedron is four half-sized (half the original size) tetrahedrons and an octahedron. Looking into an octahedron are six octahedrons and eight tetrahedrons, all half the size from its original.

Going within just forty-seven (47) steps is the proton. Going further within another sixty-six (66) steps is the Planck Length. Going out just ninety-two (92) steps is the length of the observable universe. So the Big Board- little universe, covers every step from the Planck Length to The Observable Universe using base-2 notations in no more than a total of two-hundred-and-six (206) steps. In a more recent lay-out of the Big Board, the steps are laid out similar to that of the Periodic Table of Elements. This, however, arranges the steps in columns and the steps recede, then climb, then recede, like a snake or a worm, to the Observable Universe. Similar to the Periodic Table, is the labeling of each cell. At the top of each cell shows the step, or group of steps, the reference to that step on a physical basis, then at the base of the cell is the length or number of vertices, depending on what scale the step is in. At the top of the table are the scales and other classifiers. The scales are defined as follows: the Small Scale (steps 1-67), the Human Scale (steps 68-135), and the Large Scale (steps 136-202+). (For more go to: http://utable.wordpress.com/2013/11/01/ The big board-little universe is 62×10 inches.; it is big-and awkward. The Universe Table is designed to be printed on an 8.5-x-11 paper.)

**Introduction to Chaos Theory and Fractals**

Chaos theory was introduced by Edward Lorenz from Cambridge, Massachusetts. A professor at the Massachusetts Institute of Technology; he thought up the scientific concept that small effects lead to big changes. This concept was awarded the name “the butterfly effect.” Could something as small as a butterfly flapping its wings in Brazil effect the atmosphere in ways that could later cause tornadoes in Texas? Chaos is the science of unpredictability. The point that it makes is to expect the unexpected. Most traditional sciences deal with predictable phenomena like gravity, electricity, or chemical reactions. Chaos theory deals with things that are impossible to predict and control. Many natural objects exhibit fractal properties (clouds, rivers, trees, landscapes, organs), and many of the systems in which people live show chaotic behavior. By learning about the chaotic nature of the world, it might give new insight, power, and wisdom into the Planck Length and the concept that it seems related to quantum mechanics and might help to understand the Planck Length. All systems, from the Planck Length to ecosystems, to all social systems, and all economic systems could be connected

**Fractals**: Benoit Mandelbrot introduced the study of mathematical and geometric fractals in 1985 while at the IBM Thomas J. Watson Research Lab. Fractals are seemingly infinite complex patterns that are never ending across different scales. They are created by repeating a process over and over in an ongoing loop. Driven by recursion (the repeated application of a reoccurring procedure or definition), fractals are images of dynamic systems and are the pictures of Chaos. Geometrically, they exist in between familiar dimensions. Fractal patterns are extremely familiar, since nature is full of fractals. Mandelbrot’s work has certain parallels with the studies began in the sophomore geometry class in 2011 for which Bruce Camber was the guest speaker. For more, we are now studying Stephen Wolfram’s work to define the one-dimensional cellular automaton rules. Wolfram’s wrote this article about Benoit Mandelbrot’s work, “The Father of Fractals,” in the Wall Street Journal, 22 November 2012.

**Footnotes**:

^{1} Wolfe, Joe “Q: What Is the Planck Length? What Is Its Relevance?” Ask a Mathematician Ask a Physicist, 30 Sept. 2013

**Works Cited**:

“Albert Einstein.” Albert Einstein. Web. 30 Sept. 2013. .

Camber, Bruce E. “Big Board of Our Little Universe.” Big Board of Our Little Universe. Web. 17 Nov. 2013

Camber, Bruce E. “Universe Table.” Universe Table. Web. 17 Nov. 2013.

“Max Planck – Biographical.” Max Planck – Biographical. Web. 30 Sept. 2013.

“The Planck Scale: Relativity Meets Quantum Mechanics Meets Gravity.” (from Einstein Light). Web. 30 Sept. 2013.

“Previous index Next PDF.” The Speed of Light. Web. 30 Sept. 2013.

“What Is Chaos Theory?” FractalFoundation.org RSS. Web. 30 Sept. 2013.

Wolfe, Joe “Q: What Is the Planck Length? What Is Its Relevance?” Ask a Mathematician Ask a Physicist. Web. 30 Sept. 2013

**Wolfram**, Stephen “The Father of Fractals,” Wall Street Journal 22 November 2012