How Big is Big and How Small is Small

How Big is Big and How Small is Small?

By Damon Nomad

Perspective and Perception

We all know that perception is a form of reality, and we perceive the world mostly through the limitation of our senses. As it relates to the world of physics, time, space, and motion are mostly perceived through our sense of sight. For those of us blessed to have this sense intact. We instinctively believe what we can see, but understand we are greatly limited by our unaided vision. For the world of the small, we have used ever more powerful microscopes to zoom in. For the grand scale of the universe, we rely on telescopes to virtually explore the cosmos. The images alone are not enough, we have wonderfully powerful brains that help us perceive, through reasoning, intellect, study, and analysis.

Around 1900 science began to contradict the average man's perception of the world, contradicting our intuition of what we see and experience. Our comfortable world where a clock always runs at the same rate for everyone regardless of whether they were on a moving train or not. Our space was three-dimensional, it was obvious in our rectangular homes with clocks ticking on the wall. Particles did not travel through barriers or supposedly be at multiple locations at the same time. A cat was either dead or alive regardless of what Schrodinger might have said about it. It was a wonderfully simpler view until Einstein, Heisenberg, Schrodinger, and others began to blow it all up. A world where space and time do not exist separately, replaced by four-dimensional space-time that is curved by the presence of mass or energy. The clocks don't run at the same time, particles don't exist at one location and the cat is neither dead nor alive. Unfortunately, to really understand the newer modern principles, you have to move beyond your normal perception. The elite scientists communicate about these things in elegant mathematics beyond the reach of most.

Part of the limit of our perception is the scale of things big and small, how big is the universe and how small is the smallest particle? Science fiction and movies have shown us how the scale of things is relative, if you shrink down to the size of ants things look very different. Let's consider the range of size or distance we need to cover, from the small to the large. From the science of the very large cosmology down to the science of the very small, the strange world of quantum mechanics.

We will boil this down to a simple question, how big is big and how small is small? No advanced mathematics is necessary, just a ball and some zeros.

Scaling Up Big

As a first step, we will establish a standard unit of measurement, let's go with the diameter of a baseball, it's almost the same size as a cricket ball for those who don't follow American Sports. To put an approximate number on it we will say a diameter of 3 inches or 0.25 feet. We will decide on how big is big and how small is small using this baseball as a reference frame. You can visualize holding it in your hand as you read this exciting essay. Sorry for that bit of self-promotion.

Continuing with a baseball theme, let's scale up to a baseball stadium. A heck of a lot bigger than one baseball right? Imagine a ball laying in the middle of Yankee Stadium in the Big Apple. How much bigger is the width of the stadium foundation than the diameter of the ball? More than five times, more than ten times, more than a hundred for sure. But how much bigger?

Using rough numbers we will approximate Yankee Stadium as a big square, 1000 feet on a side. This means that Yankee Stadium is 4000 times bigger than a baseball, that's the basic method we will use to say how big something is, we will compare it to a baseball. Dividing 1000 by 0.25 we get 4000. For the average person, a professional sports stadium is big. That gives you a visual image and a numerical value you can wrap your head around and demonstrates the basic method of deciding how big or small something is relative to a baseball. We all know, there are things a lot bigger than a baseball stadium.

So let's keep going, and we are going to use scientific notation very soon, you remember it and some of you may even use it if you are a student or work in engineering or the sciences. It is simple, just count how many zeros we need. For example, 4000 in scientific notation is 4 E 3, four followed by three zeros. It will work when we go small as well, for example, one-hundredth (0.01) is 1 E-02 again shifting decimal places two spots, to the left this time.

For our next step in scaling things up, we ask how much bigger is New York City as compared to our baseball? Using our simple method, we find that the approximate diameter of New York City is about fifty thousand times bigger than a baseball ( 5 E 4). Let's keep going, the straight line distance across the United States is about 784,000 feet, or six million times bigger than a baseball (6 E 6).

Now you are thinking so what, enough! A baseball is millions of times smaller than the US, not surprising you could have guessed that without reading this essay. I ask for just a little more patience, things have not really taken off so to speak. We haven't gotten off the earth yet. Think about these last few data points in your mind's eye. It is still within your grasp of perception based on experience, something you can wrap your mind around. A baseball, laying in the middle of Yankee Stadium would be invisible from above New York City while flying over in an airplane, you might see the stadium below depending on how high you were. New York while big is still quite small compared to the scale of the country, but you can still roughly perceive all of this from driving and flying. Now things are getting ready to grow beyond the perceptions of your life experiences.

As we go planetary, we find that the earth is 2 E 8 bigger than a baseball, and the solar system out to Pluto is 1 E12 bigger than a baseball. Twelve zeros and we are still just in our local planetary neighborhood, most of the planets we can see with a small telescope. Here it is without scientific notation: Our solar systems diameter is roughly 1,000,000,000,000 times bigger than a baseball, and the size of the solar system is well beyond your experience of perception. Can you perceive a number that big? Probably not, you know the solar system is big, it takes years to traverse with the speedy probes launched from earth traveling at tens of thousands of miles per hour. But the numbers are becoming incomprehensible at this point, and the solar system is quite small in comparison to how much bigger we need to go.

The next scale-up is our galaxy the milky way. Our galaxy is about 6 E 17 miles across, yes miles. This makes the milky way galaxy about 1 E 22 times the size of our fist-sized baseball.

Now for our last stop, the observable universe with advanced telescopes. This is as far as we will go, beyond this point we have speculation without observation. The most distant observed galaxy is approximately 4 E 26 miles from earth and what we will use as the definition of big. Writing out the zeroes, the size of the observable universe is approximately 400,000,000,000,000,000,000,000,000 miles.

The baseball analogy has become useless now, let's use another image. Think of the last time you looked up at the night sky and saw the moon. You know it is far away, it took three days in a rocket to get there. It's about 240,000 miles from the earth, so let's put those two numbers side by side. The distance to the moon and the observable universe.
240,0000 miles [Earth to Moon] verses 400,000,000,000,000,000,000,000,000 miles [Universe]

Kind of hard to wrap your mind around that comparison isn't it, that number does not mean anything to you. The physical laws of Newtonian physics using basic algebra and calculus work in our comfortable perception of the world of baseballs and airplanes. It even works pretty well for the celestial mechanics of the movements of the planets in our solar system. But that's when science started to run into problems with Mercury, as Newton and his predecessors discovered. Our intuition driven by perception fails us in trying to understand the forces and movement of objects at these scales. Something is missing in our simplified views of time, space, and motion. We should not be surprised given the scale of things in our universe. The mathematics developed by Einstein and others to explain this galactic scale are what is needed to truly comprehend the meaning of time and space. This mathematics is within the grasp of an elite few, which leaves the rest of us with imperfect analogies about space-time, and the curvature and the warping of space. Imperfect images and explanations which leave some skeptical as to whether the scientists really have it right.

Scaling Down

Now we will scale down, starting again with our wonderful baseball. Scaling down is a different experience than scaling up because our sense of the visual is quickly lost as things go microscopic, then to the atomic, and then to the subatomic. Our perception is of little use because things are just smaller and smaller beyond our visual resolution. This brings us to the realm of quantum mechanics and the crazy and imperceptible concepts that even the world's greatest scientists admit are beyond explanation in a common sense way. The world of the tiny does not behave the same as the world we live in.

When I scale down I am going to use meters instead of feet because subatomic physics values are nearly universally reported in references using the metric system. We will go quicker here, it's the same idea of scaling compared to a baseball.

A baseball is about 7.6 centimeters, let's approximate it as 0.08 meters ( 8 E-02). If you are not comfortable with meters, a meter is roughly a yard. What is something small that almost everyone has touched and seen? How about a grain of sand? We will use a value of 0.2 cm or 0.002 m ( 2 E -3) in diameter.

Now to the microscopic, a red blood cell. 1 E -06 m

The atomic, size of an atom. 1E-09 m

Subatomic, the size of a proton. 1 E-15 m

Fundamental particle, size of quark 1 E- 18 m

That is where we will stop, it's a bit speculative the size of the quark is not exactly known. A quark is what we will consider as small. When going down in size quantum mechanics says there is a limit as to how far you can go. A size below which nothing can be smaller is known as the Planck distance of 1 E-35 m. The reason for this limit is the subject of another essay, we will stick with the quark for our summarizing our story.


From the Biggest to the Smallest

So now we have our biggest thing, the observable universe, and our smallest thing the lonely quark. In answer to our question, how small is small we have the quark and the answer 1 E-18 meter. In answer to how big is big, we have the observable universe which when converted to meters, is 6 E29 meters. What is the range of difference in size between the big and small? As a reminder, Yankee stadium was about 4000 times bigger than our baseball, something you can visualize.

The observable universe is on the order of 1 E 47 times the size of a quark. Yes, forty-seven zeros, dare I write it out?

The observable universe is about 100,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 times bigger than a quark.

Try developing a governing set of equations that describe behaviors of interactions and forces of nature, and time that cover that range of size. Is it any wonder, our common sense views about the behavior of the universe cannot cover such a wide range?

A last point to ponder, something that is driving the cosmologists and quantum physicists crazy, well one of the things driving them crazy. According to current theories the universe once was the size of a quark, try and explain how something grew from that spec in space by 47 orders of magnitude and ended up with the majestic beauty and structure we see in the sky with our eyes and telescopes. It seems to be beyond explanation, some might say miraculous.

PS - If you find a math error; keep it to yourself unless it changes the bottom line of what is big and small.