The Scale Problem

Someone asked me a logical question in a party - Suppose you are able to fold a newspaper 100 times, how much thick would that bundle be? He also dropped us a hint - remember, each time you fold it, it doubles in thickness - so answer judiciously. Assume for now that in fact you can fold it 100 times without running out of paper. Of course we cannot work out without a calculator. But remember, it is an exercise on how your brain comprehends such a problem. We were asked to take a guess. We did, but before you read any further - pause and think - What would your guess be?

Some of us guessed that it would be as thick as a brick. One guy said 4 stories tall. Someone with a scientific outlook suggested that it might extend upto the sun from the earth, drawing many disdainful eyes. So, what was your guess? The way most people thought even after being given the hint, was that it would be 100 times (or 100x) the thickness of the newspaper. You might think that the claim that the thickness would extend upto the sun would be outrageous. But wait! it is not so strange after all. The answer is far larger than the distance upto the sun (which is approx. 150 million kms). In fact, if you could fold it would extend to the boundary of our solar system. Umm, my mistake. It would extend to the boundary of our observable universe which is approximately 46.5 billion light years across. How does that appear?

46.5 billion light years is approx = 8.8e+23 km or 8.8e+26 m = 8.8e+29 mm

1 light years is the distance traveled by light in 1 year which is 9.46e+12 kms.

(Reference: 2.3e+22 means 2.3 times 10 to the power of 23)

Let us see how much the paper folding would result to. Let us assume the width of each paper is 0.1mm. If you fold it 100 times you get the size in mm as 0.1 x 2 power 100. Remember, the width doubles after every fold. We know 2 power 10 is 1024 which in turn is approximately 1000 or 10 power 3 (or 1e+3).

2 power 100 is approximately 1e+30

So the thickness would be 0.1 times 1e+30 mm or 1e+29 mm. Our observable universe is of the scale 1e+29mm too. Quite a coincidence I must say.

Our brains are not designed to comprehend non-linearity. Any scale that is very large, or very small, for that matter, is really difficult to imagine in our heads. We may easily comprehend what a 2x or a 3x or maybe what a 10x size means, but not what 1200x means. Fortunately humans have an ability to adjust the scale and make the understanding easier. It means have a relative scale and help your brain grasp it. It means.... oh forget it. Let me explain.

How much is a million to you? Not in dollars but in terms of numbers. One way is to understand by comparison. Do you know what a second is? Of course you do. Let's use this analogy then. Start by counting numbers starting from 1 to a million (1e+6). Let us assume that you take exactly 1 second to count a number. Of course nine-hundred-and-seventy-thousand-three-hundred-and-fifty-seven for example would take much more than a second, but let's assume, that you are a mutant who can do so. The question is how much time would you take? Any guesses?

It would take around 11.5 days (divide 1e+6 by 24x3600 seconds). Hard to comprehend but yes. And that when you count non-stop 24 hours everyday. In other words, there are approx one million seconds in 11.5 days. Now you can compare 1$ to a million, and see how far that is. How about a billion then? A thousand million is a billion. How much time would you take to count a billion? I am sure that you can work it out yourself. Still the answer 31.7 years would appear very surprising.

How do you visualize a billion years then? I found it hard to explain the concept of evolution taking place over billions of years, to my mother-in-law. In this case, I tried to convert the time scale to distance. A billion years, I said, is that much of a time, that if you travelled 4 cm in a year (not kidding), you would circumnavigate the earth. Lets try this out.

Circumference of earth = 40,000 km (or 4e+9 cm)

1 billion = 1e+9
dividing it you get 4 cm

Imagine the length of the circumference of the earth and you travelling just 4 cm in an year. A billion such steps will bring you back where you started. Earth is 4.5 billion years old. That much of a time is large enough to evolve a single cell to a human or in other words, make a mountain out of a molehill.

Let us understand the expanse of our solar system. Do you know how far is the nearest star "Proxima Centauri" to us? Approximately 4.25 light years. 1 light year is a distance roughly equal to 9.46e+15 meters. But how do you imagine that distance? Let's try to compress them in a scale that is little smaller and more comprehensible. Try comparing it with the edge of our solar system. We know that the light takes approx 5.5 hours to reach pluto and takes 4.25 years to reach the nearest star. We have a comparison. Taking the ratio of the distances, Pluto : Proxima Centauri is 1 : 6769, and that my friend is quite a big deal. Pluto is not near to us. It took the probe "New Horizons" 9 years to reach pluto with the speed of 20km/second. That is pretty fast. Imagine you reaching office and even back in less than a second. When the probe began its journey Pluto was still a planet. Unfortunately it was demoted to a lesser god by the time the probe reached it, and for now has to live with the fact that it is a dwarf planet. 

If we shrink the Sun and Pluto on a meter scale to 1 cm apart, then relatively the nearest star would be 67.69 meters away from the sun. That is too much of empty space in between. The milky way, our own galaxy, is 100,000 light years across. Our modern civilization is a mere 5000 years old. In that much time, light from one end of the galaxy would have covered just 5% percentage of the distance end to end. Keeping our milky way on the same scale would span across 1600 km. The largest galaxy discovered so far is IC 1101 is nearly 2 million light years across. On the same scale as above, it would amount to a length of 32,000 kms.

No surprise that such a vast universe is mostly empty with the density of just 0.25 Hydrogen atoms per cubic meter. How much is that? Compare it to water molecule count. There are 3 atoms in water (2 Hydrogen and 1 Oxygen) so roughly it is 0.25/3 or 0.083 molecules of water per cubic meter. How much volume of average space would you need to fill a 200 ml. glass of water?

18 grams (1 mol) of water contain 6.022e+23 water molecules (Avogadro Number). 

Thus, 200 gm (or 200 ml) of water will have approx 6.686e+24 water molecules. 

That would need a volume of  8e+25 cubic meters of space. 

The volume of all the planets of our solar system is around 2.4e+24 cubic meters. So if you collect water from 33 times of volume of all planets in the solar system in an average space of the universe, you would get a glass of water. We are in fact in very a dense area of the universe with so much of water around. Universe is mostly empty space with a few particles of matter scattered around it. Matter itself is mostly empty from within too, as we shall see later. 

Let's recollect our scale for the distance ratios.

Sun - Pluto : Sun - Proxima Centauri : Milky Way Diameter = 1 cm : 67.7 meters : 1600 kms 

With things spread across so far and wide it shouldn't come as a surprise that there are billions of stars in just our galaxy - the Milky Way. The nearest galaxy to our own is Andromeda that is some 2.5 million light years away. If you compare it with the same scale it would be around 40,000 kms afar. It is nearly equal to the circumference of the earth.

Let's go inwards to the realm of the small. Not only the large scales are mind boggling, the smaller ones are not very comprehensible either. We will use the same standard exponential convention. 10 power -3 (or one thousandth) is written as 1e-3. 

Let us start with the diameter of a bacteria. Most of them are less than a micron (1e-3 mm). 1 mm = 1000 microns. If you have to pack a number of bacteria in 1 sq mm what would that number be? No it is not 1000. It is 1000 * 1000 or a million. Giving a lot of living space to bacteria, you can come down to a 10 thousand. Imagine 10 thousand bacteria dancing on the tip of a needle. Each of them is a living entity that can swim, walk and reproduce. How many bacteria can fill in your body volume? Hmm tricky one, but I suggest that you do the math yourself. (Hints : volume of human body 95 liters, volume of a bacteria 0.7 cubic micron, 1 lit = 1e+15 cubic micron). If packed nearly 1.35e+17 bacteria would fill an average human body. Much larger than all the humans that ever walked on the earth. Viruses are even smaller. Some 10 to 100 times smaller than bacteria.

Look at our Red Blood Cells (RBC). They are about 4-8 microns in diameter. Little bigger than an average bacteria, but very small nevertheless. In one cubic millimeter there are about 5 million RBC. An average human has around 25 trillion RBC in his body. A trillion is a thousand billion or a million million as you wish to call it. You know a billion seconds is 31 years. A trillion seconds is 31,000 years. That many years ago we Homo sapiens were hunting in forests and sharing the planet with Neanderthals. If you earn $1000 every second. It will take 31 years to amass a trillion dollar wealth. Even the entire US GDP (in the year 2018) is 20 trillion USD, much lesser than the total RBC in our blood.

Let's fill the volume of the RBC is the volume of Hydrogen gas. Yes the gas, and not the Hydrogen in liquid or solid form. Note that the atoms in the gases are very far apart and not in contact with each other. The question is - How many molecules of hydrogen gas would be there in the tiny volume of a Red Blood Cell (RBC)? We know 2gm of Hydrogen in STP (Standard Temperature and Pressure) occupies 22.4 liters and has 6.022e+23 molecules (Avogadro Number). The MVC (Mean Corpuscular Volume) of RBC is around 9.28e-14 liters. So the number of Hydrogen gas molecules in the volume of RBC would be 2.5 billion. 

The diameter of a Hydrogen atom is around 120 pm (picometers) or 1.2e-12 meters. It is the smallest atom available to us. Of course the atom's electrons are free to move about and do not follow a definite path. Thus an atom does not have fixed size. What we are calculating is actually Bhor's radius, which is the most probable radius of an atom. The nucleus of hydrogen atom (or a proton) is around 1.7e-15 meters. How small is that? Let's blow up the nucleus to say a size of 1 meter diameter. Do you know how big the atom would be? It would be around 700 meters in diameter. That is quite amazing. 1 meter (diameter) of the nucleus and an electron smaller than a football at distance of 700 meters (radius of 350 meters) with absolutely nothing in between. Matter as we know it is mostly empty. Just like the emptiness of the universe. The size of the atom has not been accurately measured. In fact it can never be because the electrons do not follow a definite path around the nucleus. Even if we take those into account with a lot of error margin, still matter is very empty from within.

Another example to demonstrate the tininess of an atom is by making a small dot on a paper. Now expand that dot to a diameter of  1 kms. The atom within it will now appear as a dot. 

What if you shrink the space between the atoms? Well the matter would become very tiny but would be very heavy. If you suck out all the space between all atoms in Mount Everest, the remaining matter would fit in a teaspoon. You will never be able to find a teaspoon strong enough to hold it, however. Imagine the pressure it would exert on the ground as the mass would be concentrated to a very small area. If all the space between the atoms are sucked out of you then your volume would be much smaller than a spec of salt. You are just a puffed up matter.

If you are thinking that this is fiction and is actually not possible then you are mistaken. The universe throws a lot of surprises. There are stars in the universe who are made with the matter like these. They are known as Neutron Stars. They are formed when a large sun explodes towards the end of its life and its gravity crushes the matter within. Some of these neutron stars having 1 solar mass are actually just 10 kms in radius. They spin really fast - around 500-1000 times per second. Now how to imagine that?

If you are thinking that that the neutron stars are the densest things in the universe then you are mistaken again. There is something even more dense. Something that even crushes the space within the nucleus (neutrons and protons) . Something that is so dense that it is no longer the matter that we recognize. Something that is dark and no one can see because its gravity is so strong that even light, the fastest thing possible, cannot escape it. It is called Black Hole. Almost all spiral galaxies are believed to have a supermassive black hole in its center. You can define a black hole with just three parameters - its mass, electric charge and how fast it is spinning. There are no geographical mountains and ridges in a black hole. It is a perfect sphere.

We have seen how mind boggling numbers our universe presents. Only when we compare it with something more obvious we see how strange this entire universe is. Till we evolve the sense of scale, I guess, we will have to keep doing similar comparisons to make some sense out of these numbers. Thankfully we have mathematics to our rescue.