2.1.4 - Scales of the Microcosm

2.1.4

Scales of the Microcosm

We have explored the massive scales of the macrocosm all the way to the limits of speculation, but what about the microcosm? If we can ask how "big things get" we can also ask "how small"? Very big numbers and very small numbers tend to go hand in hand. When something is very small it just means that something ordinary size is "very big" in comparison. For this reason studying very small things can also lead to very large numbers, like how many cells are in the average human body, and how many atoms exist in a drop of water. Furthermore when we go to the extremes of smallness it is often just as mindboggling as trying to comprehend the extremes of largeness.

So let's take a journey from our familiar realm to one not millions of lightyears away, but right in front of our faces! But when we look closely we discover that we ourselves are universes teeming with life and intricacies beyond imagining! This world can be quite terrifying not because it's far away but because it's right in front of us, we just can't see it. It's too small for us to see! The world is far more nuanced then our eyes or minds can perceive. None the less with the aid of microscopes we can imagine what it might be like to live in this invisible world. We will begin again with a familiar point of reference, ourselves, and then gradually shrink down many many orders of magnitude until we reach the theoretical and speculative limits of the small. Ready ...

Human

1.75 m

(1750 mm)

House Cat

0.406 m

(406 mm)

Let's begin our journey with something familiar and not too small...

Human height varies drastically but for simplicity we can approximate the average height of modern humans as around 1.75 meters. There is nothing spectacularly significant about this number. The meter was originally chosen to be 1/40,000,000 of the circumference of the earth, and was then redefined as exactly 1/299,792,458 the distance light travels in a second, a second itself being a kind of arbitrary unit that is 1/86,400 the length of an earth day. It just so happens humans generally fall somewhere between 1 and 2 of these units called meters. A meter is a handy reference for exploring the scales of the microcosm as it is more or less commensurate with our common experience.

It's worth noting that a lot of our experience as humans beings does not involve dealing with things larger than us, but usually about the same size or smaller. Take for example, your house keys, books, laptops, pens and pencils, change, your cell phone, watches, calculators, etc. These things are smaller than us mainly because we made them small for our own convenience. But objects which become too small may become difficult to manipulate. And eventually we reach objects so small that we can't even see them anymore.

Our diminutive furry companions, cats, barely reach our knee caps even when in an upright position. Here the mans foot as been scaled to match the cat. Cats are small enough to cradle in our arms (even though most cats don't enjoy this for very long). Cats probably view us as giants. Standing at about 0.406 meters (or 406 millimeters), we stand at about a ratio of 4.31:1 to them. Try imagining someone over 4 times as tall as you! When walking cats only stand up to about 0.25 meters. Their total body length however, including their tail, can be as much as 0.76 meters. Still cats are not small in a way that boggles the mind. They are small in a way that is safe and familiar. Let's go a little smaller ...

House Mouse

0.162 m

(162 mm)

The common house mouse has a body length (tip of nose to base of tail) from 75-100 mm, and a tail length from 50-100 mm. This gives it a total length ranging from 125 to 200 mm, averaging about 162 mm. Here it is scaled in comparison to the cat from the prior entry. As far as the mouse is concerned, the cat already IS a giant. Taking the ratio of the cats body length + Tail to the mouses is about 4.69:1, close to the ratio between a human and a cat obtained earlier. Being small has it's advantages though. Mice need less food to survive, and shorter life spans (they reach sexual maturity at only 50 days), shorter gestation periods (20 days), and therefore can reproduce much faster than other mammals. The downside is of course that mice are the bread and butter of many other species including cats, birds, and reptiles. The world is a big place to a mouse. Comparing them to our dimensions we are about 10.8 times their size. So they literally live about an order of magnitude beneath us! This should mean that they also weigh rough 1/1000th of our weight. In fact their nominal weight is given as 10-25g, where as human weight is about 80 kg, putting the weights at a ratio of over 2000:1.

The mouse is small enough to rest comfortably in the palm of your hand. But a mouse is still, relatively speaking, a behemoth. We are still very comfortably in the sphere of the human experience. When do we exit this sphere? Once we go to sizes to small for us to perceive. This happens roughly around the order of 0.1 mm. We still have 3 more orders of magnitude before we even reach that point.

So far our journey has been pleasant ... and furry. But we are about to start entering slightly more uncomfortable territory. To go smaller we need to start looking at the world of insects ...

Lady Bug

0.018 m

(18 mm)

If this image is already making you uncomfortable ... be warned ... it's going to get a whole lot worse before we are through. At a mere 18 mm the lady bug is almost a 10th the size of a mouse. It's so small it can it comfortably at the tip of your finger. Normally the lady bug is considered one of the less "scary" bugs. However magnified we can see it is still very much of the insect family, complete with segmented legs antenna and compound eyes. To the lady bug even the humble mouse is a giant, and a potentially dangerous one at that (mice do sometimes eat insects. Mice are not known to be picky eaters).

Insects being even smaller than mice are even more prolific reproducers. There is an estimated 1.4 million trillion insects in the world. Yes you read that right a million trillions! Told you there be some large numbers involved. And we are only getting started.

We still are talking about creatures we can see, but now we are getting pretty small (from our point of view anyway). Let's get smaller still ...

Clover Mite

0.000850 m

8.5x10^-4 m

(0.850 mm or 850 μm)

But first let's see just what a behemoth the humble lady bug is compare to this truly tiny creature:

That little red "dot" is the almost inconceivably tiny "Clover mite", which is not a mite at all in fact, but a very small arachnid. Even a penny looks absolutely gigantic next to it. It measures just under a mm in length (about 0.850 mm or 850 μm). This puts it very close to our limit of visibility and close to the border of what is defined as the "microscopic". The microscopic is by definition anything smaller than we can see with the unaided eye. We will be entering the microscopic realm very very soon.

o_0;

The tininess is starting to hurt. But we are still just getting started. Next we introduce the...

Dust Mite

0.000250 m

2.5x10^-4 m

(0.250 mm or 250 μm)

... dust mite, the first creature we encounter that is so small that it is almost impossible to see. A dust mite, if you could manage to spot one, would look like a tiny translucent spec with no discernible features. Measuring somewhere between 0.2 and 0.3 mm this creature makes the clover mite look like a mighty beast in comparison. The ratio of their body length to that of the clover mite is approximately 3.4:1.

Dust mites are microscopic arachnids that usually feed off dead skin cells. Although they don't live on people they commonly cohabitate with them, living and hiding within mattresses bedding and carpets.

They have life spans of only around 100 days and females can lay anywhere from 60 to 100 eggs in their brief lifetime. Suffice it to say these guys can really multiply. Theoretically a dust mite population could grow a million fold in just one year if they were given unlimited resources.

We have reached the threshold of human perception. Beyond this point thing creatures and objects we will discuss are simply too small for us to actually see at all...

Paramecium

0.000050 m

5.0x10^-5 m

( 0.050 mm or 50 μm)

Now we reach the truly microscopic world of single celled organisms. First up is the relatively large paramecium, with the smallest specimens around 50 μm. These are small enough to make the dust mite seem gigantic in comparison. The paramecium is about 1/5th the length of the dust mite. Paramecium were among the earliest known micro organisms, their name coined as early as 1752. These guys are relatively large compare to other cells ...

E. Coli

0.000002 m

2.0x10^-6 m

( 0.002 mm or 2 μm)

Now we reach a size so small that it is actually kind of painful to think about. The E.Coli bacterium is only about 2 μm across, making the paramecium look like a whale in comparison. It's almost vanishingly small in comparison, yet the paramecium is itself something so small it's difficult to imagine.

We have now traveled down a mere 6 orders of magnitude, but we can get much smaller still! ...

Bacteriophage T4

0.0000002 m = 2x10^-7 m

( 0.2 μm or 200 nm )

It's body consists of an icosahedral shaped head, a screw like body and several leg like appendages at the base. The dna is contained within the head.

The T4 is about 200nm tall making it approximately 1/10th the length of a typical E.Coli.

Despite it's tiny size it's dna still contains approximately 169,000 base pairs! This encodes for 289 different proteins. It's a impossibly small little self-replicating machine.

Bacteriophages are a type of virus that reproduces by injecting it's dna into a host cell causing it to replicate it. T4 is a common example of a bacteriophage, and the creature you are most likely to think of when you try to picture a virus. Interestingly T4's typically use E.Coli as a host.

Bacteriophages are among the most plentiful creatures in the biosphere with an estimated 10³¹ bacteriophages on the planet, easily outnumbering all humans, birds, fish, insects, and even bacteria combined! This is a stupidly big number. For point of comparison, there are about 10²² stars in the observable universe, and 10²⁶ drops of water in the ocean. This number dwarfs these! We might be able to just barely conceive of a million, billion, or trillion, but when we get to a number like 10³¹ which is ten million trillion trillion it's basically hopeless!

Since we are on the topic of creating very large numbers using small things, if we can the dimensions of a typical T4, how many would it take to form a volume the size of the earth? See if you can guess the order of magnitude.

If you guessed about 10⁵⁰ then you overshot by a large margin, but if you guessed about 10⁴⁰ you are almost exactly on point. The earth is about 1.08x10²¹ m³, while the typical volume of a T4 is 1.62x10⁶ nm³. However there are 10²⁷ cubic nanometers to every cubic meter! Therefore the volume of the earth is 1.08x10⁴⁸ nm³. Dividing this by 1.62x10⁶ nm³ we get 6.67x10⁴¹. Insane!

Let's go even further ...

Rotavirus

0.0000000765 m

7.65x10^-8 m

( 0.0765 μm or 76.5 nm )

Porcine Circovirus

0.000000017 m

1.7x10^-8 m

( 0.017 μm or 17 nm )

There are 9 species of rotavirus. Like the corona virus they are spherical with a "crown" of protrusions all about them. They can also cause similar symptoms. The genome of the rotavirus is a mere 18,555 nucleotides long. The diameter of a rotavirus can be as "much" as 76.5 nm, which as you can imagine from all the orders of magnitude we've already travelled down is quite inconceivably tiny.

The rotavirus is shown here beside the T4 in scale, and as you can see they fall about in the same order of magnitude. We still have about 3 orders of magnitude to go however until we get to the size of atoms. What could be even smaller than a rotavirus?

DNA Molecule

0.0000000026 m

2.6x10^-9 m

( 0.0026 μm or 2.6 nm )

The porcine circovirus, possibly the smallest known virus, clocks in at a mere 17 nanometers in diameter, making even the rotavirus look comparatively gargantuan. We still have a little over 2 orders of magnitude to get to atoms however. Let's start looking inside the cell for even smaller things ...

Francium Atom

0.000000000696

6.96x10^-10 m

( 0.696 nm or 6.96 A )

We still have a little less than one and a half orders of magnitude to go to reach atoms. This is enough however to now reach the largest atoms easily...

At last we reach the monumental smallness of DNA. Specifically DNA is extremely thin but also macroscopically long. The two coils that wrap around each other only have a diameter of about 2.6 nm, making the circovirus look large in comparison. At the same time a single DNA molecule can be as much as 3 meters long (length of dna in a single human cell).

The radius of an atom is defined as the distance from the center of the nucleus to the outer most electron shell. The size of atoms are complex results of the interactions of electrons and the positively charged protons in the center. Having more shells gives us bigger atoms, but at the same time having those shells more filled out causes the atom to shrink due to the greater attraction of those electrons with the nucleus. As we go up the periodic table of elements the size of elements therefore decreases and then drastically jumps in size in a kind of sawtooth pattern. The largest element is francium at 0.696 nm in diameter or 6.96 A. As you can see, shown to scale next to the DNA molecule, the atoms that make up its structure are considerably smaller than the francium atom shown on the right. If francium is the largest atom, what is the smallest?

Hydrogen Atom

0.00000000024

2.4x10^-10 m

( 0.240 nm or 2.4 A or 240 pm)

Muonic Atom

0.0000000000012 m

1.2x10^-12 m

( 0.0012 nm or 0.012 A or 1.2 pm )

The size difference between the largest atom (francium) and the smallest atom (hydrogen) isn't actually all that great. In fact they easily fall in a range that we can comprehend. The hydrogen atom radius is only about 34.4% of the radius of the francium atom, so the francium atom is only about 3 times larger. We've made it to atoms but how do we get to the even smaller subatomic particles, which are still about 5 orders of magnitude down from this already minuscule size?! ...

We can get there using a highly experimental exotic atom known as muonic hydrogen. Basically a hydrogen atom is just an electron orbiting a proton. What if we used a muon instead of an electron. Well we get a highly unstable atom which only lives about 2.2 microseconds. This is because the muon is itself unstable and decays in this time frame. The interesting part for our purposes however is that due to the proton-muon interaction the orbit of the muon is much much smaller than the orbit for electrons in ordinary atoms. The muonic hydrogen atom has a radius about 1/200th that of a normal hydrogen atom. If a hydrogen atom was expanded to the size of Jupiter than a muonic atom would be about 1/5th the size of the moon or 2/3rds the size of Ceres! Although the muonic atom is quite tiny in comparison it makes the gap from atoms to the nuclei of atoms manageable. We can now use the muonic atom to reach the largest atomic nuclei.

Uranium-235 Nucleus

0.0000000000000117142 m

1.17142x10^-14 m

( 0.0000117142 nm or 0.0117142 pm or 11.7142 fm )

Even if we take the smallest atom and the largest nucleus the ratio of sizes is still around 20,488 times smaller! To give a sense of scale, imagine the hydrogen atom was the size of Jupiter again. At this scale a Uranium-235 nucleus would be about four times the size of an asteroid like 951-gaspra (refer back to macrocosm article).

Comparing it to muonic hydrogren instead however, the uranium-235 nucleus is only about 102 times smaller, a much more manageable figure that we can gain a direct experience of as the image above demonstrates.

The uranium-235 nucleus itself would be composed of 92 protons and 143 neutrons. These large nuclei tend to be unstable and decay giving off radiation and slowly converting into lower elements.

Next we can explore the world of sub-atomic particles ...

Proton

0.000000000000001 m

1.0x10^-15 m

( 1.0 fm )

The proton is said to have a diameter of approximately 1 femtometer (10^-15 meters) based on the range of the strong nuclear force which binds 3 quarks together to form a nucleon. The proton is so small that about 7.2 quadrillion (7.2x10^15) of them would fit inside a single hydrogen atom. Here beside the uranium nucleus the size difference is something we can easily wrap our heads around ... don't get used to the that ... this is the last time THAT WILL HAPPEN ... -_-;

Electron

0.000000000000000001 m

1.0x10^-18 m

( 0.001 fm or 1.0 am )

The electron is held to be even smaller at only about an attometer across. This makes the proton 1000 times larger and have 1,000,000,000 times the volume! Sub-atomic particles can be given mass based on their radius of interaction.

Planck Length

0.00000000000000000000000000000000001616 m

1.616x10^-35 m

( 0.00000000000000001616 am )

The next entry basically breaks the entire scale as we suddenly jump an additional 17 orders of magnitude to reach it. This would be like if we went all the way from the electron as the size of the crab nebula and then shrunk down to the size of humans! We went 18 orders of magnitude just to get to the electron, yet we got another 17 orders to get to the planck length. In other words about half of our scale is completely devoid of anything to talk about!

This is believed to be the scale at which vibrating strings in string theory form our fundamental particles. It is also believed to be the scale at which general relativity and quantum mechanics collide preventing us from making meaningful predictions at this scale. This is also popularly regarded as the "smallest scale possible" or "smallest unit of space". This is something of a misunderstanding. Einsteins special and general relativity assume space is continuous and therefore there is no smallest possible scale of spacetime. Some physicists have proposed "digital" models of the universe in which there is a smallest discrete scale, making the world digital rather than analog, but there is no experimental evidence to support it or refute it. The scales are simply too small for us to probe directly and so we are now in the territory of mathematical/scientific speculation.

While spacetime is held to be tumultuous at this scale this doesn't mean that things might be happening at even smaller scales, we just can't describe them with our current models of physics.

So is that it? Have we reached the smallest length in science. Actually we haven't. There is one other number that goes even further down ...

Schwarzschild Radius of Black Hole with mass of the Electron

0.000000000000000000000000000000000000000000000000000000001353 m

1.353x10^-57 m

The Schwarzschild radius is the radius of the event horizon for a black hole. Theoretically any amount of mass can become a black hole if it is simply packed tight enough. So what if we tried to turn an electron into a black hole? We would have to pack it into a radius 22 orders of magnitude below the planck length!!! This may well be the smallest length ever computed in mainstream science, by Karl Schwarzchild using einstein's general theory of relativity no less! The formula for the schwarzschild radius however is relatively simple. It's just 2Gm/c^2, where G is the gravitational constant, m is the mass, and c is the speed of light in a vacuum. Since G and c are constants this means the radius scales linearly with mass. So why not chose a super light particle and then you can get an even smaller radius...

Schwarzschild Radius of Black Hole with mass of the Neutrino

0.000000000000000000000000000000000000000000000000000000000000001 m

10^-63 m

The electron mass is about 10^-31 kg while the mass of a neutrino is on the order of 10^-37 kg. Since the neutrino is about one millionth of the mass is schwarzschild radius should be about one millionth of that of the schwarzschild radius for the electron. This gets us to a whopping 63 orders of magnitude below our scale! Can we go further? Well not really with this idea. The problem is finding super light sub-atomic particles that aren't massless. The neutrino appears to be the lightest mass particle. If we plug a massless particle into the equation we just get 0. To a googologist this is just as useless as getting infinity. We want to try to get as close to zero without actually reaching it.

So that being said, with no way to derive a smaller figure, and no way to experimentally prove anything below this scale, or significantly above in fact, we are left with only our flight of imagination to continue. Perhaps just as atoms are composed of electrons and nucleons, and nucleons are composed of quark triplets, perhaps its sub-atomic particles all the way down. Perhaps spacetime is infinitely divisible just like the real number line. Some find this line of reasoning dubious. They hold that nature can not contain an actual infinity. But there doesn't appear to be an actual argument against it other than personal incredulity. Simply because it seems impossible to imagine or leaves the ultimate question of what the universe is fundamentally composed of unanswerable not just in practice but in principle people will tend to feel uncomfortable with the idea. But the universe owes us no particular comfort. It could just be fundamentally unknowable. On the flip side maybe after going thousands, millions, billions, trillions, or even more orders of magnitude down ... somewhere down in those unfathomable depths is the heart of reality. Nobody really knows. But for now science is pretty much restricted to scales well above the planck length as our particle accelerators simply don't have the energy levels required to explore deeper. Furthermore we may never produce a particle accelerator with enough power even if we create a ring around the entire solar system! Perhaps we are hitting a dead end for human exploration of the very very small, or perhaps some heretofore unthought of method might allow us to experimentally prove the existence of even smaller things.

In any case I think we are done here for now. Just to finish things off, what if we were to take this tiny radius and try dividing the size of the observable universe by it. How many of these super minuscule black holes could fit? Well very roughly there is a difference of 90 orders of magnitude from this radius to the size of the observable universe. So cubing this we have about 10^270 of these black holes fitting in the observable universe! As promised we got a pretty large number out of this journey to the very very small. Of course mathematically we can devise much much larger and smaller numbers with ease without the need for an actual physical thing to represent it, and that's kind of the point of all this. Mathematics goes much much further out into the number line that anything in the known universe could ...

so read on if that intrigues you ...

NEXT >> 2.1.5 - Everyday Large Numbers for a Modern World

Sources:

https://en.wikipedia.org/wiki/Escherichia_virus_T4

https://en.wikipedia.org/wiki/Bacteriophage

https://www.thoughtco.com/atoms-in-a-drop-of-water-609425

https://en.wikipedia.org/wiki/Rotavirus