Kurzgesagt – In a Nutshell
Thanks to our experts —
Prof. James Gurney
Georgia State University
Prof. Matthew Caplan
Illinois State University
Sources – The incredible Vastness of the Very Small
– The universe is pretty big and very strange. Hundreds of billions of galaxies with quintillions of stars and planets and in the middle of it all there is earth, with you and us.
The Hubble Space Telescope was used to conduct the Ultra Deep Field survey, giving us our first useful estimate of the total number of galaxies in the Universe. The number it suggested was 200 billion.
#NASA (2016: Hubble Reveals Observable Universe Contains 10 Times More Galaxies Than Previously Thought
Quote: "One of the most fundamental questions in astronomy is that of just how many galaxies the universe contains. The landmark Hubble Deep Field, taken in the mid-1990s, gave the first real insight into the universe's galaxy population. Subsequent sensitive observations such as Hubble's Ultra Deep Field revealed a myriad of faint galaxies. This led to an estimate that the observable universe contained about 200 billion galaxies."
It has been suggested that there are up to 10 times more galaxies in the Universe than we have observed so far with telescopes like the Hubble Space Telescope, bringing their number up to 2 trillion.
#Conselice, C. J. et al. (2016): THE EVOLUTION OF GALAXY NUMBER DENSITY AT z < 8 AND ITS IMPLICATIONS. The Astrophysical Journal, Vol. 830 (2)
https://iopscience.iop.org/article/10.3847/0004-637X/830/2/83
Quote: "The end result of this is that there are at least 2 × 10^12 (two trillion) galaxies in the currently visible universe, the vast majority of which cannot be observed with present-day technology as they are too faint."
However, a more recent study using the data from the New Horizons mission suggests that we can already see half the galaxies there are in the Universe with telescopes like the Hubble Space Telescope, which suggests that the real number is about 400 billion.
#Lauer, T. R. et al. (2020): New Horizons Observations of the Cosmic Optical Background. The Astrophysical Journal, Vol. 906 (2)
https://arxiv.org/pdf/2011.03052.pdf
Quote: "If the COB is due solely to the collective light from faint galaxies, then the 2σ difference between our derived IGL and our estimate of the COB would imply a factor of ∼ 2 under-count of galaxies even at apparent magnitudes well within the grasp of current
telescopes"
There are between 100 billion to 400 billion stars in our Milky Way, which is a pretty big galaxy. The other nearby galaxies are smaller, like the Triangulum galaxy that only has 40 billion stars or the Carina Dwarf Spheroidal Galaxy that has a million times less mass than our Galaxy.
#NASA (2015): How many stars in the Milky Way?
https://asd.gsfc.nasa.gov/blueshift/index.php/2015/07/22/how-many-stars-in-the-milky-way/
Quote: “There are different models for estimating the number of stars in the Milky Way and the answers they give differ depending on what is used as the average mass of a star. The most common answer seems to be that there are 100 billion stars in the Milky Way on the low-end and 400 billion on the high end.”
#ESA (2019): Hubble takes gigantic image of the Triangulum Galaxy
https://esahubble.org/news/heic1901/
Quote: “The NASA/ESA Hubble Space Telescope has captured the most detailed image yet of a close neighbour of the Milky Way — the Triangulum Galaxy, a spiral galaxy located at a distance of only three million light-years. This panoramic survey of the third-largest galaxy in our Local Group of galaxies provides a mesmerising view of the 40 billion stars that make up one of the most distant objects visible to the naked eye.”
#de Boer, T. J. L. et al. (2014): The episodic star formation history of the Carina dwarf spheroidal galaxy. Astronomy & Astrophysics, Vol. 572 (A10)
https://www.aanda.org/articles/aa/full_html/2014/12/aa24119-14/aa24119-14.html
Quote: “Carina is one of the smaller “classical” dSphs, with a total (dynamical) mass of ≈3.4 × 10^6 M⊙ within a half-light radius of rh = 250 ± 39 pc, and an absolute visual magnitude of MV = −9.3”
So if we multiply the number of stars per galaxy by the number of galaxies in the Universe, we can expect to have between 200 quintillion and 800 sextillion stars in the Universe!
The Miniature Realm
– You are the size of a grain of sand just 2 mm tall, standing on a blade of grass that seems as tall as an eight storey building to you. A square meter of lawn is now a dense metropolitan area, with 100,000 blades, or two Manhattans worth of grass towers. From your new tiny perspective, the park that you could quickly stroll through before, is now the size of France. Crossing it would take at least a week.
If you stood at an average 1.7 meters tall, your height would be 1.7 millimeters if you were shrunk a thousand-fold.
From your perspective, a rather short 2.5 cm long blade of grass would appear to be 25 meters tall. That’s the equivalent to an 8 storey building.
These grass blades can be packed very tightly. A ‘high quality lawn’ can have 100,000 blades of grass in each square meter.
#Wolf Garten (retrieved 2022): Creating a beautiful lawn
https://wolfgarten-tools.co.uk/media/the-gardening-year/may
Quote: “In addition to mowing, scarification, fertilising and watering, we would advise choosing a high quality lawn seed that covers on average 100,000 blades per square metre.”
According to the Primary Land Use Tax Lot Output database from New York, there are about 43,000 buildings with borough code Manhattan. Two Manhattans worth of buildings would add up to 86,000 buildings, which comes close to the number of grass blades per square meter.
#City of New York (2022): Primary Land Use Tax Lot Output
Also from this perspective, a 1000m long park will appear to be 1000 km long. This is a little more than the width of France. At a 5 km/h steady walking pace, it would take 8.33 days or just over a week to cover this distance.
#CalcMaps (2015): Map distance calculator
– Human sized humans loom over you, 4 times taller than the Empire state building, their steps falling from horizon to horizon. A bee the size of a helicopter lands near you, making the ground shake, as its hairy carapace vibrates with each wingbeat. You try to escape but are barely able to move because the air is so… gooey.
A regular sized human of 1.7 meters would appear to be 1700 meters tall in this 1000x perspective. That is 3.8 times taller than the Empire State Building.
#Encyclopedia Britannica (2020): Empire State Building
https://www.britannica.com/topic/Empire-State-Building
Quote: “At the time of its construction, there was fierce competition to win the title of tallest building in the world. The Chrysler Building claimed the title in 1929, and the Empire State Building seized it in 1931, its height being 1,250 feet (381 metres) courtesy of its iconic spire, which was originally intended to serve as a mooring station for airships. A 222-foot (68-metre) antenna was added in 1950, increasing the building’s total height to 1,472 feet (449 metres), but the height was reduced to 1,454 feet (443 metres) in 1985 when the antenna was replaced. ”
Honeybees are usually around 12 millimeters long. If we scale them up by a factor 1000, they would appear 12 meters long. This is comparable to the 12 meter length of the Bell 407 helicopter.
#Encyclopedia Britannica (2022): Honeybee
https://www.britannica.com/animal/honeybee
Quote: “A. mellifera is about 1.2 cm (about 0.5 inch) long, although size varies among the several strains of this species.”
#Flugzeuginfo.net (retrieved 2022): Bell Helicopter Bell 407
– Before you clicked the button air resistance was barely noticeable – but as you are a thousand times smaller, it is as if the air has become a thousand times denser. It feels like you are moving through honey.
In fluid mechanics, the effect of scale is codified in the Reynolds number (R). For an object of linear dimensions L moving through a fluid at a relative speed v, the Reynolds number is given by
R = Lv / K
where K is the “kinematic viscosity” of the fluid. The Reynolds number is a dimensionless quantity that measures the relative importance of inertial forces (“ability to keep on moving if nothing changes”) to viscous forces (“friction”) inside the fluid.
If R is bigger than 1, inertial forces dominate: given an initial velocity, you’ll keep on moving in the fluid for a while. But if R is much smaller than 1, “[i]nertia plays no role whatsoever. If you are at very low Reynolds number, what you are doing at the moment is entirely determined by the forces that are exerted on you at that moment, and by nothing in the past!”, as explained long ago in this classic paper of E. M. Purcell:
#Purcell, E. M. (1977): Life at low Reynolds number. American Journal of Physics, Vol. 43 (3)
http://www.damtp.cam.ac.uk/user/gold/pdfs/purcell.pdf
Let’s consider a human (characteristic length scale L1 = 1m) standing in the park under a light breeze at v = 1 m/s. The kinematic viscosity of the air at 20 ºC is KA = 1.5·10–5 m2/s, as can be obtained from e.g. here:
#The Engineering Toolbox (retrieved 2022): Air - Dynamic and Kinematic Viscosity
https://www.engineeringtoolbox.com/air-absolute-kinematic-viscosity-d_601.html
Therefore we get a Reynolds number of about
R1 ≈ 7·104
i.e. much bigger than 1. Inertial forces dominate and standing in the breeze doesn’t require much effort from our human, as we all know.
But things change after becoming 1000 times smaller. Your characteristic length scale is now L2 = 10–3 m; therefore, if you are to stand under the same breeze of v = 1 m/s, your Raynolds number will be 1000 times smaller:
R2 ≈ 70
Inertial forces still dominate, but the relative importance has become 1000 times smaller.
Under which conditions would a normal human (L1 = 1m) experience a Reynolds number of 70? If we consider the same breeze, we would need an “air” 1000 times more viscous. What fluid has a kinematic viscosity 1000 times higher than air? The answer turns out to be (cold) honey:
#Nayik, G. A. et al. (2015): Physico-chemical, rheological and sugar profile of different honeys from Kashmir Valley of India. Arabian Journal of Chemistry, Vol. 12 (8)
https://www.researchgate.net/publication/281280559_Physico-chemical_rheological_and_sugar_profile_of_different_honeys_from_Kashmir_Valley_of_India
Values listed above refer to the dynamic viscosity D of different honeys. The dynamic viscosity is related to the kinematic viscosity by the formula K = D/ρ, where ρ is the density of the fluid.
Honey is denser than water. Taking ρ ≈ 1500 kg/m3 as a reference value for honey (any value of the order of 103 kg/m3 will do), we see that for a wide range of values above we get kinematic viscosities of the order of
KH ~ 10–2 m2/s
i.e. 1000 times bigger than the viscosity of air. Therefore, our normal-sized human in honey will experience a Reynolds number similar to that of our millimetre-sized human in air.
– Flying Insects like bees use this to their advantage. Their wings are not made for gliding but like paddles that row through the air. Scaled up to the human size, the bee would outrun a Concorde Jet – except it couldn’t even take off because it would be too heavy for its wings.
Bees' flight is fascinating. They beat their wings 230 times a second. They also rotate in a way to create vortices in the airflow to generate much more lift than a simple flat wing.
#Engels, T. et al. (2018): Helical vortices generated by flapping wings of bumblebees. Fluid Dynamics Research, Vol. 50 (1)
Quote: " Airplane wings are smooth and use airfoil shapes designed to produce lift from an attached flow which is accelerated on the suction side. Flow separation (stall) limits the range of angle of attack in which these airfoils are useful. By contrast, insect wings feature sharp edges, essentially flat profile and large angles of attack. Under these conditions, flow separation is inevitable and large amounts of vorticity are generated at the leading edge. This vorticity forms a strong vortex which moves with the wing and detaches only at the stroke reversals. It has been suggested that insects can capture it at early times in the following half-stroke to provide an additional benefit [26]. Some insects clap their wings together and the subsequent opening motion creates a fluid jet which also provides additional forces. This mechanism is known as clap-fling-sweep [43] and it has recently been revisited [21]. Dragonflies and some other species can control their four wings independently
and have arranged them in a configuration that allows aerodynamic interaction between fore- and hindwing. This interaction depends on the phase difference in their kinematics and can contribute to force production as well"
#Taylor, G. J. et al. (2013): Vision and air flow combine to streamline flying honeybees. Scientific Reports, Vol. 3 (2614)
Supplementary material: https://static-content.springer.com/esm/art%3A10.1038%2Fsrep02614/MediaObjects/41598_2013_BFsrep02614_MOESM1_ESM.pdf
#Bomphrey, R. J. et al. (2010): Smoke visualization of free-flying bumblebees indicates independent leading-edge vortices on each wing pair. Experiments in Fluids, Vol. 46 (5)
Quote: " We find that bumblebees, in common with most other insects, exploit a leading-edge vortex. However, in contrast to most other insects studied to date, bumblebees shed both tip and root vortices, with no evidence for any flow structures linking left and right wings or their near-wakes"
A bee flies up to 28 km/h depending on whether it is loaded with nectar. That’s equivalent to 7.8 m/s.
#The British Beekeepers Association (retrieved 2022): How fast can honey bees fly?
https://www.bbka.org.uk/faqs/how-fast-can-honey-bees-fly
Quote: "The normal top speed of a worker would be about 15-20 mph (21-28 km/h), when flying to a food source, and about 12 mph (17 km/h), when returning laden down with nectar, pollen, propolis or water."
A 1.2 cm long bee that travels at 7.8 m/s is covering 650 body-lengths per second. A human attempting to do the same with their 1 meter scale stride length would need to travel at about 650 m/s or 2340 km/h. That’s faster than a Concorde!
#Encyclopedia Britannica (2022): Concorde
https://www.britannica.com/technology/Concorde
Quote: “The Concorde had a maximum cruising speed of 2,179 km (1,354 miles) per hour, or Mach 2.04 (more than twice the speed of sound), allowing the aircraft to reduce the flight time between London and New York to about three hours.”
– You’ve entered the microscopic realm and are now less than 2 micrometers tall, about the size of an e coli bacteria. From your new tiny perspective, the park you started in is now a million kilometers wide to you – if you walked non stop it would take some 25 years to cross it.
If you shrunk a regular sized human of 1.7 meters down by a factor 1,000,000 they would appear to be 1.7 micrometers tall. For comparison, an E.Coli bacterium is 2 micrometers long.
#Riley, M. (1999): Size Limits of Very Small Microorganisms: Proceedings of a Workshop. National Academies Press
https://www.ncbi.nlm.nih.gov/books/NBK224751/
Quote: "Escherichia coli is a typical gram-negative rod bacterium. Its dimensions are those of a cylinder 1.0-2.0 micrometers long, with radius about 0.5 micrometers."
A 1000 meter long park would seem to be 1,000,000 km long from a million-fold smaller perspective. At an average walking speed of 5 km/h, it would take 200,000 hours or 22.83 years to cover this distance.
– It is hard to grasp just how huge the microscopic world is to its tiny inhabitants. The giant bee that was close a moment ago, is now the size of Mt. Everest, towering high into the sky – but alive, humming and vibrating.
Meanwhile, the 1.2 cm long bee would now appear to be 12 km long and stand about 8 km tall. For comparison, the tallest mountain, Mt. Everest, is 8.85 km tall.
#Encyclopedia Britannica (2022): Mount Everest
https://www.britannica.com/place/Mount-Everest
Quote: “Reaching an elevation of 29,032 feet (8,849 metres), Mount Everest is the highest mountain in the world.”
– The air here feels even almost solid to you, on the human scale it would be as viscous as lava, extremely hard to push through.
We can repeat the same computations as before but for a scale of L3 = 10–6 m. At this size, the breeze of 1 m/s will imply that our micro-human will experience a Reynolds number of
R3 ≈ 7·10–2
At the human scale (L1 = 1 m), experiencing such a Renolds number would require moving in a fluid 1,000,000 times more viscous than air, or 10,000,000 times more viscous than water (the kinematic viscosity of water is KW = 10–6 m2/s). Such extreme values are typical of some magmas:
#Nelson, S. A. (2015): Volcanoes, Magma, and Volcanic Eruptions
http://www2.tulane.edu/~sanelson/Natural_Disasters/volcan&magma.htm
Quote: “Rhyolitic magmas tend to have even higher viscosity, ranging between 1 million and 100 million times more viscous than water.”
– The blade of grass now expands so far you can’t see its edges, stretching as wide as Paris would to a regular sized human. You see valleys that look like dried up riverbeds, dead patches like deserts and giant craters left behind by voracious aphids.
A 2.5 cm long blade of grass at this scale would appear to be 25 km. This is similar in size to the city of Paris, if we define its edges as the Paris Super Peripherique A86 road that encloses it in a 20 x 25 km ring.
Wikipedia (retrieved 2022): Boulevard Périphérique
https://en.wikipedia.org/wiki/Boulevard_P%C3%A9riph%C3%A9rique#/media/File:Les_rocades_de_Paris_2.png
You can measure its size using this map tool:
#CalcMaps (2015): Map distance calculator
https://www.calcmaps.com/map-distance/
To get an idea of what a blade of grass would look like, imagine zooming into the following image with a microscope:
#LSU College of Agriculture (retrieved 2022): Are chinch bugs sucking the life from your lawn?
The damage is often done by gass-eating aphids:
#Salisbury, S. E. & Anderson, N. P. (2022): Grass Seed Pests. In: Kaur, N. (ed.) Pacific Northwest Insect Management Handbook
https://pnwhandbooks.org/sites/pnwhandbooks/files/insect/chapterpdf/legume-grass-field-seed.pdf
Quote: “Aphids remove plant sap, secrete honeydew, and mechanically damage leaf tissue and developing seed heads. Light seed and reduced yields can occur, although not very often.”
– But if you look closely, this is not terrain. These are rows of individual cells, each the size of a house with hard exteriors like glass shells. Every few cells, there are huge openings called stomata, like mouths, sucking in air and blowing out oxygen.
#McKown, K. H. & Bergmann, D. C. (2018): Grass stomata. Current Biology, Vol. 28
https://www.cell.com/current-biology/pdf/S0960-9822(18)30710-3.pdf
Quote: "When plants began to colonize land over 400 million years ago, the change in scenery necessitated adaptations to withstand dry conditions. The nearly universal solution was to waterproof the aerial surfaces with a waxy cuticle and interrupt this barrier with stomata (literally ‘mouths’). At their simplest, stomata consist of epidermal guard cells flanking a pore and overlying an airspace in the internal photosynthetic tissue. Stomata are adjustable valves through which atmospheric gases like CO2 or, unfortunately, ozone enter the plant. Stomata also serve as exit points for water that has journeyed from the soil to the plant."
One unique feature of plant cells is their protective cell wall. This rigid structure grants strength to materials like wood.
#Wayne, R. O. & Whaley, W. G. (2019): Plant cell
– Suddenly the gigantic bee begins to move – a construct made of rigid pieces that slide against each other, like a suit of armor. It takes off to escape a drop of water the size of an Asteroid, that fell from another blade of grass and is now rushing at you at breathtaking speeds.
The bee’s organs are held inside an external carapace called an exoskeleton. Like other insects, it is made of chitin modified with sclerotin to become as hard as a suit of armor.
The abdomen, for example, it is made of 12 segments that slide against each other thanks to an ingenious evolved solution involving internally folded membranes. These are labelled AS1, AS2, AS3, AS4, AS5, AS6 alongside their belly counterparts (un-labelled) in the following diagram:
#Zhao, J. et al. (2016): Critical Structure for Telescopic Movement of Honey bee (Insecta: Apidae) Abdomen: Folded Intersegmental Membrane. Journal of Insect Science, Vol. 16(1): 79
A drop of water would be about 3.5 mm wide. From a 1,000,000x perspective, it would appear to be 3.5 km wide instead. For comparison, the asteroid that created the Chixculub crater and caused the dinosaur extinction was about 10 km wide.
– You brace for impact but instead of feeling a strong punch you just get sucked in. You try to swim but the water feels thick and sticky and holds onto your limbs like glue. Air's molecules are free spirits while water molecules act more like social creatures that group together whenever possible. They pull on each other and create a relatively strong cohesive force that traps you. You can’t help it but you are still moving, tumbling in all directions, helplessly dragged along by an invisible current.
The water molecule is polar, meaning it has a clear positively charged end and a negatively charged end. Like so many magnets, these molecules pull on each other to create a cohesive force. One of the results of this pull is a very strong surface tension, far stronger than that of liquid made of non-polar molecules like oil.
#USGS (2018): Adhesion and Cohesion of Water
https://www.usgs.gov/special-topics/water-science-school/science/adhesion-and-cohesion-water
Quote: “Water is highly cohesive—it is the highest of the non-metallic liquids. Water is sticky and clumps together into drops because of its cohesive properties, but chemistry and electricity are involved at a more detailed level to make this possible. More precisely, the positive and negative charges of the hydrogen and oxygen atoms that make up water molecules makes them attracted to each other. If you've played with bar magnets you will know that the north pole of one magnet will repel the north pole of another magnet, but it will attract the south pole of another magnet. Opposite magnetic poles attract one another much like positively charged atoms attract negatively charged atoms in water molecules.”
#Hauner, I. M. et al. (2017): The Dynamic Surface Tension of Water. The Journal of Physical Chemistry, Vol. 8 (7)
https://pubs.acs.org/doi/10.1021/acs.jpclett.7b00267
Quote: "The surface tension of any liquid is positive, which can be understood from the fact that on average a molecule on the surface has fewer neighbors and hence fewer attractive van der Waals interactions than a molecule in the bulk. It therefore costs energy to create new surfaces and indeed, reasonable estimates of the surface tensions of apolar liquids can be obtained in this way."
– Floating in this miniature lake are tens of thousands micro-organisms. They take on many forms – viruses the size of tennis balls float around you aimlessly, others like the bacteria Euglena oxyuris cells pass you like freight trains.
Untreated water outdoors is filled with microorganisms. 1 microLiter (μL) is the volume of a spherical drop of water with a diameter of 1.24 mm.
#de Carvalho, C. C. C. R. (2018): Marine Biofilms: A Successful Microbial Strategy With Economic Implications. Frontiers in Marine Science, Vol. 5
https://www.frontiersin.org/articles/10.3389/fmars.2018.00126/full
Quote: "However, it has also been shown that diatoms can attach directly to clean surfaces such as stainless steel and glass just after a few hours of immersion (Cooksey and Wigglesworth-Cooksey, 1995). Nevertheless, considering that 1 μL of surface seawater may contain ca. 10,000 viruses; 1,000 bacteria; 110 cyanobacteria; 10 eukaryotic algae and 10 protists (Azam and Malfatti, 2007), not only cell-surface but also cell-cell interactions should influence adhesion."
Viruses mostly vary in size from 20 nanometers to 400 nanometers. This would translate to 2 cm up to 40 cm from your 1,000,000x reduced perspective.
#Encyclopedia Britannica (2022): Virus
https://www.britannica.com/science/virus/Size-and-shape
Quote: “Most viruses vary in diameter from 20 nanometres (nm; 0.0000008 inch) to 250–400 nm; the largest, however, measure about 500 nm in diameter and are about 700–1,000 nm in length. Only the largest and most complex viruses can be seen under the light microscope at the highest resolution. Any determination of the size of a virus also must take into account its shape, since different classes of viruses have distinctive shapes.”
Microthrix parvicella bacteria can be 400 micrometers long. At this scale, that’s 400 meters long!
#Gu, Y: et al. (2018): A novel fluorescent long-chain fatty acid-substituted dye: labeling and biodegrading of Microthrix parvicella; RSC Advances, Vol. 8 (62)
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9088196/
Quote: “The distinct morphotype of filamentous bacteria species and M. parvicella in activated sludge were studied with Gram staining and FISH probes (Fig. 7). Fig. 7a shows filamentous bacteria identified by Gram-positive staining. They have unbranched long and curly filaments with diameters that are approximately of 0.7–0.9 μm and a length of 200–400 μm. This agrees with the distinct morphotype of M. parvicella.”
There are also many larger bacteria. Thiomargarita magnifica is 9000 micrometers long.
#Volland, J. M. et al. (2022): A centimeter-long bacterium with DNA contained in metabolically active, membrane-bound organelles; Science, Vol. 376 (6600)
https://www.science.org/doi/10.1126/science.abb3634
Quote: “Cells of most bacterial species are around 2 micrometers in length, with some of the largest specimens reaching 750 micrometers. Using fluorescence, x-ray, and electron microscopy in conjunction with genome sequencing, we characterized Candidatus (Ca.) Thiomargarita magnifica, a bacterium that has an average cell length greater than 9000 micrometers and is visible to the naked eye.”
– But most look like oily jellyfish the size of a car, sporting long tentacles that act as super charged propellers. Despite the water holding onto them like glue, some move hundreds of body lengths per second, equivalent to a person shovelling through mud at over 600 km/h.
#Water Quality Association (retrieved 2022): Bacteria & Virus Issues
https://www.wqa.org/learn-about-water/common-contaminants/bacteria-viruses
Quote: “Bacterial cells range from about 1 to 10 microns in length and from 0.2 to 1 micron in width. They exist almost everywhere on earth. Some bacteria are helpful to humans, while others are harmful.”
Flagella are the whip-like extensions that allow bacteria to overcome the challenge of moving quickly through water at the microscopic scale.
#Nedeljkovic, M. et al. (2021): Bacterial Flagellar Filament: A Supramolecular Multifunctional Nanostructure. International Journal of Molecular Sciences, Vol. 22 (14)
https://www.mdpi.com/1422-0067/22/14/7521/pdf
Quote: “Bacterial flagella are appendages on the cell body that provide motility. The number of flagella per cell varies depending on the species. Monotrichous species have only one flagellum at the pole (e.g., Pseudomonas aeruginosa); lophotrichous have multiple flagella at the same pole (e.g., Helicobacter pylori); amphitrichous have one flagella at each pole (e.g., Campylobacter jejuni); and peritrichous bacteria have multiple flagella in all directions (e.g., Escherichia coli, Salmonella enterica) [1]. Flagella consist of three morphologically distinct sections: a membrane-embedded basal body, a hook and a filament [2]. The basal body is a complex assembly that functions as a motor powered by a proton gradient; its rotation generates torque. This torque is transmitted through the hook to the filament that serves as a propeller, creating thrust and pushing bacteria through liquid environments [3].
Counterclockwise rotation of the motor provides smooth forward swimming, and in peritrichous bacteria filaments form a bundle on one pole of the cell [4]. Clockwise rotation of some of the flagella results in unraveling of the bundle and specific tumbling movement. Filaments are much longer than the cell body and during smooth swimming they adopt a left-handed supercoiled corkscrew shape, while clockwise rotation changes the handedness”
Bacteria are extraordinarily fast considering how much resistance they face. They can cover distances a hundred times greater than their own body length per second. If a 1.7m tall human did the same, they would reach speeds of 170 m/s or 612 km/h.
#Luchsinger, R. H. et al. (1999): Bacterial Swimming Strategies and Turbulence. Biophysical Journal, Vol. 77 (5)
https://www.sciencedirect.com/science/article/pii/S000634959977075X
Quote: “While most marine bacteria are capable of motility, it is used only intermittently. Their swimming speed can reach more than 100 body lengths per second (Mitchell et al., 1996), suggesting that motility is important for some of the environmental niches that marine bacteria occupy.”
– However bacteria weigh so little and water is so viscous that they have basically no inertia – there is no gliding on this scale. The result is a weird jerky motion that’s hard to keep track of. Maybe we can learn more about this strange motion if we go even deeper.
The performance of these little flagella is very impressive considering what they have to move through:
#Purcell, E. M. (1977): Life at Low Reynolds Number. American Journal of Physics, Vol. 45 (3)
http://www.damtp.cam.ac.uk/user/gold/pdfs/purcell.pdf
Quote: “In water where the kinematic viscosity is 10^-2 cm/sec these things move around with a typical speed of 30 micron/sec. If I have to push that animal to move it, and suddenly I stop pushing, how far will it coast before it slows down? The answer is, about 0.1 angstrom. And it takes it about 0.6 microsec to slow down. I think this makes it clear what low Reynolds number means. Inertial plays no role whatsoever. If you are at very low Reynolds number, what you are doing at the moment is entirely determined by the forces that are exerted on you at that moment, and by nothing in the past.”
#Yates, G. T. (1986): How Microorganisms Move through Water: The hydrodynamics of ciliary and flagellar propulsion reveal how microorganisms overcome the extreme effect of the viscosity of water. American Scientist, Vol. 74 (4)
https://www.me.psu.edu/cimbala/me521/Yates_microorganisms_article_27854249.pdf
Quote: “The typical viscous forces which act on a human swimmer are at least 6 orders of magnitude smaller than the inertial forces. To appreciate the nature of the viscous forces experienced by microorganisms, we would have to swim in a fluid one million times more viscous than water. Even in a pool filled with honey the relative strength of viscous to inertial
forces would be two orders of magni tude smaller than it is for microorganisms swimming in water, and a more appropriate fluid would be molasses or even tar.
Whenever the Reynolds number is small, the rate of change in momentum over time can be neglected relative to the viscous and pressure components, and time becomes simply a parameter. This means that the fluid motion generated in response to a microorganism's movement depends on its instantaneous velocity, and not on its velocity at any previous instant or on the rate of change of any other quantity with time. Furthermore, because the fluid inertia (mass times acceleration) is negligibly small, any external force must be balanced by the viscous stress and the pressure in the fluid.”
– You’ve become the size of a molecule, just under two nanometers wide. At your new tiny scale, the droplet now seems as big as the Moon to a regular human. The blade of grass it rests on could reach from the tip of Alaska to the end of Australia, and the park is now the size of the Solar System – but instead of mostly empty space, it is filled with stuff.
At this scale, you are 1,000,000,000 times smaller than your original size. If you were 1.7 meters tall before, you are 1.7 nanometers tall now.
Scale of some common biological molecules
#Milo, R. & Phillips, R. (2015): How big are biochemical nuts and bolts? In: Cell Biology by The Numbers
https://www.dropbox.com/s/gvpleqtcv8scro4/cellBiologyByTheNumbersJuly2015.pdf?dl=1
If a droplet of water was a mere 3.5 mm wide, it would look like a 3500 km wide sphere to your nanometer-sized self, similar to the size of the actual moon.
A blade of grass that is 2.5 cm long would appear to be 25,000 km long, which is the straight line distance from the northern tip of Alaska to the southern tip of Australia.
#CalcMaps (2015): Map distance calculator
A 1 km wide park scaled up by a factor of a billion would appear to be 1 billion km wide. That is equivalent to 6.7 times the distance from the Earth to the Sun. This park could have one edge against the Sun and the other extending past Jupiter, which orbits at 778 million km from the Sun.
#NASA (2021): Planetary Fact Sheet
– Everywhere you look, there are uncountable numbers of molecules and atoms. The rigid walls of the grass cells beneath you are clearly vibrating, rippling with waves of energy. The water droplet contains nearly sextillion water molecules that are all in motion.
A 3.5 mm wide droplet would have a volume of 22.45 mm^3.
#CueMaths (retrieved 2022): Volume of Sphere
https://www.cuemath.com/measurement/volume-of-sphere/
Quote: “If the radius of the sphere formed is r and the volume of the sphere is V. Then, the volume of the sphere is given by:
Volume of Sphere, V = (4/3)πr3”
At a density of 1 g/cm^3 or 0.001 g/mm^3, we have a droplet mass of 0.02245 grams. We can use this mass to find out how many molecules of water it contains.
The molar mass (how much one mole of water masses) of water is 18 grams per mole. There are therefore (0.02245 / 18) = 0.0012472 moles of water in this droplet.
One mole is 6.022 x 10^23 particles, so we have in here roughly 0.0012472 x 6.022 x 10^23 = 0.751 x 10^21 or 0.75 sextillion water molecules.
Molar mass:
#WebQC (retrieved 2022): Molar Mass, Molecular Weight and Elemental Composition Calculator
https://www.webqc.org/molecular-weight-of-water.html
#Encyclopaedia Britannica (2020): Mole
https://www.britannica.com/science/mole-chemistry
Quote: “The mole designates an extremely large number of units, 6.02214076 x 10^23.”
– Water is actually a storm of H2O molecules smashing into each other hundreds of trillions of times a second. Each of them is moving at speeds of around 2300 km/h and bombard their surroundings mercilessly, sending small objects hurtling in all directions. This is the source of the invisible current that you noticed when you were a thousand times larger
You can estimate the velocity of the molecules in water of a certain temperature using this equation:
Molecule Velocity^2 = 1380 x (273 + Temperature)
The velocity is in meters per second and the temperature in degrees Celsius.
At 22°C, the result is 638 m/s. That’s 2297 km/h!
#NASA (retrieved 2022): Exploring Energy and Temperature
https://spacemath.gsfc.nasa.gov/planets/91Mod11Prob1.pdf
Quote: “A simple formula gives us the average speed, V, of water molecules in meters
per second (m/s) for a given temperature in degrees Celsius, T:
V^2 = 1380(273+T)”
Because there are so many molecules so close to each other and travelling at high speed, there is a tremendous number of collisions between them every second.
# Feynman, P. R. et al. (2011): The random walk. In: The Feynman Lectures on Physics I
https://www.feynmanlectures.caltech.edu/I_41.html#Ch41-S4
Quote: “The reader may easily verify that the number of collisions a single molecule of water receives in a second is about 10^14, so in a hundredth of a second it has 10^12 collisions, which is a lot!”
Each collision has a chance to create or destroy a temporary bond between water molecules.
#Chaplin, M. (2007): Water’s Hydrogen Bond Strength. London South Bank University
https://arxiv.org/ftp/arxiv/papers/0706/0706.1355.pdf
Quote: “Hydrogen bond lifetimes are 1 - 20 ps, whereas broken bond lifetimes are about 0.1 ps (Keutsch and Saykally, 2001). Broken bonds generally reform to give same hydrogen bond; particularly if water's other three hydrogen bonds are in place. If not, breakage usually leads to rotation around one of the remaining hydrogen bonds (Bratos et al., 2004) and not to translation away, as the resultant 'free' hydroxyl group and ‘lone pair’ are both quite reactive.
– Scaling this speed up to the human scale is impossible, as a human sized molecule would be 2000 times faster than the speed of light.
If we scale up the movement of molecules by a factor of 1 billion, then the 638 m/s of water molecules would appear to be 638,000,000,000 m/s.
The speed of light is about 300,000,000 m/s. So this scaled-up molecule would be zipping around at a physics-breaking 2126 times the speed of light.
– All this furious motion comes from heat. Heat is a bit abstract at our human scale, where you touch something and get a vague sense of whether it is hot or cold. But down here, you really feel what ‘heat’ is: the motion of molecules, vibrating, twisting and colliding like inside a furious ballpit. When these molecules lose heat, they move more slowly and collide less often. When they gain heat, they speed up and smash together with renewed fervour. Temperature is basically the measure of the average speed of the fantastic dancers performing all day.
This description of the nature of heat is based on Kinetic Molecular Theory.
#Purdue University (retrieved 2022): The Kinetic Molecular Theory
https://chemed.chem.purdue.edu/genchem/topicreview/bp/ch4/kinetic4.html
Quote: “The kinetic molecular theory can be used to explain the results Graham obtained when he studied the diffusion and effusion of gases. The key to this explanation is the last postulate of the kinetic theory, which assumes that the temperature of a system is proportional to the average kinetic energy of its particles and nothing else. In other words, the temperature of a system increases if and only if there is an increase in the average kinetic energy of its particles.
Two gases, such as H2 and O2, at the same temperature, therefore must have the same average kinetic energy.”
#Elliott, J. R. & Lira, C. T. (2012): The Molecular Nature of Energy, Temperature, and Pressure. In: Introductory Chemical Engineering Thermodynamics
https://www.informit.com/articles/article.aspx?p=1845240&seqNum=2
Quote: “Energy is a term that applies to many aspects of a system. Its formal definition is in terms of the capability to perform work. We will not quantify the potential for work until the next chapter, but you should have some concept of work from your course in introductory physics. Energy may take the form of kinetic energy or potential energy, and it may refer to energy of a macroscopic or a molecular scale.
Energy is the sum total of all capacity for doing work that is associated with matter: kinetic, potential, submolecular (i.e., molecular rearrangements by reaction), or subatomic (e.g., ionization, fission).
Kinetic energy is the energy associated with motion of a system. Motion can be classified as translational, rotational, or vibrational.
Temperature is related to the "hotness" of a substance, but is fundamentally related to the kinetic energy of the constituitive atoms.
Potential energy is the energy associated with a system due to its position in a force field.”
– On average a molecule in the air travels for about 60 nanometers, which is about the length of a hockey rink if it were the size of a human. If we were to compress all the molecules and atoms buzzing around in the room you are watching this in, they would only fill about 0.1% of its volume. 99.9% of the space around you is a vacuum, you just don’t notice it. Which also means that every time you take a breath, you breathe in mostly nothing with a few atoms.
Air is empty enough that the average nitrogen N2 or oxygen O2 molecule can travel 60 nanometers before hitting something. At this scale, it is like travelling through 60 meters of empty space, which is the length of a standart ice rink in North America (200 feet).
#NFL.com (2009): Not every 200 foot by 85 foot NHL rink is the same
https://www.nhl.com/news/not-every-200-foot-by-85-foot-nhl-rink-is-the-same/c-501626
Quote: "The standard ice rink in North America measures 200 feet long by 85 feet wide. And every NHL game in North America is played on a standard-size rink, which means conditions in every building should be identical."
#Encyclopaedia Britannica (2007): mean free path
https://www.britannica.com/science/mean-free-path
Quote: “The constant μ is the mean free path and is the average (mean) distance traveled by a molecule between collisions. The mean free path of an oxygen gas molecule under a pressure of 1 atmosphere at 0° C is about 6 × 10^-6 cm (2 × 10^-6 inch).”
#J. H. Williams (2015): Materialism: What is there between atoms and molecules? In: Order from Force: A Natural History of the Vacuum
The actual volume occupied by the air molecules depends on the exact size that we take for the nitrogen molecule. Other estimates for the effective size of the molecule, like the ones given here:
#Vallejos-Burgos, F. et al. (2018): Air separation with graphene mediated by nanowindow-rim concerted motion. Nature Communications, Vol. 9 (1812)
https://www.nature.com/articles/s41467-018-04224-6
yield a larger proportion, so we’ve chosen 0.1% as a representative number for the relative volume occupied by the air molecules of a room.
– At your size of under 2 picometers, scale starts to lose its meaning. A human would be nearly 2 billion kilometers tall relative to you, so large they could stretch their arms from the Sun to Saturn.
If you shrunk a regular sized human of 1.7 meters down by a factor 1,000,000,000,000 they would appear to be 1.7 picometers tall. From this perspective, a regular sized human is 1.7 billion kilometers tall. They’d easily be able to stretch their arms from the Sun to Saturn, across a 1.43 billion km distance.
#NASA (2021): Planetary Fact Sheet
– An atomic nucleus would be the size of a grain of sand you could hold on the tip of your finger. That grain holds 99.97% of the atom’s mass.
An atomic nucleus is an incredibly dense dot at the center of an atom. Its radius can be estimated using the following equation:
Radius = 1.2 x 10^-15 x A^(1/3)
The radius is in meters. A is the Atomic Number of the atomic. For carbon it is 12.
A carbon atom has a nucleus with a radius of 1.2 x 10^-15 x 12^(1/3) = 2.7 x 10^-15 meters
#Tsui, O. K. C. (2016): Class 40: Nuclear Radiation. Department of Physics Lecture Notes. Boston University
http://physics.bu.edu/~okctsui/PY106_lecture_notes/Class40_nuclear%20radiation.pdf
Quote: “We know that atoms are a few angstroms, but most of the atom is empty space. The nucleus is much smaller than the atom, and is typically a few femtometers. 1 fm = 1 x 10^-15 m.
The nucleus can be thought of as a bunch of balls (the protons and neutrons) packed into a sphere, with the radius of the sphere being approximately:
r = 1.2 x 10-^15 A^(1/3)
R radius in meters, A atomic number”
At this 1,000,000,000,000x scale, 2.7 x 10^-15 meters would appear to you as 2.7 millimeters.
That carbon atom is made up of 6 protons, 6 neutrons and 6 electrons. The protons and neutrons are held inside the nucleus while the electrons occupy up the rest of the atom.
A neutron is about 0.13% heavier than a proton, and an electron is 1836 times lighter than a proton. If we set the proton mass to equal 1 unit, then we get:
Ratio of nucleus mass to atom mass = (Proton + Neutron) / (Proton + Neutron + Electron) = (6x1 + 6x1) / (6x1 + 6x1 + 6 x 1/1836) = 0.99972
That means 99.972% of the mass is contained in the nucleus.
#NIST (retrieved 2022): Neutron-Proton Mass Ratio
https://physics.nist.gov/cgi-bin/cuu/Value?mnsmp
Quote: “ Numerical value 1.001 378 419 31”
#Max Planck Institute for Nuclear Physics (2014): Electron on the scale
https://www.mpg.de/7961020/electron-mass
Quote: “The result is a fantastically precise number revealing that an electron has a mass 1/1836.15267377 that of a proton. Stated in kilograms, the electron's mass is around an unimaginable 10-30 kilogram, or thirty zeros after the decimal point. Although the electron truly is a lightweight, it plays a heavyweight role in nature.”
– The rest, a sphere of influence about as large as the Eiffel tower from your perspective, is filled with an electron cloud. That’s basically all the places where electrons might be at any given moment in time. Electrons are shapeshifters that morph around outside a nucleus, creating a new and vibrating mess of different shapes with every new moment.
The atomic radius can be defined in many ways.
The van der Waals radius is a sort of ‘exclusion zone’ that other atoms cannot enter (under normal conditions).
A carbon atom has a van der Waals radius of 1.7 x 10^-10 meters. That forms a sphere with a diameter of 340 picometers.
#Yale University (retrieved 2022): Van der Waal's Radii
http://ursula.chem.yale.edu/~chem220/chem220js/STUDYAIDS/vanderwaalsradius.html
#Cooley, A. (2020): Covalent Bond Distance, Radius and van der Waals Radius
Quote: “Covalent bond distance, covalent radius, and van der Waals radius are used to describe the size and distance between atoms. Covalent bond distance refers to the distance between the nuclei of two bonded atoms. Covalent radius is half of the internuclear separation between the nuclei of two single-bonded atoms of the same species (homonuclear). While van der Waals radius is used to define half of the distance between the closest approach of two non-bonded atoms of a given element.”
– Unlike the graceful motion of planets, the atomic nuclei are chaotic blurs. They bulge, roll, quiver and breathe. They hold back the same energy that powers nuclear bombs and doesn’t let them sit still. Even when they are stuck together they manage to vibrate sextillion of times a second.
The nucleus of an atomic is often described as a drop of ‘quantum liquid’. Like a liquid, it can bend and change shape under the influence of external or internal forces, and wobble when it ejects a part of itself or gets struck by something. These wobbles are extremely fast. Their frequency is on the order of 10^21 Hertz, which is 1000 billion billion vibrations per second.
#CNRS (2012): he atomic nucleus: fissile liquid or molecule of life?
https://phys.org/news/2012-07-atomic-nucleus-fissile-liquid-molecule.html
Quote: “The atomic nucleus is generally described as a drop of quantum liquid with a diameter of around a million billionth of a meter. In particular, such liquid-like behavior explains nuclear fission, and applies especially to heavy nuclei, i.e. nuclei that contain a large number of nucleons (neutrons and protons)”
#Bertsch, G. F. (1983): Vibrations of the Atomic Nucleus. Scientific American, Vol. 248 (5)
https://archive.int.washington.edu/users/bertsch/general_interest/scientific_american_1983.pdf
Quote: “The nucleus can quiver, ring or even "breathe"”.
Quote: “If a nuclear vibration is to be excited, the first and simplest condition to be met is that the energy imparted to the nucleus must be equal to the energy associated with the vibration. The nucleus vibrates at extremely high frequency; the energy of the vibration is equal to the frequency multiplied by Planck’s constant, and so the energy is also comparatively high. A typical vibrational frequency is 5 x 10^21 hertz, which corresponds to a vibrational energy of 20 million electron volts (MeV)”.
– We have reached the bottom, the border between reality and unreality. The scale here is the Planck length, which is the distance light travels in a Planck Time. Planck time is the time it takes light to travel a Planck length. Ah ok. None of our models of the universe make sense at scales smaller like this, so for now, this is it.
The Planck length is defined as the only quantity with units of length that one can get by combining the fundamental constants of nature c (the speed of light), ℏ (Planck's constant) and G (gravitational constant). Such a combination is:
LP = sqrt (ℏG/c3) ≈ 1.6·10–35 m
#Swinburne University of Technology (retrieved 2022): Planck Length
https://astronomy.swin.edu.au/cosmos/p/Planck+Length
Quote: “The Planck length is the fundamental unit of length in the system of Planck units. It has the value:
lP = 1.62 x 10^-35 m”.
In a similar manner one can define the Planck time: i.e. the only quantity with units of time that one can get by combining the fundamental constants of nature. This is given by
TP = sqrt (ℏG/c5) ≈ 2.2·10–44 s
Since LP = cTP, both are related by the following property: the Planck length is the distance that light travels during a Planck time
– We think that down here, particles bubble into existence and then spontaneously disappear, creating a quantum foam of unimaginable energy. Can we go even smaller? We don’t know. It is time to return.
The Planck scale is at the limit of our understanding. We don’t really have an idea of what it looks like, or how things work at this ultra-small scale.
Physicists think that, to describe the universe at these or smaller distances, one needs a quantum theory of gravity. Since until now no such theory is known, our current theories of fundamental physics stop working at these distances.
#Symmetry Magazine (2016): The Planck scale
https://www.symmetrymagazine.org/article/the-planck-scale
Quote: “The Planck scale is the universal limit, beyond which the currently known laws of physics break. In order to comprehend anything beyond it, we need new, unbreakable physics.”
The nature of space and time at the Planck scale should be related to the structure of the vacuum, i.e. with the structure of empty space itself. According to quantum physics, the vacuum should have some intrinsic energy. When this energy is computed using the best fundamental theories that we have, one gets an enormous amount.
#Carroll, S. M. (1999): The Cosmological Constant. Living Reviews in Relativity, Vol. 4 (1)
https://arxiv.org/pdf/astro-ph/0004075.pdf
Quote: “If we are confident that we can use ordinary quantum field theory all the way up to the Planck scale [...], we expect a contribution of order ρPl Λ ∼ (1018 GeV)4 ∼ 2 × 10110 erg/cm3.”
This value is so different from what we get from experimental observations that it has been called the biggest mismatch ever between theory and reality in physics. It’s the origin of the cosmological constant problem.
#Carroll, S. M. (1999): The Cosmological Constant. Living Reviews in Relativity, Vol. 4 (1)
https://arxiv.org/pdf/astro-ph/0004075.pdf
Quote: “As we will discuss later, cosmological observations imply |ρ(obs)Λ| ≤ (10−12 GeV)4 ∼ 2 × 10−10 erg/cm3 “