Pay attention to your breath. Breathe in through your nose…

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And breathe out through your mouth…

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I hope you feel a little bit more calm. All humans breathe to supply oxygen to our demanding bodies, a process necessary to sustain the chemical miracle called life. Across the world, breaths are taken every second of every minute of every day, constant exchanges of oxygen and carbon dioxide with the atmosphere around us. Invisible yet omnipresent, breaths haunt us. Always there, never seen.

Consider the moments during which these breaths are taken. Last breaths. First breaths. Two breaths, perhaps in sync, before a kiss at a wedding. A breath before conquering… or bombing… a job interview. A deep breath over food simmering on the stove, followed by a wide smile and an “mmm…” A broad-chested breath in, followed by an even bigger exhale upon entering home after a stressful work week. Breaths, simply because they’re so damn necessary, are a simple commonality of the human experience. For wherever people go, and whatever people do, breaths follow. They’re corporeal ghosts of the human condition just floating about the atmosphere.

A pause for a brief math problem: as a brief exercise to help conceptualize the true magnitude of the global miracle of breathing, it’s fairly straightforward to approximate the number of breaths taken per minute by all human beings. The average breathing rate is roughly 12 to 18 breaths per minute for an average adult. Averaging that range to 16 breaths a minute, and approximating that there are 8 billion people on earth in the year 2023, we get roughly 128 billion human breaths taken every minute across the globe. With 525,600 minutes in a calendar year, that’s roughly 67 quadrillion breaths per year, a number several orders of magnitude greater than the number of galaxies there are in the observable universe.

The question of how many breaths are taken around the globe per minute is an example of a “Fermi problem.” Such problems are named after Manhattan Project nuclear physicist Enrico Fermi, who (questionable ethical contributions to science and society aside) was supposedly quite good at approximating answers to questions that may have been otherwise unanswerable. Fermi questions have been used in everything from student science competitions to the formal approximation of the number of intelligent civilizations in the Milky Way galaxy. Fermi problems use simple data, such as a rough estimate of the human population and the average adult breath rate, to approximate answers to exceedingly complex questions often dealing with extremes in magnitude. Hence, they are also known as ‘order-of-magnitude problems,’ which emphasizes the logic’s ability to get within an order of magnitude of the true answer.

Now, think again about the solution of 128 billion breaths taken per minute across the globe. While it may have seemed precise enough, the solution is, at best, an approximation. For one, the assumption that we can “average” the standard range of breaths per minute for an adult could be faulty. An average works best assuming a non-skewed distribution within the range. Then consider that children may have a different breathing rate, people might breathe more slowly when breathing with assistance, and breathing rate in general is affected by environment and physical activity. Even the number 8 billion is an approximation used to simplify the calculation. The true number of people is slightly more than 8 billion and is constantly in flux, with an average of over one death and four births per second. 

Another such example of a Fermi problem is very intimately related to breaths. In fact, it inspired a whole book on the human-cultural ties to the scientific and human history of the atmosphere, and the problem and book share a title. The book is Caesar's Last Breath: The Epic Story of the Air Around Us, written by scientific essayist Sam Kean and published in 2017. The problem is simply known as “Caesar’s Last Breath,” and it goes as follows:

 On the Ides of March of 4.4. B.C.E., 60 conspirators plotted to assassinate Julius Caesar in full view of the senate floor. First, senator Tillius Cimber was to distract Caesar by handing him a petition. The other conspiratorial senators would then cluster around the pair to voice their opinions, at which point they would attack with their daggers.

When Cimber acted on the plan, although Caesar first tried to cast Cimber away, Cimber was able to tear down Caesar’s toga to his lower back. This gave time for Publius Servilius Casca, the plebeian tribune of Rome, to strike the first dagger in the revealed side of Caesar’s shoulder. Soon, and according to plan, Caesar was surrounded by attacking senators, striking with daggers from all sides. In the end, Julius Caesar suffered 23 stab wounds. It is likely that he died from simply bleeding out on the senate floor. But just before he did, he emitted one last breath. Just what became of it?

An average human breath occupies roughly a half liter in volume, a tiny blip in the atmosphere emitted from Caesar’s body. We will compare the size of this meager breath to the grand size of the atmosphere. First, we must calculate the atmosphere's volume using estimates of its thickness along with the earth’s radius. Chemistry has an equation called the ideal gas law, which allows us to find the number of gas molecules in the atmosphere by relating that value to the atmosphere’s temperature, pressure, and volume. In line with Fermi’s technique of simplification, the ideal gas law is so foundational in the field of chemistry that it is taught in many introductory classes, and we need nothing more complex to solve the problem. 

We can perform the same calculation for Caesar’s breath using approximations of the temperature and pressure during Caesar’s time of the Roman Republic. The final answer is the fraction of the number of molecules in Caesar’s breath versus the overall number of molecules in the atmosphere. We compare this to the half-liter human breath size. On average, the average human would inhale just over one molecule of air emitted from Julius Caesar’s lungs every time they breathe. 

The molecules of Caesar’s dying breath have dispersed around the globe. He haunts us through the specters of chemistry and physics.

Unfortunately, there are limitations to this line of thinking that complicate the simple answer of one molecule per breath. The true answer would require knowledge of Caesar’s breath capacity, the exact weather at the time when Caesar died, the exact capacity of every human breath, and more, meaning it is essentially impossible to calculate if not for the Fermi estimation.

Perhaps the biggest limitation, though, is our assumption that the atmosphere is a closed system. This assumption means that we can employ the Law of Conservation of Matter, meaning that the amount of matter in the atmosphere would be constant across time. The atmosphere, however, is a very open system. It is constantly changed by us humans as its tumultuous tunnels of wind disperse the molecules exchanged within our lungs across the globe. And we have been exchanging air with the atmosphere for over far longer than the two millennia since Caesar kicked the bucket on the senate floor. Each time the alveoli within a human lung convert oxygen to carbon dioxide, or photosynthesis takes place within a plant’s chloroplasts, or rocks react to rainwater, the molecules in the atmosphere are changed, exchanged, absorbed, and added to. The atmosphere has no truck with a meager law of matter conservation. The exact calculation of Caesar’s breath spreading throughout the atmosphere would require an intimate knowledge of the matter exchanges between it and every other earth subsystem. What you, I, or Julius Caesar breathed out could now be part of a rock on the opposite side of the planet.

Effectively, this all means that we are not breathing in the exact molecules that Caesar breathed out. But molecules can be broken down into atoms, and atoms can be broken down into smaller pieces, including protons and neutrons. And these protons and neutrons can be broken down even further into particles called quarks, assumed to be among the very most fundamental building blocks of life, unable to be broken down further. If we do not share the same molecules with Julius Caesar and the neighborhood rock, then we might share the same atoms. And if we do not share the same atoms, then we may share the same protons or neutrons. And if not protons or neutrons, then we must share the same quarks–which, in theory, have existed since the birth of the universe.

It is common to say we are stardust. This is literal. We are stardust, and we are each other’s breaths. 

If we are the stuff of stars, then we also change the stuff of stars. We warp it, breathe it, cause it to react and form new chemical compounds through our process of daily existence. We distribute it on a planetary scale. Although small on the grand stage of the universe, life begins to seem pretty powerful.

And life is cyclical. The best example is a forest: in a meadow, tree saplings might grow quickly, searching for light. Then, the trees will reach height and slowly die off, until only the trees that were able to compete for the most light remain. Finally, these old trees start to die—randomly, perhaps from disease or natural disaster, forming small patches of sunlight in the aged forest. And in that patch, the process begins again. Or maybe a wildfire comes through, and the process starts from the beginning, using the old forest as fuel for regeneration.

Caesar’s last breath is a legacy. Sure, he might be written in the history books, but his breath imbues him into the regeneration of the forest. His breath is his ashes, the forest his urn. And in the process of growing, living, dying, so are we. 

Physically, the dead sink into the ground, becoming part of the soil and hence fertilizer for the living. This body-fertilizer will remain stuck in the forest on human timescales. On the other hand, chemistry and physics imbue a breath into the globe’s pores. Caesar’s last breath hasn’t just been confined to one forest— it’s fertilized the world.

Of course, not only Caesar emits a last breath. Our bodies are not just urns for a dictator of the Roman Empire. We are urns for the breaths of our ancestors—parents and grandparents who came before us, dead pets, past leaders who fought for justice, even the problematic Fermi. We are constantly cloaked in the embrace of the breaths of the almighty. Just like the forest, our corner of the universe works in cycles: seasons, life and death, tides rising and falling with the moon’s gentle influence, fires and volcanoes and ice ages and extinctions, the post-extinction explosions of life that follow, the circulation of blood powered by the human heart, Earth’s rotation about its axis, and the cycles of water and carbon and nitrogen known as Earth’s biogeochemical cycles. Breaths bring our lives into the biogeochemical realm and put us in sync with the rhythm of the world.

The atmosphere is a cemetery with billions of empty graves. Feel a breeze? Caesar haunts you. We will become one with the grand tale of life, and with physics, our breaths will haunt the rest of time.