Growing Up in the Milky Way: The History of Science as Real Magic

Science is not the opposite of magic. It is magic that works — an incantation with the right ingredients that produces a predetermined outcome, reliably, by anyone who follows the recipe. The hand axe was the first spell. The gravitational wave detector is the latest. Between them lies the story of how humanity learned to see what the universe is actually made of.

This post traces that arc: from 3.3 million years of stone tools to the Cosmic Microwave Background, from Anaximander's rain cycle to Einstein's spacetime, from the naked eye and a clock to detectors sensitive enough to feel two black holes collide a billion light-years away.

Growing up in the Milky Way Galaxy (with apologies to Timothy Ferris)

Watching the stars was the way Humanity figured out where it was in the cosmos.

You can't tell if you don't look

Science is an ever growing body of knowledge that continues to expand exponentially. This points to a singularity sometime in the near future. We've built machines that can imitate speech (passing the Turing test^[!], very soon passing it reliably.) And it all started with a singularity in the past where Humanity learned to speak.

When did science start? Did it start in pre-history? Before people could write? Does science include the ability to make stone tools and slowly improve them so they are easier to use and more efficient? Of course. Science is the observation of objective reality and the ability to construct models with reproducible predictions. Take two rocks of a certain type and shape and bang them together this way and you get a hand axe. It’s magic! Not a miracle but real magic. An incantation with the right ingredients can use the recipe to get a predetermined outcome. Real Magic.

Granted, it’s not very complicated Magic. Two types of stone and one procedure to figure out where to knock them together. Any old monkey could do this. It probably doesn’t require any complicated language to make these simple hand tools. These tools are the earliest example of science that have survived to the present day. These tools evolve very slowly without changes for hundreds of thousands of years. The changes are glacial at this first attempt at technology 3.3 M yrs ago, possibly by Australopithecus, but certainly Homo Habilis, ancestors of Homo Erectus,

Mode I: Homo Habilis. The Oldowan Industry

The rate of evolution is slow. Almost as slow as the evolution by natural selection of our ancestors. And it proceeded by the same process, small random changes, like a game of telephone tag over the ages.

The ancestors that created the next type of tools were not our current species, Homo Sapiens, they were Homo Erectus and they spread all over the world from Africa. Their first stone tools are found about two and a half million years ago. And they slowly evolve for about a million years. We find and are able to date them all over the world. About a million years ago the rate of change of new tools starts to exponentially speed up. It's obvious at this point that we can talk. Not just name things, but actually converse. Acheulean tools, which are much more complicated, require complex planning and make butchering much simpler.

Mode II: Homo Erectus: The Acheulean Industry

Then the next few modes of stone tools come quickly. This is all still during pre-history, before we can write, but after we could talk.

Mode III: Homo Sapiens: The Mousterian Industry

Then we learn how to make clothes: needles, cloth and thread. This occurs about a hundred thousand years ago and allows Homo Sapiens to spread throughout new environments around the whole world.

Mode IV: The Aurignacian Industry

This seems like the beginning of science to me. The slow growth through practice and observation of the ability to understand and control the environment by using a constructed apparatus: a tool. People are still nomadic at this point. Homo Sapiens migrate to North America around 30,000 years ago and continue to evolve stone technology [Clovis points] separately from the rest of the world. It’s still a pre-literate society at this point, but the Sapiens in America learn to write a few thousands of years ago. Until then, everything is passed down by word of mouth. Game is plentiful. Counting coupe is useful. All is well.

The next innovation to occur, that changed how we defined ourselves as human beings, was the domestication of plants and animals: Farming, ranching. (Eventually Feudalism: domestication of people.) [Sometimes we forget what has changed and how much it has changed.] Owning and controlling land was now valuable and could be used to support many more people than before domestication. And defense is easier than offense. You need about three times the number of attackers to defenders to ensure victory. So large settlements became larger for their own survival.

Above the size of a few hundred people a small town could run without anyone in charge except one priest being literate enough to resolve inheritance issues, but a city needed to have records, taxes, tithes, laws, rights, ownership, responsibilities and duties. They needed a way to record these transactions, debts and payments, interest rates and money. Accounting was required to run a city. Accounting had to be reliable and reproducible, just like science. It’s a tracking of the promises made, kept and owed between human beings.

It’s marks on a slate. How much are your taxes this year? Ten bushels of wheat to be delivered on October 15th at the central Grainery. And if you didn’t deliver it, you got fined. You couldn’t deny your bill, it’s right there in front of you, a picture that tells you what you have to deliver. This one small thing makes large permanent cities possible. They start in the Middle East and spread this system around the world.

Not just accounting, but also literacy. A recording of the rules that govern a citizens rights and duties. Literacy allows the Real Magic to be written down and accumulated in books. What is this Magic? It’s the formal system of universal computation. How to write down the information needed to complete any recipe for a predicted outcome from baking a cake to building a car to designing a government to specifying a moral or legal code. There’s an incantation on how to do it right and fairly. Science is the same way, but science is limited to the investigations and observation of objective reality. There are actually three branches of science that follow the same methods:

1. The study of atoms (models of objective reality: Physics.)
2. The study of life (emergent behavior: Biology)
3. The study of formal systems (possible models: Mathematics.)

Accounting (a formal system) continues to evolve to become more sophisticated. Double-entry bookkeeping becomes the norm. Except for accounting, the ancient Greeks attribute everything to the gods. Zeus throws thunderbolts. Neptune makes the winds blow. It’s an easy explanation… except things happen in cycles. There are up to four distinct seasons everywhere. The moon cycles overhead every thirty days. There are eight other lights in the sky that move along the same general path. These wanderers (or planets [derived from the Greek]) follow their own cycles. And there were eclipses. And they suggested patterns. There are hints that Stonehenge and other historical buildings recognize the alignment of the stars and the seasons.

It was critical to know when to plant and when and how to fertilize and harvest. This has only gotten more sophisticated over the centuries. including further domestication of plants and animals through the direct modification of their genes. Farmers use satellite photos supplemented by ground measurements to determine pesticide, water and fertilizer dosage. The methods of Science play a huge part in this progressive drive to make our lives more efficient.

But science is more than that. The scientific method is a way to translate measurements into meaning. Or at least into predictions that allow you to argue about the meaning. The first recorded use of the “Accounting” method of explanation was in 600 BCE by Anaximander, the Macedonian, who proposed the rain cycle and what causes earthquakes.

He was basically correct in the first instance and deeply wrong (wind) on the second. But it was a model, a blameless model (not that one of their gods did it, which was their previous explanation.) It’s amazing that the first models were wrong half the time. Actually science is almost always wrong. It just gets more accurate as time moves on. Aristotle. Copernicus. Brahe. Kepler. Galileo. Newton. Laplace. Einstein.

This the beginning of Science with a capital 'S.' Financed by city-states, more for war than anything else. Increasing the “kill-radius” of weapons to the ridiculous point of destroying the entire planet! Finally in the 80s the realization of nuclear winter got everyone to stop practicing nuclear explosions in the open air. See, humans can be sensible sometimes!

What makes up the Scientific method? It’s a model of the ‘situation’ or formally a theory is a set of universal laws, recipes, or formulas that can be used to make predictions. The genius Newton showed us how to build a model from scratch. How to define the terms, define the experiments, the measurements and predictions and publish his formulas so that anyone can use them to reproduce his results and generate new results. And he showed that his predictions were more accurate than the old Greek system (which was pretty damn accurate.) The giant’s shoulders that Newton stood on included Kepler.

But the real giant was Copernicus. He took the model presented by the Greeks, that was highly accurate, by the way, (as demonstrated by the Antikythera) and proposed it was not the simplest way to look at the solar system. Humanity was not at the center of the universe, there were no spheres within spheres within spheres (although that was an ever more complicated but quite accurate model.) The simpler view was that the Earth was a planet like all the wanderers and they all circled the sun. And furthermore, the Earth was a top, always pointing in one direction. To first order it explained everything: The seasons, the Sun's orbit, the other planets' orbits.

The Earth 'circles' the Sun as a top

But it wasn’t as accurate as the old Greek model. Careful measurements showed significant deviations (Circles vs. ellipses, for instance.) Measurements made by Tyco Brahe were used by Kepler to propose his three laws of planetary motion. He examined the data and noticed systematic deviations that could be explained by assuming that the orbits were ellipses not circles. This is a replacement of the Greek model centered on the Earth. It was a solar centric model. Brahe had his own model where the sun went around the earth and all the other planets went around the sun. The interesting thing is they are all correct. The math works out wherever you pick the center to be. Some choice of coordinate systems are more illustrative than others. There’s just one objective reality and you can assume the universe rotates around any point in it, or not.

That’s not to say that scientists weren’t proponents for their own views. Tycho made his observations without the aid of any optics. He used accurate timing and long deep shafts to determine the angle to make accurate measurements. He may be the first scientist in the modern sense of the word. He was financed by the city-state, as was Kepler.

Kepler had the years of Brahe’s measurements to use and he had optical telescopes to verify and reproduce the measurements. Brahe’s measurements were accurate up to one arcminute. That’s about a thirtieth of the width of the moon or about the amount the moon moved in one minute. Kepler proposed three laws of planetary motion based on Brahe’s measurements of Mars.

1. The planets orbit the sun, which is at the focal point of the elliptical orbit.
2. Equal areas are swept out over equal times.
3. The orbital period squared is proportional to the semi-major axis cubed.


Kepler published his laws of planetary motion — the first two in *Astronomia Nova (1609) and the third in *Harmonices Mundi* (1619) — and they were immediately recognized as a triumph of observational astronomy. Newton reads the published laws and derives them using mathematics he invented for the purpose (calculus) but doesn’t bother to publish the discovery: he’s a poor graduate student and can’t afford to. Halley (of Halley’s Comet) hears this and offers to pay to publish the greatest scientific work of all time: "The Mathematical Principles of Natural Philosophy" that explains the first Scientific theory: Newton's Laws of Motion. He showed that all three of Kepler's laws could be deduced in his system from one fact: that bodies attract each other proportional to their mass and inversely proportional to the square of the distance between them.

This theory of gravity proposed by Newton is highly accurate, but not exact. All physics' theories are incorrect in some small form or manner. There were nagging questions about how much gravity affected starlight and how fast Mercury’s orbit precessed that told us it wasn’t exact. It took the genius of Einstein to build a more accurate model that looked like Newton’s law in all known physical situations and in other weird situations (high speeds, large masses) it looked different. It explained all the anomalies and predicted a bunch more things tangential to gravity (spacetime.) Einstein’s theory of gravity has been tested to an amazing degree and it has never been found to be wanting.

As in the most respected physics' theories, it’s eminently falsifiable. And like the best theories, it has passed all of its tests. Everywhere we look we see Einstein’s theory playing out in the cosmos. Before Einstein’s time the naked eye and an accurate clock were used to make astronomical measurements. All you can see are planets and stars, maybe some meteoroids or comets, just photons and nothing else.

Astronomy has gone far beyond just photons. First we noticed that there are things beyond stars. Other galaxies like our Milky Way Galaxy, full of stars, and that they were traveling away from us: the further away, the faster they were going. The universe was at least a million times, if not a billion times bigger than we had ever imagined. Here’s the galaxies the Hubble Space Telescope sees in an area the size of a dime held at arms length.

Galaxies seen by the Hubble Space telescope in a patch of sky the size of a dime at arm's length.

For the Fermi paradox corollary — given all of this, where is everyone? — see: Where Are They? The Aliens, I Mean. (2016).

And if you extrapolate back in time, at one point all the galaxies are on top of each other in one ‘Big Bang’, as the theory's biggest opponent described it, attempting to ridicule the idea (Fred Hoyle, a distinguished astronomer, scientist and popularizer of the day.) We measure how far away these galaxies are and how fast they are traveling and find the distance is proportional to the velocity. The only way to explain this is to assume the Big Bang actually happened.

But it also tells us to look for some other things: this Big Bang was first just a huge cloud of very hot and dense energy. Eventually, it expanded so that it was transparent to light, about 300,000 years after it started, and we can see that signal everywhere in the sky. 

Cosmic Microwave background, differences from a perfect black body.

We see a black body radiation curve that has very little variation from the ideal measurements. Surprisingly little difference from the prediction. This little difference needs an explanation and the explanation is called "Inflation." Alan Guth (a physicist at Cornell, Stanford and now at MIT) proposed this in 1979. (The year I graduated from Stanford with a BS in Physics. It was a bigger story than the Nobel prize my teacher had just won for discovering the J/Psi particle.) When the universe was very small to have such small temperature differences in different parts of the sky is unexpected, unless all the particles were in communication with each other during the process.

Deviations from the perfect black body radiation curve.

Inflation makes this happen. When the universe is one-hundred billion times as small as a proton (some 10^(-26) meters, which is smaller than anything we can probe these days, light can travel across the entire universe in less than a picosecond. To make sense of the smooth observations, the Universe must have inflated to about a meter in 10^(-35) seconds. Much faster than the speed of light. The idea is that the Universe spills out of contact for a brief instance as it expands, cooling to 10^27 to 10^22 degrees Kelvin (extremely hot!) And there must be some mechanism to reheat the Universe with radiation. The current favored idea is that the gravitational energy is converted to particles and reheats the plasma as it continues to expand (now at sub-light speeds.)

It's an amazing story. Just by looking in great detail at the world around us, we've basically locked down a reasonable theory of the origin of the Universe. There's still many details to be discussed, but any new model will have a hard time replacing the "Big Bang" with an explanation that meets all the facts we observe. 

And how has astronomy changed since it started? First, it was recorded by the human eye and a clock. You can't make sense out of the heavens without a clock. Next, the invention of the telescope, which greatly expanded the accuracy of observations, really pushed the envelope and allowed us to finally see the other galaxies out there. And we've been continuing to get better at looking at all those photons over the years. As my physics thesis supervisor (Vernon Hughes) used to say back in the 1980s, "Astronomers only look at photons." That was true then, but has been blasted to smithereens by the advances in particle detections and gravity waves^[2]. We keep looking at the stars and finding out what they are sending us. It's estimated that one-third of the matter in our galaxy originated in a different galaxy. It's an amazing world out there!

And Science really is Magic

Thanks for reading,

 - Dr.Mike

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^[1]↩ The Turing test, originally called the imitation game by Alan Turing in 1949, is a test of a machine's ability to exhibit intelligent behaviour equivalent to, or indistinguishable from, that of a human. 

^[2]↩ A gravitational-wave detector (used in a gravitational-wave observatory) is any device designed to measure tiny distortions of spacetime called gravitational waves. Since the 1960s, various kinds of gravitational-wave detectors have been built and constantly improved. The present-day generation of laser interferometers has reached the necessary sensitivity to detect gravitational waves from astronomical sources, thus forming the primary tool of gravitational-wave astronomy.

Capitalism is not inherently bad, unethical or immoral. Limited Liability Companies? They are the engines that pollute the world.

Capitalism is not the villain of this story. It is a tool — a method for harnessing individual self-interest toward collective benefit. Used well, it has lifted more people out of poverty than any system in human history.

The villain is the Limited Liability Corporation: a legal structure that removes personal consequence from business decisions, systematically incentivizes amorality, and — absent strong regulation — will lie, cheat, pollute, addict, and kill whenever doing so improves the quarterly return. This post is about why, and what to do about it.

What is wrong with capitalism?

I blame Limited Liability Corporations.

Let’s be clear: Corporations are not human beings. Corporations have no conscience. Corporations are immoral, doing whatever it takes to survive. Capitalism tried to harvest that greed to benefit all mankind. Most would say it's been successful; however, without regulations, Corporations will do anything to improve their profits. They will lie, cheat and steal. Corporations will force their costs on anyone they can, refuse to clean up their messes and kill people if it means they will make more profits. They have no other incentive without regulations. As Limited Liability Corporations the owners have no fear of retribution from the state, except to lose their investment. It’s a perfect storm of corruption.

Capitalism is designed to harness this immoral greed and attempt to benefit mankind. And it works surprisingly well. The number of people in abject poverty has fallen absolutely in the last few decades. [URL 1] Corporations don’t give a shit about mankind except that’s where some of their profits lie. Corporations have no structural incentive to care about mankind. Did the tobacco corporations care that they were killing their customers? Only in the sense that they had to get new, younger customers hooked to keep their revenue stream going. It cost too much to keep the older ones alive, it was cheaper to lie, obfuscate and preach doubt to recruit the younger generation to their addictive death-dealing drugs. Same for the opioid crisis. Same for the fossil fuel business.

Think about the coal business. In the US it’s dwindled to 40,000 workers. Burning coal causes almost 100,000 early deaths every year in the US alone, as documented by this global warming/climate changing denier administration [URL 2]. Burning coal puts more radioactivity in the atmosphere every day than the nuclear industry has ever put into the atmosphere. [URL8] And that includes Chernobyl. It’s burning the globe to a crisp, but the industry doesn’t care. Why should they? They still make profits. These companies are immoral. They will bribe politicians and lie to the public to continue making profits.

The market is not ‘self-correcting’ when it comes to death and morals. The market only moves towards more profits for the corporation. Can we pollute and make things cheaper while killing hundreds of thousands? Sure! Let’s do it! The calculus is straightforward: pollute and profit. Remember, Limited Liability Corporations were invented to enslave the world. Who ran the colonialist empires? Corporations. You could invest your money, with no consequences for immoral acts the corporations used that money to commit. And you would get a good return on that investment! It's amazing how much money the colonizers could extract.

Corporations don’t care about their employees except as delivering a service at a cost. If they weren’t forced to provide healthcare and safe working places, they wouldn’t. If they weren’t forced to not use child labor. They would [URL 3]. They are immoral. And immoral beings are evil. Corporations are the equivalent of psychopaths, all of them. Without regulations we saw what happened. They polluted our air, they polluted our waters, they sold poisonous food, they peddle quack remedies that at best are harmless, at worst cause death and destruction. They lie, cheat and steal. They were invented to enslave the world. Think about that for a second. Have they changed their ways? No. They’ve only responded to raw naked power. They fight for monopolies, they collude to increase their profits. For anyone to think unfettered, 16th century, Colonial Capitalism is the right economy for the world is out of step with history and freedom. This would enshrine and increase economic inequality, not a good thing [URL 4]

For the analysis of how corporate capture of government produced the specific political conditions of the Trump era, see: The Power of Trump (2020).

That’s right. The stand of the Republican Party to unconditionally support "Free Enterprise" is not a good thing for the world, let alone for America. The party has been coopted by the corporations which has allowed the amoral corporations to coopt the regulatory departments. The industry staffed government regulatory department allow whatever industry requires to increase profit. These corporations blame everyone else for their own problems and pollutions, force society to pay their cleanup and infrastructure costs while extorting monopoly rents from the masses for profits and extract public subsidies for their losses. Cruel capitalism for the masses, socialist welfare for the corporations.

Why are corporations amoral? Because they are Limited Liability. The state has promised not to prosecute you beyond taking your investment. You are not at risk from the state for anything the corporation does. This makes investment an actual business, not an unlimited risk that nobody could afford to take. Further, the current legal theory is that these corporations exist only to return the maximum profits to their investors. [URL 5] This implies that the corporations are incentivized to lie, cheat and steal to improve profits, as fines are hardly ever more than profits. This means that shareholders could sue the company if it does something that reduces profits, for any reason. There is an inherent incentive for the corporation to be amoral or even immoral.

For the Friedman Doctrine as practiced by the Wall Street Journal's opinion page, see: Wall St. Journal Nails It Again (2020).

Given this fact, it is obvious that we do not want these amoral corporations corrupting government.. or public opinion. They should be disbarred from donating to politicians, political action committees and, just like churches, they must be forbidden from recommending politicians or political questions or lose their limited liability. They are different from humans: they can only be bankrupted or disbanded by the state, and they can be reincarnated by anyone. They can be restricted from advertising on the air waves to the public. They can be forced to label their products as harmful and addictive. They pay taxes at different rates than humans.

To stop the damage that immoral corporations are inflicting on the world, we would need to reverse the Citizen's United decision at a minimum. [URL 6] We may need to increase our ability to control advertising, especially false claims. We may have to change campaign finance laws. This list is neither complete nor sufficient. To make the government stop ignoring the will of the people, we must change the way government is financed. This will require a long, dedicated effort. You can support this work done by the smartest person I know on constitutional law: Lessig.org [URL 7] the leading constitutional law scholar on campaign finance reform.

Thanks for reading,

-Dr. Mike

For the full analysis of how Citizens United functions as legalized corruption, see: Political Corruption (2025).

A Modest Proposal for Reparations

Reparations for slavery and the systematic ghettoization of Black Americans are not a gift — they are a debt. The question is not whether to pay it, but how to calculate it fairly and deliver it efficiently.

This post proposes a specific mechanism: a DNA-ancestry-based tax credit, calibrated using the Rule of 72, sized to close the documented 7:1 racial wealth gap in three generations. It would cost less than the 2017 Tax Cuts and Jobs Act.

 America needs to forgive itself for its original sins of slavery and ghettoization of Native Americans. To claim that was a different time and a different set of people is disingenuous. The government aided and abetted these travesties and they need to help alleviate them. So the question is how to do this fairly?

Here’s a simple proposal. First we must make some observations: with DNA tracking, we know exactly who your ancestors were. With DNA ancestry records — increasingly available through commercial testing — we can establish proportional ancestry and scale the benefit accordingly, without requiring case-by-case documentation of historical harm. Your ancestors were either discriminated against or not. And we know the proportions, so we can base the reparations on a standard legal ground. The idea is to take each grouping of families and compare their wealth to the average. We can figure out what the growth rate would need to be to match the wealth of the groups at any time in the future. Pick a time frame to eliminate this systemic racism. Maybe as long as it was in place? About 350 years? So say we plan to be an egalitarian society at that point? That seems ridiculous! How about reversing the last 55 years of streaky relative decline since the Voting Rights Act was passed? That was almost 3 generations ago, about time to fix it.

What additional revenue would this group need to match wealth in a few centuries? Today’s ratio of wealth white to black is about 7 to 1. [1] So we need approximately 3 doublings to fix this. (2³ = 8, so three doublings eliminates a 7:1 gap) So to fix it in three generations we have to double it every 20 years. This would mean an extra 3.5% income every year. (By the Rule of 72, doubling every 20 years requires a 3.5% annual growth premium. We can fade it out over the three generations, so start with double that amount and fade it to zero of a half century. How do we get this income to the right people? Tax credits. Give the reparations to the needy as a 7% tax credit, a form of Universal Basic Income for all those who were discriminated against. We can afford it. It would end up costing us less than the Republican’s latest tax scam for the rich. And it would go a far way towards making income inequality disappear. 

The underlying math — the Rule of 72 and doubling-time calculations — is explained in detail in: Hacks of System Engineering to Understand Exponential Growth (2021).

This is intentionally brief. The full calculation is left to the reader as an exercise. Note this fixes income not wealth. It would only be a down payment on the reparations.

thanks for reading,

 -Dr. Mike

For the ethical framework underlying this proposal — that the Declaration's promise of equality creates a positive obligation to remedy systematic inequality — see: The Declaration of Independence as the Foundation of Modern Ethics (2017).

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[1] https://en.m.wikipedia.org/wiki/Racial_inequality_in_the_United_States#Racial_wealth_gap

Hacks of System Engineering to Understand Exponential Growth and Vaccine Panic.

Two mathematical tools account for most of what the average person needs to reason about exponential processes: the Rule of 72 (how fast does doubling happen?) and the Birthday Paradox (what are the odds that anyone in a group has a given property?). Both tools are simple enough to run in your head. Both were essential for understanding why COVID-19 was always going to be catastrophic without early intervention.

This post teaches how to use both tools, shows where they come from, and applies them to the two questions that mattered most in early 2020: how fast would the virus spread with no action, and how safe was it to attend a vaccinated outdoor event under Delta variant conditions in late 2021?

Note: This post was written in October 2021 during the Delta variant wave. The specific COVID statistics reflect that period. The mathematical tools — Rule of 72 and Birthday Paradox — are timeless and apply to any exponential growth context.

Okay! Today I get to play armchair epidemiologist! But first, let's all learn the Rule of 72.

The Rule of 72 is a way to understand exponential growth. Exponential growth is something we see all the time in information technologies. Moore's law [1] is an example. It says that every two years the number of transistors you can fit into a fixed area doubles. 

This picture is updated from one of Kurzweil's books: [2]  "The Singularity is Near." It shows several things that will become very useful to be able to predict exponential growth rates. There's a linear X-axis that goes in terms of years (it could be any standard increment of time, days, hours, weeks, months, etc.) Then there is a Y-axis that is given in powers of 10. Each tick represents 10 times the tick below it. Thus a straight line in this graph is actually an exponential growth curve. For every increment of time you get a multiplicative increase in what is measured.

It turns out this is also the way interest works and there's a well known hack for interest rates called the Rule of 72. You take the interest rate and divide it into 72 and that tells you how long it takes for your money to double. Let's do an example.

Assume you get 12% (Interest) on your $100 (Principle) deposited in a bank (it happened in the 80's, just sayin') that means after the first year your hundred dollars turns into $112 dollars. The formula is 

P * (1 + I) = P', 

where P' is the principle after one year (the same time the interest rate is quoted over.) To find out what happens after N years you just have to keep multiplying by 1 + I:

P'(N) = P * (1 + I) ^ N.

Where the " ^ N " (to the Nth power) just means you multiple (1 + I) together N times: (1 + I) * (1 + I) * (1 + I).... N times. 1.12 * 1.12 = 1.25, 1.25 * 1.12 = 1.40, 1.40 * 1.12 = 1.57, 1.57 * 1.12 = 1.76, 1.76 * 1.12 = 1.98 (essentially doubled!) How many times did we have to multiply 1.12 times itself to get to 2? 6. Six times. (You can count them.) And it continues into the future. If you want to double your money again, it's six more years. So now you have four times as much money. Or double it again, now we have 8 times as much money after six more years. How long did that take? We had three doublings (2, 4, 8) and each one took six years. So we have 8 times as much money in 3 * 6 = 18 years if we earn 12% interest. 

Now where does the Rule of 72 come in? It's a shorthand way of figuring out the doubling time. To figure out how many periods it takes the return to double you can just divide the interest rate for a period into 72 and that tells you how many periods it takes to double. 72 divide by 12 is 6. So it takes six years (the period) to double your returns. 

For the general toolkit of Fermi estimation — how to get useful answers from almost no information — see: How to Estimate Anything From Nothing (2016).

Lets' look at Moore's law = the thing that doubles takes 2 years. 72 divided by I = 2, so I must by 36%. Every year the engineers are forced to increase the number of transistors in a fixed area by 36%. Now Moore was being conservative as this means every year the width and length of a transistor must get smaller by only 20%. Why is this so? If a transistor's side is 80% as big as it was last year, then the area it takes up is 80% * 80% = 64% as much space, or 36% less area. So by making transistors lengths 20% smaller in two dimensions we get twice as many in a fixed area every two years.

What else can use this exponential growth hacks on? 

Well, the most disturbing thing in the last few years I used this rule on was the outbreak of the Covid-19 Virus last year. 

What does exponential growth mean for the Covid-19 Virus? 

Back in late January 2020, I saw the Chinese building field hospitals that could handle thousands of very sick people. They thought this virus was going to be very bad (and they were right.) But how bad? Even in China at this time there were only a few hundred deaths, but they obviously expected a whole lot more... on January 23 the Chinese reported 250+ cases confirmed in the laboratory. 2 ^ 8 = 256. This means that the number of infections had doubled 8 times. The earliest known cases were from a Doctor who reported the deaths of 4-8 patients in mid-December, and caught the disease and died himself. (Unbelievably sad that he was ignored by the authorities - one of the problems that a one-party state makes worse is hiding relevant information so they won't look bad...) 

It turns out that cases were popping up in other places. Nobody had stopped travel from Wuhan China, and it was spreading across China (Beijing, Shanghai.) What would stop it from coming to the US. NOTHING. It was already here. You can just look at the flights out of Wuhan to the US, assume the infection rate was the same as in China and you would likely conclude that a few infected people from China were already here in the US. And living in a town that is more than half Chinese, it was likely someone had the virus and lived less than two miles from my house. So now the critical number becomes how fast are people infecting others? How many people do they typically infect over the course of the disease? The dreaded R0. 

If you want to find out what the professionals were doing at this time, I suggest you read Michael Lewis's book "The Premonition: A Pandemic Story". He goes into great detail about how some of the smartest doctors in the land were looking at the tragedy that was about to unfold and what they were trying to do about it. Myself, I'm not a professional medical person, I'm just a lowly physicist who pretends to do system engineering. So I sat down in late January and said, "What's going to happen in my town?" I was already confident that there were infected people in my town. We already have an estimate of how fast it's spreading in China and they went into a serious lockdown about a week later, trying to stop the spread of the disease. And they did, but not until almost 10,000 people died. 

Looking at the doubling rate of 8 times in about 7 weeks scared the piss out of me. This disease was doubling every week. Every week. If we did nothing we can use the rule of 72 to figure out when everyone in the US would catch the disease. 

Let's practice! 

Assume that only one person from China got to my town with the infection. During that week they spread it to two people. Then the next week those two each spread it to two more people. How many doublings do we need to get to 350 Million? We can use some shorthand knowledge of powers of 2. 2 ^ 10 = 1024. If you double something ten times you get one thousand times more of it. Continue it ten more times and you get a million times more of it. (One thousand times one thousand is a million.) We need to double it to 350, which is about 9 more times. It doubles every week. In 29 weeks everyone in the United States would likely have gotten Covid-19 if we did nothing. So by the time my birthday rolled around in 6 months, 3.5 million people would be dead in the US. (The death rate at that time was estimated to be around 1-1.5%, but maybe as high as 3%...) 

This did not make me very happy. In fact, it threw me into a panic. I had just returned from a trip to India through Singapore, and they weren't fooling around. Everyone was checked for a fever when they got off the plane. If you had one you were taken away and quarantined for two weeks. In the US, the government was laughably lackadaisical about the whole thing. "It'll just go away. It'll be a miracle. One day you'll wake up and it'll  be gone." (And lots of your friends will be dead, I was thinking.) 

But some instinct kicked in finally and California declared a lockdown in early March. A week or two later the New York area did the same. That two week delay caused havoc with people's lives in New York. While the doubling rate in California had been basically halved two weeks earlier, New York just missed the bus by that much and the virus raged through the population for multiple weeks until their lockdown stopped the spread and tens of thousands of people had died. For the entire month of February plus a week of January and a week of March went by and the federal government did nothing. 6 doubling times. 2, 4, 8, 16, 32, 64. Where there had been one case in my town, there were now 64. If we had waited that extra two weeks there would have been 256. If we waited as long as New York it would have been 1000 cases. If we had waited an additional 6 weeks, every one in my town would have been sick (2 ^ 6 is 64 or 64,000 cases, but there's only 50,000 people in my town.)  It's just the mathematics of exponential growth.

For the political arguments deployed to resist lockdowns — and why they were wrong — see: The Lockdown Rebellion vs. The Covidiot Death Parties (April 2020).

The amazing thing is that for the first time in history a lockdown actually worked and prevented this disaster from happening. My entire town didn't get the virus (and thank god for that

For the analysis of California's specific COVID response failures and the federal government's abdication, written in July 2020, see: What's Wrong With California? Utter Failure in Handling COVID-19 (2020).

So now what other equation can we learn how to use to help us in these perilous times? We can learn how to solve the birthday paradox. 

Solving the Birthday Paradox, for Fun and Safety!

What is the birthday paradox? It's how you calculate the odds that anyone has the same birthday in a group of people. It's very similar to the Rule of 72. Here's the way to reason about this situation. If there are two people in a room, what are they odds that they have the same birthday? My birthday is on one particular day. What are the odds some random person has that same birthday? The probability, Pb,  is one out of 365 or about a 0.3% chance. What are the chances that person doesn't have the same birthday as me? 364 / 365, or 99.7% chance our birthdays don't match or (1 - (1 / 365)) or in algebraic form: 

P(no common birthday) = (1 - Pb)

for N people we can multiply the probabilities together, every time we introduce another person, we just have to multiply the odds for having none of the same birthdays together. This tells us the probability that no birthdays overlap, so the odds that anyone's birthday is the same is just 1 - P(no common birthday). Actually, the birthday problem is worse than the Covid-19 problem because it has a limited number of dates for birthdays, so the odds of the third person is (1 - 2/365) that they don't have the same birthday as the first two, and the odds that the fourth person doesn't have the same birthday as the other three are (1 - 3/365) and so on.

For Covid-19 it's much easier. We can look at the infection rates in the area surrounding an event, we can see how many people will be at the event and we can calculate what are the odds that anyone has Covid-19 at the event. Let's try a sample!

I wanted to go to a concert in Napa last month. The venue was outside (smaller chance of Covid-19 spreading over the air as the viral molecules, typical encased in airborne water droplets, get blown away quickly when you're outside.) The venue was small, about 500 people. The number of cases per day that were being found in Napa was about 10 per 100,000 people. A case lasts bout ten days, during which time we will assume the person was infectious the whole time. This means there were 100 people with Covid-19 in Napa at that time. So the probability of someone having Covid-19, Pc, was about 100/100,000 or 0.1%. (Yes, that's only the tested people, so probably more had it than that... but I was planning on wearing my mask which reduces the odds of catching the virus by about a factor of 10, so I felt my estimates were safe.)  So the odds of any one of the concert goers not having covid-19 was (1 - Pc). The odds of two people not having Covid-19 was (1 - Pc) * (1 - Pc). The odds of nobody in N people having Covid-19, P(N), is:

P(N) = (1 - Pc) ^ N.

And the odds of somebody (or more than one person) having Covid-19 is 1 - P(N).

Look familiar? Look at the first equation in this blog. Remember the caret symbol is exponentiation. You multiply the stuff in parenthesis with itself N times. These equations are just two sides of the same coin. So how does this work out for the concert? I can use my handy-dandy scientific calculator to give the answer:

P(500) = (1 - 0.001) ^ 500 = (0.999) ^ 500 = 60%

1 - P(500) in Napa at that time was 40%. The odds that someone at the concert had Covid-19 was 40% if they let random people in the concert. However, they did not let random people in to the concert, you had to show your proof of vaccination. How does this affect the odds that someone at the concert has Covid-19? For that we have to calculate what are the odds a vaccinated person has Covid-19, in other words, how prevalent breakthrough cases are. There's a lot of conflicting data on this, but the odds of a vaccinated person having Covid-19 started out as 20 to one when compared to a non-vaccinated person. This is what the vaccine trials showed. When they say the vaccine is 95% effective it means that for every 100 unvaccinated people that would have got the virus only 5 people actually got it. Twenty to 1. At this time the delta variation had started to spread across the US and the vaccines are much less effective against that variant. Looking at the numbers it was from 6 times to 2.5 times less likely to catch the delta variant (depending on whose numbers you believe, how much of the delta variant is actually out there when you do the test and lots of other issues. The number that got me, though, was that the hospitalization rates and death rates were still down by a factor of 10, even with the Delta variant. You might catch it, but you weren't going to die from it.)  So what are the odds of someone having Covid-19 at the concert if everyone is vaccinated? The odds of a single vaccinated person having it are 10x less (with masks and not so much Delta around...) than a vaccinated person. So we do the same calculation, only this time the odds of having Covid-19 are one out of a hundred thousand. This means the P(500) for vaccinated people, the chance that anyone at the concert has Covid-19 is not 40% but it's actually closer to 1 - (0.9999) ^ 500 = 1 - 95% = 5%. The odds that one or more vaccinated people at the concert had Covid-19 was 5%. If this model was correct, and we think it was conservative, (the odds are likely to be less, in our opinion) 95% of the time nobody who went to the concert had Covid-19. Even if a few people at the concert had Covid-19, I was wearing a mask which reduced the odds even more. Even if the one person who had Covid-19 was sitting close to me, I was unlikely to get the disease since we were outdoors and the wind was quickly blowing any virus they were emitting away. I felt safe. 

And I was safe. This is why California is looking to reduce its mask mandates. So few people have Covid-19 that the odds you are going to catch it if most people are vaccinated is miniscule. Would I go to a concert that had ten times as many people? I have. Would I got to a theme park that had another ten times as many people? I'm planning on doing it in a few weeks when it looks like the infection rate will be down by another factor of ten. Have we finally whipped this virus? If everyone got the vaccine, then yes we could whip this virus. If Covidiots continue to resist the vaccine they will continue to die at a higher rate than the vaccinated are dying. It's a chance to see how evolution actually works. Those who insist on virtue signaling (My immune system is strong enough to naturally survive the virus! I don't need any help... Yeah, you do, and the vaccine basically helps your body make it's own mono-clonal antibodies that help you fight the infection.) Get Vaxed.

Get Vaxed! Take the vaccination. Don't try to 'man up' and show how 'fit' you are and end up infected and dying like a whole host of right-wing radio talk show hosts did last month. And if you think it's better to not get vaccinated because your god given immune system should protect you... Please remember, God also gave you a brain, and we've used it to figure out how to create and test for safety in record time a vaccine that works. To not take advantage of this scientific breakthrough is why evolution works at all. If you aren't adaptable, nature will just take you out and your descendants will be non-existent. That's how evolution works. Be smarter than the dinosaurs, don't let a preventable accident cause your death (in their case their space program was woefully inadequate to prevent dying from a meteor impact, in our case our modern medicine used science to invent a vaccine in record time.) 

Get vaccinated. Save lives. 

Not to mention, the sooner we limit the number of cases we limit the number of variations that can evolve and be more infectious or more deadly. We need to reduce the dreaded R0 below 1. The further below the better.

Get vaccinated. Stop mutations. Save lives.

And for what it's worth — this was not abstract for me. My gut blew out the day California declared lockdown... , If we hadn't, and my gut had held together a little longer, I'd have been out of luck as the hospitals would be full of Covid-19 patients and I'd be in the ground, dead from neglect.

I didn't die. The doctor said that if my disease had occurred thirty years earlier, I would have been dead. And the recovery time was about six weeks. When my grandmother had the same disease, they caught it earlier, so she lived, but her recovery time was a year. Thank the Lord for small incremental improvements in medical care over the last sixty years. It's all about information, and information changes can keep improving for a long time at a steady rate and the accumulated improvements really do save lives.

Thanks for reading!

 -DrMike

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[1]https://www.flickr.com/photos/jurvetson/31409423572/ Steve Jurvetson's photo of Ray Kurzweil's exponential growth chart

[2] https://en.wikipedia.org/wiki/The_Singularity_Is_Near

It's been fixed! There is no longer an insane narrowing of the 280 Freeway in Santa Clara County.

UPDATE: BREAKING NEWS (8/15/21): They painted the final lines and messed it up again at Pagemill road (going down to three lanes and taking one as a dedicated exit and barely fixed the entry of the diamond lane... so it's still a little bit screwed up at one place... backups will not be eliminated entirely. DOH!

It seems the California Department of Transportation has taken my suggestions made four years ago and decided to repave the 280 freeway so that there are four continuous lanes for the entire trip between San Francisco and San Jose. 

My last report on this crazy backup causing 'improvement' was here: https://www.wiigf.com/2016/11/a-proposal-to-fix-unnecessary-freeway.html.

I pointed out that there was space to stop the backups and increase the traffic flow and was flabbergasted as to why they had designed the freeway to impede traffic flow. Now they've gone back and repaved it and fixed it. I have no need to repaint the lines anymore as the DOT has done it themselves! When they resurfaced the road this month, they repainted the lanes to increase the traffic flow. Finally!

I'm not sure if it was an accident or not. Maybe the theory of queueing has penetrated into traffic engineering. Seeing what they are doing to other nearby entrances and exits, I'm not so sure.

Anyway! Let's celebrate one more free-flowing freeway! Rebuilt like it was supposed to be built in the first place to maximize the movement of human beings. 

Thanks for reading.

 DrMike

 30 July 2021