Super fact – Super Facts

Time is a Fourth Dimension

Super fact 58 : In relativity, time is considered the fourth dimension, inseparable from the three spatial dimensions to form a four-dimensional continuum called spacetime. Adding time as a fourth dimension, not (x, y, z), but (x, y, z, t), results in spacetime measurements called spacetime intervals that all observers can agree on.

Before relativity the distance between two points was the same for all observers. The distance between points is calculated using the Pythagorean theorem: (d^{2}=x^{2}+y^{2}+z^{2}). You calculate the distance between two end points in a coordinate system using Pythagoras theorem because the points make right angled triangles along the x-axis, y-axis and z-axis. See the picture below.

The image shows the formula for Pythagoras theorem in two and three dimensions and Pythagoras theorem applied to the distance between two points. — Pythagoras theorem in two and three dimensions which also apply to the distance between two points. The points are indicated in red.

Let say you add another coordinate system (x’, y’, z’). The new coordinate system could be translated and rotated compared to the first one. The values of (x, y, z) and (x’, y’, z’) would be different and yet the distance between point-1 and point-2 would be the same. Well as long as you don’t change units, like using meters in one coordinate system and feet in the other. The distance between the points would be a so-called invariant. Now imagine that you forgot to include one coordinate in Pythagoras theorem, for example, y and y’ or x and x’, then your calculation for the distance would be different for the two coordinate systems. We need all coordinates, or all dimensions. See the picture below.

The picture shows two different coordinate systems. One is rotated and translated compared to the other. There are also two points and the distance between them is indicated. The formula for Pythagoras theorem is shown for both coordinate systems. — Pythagoras theorem is used to calculate the distance between two points from two different coordinate systems, with different coordinate values for the points. You still have the same distance for both coordinate systems. The points are indicated in red.

In relativity the length of objects, as well as the time between events is relative and varies from observer to observer. In other words, distance and time varies from coordinate system to coordinate system. However, if you add time to the three space dimensions and calculate the distance between events using the Pythagorean theorem for intervals (between two events): or (s^{2}= x^{2}+y^{2}+z^{2} – t^{2}) or (where the ‘t’ represents time in appropriate units), then the difference between different observers vanish. The interval is the same for all observers. It is a so-called invariant. The formula for the spacetime interval comes in a few different forms. One for distance like intervals (space distance bigger than time) (s^{2}= x^{2}+y^{2}+z^{2} – t^{2}), and one for time like intervals (time is bigger than the space distance) (s^{2}= t^{2} – (x^{2}+y^{2}+z^{2})). There is also one that includes the imaginary number (s^{2}= x^{2}+y^{2}+z^{2} + (it)^{2}). See below.

The image shows three formulas for the spacetime interval Euclidian: “(s^{2}=x^{2}+y^{2}+z^{2}+(it)^{2}”. For Time like intervals, the standard form: “(s^{2} = t^{2} – (x^{2}+y^{2}+z^{2}))”. For distance like intervals: “(s^{2} = ((x^{2}+y^{2}+z^{2}) – t^{2}))”. — The three formulas for the spacetime interval above all assume that the unit used for time is the time it takes light in vacuum to travel the distance unit used. If that is meters, it would be the time it takes light to travel one meter. The top formula is the Euclidian form of spacetime. It contains only the ‘+’ operator at the expense of adding the imaginary number (square root of -1) in front of the time coordinate. The second form is typically used with time like intervals and considered the standard form. The third form is used when the distance between two events is larger than the time distance, or distance like intervals.

The interval concept was developed, not by Einstein, but by Hermann Minkowski (a few years after special relativity) and is often referred to as Minkowski space. Time is like a space coordinate but the opposite signs in the equation make it different. Based on articles I found it appears that the opposite signs (minus vs. plus) means that you cannot move “backwards” in time as you can in a space dimension.

I admit that this is a very abstract super fact, but it basically means that if you add time as an extra coordinate to the three space coordinates x, y, z you get something, the spacetime interval, that everyone regardless of speed, orientation, etc., agrees on, despite relativistic length contraction and despite time dilation and non-simultaneity.

Time Expressed in Appropriate Units

I would also like to explain what I mean by (where the ‘t’ represents time in appropriate units), as I stated in the above. For physical formulas to work they need to be expressed in consistent units. For example, you can’t use kilometers for the coordinate x, and miles for coordinate y, not without adding a constant to adjust for it. For the formula (s^{2}=x^{2}+y^{2}+z^{2}-t^{2}) to work you need to express time in a unit that corresponds the time light travels in one meter if x, y and z are expressed in meters. If you express x, y, and z in meters and express time in seconds you must adjust the formula with the constant c = 299,792,458, the speed of light in meters per second, so you get (s^{2}=x^{2}+y^{2}+z^{2}-(ct)^{2}). See the picture below.

The image shows the formulas for the spacetime interval with the constant representing the speed of light in vacuum “(s^{2}=x^{2}+y^{2}+z^{2}+(ict)^{2}”, “s^{2}= (ct)^{2} – (x^{2}+y^{2}+z^{2})” and “(s^{2}=x^{2}+y^{2}+z^{2}-(ct)^{2}”. — If you measure the space coordinates in meters and the time in seconds you must adjust the units to match by inserting the speed of light in vacuum c = 299,792,458. The three forms of the space interval now have the constant c attached to the time coordinate.

Time Like Space Intervals

The formula for time like intervals is typically used for the situation where the time component is larger than the space component, which also means that it is possible to physically travel between the two events forming the space interval. As you can guess, that is a pretty normal situation. Let’s say you are watching TV and having a pizza. Your sofa is your coordinate system. You turn on the TV and 100 seconds later you move 2 meters to get a slice of pizza. Let’s calculate the spacetime distance between those two events.

The space component is easy, that’s 2 meters. However, if we express time in the time it takes light (in vacuum) to travel one meter we get 100 times 299,792,458. If you express time in seconds, you adjust it using the constant c = 299,792,458, and again you multiply 100 with 299,792,458, which is 29,979,245,800. So, the distance in time is almost 15 billion times larger. You really did not move far in space, but you moved very far in time. Now ask yourself. Are you spending your time well?

The Minus in Front of the Time Coordinate

There is one obvious difference between time and the space coordinates. In a coordinate system you can walk forward, along let’s say, the x-axis and then walk back the same way. You can walk back and forth as many times as you want, no problem, but you cannot do that with time. Time may be a space-time coordinate, but it is different from the other three coordinates in that way, and that’s where the opposite signs in the formula for the space-time interval comes in. This is beyond the scope of this super fact blog post, but you can read more about this here and here.

Other Relativity Related Superfacts

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The Bermuda Triangle the Big Non-Mystery

Super fact 56 : The Bermuda Triangle mystery is a myth. There is not a higher risk of disappearances in the Bermuda Triangle. To be specific, disappearances do not occur in the so-called Bermuda Triangle, or Devils Triangle, with any higher frequency than in other comparable regions of the ocean. The “mystery” of the Bermuda Triangle is largely a manufactured one, perpetuated by sensationalized accounts that often misrepresent the facts and downplay the role of natural hazards like storms.

The number of ships and aircraft reported missing in the Bermuda Triangle is not significantly greater, proportionally speaking, than in any other part of the ocean. The U.S. Coast Guard, along with NOAA, the U.S. Navy, Lloyds of London who pays out insurance for ships and aircraft lost/missing at sea, and other organizations do not recognize the Bermuda Triangle as a unique or mysterious geographic hazard. They emphasize that this is a highly traveled area where the losses are consistent with natural phenomena such as strong storms, the Gulf Stream, human error, and poor navigation, rather than any mysterious forces.

Considering all this, the number of disappearances and accidents is what you’d expect. The Bermuda Triangle isn’t any more mysterious than the Greenland square, the New Zeeland circle, or the Azores Octagon, that I just made up. I consider this a super fact because it is very likely true, and yet surprising to many people who are convinced that there really is a mystery. Furthermore, it is important because it is such a well-known myth.

The Bermuda triangle has one corner in Bermuda, one in Puerto Rico and one around Miami, Florida | The Bermuda Triangle the Big Non-Mystery — The Bermuda Triangle: It is approximately defined as a triangle Florida, Bermuda, and Puerto Rico. There is no exact definition. Alphaiosderivative work: -Majestic-, Public domain, via Wikimedia Commons.

Bermuda Triangle Mysteries

Just because the risk of disappearances of planes and ships is not higher in the Bermuda Triangle, does not mean that there aren’t mysterious disappearances and mysterious phenomenon occurring in the Bermuda Triangle. Some notable disappearances are USS Cyclops (1918), Flight 19 (1945), Star Tiger and Star Ariel (1948–1949), and the Witchcraft (1967). However, there are mysteries and mysterious phenomena occurring all around the world.

One of the mysterious phenomena occurring in the Bermuda Triangle is ocean swirls, and rogue waves, and methane burps might be another problem, but it is far from unique to the Bermuda triangle, and there are no magnetic anomalies in the Bermuda triangle as often alleged.

Ocean swirls frequently occur all over the world with some famous hotspots for ocean swirls by the coasts of Japan, Norway, Italy, Scotland, and Maine, USA. The ocean swirls in the Bermuda triangle might be due to movement of water between landmasses and/or the Gulf stream, but this is under investigation. There are no known giant or permanent ocean swirls in the Bermuda triangle.

A giant deep ocean swirl in the middle of the blue ocean. — Ocean swirl allegedly in the Bermuda Triangle Asset id: 1158148882 by PHOTO JUNCTION

As mentioned, another mysterious phenomenon is methane bubbling to the surface of the ocean. However, as can be seen in the maps in this National Geographic blog post the source of these methane burps of death aren’t typical to the Bermuda triangle. The methane hydrate field in the first map of the National Geographic blog post is mostly outside of the Bermuda triangle stretching from Cuba and up along the Florida coast. The second map, the world map, shows that these methane hydrate fields exist all around the world.

Our Honeymoon in Bermuda

Below are some old photos from our honeymoon in Bermuda in August of 1991.

The Bermuda Triangle the Big Non-Mystery

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Digging Up Super Facts

Daily writing prompt

What change, big or small, would you like your blog to make in the world?

View all responses

I have two blogs, Leonberger Life and this one called Superfactful.

Leonberger Life feature amusing and heartwarming stories about our late Leonberger dog Bronco, as well as other Leonbergers. It also has a lot of information about the Leonberger breed, the history, care, training, Leonberger organizations, etc. I also wrote a Leonberger book, which I am featuring in the sidebar. With my Leonberger Life blog I want to spread information about the Leonberger breed, a rare, large, furry, friendly and fun dog breed and also bring attention to my book The Life and Times of Le Bronco von der Löwenhöhle, stories and tips from thirteen years with a Leonberger.

The goal of Superfactful, which is this blog, is to create a long list of facts that are important, not trivia, and that are known to be true and yet are either disputed by large segments of the public or highly surprising or misunderstood by many. I call these kinds of facts Super facts because they could potentially be very impactful on how we view the world.

Humans have accumulated an enormous amount of knowledge. Science does not know everything, but it knows a lot. The same cannot be said for us as individuals. We know next to nothing and harbor a lot of false beliefs. I think that is pretty much true for all of us, including me, but we may not know it. With this blog I am trying to correct some of that, at least regarding important facts. In addition, along the way, I am hoping to learn a lot myself and have some of my own false beliefs corrected.

In short, the change I would like to make in the world with my blog is to correct as many false beliefs as possible and educate my readers, and myself, about facts that are both important and mind blowing.

A blue brain is splitting up into pieces. — Smash your old beliefs with new surprising facts, super facts. Expand your mind. Shutterstock ID: 1685660680 by MattL_Images

What is a Super fact?

A super fact is:

An important fact but it can be simply stated.
Very surprising, shocking, widely disputed, misunderstood, or mind-blowing.
Yet it is true with a very high degree of certainty.

The first two criteria are subjective. The last criteria can be determined from longstanding scientific consensus, my own expertise and education (valid for just a few topics), agreement between multiple reliable online or offline sources such as agreement among research papers, reputable scientific organizations, NASA, NOAA, Pew Research Center, Our World in Data, etc. I should say I also link to less academic sources such as Wikipedia, but I do not solely rely on them.

You can read more about how I choose super facts here. Also, I am open to challenges based on good data (not opinions), as well as questions. With that I don’t mean that you cannot give your opinion in a comment. I just won’t update or remove a super fact based on an opinion. In addition, I am happy to receive suggestions for super facts. I am trying to collect a few hundred super facts and need all the help I can get. In the end I want to pick the 100 best ones. I might use a poll for that.

Sometimes a super fact involves doing myth busting of a popular myth and sometimes a super fact is stating something that is well known but disputed by many. In these cases, the evidence is likely to be conclusive, but the fact is still disputed by those who don’t know much about the evidence, or don’t want it to be true. In this case, I will include a substantial amount of evidence, and it might be lengthy. People get bamboozled all the time, and that includes me. It is not easy to admit that you have been bamboozled. You can read about my own experiences with that here.

Picture shows a scale held by a pointing finger. Fact is on the left shown as a bright light bulb. Myth is shown on the right as grey ball | Digging Up Super Facts — Fact or myth. Shutterstock Asset id: 2327968607

Examples of Super Facts

At the time of posting this I have made plans for 150 super facts and so far, I have posted 54. I will post a lot more. Below I am listing a few of my first 54 super facts.

Superfact 5 : Two events may be simultaneous for some but not for others. Click to visit post.

Two events may be simultaneous for some but not for others. This means that two events that are simultaneous to an observer may happen at different times to other observers. If two lamps A and B turn on at the same time according to observer #1, lamp A may turn on first for observer #2, and lamp B may turn on first for observer #3. All three observers are correct because time is relative.

The picture shows three pairs of lamps and three people. The setup is used to show three situations. — Amy is traveling at a high speed to the left compared to two lamps A and B. Alan is standing still compared to the lamps. Adam is traveling at a high speed to the right compared to two lamps A and B. Alan turns on the lamps at the same time. After considering the travel time of the light she sees, Amy concludes that lamp B turned on first. After considering the travel time of the light he sees, Adam concludes that lamp A turned on first. I should add this non-simultaneity can only happen if the lamps are separated by a distance.

Superfact 7 : Poverty and child mortality has been sharply reduced worldwide. Click to visit post.

Extreme poverty as well as child mortality has been sharply reduced the world over. The countries that are the worst-off today are still better off than the countries that were doing the best at the beginning of the 19th century. Over the last 20 years extreme poverty and child mortality have continued to decline sharply.

This graphics contain two graphs one for the world (blue) and for the United States (red) | Digging Up Super Facts — Child mortality in in the world since 1950. The spike you see around the end of 1950 to 1960 is the great leap forward famine in China. In 1950 the child mortality rate was 22.7% and in 2023 the child mortality rate was 3.6%.

Superfact 25: Global Warming is Happening and is Caused by us – click to visit post.

Global warming is happening. Or if you call it Climate Change or Climate Disruption is indeed happening. And it is happening very fast. We also know that it is caused by us primarily as a result of our burning of fossil fuels. There is a long-standing scientific consensus on these two facts because the evidence is conclusive. Check the evidence in the post.

Superfact 28: That Earth is round was well known long before Columbus – click to visit post.

That Earth is round, or spherical (or closely spherical) had been known for at least a couple of thousand years by the time Columbus set sail. Columbus did not set sail to prove that earth was round, and he knew it was round.

On the left a wheel with spokes. On the right there is a sphere and sun rays that hit in two places. One without a shadow and one with a shadow | Digging Up Super Facts — Illustration of the measurement of the Earth circumference by Eratosthenes (2,300 years ago). On June 21st there is no shadow in Syene/Aswan but there is one in Alexandria. Asset id: 2319651251 by Javier Jaime

To see a list of the Super Facts click here

Ukraine Gave up Thousands of Nuclear Warheads

Super fact 47 : In 1991, after the dissolution of the Soviet Union, Ukraine became the third largest nuclear power in the world after Russia and the United States. Ukraine held about one third of the former Soviet nuclear weapons and delivery systems. Ukraine agreed to transfer these weapons to Russia for dismantlement in exchange for economic compensation and assurances to respect Ukrainian independence and borders.

After the dissolution of the Soviet Union in 1991, Ukraine inherited an estimated 1,700 to 1,900 strategic nuclear warheads, 130 UR-100N intercontinental ballistic missiles (ICBM) with six warheads each, 46 RT-23 Molodets ICBMs with ten warheads apiece, and an estimated 2,650-4,200 tactical nuclear weapons. It should be noted that these nuclear warheads were not under Ukrainian control.

In 1994, Ukraine agreed to transfer these weapons to Russia for dismantlement in exchange for economic compensation and assurances from Russia, the United States and the United Kingdom to respect Ukrainian independence and sovereignty within its existing borders. These political agreements are referred to as the Budapest Memorandum.

These events are relevant to what is happening in Ukraine today, and yet it is seldom discussed, and many people are unaware of or have forgotten about this history. It also comes as a surprise to many that there are nuclear states who have relinquished their nuclear weapons. This is why I call this fact a super fact.

Atomic bomb explosion in a city - nuclear attack on a crowded city – 3D rendering of a mushroom cloud encompassing a city with skyscrapers. — Nuclear bomb dropped on a big city. Shutterstock, asset id: 2188083835 by CI Photos.

Nine Nuclear States

There are nine nuclear states in the world as of 2025 according to the Federation of Atomic Scientists. There are 12,331 nuclear warheads including 9,600 in active military stockpiles.

Russia – 5,449 warheads
The United States – 5,277 warheads
China – 600 warheads
France – 290 warheads
United Kingdom – 225 warheads
India – 180 warheads
Pakistan – 170 warheads
Israel – 90 warheads
North Korea – 50 warheads

There are also countries that are hosting nuclear warheads owned by other countries.

Italy (the United States) – 35 warheads
Turkey (the United States) – 20 warheads
Belgium (the United States) – 15 warheads
Germany (the United States) – 15 warheads
Netherlands (the United States) – 15 warheads
Belarus (Russia) – ? warheads

Nuclear bomb, chemical weapons, missile defense, a system of salvo fire. — Four missiles aimed at the sky at sunset. Shutterstock, asset id: 2131803989 by Hamara

Four nations that relinquished their nuclear weapons programs

The four nations that relinquished their nuclear weapons programs are Belarus, Kazakhstan, South Africa, and Ukraine. Belarus, Kazakhstan, and Ukraine returned their inherited nuclear weapons to Russia after the dissolution of the Soviet Union. However, it should be noted that in 2023 Russia began deploying tactical nuclear weapons to Belarus. However, Belarus does not currently possess its own nuclear weapons. South Africa voluntarily dismantled its nuclear weapons program in 1991.

In 2023 Russia began deploying tactical nuclear weapons to Belarus.

Russia and Ukraine as well as the borders of Russia and Ukraine are colored and look different from other countries. — Russia (green) and Ukraine (brown) map on a world map. Ukraine is about the size of Texas. Russia is about twice the area of the United States. Belarus is the country located immediately to the north of Ukraine and Kazakhstan is the big country immediately to the south of Russia and east of Ukraine. Shutterstock Asset id: 2121271067 by buraktumler

How to Build a Nuclear Bomb

This section is just some extra reading that is only somewhat related to the topic. However, since it is an interesting topic somewhat related to the topic I might as well explain how to build a nuclear bomb. Don’t worry I will not present any engineering details, only general principals, which is all I know, and which are already all over internet. Besides if I were to give detailed engineering instructions some peacenik hippie might have a hissing fit and swear in the comment section (that was a joke).

Anyway, the main idea behind a nuclear fission bomb is to achieve a runaway chain reaction. A fusion bomb, or a so-called hydrogen bomb is different. To create a fission bomb you are not looking for the most radioactive materials there are. You are looking for a fuel which you can use to create a runaway chain reaction, and which is also stable enough to make a bomb possible, in other words not too radioactive. Basically, the fuel must be just right. The primary fuels used in fission bombs are uranium-235 and plutonium-239. These isotopes undergo fission when struck by neutrons, releasing a massive amount of energy in a chain reaction.

The image shows a Uranium atom on the left arrows in the middle and an alpha particle, a gamma ray, a proton, a neutron, and an electron on the right | Ukraine Gave up Thousands Nuclear Warheads — Uranium-235 and Uranium-238 are both not very radioactive and can be used for radiometric dating that stretches millions and billions of years. Uranium-235 is also the “just-right” uranium isotope that can be used for bomb making. Shutterstock Vector ID: 2417370135 by grayjay

I should explain, isotopes are different forms of an element. For example, hydrogen comes in three different forms, a nucleus with just a proton, a nucleus with one proton and one neutron (deuterium), and a nucleus with one proton and two neutrons (tritium). Isotopes for the same element are chemically identical but have different atomic weight and they may or may not be radioactive.

The three isotopes of Uranium are uranium-234, uranium-235, and uranium-238. The one we need is uranium-235, which has 92 protons and 143 neutrons in the nucleus. The isotopes of Plutonium include Pu-238, Pu-239, Pu-240, Pu-241, and Pu-242 but there are others. The one we need is plutonium-239, which has 94 protons and 145 neutrons in the nucleus. There are more than 3,500 known isotopes of which 3,000 are radioactive.

During Uranium-235 fission, an average of 2.5 neutrons are released. Specifically, the fission of U-235 typically releases 2 or 3 neutrons, with the average being close to 2.5. During the fission of plutonium-239, an average of 2.9 neutrons are released (depending on the energy of the incident neutron). The important thing for bomb making is that one atom/nucleus releases enough neutrons so that the neutrons from one nucleus cause more than one fission. For example, a nucleus releases three neutrons and two of those neutrons cause two more fission events, which in turn cause four fission events, etc. 2, 4, 8, 16, 32, 64, a trillion…

From left to right : a neutron strikes a uranium nucleus, and it breaks apart into a Krypton and Barium isotope and release three neutrons, which strike three uranium nucleuses, causing three fission events and releasing nine neutrons in total, etc | Ukraine Gave up Thousands Nuclear Warheads — Illustration of nuclear chain reaction. Uranium-235 fission. Unfortunately, the picture has an error in it. It is 92 protons not 95 in a U-235 isotope. – Shutterstock asset id: 73714504 by Mpanchenko.

By putting together enough U-235 you can make it so that one fission event will result in more than one additional fission event. This is called the critical mass. The critical mass for U-235 is 47 kilograms (104 pounds). Theoretically, you can achieve this by taking a 24-kilogram half sphere of U-235 in your right hand and a 24-kilogram half sphere of U-235 in your left hand and bring them together. You will achieve a limited chain reaction for a nano second, but you will just blow the two halves apart and kill yourself, but your city will survive. This is called a fizzle. To make most of the 48-kilogram mass undergo fission you have to force them together long enough for the chain reaction to complete (or almost complete). This requires force and precise calculations. See the illustration below.

A piece of arrow shaped uranium-235 is shot from the left to the right to collide with a larger half-sphere-shaped uranium-235 core thus achieving critical mass forcefully and quickly | Ukraine Gave up Thousands Nuclear Warheads — Components Inside of Uranium Nuclear Fission Bomb illustration – Shutterstock asset id: 2271462995 by BlueRingMedia.

Another difficulty is obtaining nearly 100% U-235 from natural uranium. 99% of the Uranium you find in nature is U-238. U-235 and U-238 chemically identical so extracting U-235 from natural uranium is difficult. However, U-235 is slightly lighter than U-238 so you can use centrifugal separation as you do to separate cream from milk. What is typically done is using a uranium compound, uranium hexafluoride, heat it into gaseous form and then utilize centrifugal separation to extract the uranium hexafluoride with U-235 isotopes. After that you can chemically extract the uranium, which is now U-235.

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The Surprising Butterfly Effect

Super fact 40 : In chaotic systems the so-called butterfly effect means that a small change in initial conditions, such as a butterfly flapping its wing in Brazil, can lead to large, unpredictable changes in a system’s future, such as the appearance of a tornado in North Texas. However, that does not mean that the butterfly directly caused the tornado. It should also be noted that chaotic systems can contain predictable patterns and external forcings can certainly make aspects of a chaotic system behave in a predictable manner.

The first part of Super Fact 40 describes a well-established phenomenon that is often surprising to people who have not heard about it before. The second part (following the word “However”) address a few common misconceptions about the butterfly effect. The butterfly effect is a surprising and widely misunderstood phenomenon and therefore I consider the information in bold above to be a super fact.

The Surprising Butterfly Effect — Photo by Cindy Gustafson on Pexels.com

The Butterfly Effect and Unpredictability

The butterfly effect is the sensitive dependence on initial conditions in which a small change in one aspect of the system can result in large differences later. A butterfly flapping its wing in Brazil, leading to the appearance of a tornado in North Texas, is one example. The butterfly effect is an aspect of chaos theory.

However, it is important to understand that the butterfly is not directly causing the tornado. It is the wing flaps of trillions of butterflies, the wing flaps of 50 billion birds, the barks of 900 million dogs, all the waterdrops in the world, and all the bushes and trees, etc., which together provide the initial conditions for the world’s weather system.

Remove one butterfly, anyone of them, or the bark of a dog, and you may or may not have a tornado in north Texas on a certain date. It isn’t the butterfly causing the tornado. Any tiny change in the initial conditions will eventually lead to a large difference in the system later. This is how the Butterfly Effect provides unpredictability.

A large well-formed tornado over the plains. — Did a butterfly do this? Stock Photo ID: 2369175167 by g images.com.

The Butterfly Effect and Predictability

Because of the butterfly effect you may not be able to predict whether it is going to rain at 1:00PM next Thursday, but you can still safely predict that Dallas, Texas, will on average be cooler in January than in July. That’s largely because the sun will heat Dallas, Texas more in July than in January. We know that if you add carbon dioxide, or other heat trapping gases, to the atmosphere it will on average get warmer. External forcings make aspects of chaotic systems predictable. You sometime hear the argument that “climate is chaotic and cannot be predicted”. This is a myth that is debunked here.

In addition, chaotic systems can feature predictable patterns, even though chaotic systems are considered unpredictable. Chaos theory demonstrates that within the apparent randomness of chaotic systems, there are underlying statistical patterns, self-similarity, fractals, and interconnection.

I once created a robot control system for which the robot was shaking a little bit. The tool tip was moving in a little circle and did not get to where it was supposed to be. The reason was that the presence of static friction made the control system I was using a chaotic system.

However, the robot didn’t randomly go all over the place. It was moving quickly in a small circle. It was chaotic, and its exact motion was unpredictable, but there was an underlying statistical pattern. Another example is, fractals, which are geometric patterns that emerge from chaotic processes described by chaos theory. Fractals feature self-similar patterns repeating at different scales. They can visually represent the complex behavior of these systems. See an example below.

A 450 layer fractal | The Surprising Butterfly Effect — This is a file from the Wikimedia Commons Wikipedia. Simpsons contributor at English Wikipedia, Public domain, via Wikimedia Commons.

Edward Norton Lorenz

In 1961, Edward Norton Lorenz was using a computer to simulate weather patterns by modeling 12 variables (heat, wind, etc.). After finishing one simulation he wanted to see it again but to save time he started it in the middle using the saved variables at the point. To his surprise his simulation ended up with completely different weather. He realized that the computer the saved data had tiny errors from the computer rounding off the numbers. For example, 3.145787 instead of 3.1457872. That small difference was enough to eventually result in completely different weather.

Lorenz was not the first person to realize that, so called, non-linear systems can be extremely sensitive initial data. This realization goes all the way back to the mathematician Henri Poincaré in the 19^th century. However, he was the founder of modern chaos theory and coined the term the Butterfly Effect.

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What do you think about the Fractal above?