Surprising fact – Page 6

Polar Nights Are Phenomenal and Cover a Large Area

Super fact 69 : Polar nights happen in the regions north of the Arctic Circle and south of the Antarctic Circle. Together the two polar regions cover an area double as large as the United States, and it includes a population of millions of people. During a polar night the sun does not rise for days, or months, up to six months. The polar nights are special, not just for the daytime darkness, but for the spectacular aurora borealis, the starlit skies, the extremely clear air, and the fact that people often lose track of time during polar nights.

It should be noted that even though the sun never rises during the polar night it may not be totally dark during the entire polar night. Part of the time you may get some light in the middle of the day, what is called polar twilight (nautical twilight or astronomical twilight). Nautical twilight tends to be bluish as shown in the photo below.

Characteristic nautical (blue) polar twilight in Longyearbyen, Svalbard, Norway. Bjørn Christian Tørrissen, CC BY-SA 3.0 <https://creativecommons.org/licenses/by-sa/3.0>, via Wikimedia Commons

When our kids were young, we visited the Ice Hotel in Jukkasjärvi in northern Sweden. We went in March, during the American spring break, which was not during the polar night, so we did not experience it then. The polar night in Jukkasjärvi will begin December 5, 2025, or December 10, 2025, depending on how you count, and end at the beginning of January. However, the Ice Hotel was a fun way to experience a polar region. When I did my Swedish army service I spent the winter in the forest around that area, and even further north, and I experienced the Polar Night firsthand,

Photo is of a large ice table and ice chairs located in the middle of the lobby, which is filled with tall pillars made of ice at the ice hotel. Polar Nights Are Phenomenal and Cover a Large Area — The lobby of the ice hotel. Our kids are sitting around a table of ice. A chandelier of ice is hanging above the table. The light is from fiber optics, so the ice won’t melt.

During my army service we stayed in tents in the forest far away from villages and cities and we often did not use any lights. The result was that it was incredibly dark most of the time. It was so dark that I could not see my own hand if I held it right in front of my face. The upside was that the sky was filled with thousands of stars on a clear night. It is estimated that you can see between 2,500 to 5,000, even 10,000, stars in the sky with the naked eye if you have good vision and it is completely dark. You could also see the Milky Way Galaxy very clearly and easily, as well as the Andromeda Galaxy, meteors, comets, satellites, not to mention the most spectacular aurora borealis that you can see anywhere on earth. That certainly beats the 14 stars and 3 planets that you can see with the naked eye on a clear night here in Dallas, Texas, where I live now.

Stars in the sky. Blue night panorama, a universe full of stars, nebula and galaxies | Milky way sky on dark background, and a lot of stars. — Milky way sky on dark background, and a lot of stars. Asset id: 2524020369 by MR.PRAWET THADTHIAM

On one occasion I was standing guard outside the camp. We were pretending that we were at war with the Soviet Union. It was minus 40 degrees, and they forgot about me. I stood there alone in the darkness and the extreme cold for more than three hours. This was considered dangerous. The rule was no more than one hour out in the cold at a time. Once they discovered that I had been there for more than three hours they came to get me, and they apologized profusely. They were happy that I was not hurt. However, I did not mind, because my night sky view was spectacular. The night sky was so spectacular, it was to die for.

If it is cloudy and you are far from any civilization it may be pitch black in the afternoon. Even if it is not cloudy but the moon is not out it will be pitch black later in the afternoon. In addition, the air is very clear during the polar night, because the cold sunless Arctic air often contains almost no moisture and the visibility stretches for 100 miles. If you want to remain unseen in this situation it is very important to understand that even the tiniest light will be very visible, far away. Our platoon did an experiment. Someone lit a cigarette about one mile away and we could very easily see the bright cigarette even through the forest. The lesson was, No Cigarettes, and No lights!

Stunning aurora borealis lights up the night sky with vibrant colors, creating a mesmerizing natural display in the Arctic wilderness. | Polar Nights Are Phenomenal and Cover a Large Area — Aurora Borealis are often spectacular in the polar regions, especially during the polar night. Asset id: 2499746583 by HappyVibeArt

I consider this a super fact because the polar night is a surprising phenomenon to those who do not know much about the arctic and my claims above are true and important since the polar regions are large and under threat from climate change / global warming. The polar regions and thereby the polar night also cover an area that is not small, and four million people live there.

Partial map of the northern hemisphere focused on the polar circle. — The Arctic Circle, at roughly 66.5° north, is a commonly-accepted boundary of the Arctic waters and lands. CIA World Fact Book, Public domain, via Wikimedia Commons.

The Midnight Sun

I can add that in the summer (northern hemisphere) you have the opposite phenomenon with the midnight sun. Just like the polar night can cause people to lose track of time so can the midnight sun. When we visited northern Sweden with our niece Jessica for my brother’s wedding, we had what is called midnight sun. Jessica was nine years old at the time. A couple of hours after we had gone to sleep Jessica woke me up “wake up Thomas, it is time to go out and play”. I protested, no it is not, it is one a clock at night and we are going to sleep. Jessica protested, she said, “No the sun is up, it is morning it is time to play”.

So, I said “Jessica, let me explain something to you. This is northern Sweden in June and here the sun is up in the middle of the night, and we are not going outside to play in the middle of the night.” After that my wife went out to play with Jessica, but I refused to go out and play at one o clock at night.

Ice Hotel Photos

Finally, I just wanted to include a few of my photos from the Ice Hotel, just for fun.

He is standing and kicking with his feet to move forward with the sled. He is passing in front of the ice theater. | Polar Nights Are Phenomenal and Cover a Large Area — Our son David with one of the kick sleds called “spark”.

A photo of an ice table and 7 instruments made of ice standing on an ice stage. — This is a photo of the ice instruments standing on the ice stage.

My wife and three children are sitting on a dog sled and looking into the camera. — We are going on a dogsled tour. The ice theater is in the background, and you can see part of the ice hotel on the right.

The kids had a lot of fun during the dogsled tour. | Polar Nights Are Phenomenal and Cover a Large Area — You can see ten dogs and the sled with us on top of it. On the left is a kåta, a movable Sami structure (indigenous arctic Scandinavian people). It is like a Tipi.

To see the other Super Facts click here

Infinities Come in Different Sizes

Super fact 68 : Infinities come in different sizes. Some infinities are larger than others. In fact, it is possible to create an infinite number of different-sized infinities. Some infinities are countable and others are uncountable.

How can infinity be of different sizes? Infinity is infinity after all. Infinity is an absolute, an absolute we cannot imagine. However, at the end of the 19^th century a German mathematician by the name Georg Cantor demonstrated that there exists a variety of infinities, and some are larger than others. In fact, his theorem implies the existence of an infinity of infinities. For those who have not heard of this concept before it is surprising and yet it is true, and it is also kind of an important thing to know about mathematics. This is why I consider this a super fact.

Let me explain. Think about all natural numbers, 1, 2, 3, 4, 5, …, a billion, a quintillion, … You can keep counting forever. There are infinitely many natural numbers. Now take two natural numbers, the numbers 3 and 4. How many numbers are in between 3 and 4? Well, you have 3.5, 3.55555, 3.016893, and you have the number pi, 3.14159265358979323846, and infinitely many other numbers with infinitely many decimals. 3 and 4 are only two numbers but you have infinitely many decimal numbers, or so called real numbers, between 3 and 4. That is true for the two numbers 4 and 5 as well, and 5 and 6, and all natural numbers. Even though there are infinitely many natural numbers there are infinitely many more real numbers.

For the explanation of the mathematical proof see this article in Scientific American. This article explains why there are an infinite number of different-sized infinities.

A neon infinity metaverse symbol. 3D illustration | Infinities Come in Different Sizes — Infinity Asset id: 2118543950 by Sahara Prince

I should say that today Saturday November 8, 2025, is National STEM Day, or STEAM Day. Stem stands for Science, Technology, Engineering, and Mathematics. The extra ‘A’ in STEAM stands for Art. Therefore, I thought that posting something related was appropriate for today, in this case it is Mathematics. I have to admit I am posting it a little bit late in the day though.

To infinity … and beyond!

Madrid, Spain; 05-14-2024: Large figure of the famous character Buzz Lightyear from the movie Toy Story in an exhibition called Pixar World about the studio's films. — Buzz Lightyear “To infinity … and beyond”. Shutterstock Asset id: 2464838811 by MSCT Pics

As mentioned, infinites can have different sizes. How about, for example, the set of all natural numbers and the set of even numbers. After all, it seems like the set of all even numbers should be half as big. As it turns out they are the same size. That’s because for all natural numbers 1, 2, 3, 4, 5 you can always find a buddy, 2, 4, 6, 8, 10 among the even numbers. This is called a bijection. However, you cannot do this with natural numbers versus real numbers. You cannot pair up a natural number with the real numbers because there are infinitely many for each single natural number.

The set of all real numbers is really a larger infinity than natural numbers, which by the way is the smallest infinity. The sets of natural numbers and, for example, even numbers are called countable infinities, and the set of real numbers are called uncountable infinities. In addition, there are infinitely many different sizes of infinities that are larger than both natural numbers and real numbers.

Another interesting, related issue is that Cantor tried to prove that there are no infinities with sizes that lie in between the natural numbers and the real numbers. This was called the continuum hypothesis. As it turned out this hypothesis is not possible to prove (you can prove that it cannot be proved) at the same time as it is not possible to disprove. It is related to Gödels incompleteness theorem but is not the same. However, it is another absolute limit to human knowledge, things we know that we can never know.

If you want to have your mind blown, watch this 8 minute YouTube video about infinities of different sizes and the continuum hypothesis.

Other Mathematics Related Super Facts

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More than half of Internet Traffic is Bots

Super fact 66 : Bots make up more than half of all internet traffic surpassing human activity for the first time in 2024. The 2025 Imperva Bad Bot Report found that bots accounted for 51% of all web traffic. Human activity accounted for 49% of all internet traffic, malicious “bad bots” accounted for 37%, and 14% of traffic comes from “good bots,” such as search engine crawlers.

What made me look up this information is that I’ve recently seen a lot of idiotic and inflammatory comments, as well as inappropriate laughing emojis on social media pages and posts. The pages that seem to be targeted the most by this abuse seems to be pages related to things like climate change, clean energy, EV cars, evolution, vaccines, modern medicine, modern physics, geopolitics, information on political issues, etc. However, those topics might reflect my interests. Perhaps all topics are targeted by this growing strange abuse.

It looks like those among us who slept through the science classes in high school now think they are the real experts and have declared war on all science nerds. The question that arose in my mind is, are these keyboard warriors humans or are they bots? It is true that Artificial Intelligence does not (yet) demonstrate true independent intelligence, but that is true for many people as well. So, how do you know the difference? Anyway, that is the background to why I investigated this issue.

I looked it up and found that bot traffic has been increasing and is now the majority of all internet traffic according to, for example, the 2025 Imperva Bad Bot Report. You can read more here. Facebook is not an exception. 40% of all posts are machine generated. These bots/fake accounts spread a lot of misinformation, inflammatory comments, and some are scams.

Many people are also spreading false information and some of them are scammers, but the fact that machines do it as well add to the problem. It is also very common for bots and fake accounts to leave reactions on Facebook posts, which might be what I saw, but I am not sure. What is certain is that I have come across a lot of false information on Facebook, as well as scams and deep fakes, and Meta/Facebook is obviously not able to clean it out. There are also bots that are remotely controlled ransomware, computer viruses, spyware, and other malware.

A picture of giant computer screen with a red warning triangle containing an exclamation mark. Underneath the triangle is the text “System Hacked” | More than half of Internet Traffic is Bots — Warning of a system hacked. Virus, cyber attack, malware concept. Asset id: 1916985977 by Sashkin

Why I consider this a super fact is because it appears to me that people underestimate the influence of malicious bots. If you had asked me before I looked this up how common bot traffic was, I might have said a few percent. After all streaming, youTube, gaming, etc., require a lot of bandwidth. Considering all the fake stuff and nonsense that is spreading partially with the help of bots, this is dangerous. We know the bots make up more than half of all internet traffic, and bot traffic is growing faster than human traffic, it is important information, and I think it is surprising information to a lot of people, thus making it a super fact.

Fake Nonsense on Facebook

This section is not directly tied to the super fact above, but it concerns a related topic and is based on my personal experience with the social media platform that I have used the most, Facebook. Instagram seems to be even worse, but I am not using it as much. Why I am bringing this up is because increased bot traffic and the increased presence of fake accounts and deep fakes on social media can make this a lot worse. Combined with our gullibility and lack of critical thinking as well as the failure of social media platforms to keep after this, we are facing a serious threat.

Gullible Planet

It is well known that there are a lot of nonsense posts on Facebook (and elsewhere). The fact that we so easily fall for it and don’t check with reliable sources is a big problem. When I see something fake, I often post corrections, for example, using sites like snopes. Sometimes people are grateful, sometimes they get angry, and I’ve even been blocked and lost friends just by posting a snopes link. A lot of the fake stuff is posted by people, but a lot of posts, comments and reactions are posted by bots, and this is becoming more common. With increased malicious bot traffic, AI and deep fakes, we must improve our critical thinking skills.

Below are some examples of fake stuff I’ve come across on Facebook

Did you read that viral article on Facebook claiming that they found 20 feet humanoid skeletons in Turkey? The article stated that archeologists think that they might be fossilized Nephilim, the giants mentioned in the Old Testament. If so, did you doubt the accuracy of the article? If you did, you did good. It was based on an article in a satirical website called World News Daily Report. However, judging from the comment section, including the comments of some of my friends, most people didn’t doubt the article’s accuracy.

How about the story from a purported science magazine that scientists had just discovered that the Easter Island statues/heads have bodies/torsos below the ground. The article stated that this was a revolution in archeology that forced a reevaluation of history. The commentors were amazed over this discovery and some pointed out that not realizing this sooner was a big failure on the part of archeologists and scientists. Well, that the Easter Island statues/heads have bodies/torsos below the ground has been known all along.

How about the story about the lunch lady named Aileen G. Ainuse who poisoned the water supply at Sunnydale High School in Goobersville, Indiana, killing over 300 students and staff. It was accompanied by a scary photo of a starving lady. The readers were shocked and appalled, but not many bothered to verify the story, for example, with the help of snopes. The story was false.

Another article stated that the fact that there were no stars in the black sky in a photo allegedly taken on the moon was proof that the photo was fake and that the astronauts were never on the moon. First of all, it was day, the sun was out. When the sun is out it is very difficult to see the stars because the sun’s light is a million times brighter than the light from the stars and in addition the bright sunlight reflected off the surface of the moon dims the stars. In addition, the cameras used had short shutter speeds for picking up the bright light, not faint stars. Seeing stars in a daytime photo taken on the moon is not something you should expect. Several commentors pointed this out but most other commentors didn’t pay attention and were fooled.

I’ve also seen the opposite, people refusing to believe a true story because they fundamentally misunderstand something. Below is a youTube video showing an animation composed of actual satellite photos by NASA. Many commentors seeing this video insisted that it was a hoax because the back side of the moon is dark. But it is not. When the side of the moon that is turned towards us (the near side) is dark (a new moon) the back side reflects the sun’s light (like a full moon). The backside (far side) of the moon also looks different from the side turned towards us. In the video below the sun is behind the camera and shines on earth as well as the backside of the moon.

A final example is a deep fake Ad featuring Meryl Streep and Dr. Sanjay Gupta promoting an Alzheimer’s cure. I saw it on Facebook several times over a period of several weeks. It looked very real to me, but something felt off, so I fact checked. It turned out that Meryl Streep and Sanjay Gupta had nothing to do with the video. They were AI generated likenesses promoting a scam product. The video used all the typical polemic tricks such as “a cure that the billion dollar companies don’t want you to know about”, “buy now before they take our website down”…. We need to get better at protecting ourselves and believing 20 feet skeletons on Facebook are real is not the way to do it.

To see the other Super Facts click here

United States is Tornado Country

Super fact 65 : The United States covers approximately 6.14% of Earth’s total land area. Despite that approximately 75% of the world’s tornadoes occur in the United States, making the risk of a tornado in the US 46 times higher than in the rest of the world. Tornadoes are particularly common in Tornado alley. Canada is the country with the second-highest number of tornadoes globally.

Our neighborhood was ravaged by an EF3 tornado in the evening of October 20, 2019. Today is the six year anniversary of that event. One year ago, I was interviewed by the NBC about this event. Yes, I was on TV. NBC had found me via my Leonberger dog blog. I’ve included the clip below.

It appears that the most common statistics stated by reliable sources is that approximately 75% of the world’s tornadoes occur in the United States (click here, here, or here). As stated, 75% is an approximate number and another common number is 90%. 90% might be overstated because smaller tornadoes are underreported in the rest of the world. Whether the correct number is 75% or 90% it is clear that the United States has the most tornadoes. It is the Tornado Country of the world.

It is quite astonishing that one country, the United States, has such a high concentration of this extreme weather phenomenon. Most Americans know that their country is special in this regard. I’ve met a lot of Europeans (I am from Europe / Sweden myself) who thinks tornadoes is just Hollywood thing. However, it is a real and very common phenomenon in the United States, and that is surprising to much of the rest of the world. It is also an important extreme weather phenomenon that kills a lot of people and causes billions of dollars in damage every year. This is why I consider “United States is Tornado Country” a super fact.

A large well-formed tornado over the plains | United States is Tornado Country — A tornado. Stock Photo ID: 2369175167 by g images.com.

Tornado Alley

Tornadoes can happen anywhere, but as mentioned they are more common in North America and especially in tornado-alley. Despite tornado alley’s small size, a quarter of all significant tornadoes in the world occurred there according to a study (1921 – 1995). The extent of tornado alley includes north Texas, Oklahoma, Kansas, Nebraska, South Dakota, Iowa, and the corners of Minnesota, Wyoming, Colorado and New Mexico. Dallas, where we live, is in Tornado Alley.

Dixie Alley

Another region with a lot of tornadoes is Dixie Alley. Dixie Alley stretches from eastern Texas and Arkansas across Louisiana, Mississippi, Tennessee, Alabama, Georgia, and mid to western Kentucky to upstate South Carolina and western North Carolina; the area reaches as far north as southeast Missouri.

Dixie Alley has fewer tornadoes than Tornado Alley, but they tend to be deadlier and more violent than in Tornado Alley. The tornadoes in Dixie Alley occur year-round, at night, and the Tornadic storms in Dixie Alley are often high precipitation supercells due to an increase of moisture from proximity to the nearby Gulf of Mexico. The heavy rain makes it difficult to detect the tornadoes. Dixie Alley also has more mobile homes, homes tend to not have basements, and it is more densely populated than Tornado Alley. NOAA records indicate that the majority of deaths from tornadoes in the US have occurred in Dixie Alley.

Map of the United States showing Dixie Alley in red. It includes parts of eastern Texas and Arkansas, Louisiana, Mississippi, Tennessee, Alabama, Georgia, and mid to western Kentucky to upstate South Carolina and western North Carolina and southeast Missouri. — Map showing Dixie Alley. Bhockey10, CC BY-SA 3.0 <https://creativecommons.org/licenses/by-sa/3.0>, via Wikimedia Commons.

Tornadoes and Hurricanes Are Different

Another related issue that has been brought to me is that many Europeans do not know the difference between a hurricane and a tornado. This is not strange since both of these extreme weather phenomena are less common in Europe.

A hurricane is a big rotating storm system originating in the ocean that sometimes makes landfall and devastates our coasts. Hurricanes are big, thousands of square miles and even a million square miles. A hurricane has wind speeds of 74 miles per hour (mph) or higher. That’s 119 kilometers per hour or 33 meters per second. They are called hurricanes in the Atlantic and the northeast Pacific, and Typhoons in the northwest Pacific, and otherwise just cyclones.

Satellite photo of Mexico, southeastern United States and the Caribbean. A hurricane is approaching from the east. | United States is Tornado Country — Satellite photo of hurricane approaching Cuba and Florida. Stock Photo ID: 2202605185 by Emre Akkoyun.

A tornado is a violently rotating column of air that is in contact with both the surface of the Earth and a cloud. They originate with thunderstorms, especially supercells, and are more of an inland phenomenon even though hurricanes can sometimes generate tornadoes. Tornadoes are much smaller than hurricanes in area, almost always much smaller than a square mile, or just 10 or 30 meters across. The reason they can be as deadly as hurricanes is that they tend to have stronger winds and they appear and disappear quicker thus taking people by surprise.

A threatening supercell with lightning — A so called supercell but without a tornado. Stock Photo ID: 1768468151 by Laura Hedien.

A nice photo of a threatening Tornado. | United States is Tornado Country — An F5 tornado in Canada. From Wikimedia Commons.

So, in summary, Hurricanes cover a large area and lasts a long time. They come from the sea and devastate coastal areas and are less common than tornadoes. Tornadoes are formed on land from thunderstorms, typically supercells. They are particularly common in tornado-alley. They cover a small area and go high up into the air forming a funnel and are commonly more intense. They are also more common, and they can surprise people. Therefore, on average, tornadoes cause more deaths in the US than hurricanes. You can read more about it here.

Our Tornado Experience

As mentioned, on October 20, 2025, our neighborhood was ravaged by an EF3 tornado. I was sitting outside in the backyard drinking a beer (Yellow Rose from Lone Pint brewery) when the sirens went off and my cell phone started screaming tornado warning “seek shelter immediately”. I did not see any problem, so I thought it was just one of unnecessary warnings again and I wanted to finish my beer. However, I decided to walk back into the house. About one minute later there was a lot of noise and a piece of the neighbor’s roof flew into our chimney, smashed it and it started raining bricks.

The house shook and thought to myself, “that was some thunderstorm”. I opened the door to the backyard, and I was astounded when I saw the devastation. My backyard was covered in debris, bricks, trees, garbage, my grill had flown off, several of our neighbors’ houses were just rubble. I guessed it must have been a tornado. My wife Claudia told me to go and check on her parents who lived about a mile away. I started driving my Toyota Prius, but I did not get very far because a neighbor’s roof was lying across the road. We live on a circle, so I decided to go the other way instead but about 30 trees lying across the road stopped me. So, I started walking but four firemen stopped me and told me that it was too dangerous to walk around. I had to go home. We would later learn that my wife’s father was injured. I am describing this in the NBC interview below.

We lost power for four days, we needed a new roof and a new chimney, the wiring and piping (for air conditioning) in the attic was destroyed, the fence and garage door were broken. It cost $50,000.00 to repair, nut our house stood, the frame was fine, and luckily our insurance paid for all of it, except for the parasol the flew away. We had no proof it ever existed. Not everyone was that lucky, especially those whose houses were destroyed. Unfortunately, our Leonberger dog Bronco had a heart failure one week after this. However, he recovered. But we are guessing that the fact that we did not have air conditioning for a while could have affected his heart. It was a tough day for Bronco, our Leonberger Dog. <<Link-13>>

A neighbor’s house with the roof ripped off. — A neighbor’s house the morning of October 21^st, 2019.

Debris on the street from a neighbor’s destroyed house. — Another neighbor’s house. This house was about 100 yards from our house. It took a direct hit from the tornado.

This house is totally destroyed. — Another neighbor’s house (a bit further away from us).

My wife Claudia is walking among the debris in my in-laws house. — Inside Claudia’s parents’ house. This was the morning after. We are walking into their house to check on them (that’s my wife).

The entire top of this house is destroyed. — This is the next-door neighbor of Claudia’s (my wife) parents.

The house is completely flattened. A large tree is destroyed. It has no branches. | United States is Tornado Country — Destroyed house in the neighborhood.

A big nice-looking house destroyed by a tornado. — Another house in the neighborhood.

A photo taken from the inside of a destroyed office. — The remains of the veterinary clinic where we used to take our dogs. Luckily there were no animals staying overnight at this time.

Crashed cars and destroyed stores. | United States is Tornado Country — View of the shopping center in our neighborhood.

The Gap store has its entire backside ripped off. — A store at a nearby shopping center

The yellow school bus is wrapped around a tree. — This was a school bus belonging to the school where my boys went.

Trees are ripped up, vehicles are crushed. | United States is Tornado Country — Streetview from the neighborhood.

My Interview with NBC

As I mentioned, last year NBC interviewed me about this event. They said that they had been reading my Leonberger blog and were interested in a post that I had made about the tornado in Dallas on October 20, 2019. They wanted to interview me and asked for permission to use a few of my photos.

A photo of a TV above the fireplace. Thomas Wikman on the TV screen. — Seeing myself on the TV was a bit surreal. Click on the image to see the interview. My interview starts at 1 minute and 11 seconds.

A photo of my TV above the fireplace. I am on the TV screen. | Thomas Wikman NBC Interview — Here I am with NBC journalist Katie Blake. Click on the image or here to see the interview. My interview starts at 1 minute and 11 seconds.

I would like to stress that there are thousands of people whom they could have interviewed. Thousands of people who probably would have been better candidates. What made all the difference was the post I made on my blog. Therefore, fellow bloggers, perhaps this is an example of the fact that blogs bring attention.

To see the other Super Facts click here

Eulers Polyhedra Math Magic

Super fact 64 : V – E + F = 2 is true for all convex polyhedra, where V is the number of vertices (corners), E is the number of edges, and F is the number of faces of a convex polyhedron. This is called Euler’s formula for polyhedra.

Convex polyhedra are polyhedra without holes, or it has no internal angles larger than 180 degrees. To express that exactly; A convex polyhedron is a three-dimensional solid with flat polygonal faces where a line segment connecting any two points on its surface or interior remains entirely within the solid. I can add that convex polyhedra consists of convex polygons as their faces. All its faces are convex polygons.

On the left are three convex polygons and on the left are three concave polygons. — The difference between convex and concave polygons. Notice that the concave polygons on the right have “holes” in them, or inner angles larger than 180 degrees, and that you can draw a line starting and ending inside the polygon that goes outside the polygon.

It is easy to understand that there are an infinite number of convex polyhedra. All pyramids (with convex polygons for bottoms) are convex polyhedra whether the bottom is a square/rectangle, or a convex pentagon, hexagon, heptagon, octagon, or any kind of convex polygon, even with trillions.

A hexagonal pyramid with differently colored faces | Eulers Polyhedra Math Magic | Euler's Polyhedron Formula — A hexagonal pyramid (it is convex). Tomruen at en.wikipedia, Public domain, via Wikimedia Commons.

Examples of convex polyhedra are cubes, boxes, a tetrahedron, pyramids of various sorts, an octahedron, dodecahedron, icosahedron, but there are infinitely many.

As an example, take a cube (like a dice). It has six faces (F = 6), eight vertices (corners V = 8), and 12 edges (V – E + F = 2 or 8 – 12 + 6 = 2). A tetrahedron has four faces (F = 4), four vertices (V =4), and six edges (E = 6) (V – E + F = 2 or 4 – 6 + 4 = 2). A pyramid (with a square bottom) has five faces (F = 5), five vertices (V = 5) and eight edges (E = 8) (V – E + F = 2 or 5 – 8 + 5 = 2). An octahedron has eight faces (F = 8), and six vertices (V = 6), and twelve edges (E = 12) (V – E + F = 2 or 6 – 12 + 8 = 2). A dodecahedron has twelve faces (F = 12), and twenty vertices (V = 20), and thirty edges (E = 30) (V – E + F = 2 or 20 – 30 + 12 = 2). An icosahedron has twenty faces (F = 20), and twelve vertices (V = 20), and thirty edges (E = 30) (V – E + F = 2 or 12 – 30 + 20 = 2). Euler’s Polyhedral Formula works for all of them and for all the infinite number of convex polyhedra that exists.

Line of colorful Platonic solids on a grey surface (polyhedra - tetrahedron, cube, octahedron, dodecahedron and icosahedron, 3d render, banner. — Tetrahedron, cube, octahedron, dodecahedron and icosahedron, Shutterstock Asset id: 2138211027 by Dotted Yeti

I consider this a super fact because it is quite amazing and surprising that the number of faces, vertices, and edges follow the same formula for all convex polyhedra. At least it is surprising if you haven’t seen it before. I should say that Leonhard Euler’s most celebrated discovery is probably his discovery of the Euler Number and the associated Euler’s Formula.

More Polyhedra Examples

The three first examples below are convex polyhedra so Euler’s Polyhedral Formula apply. The last two examples are not so Euler’s Polyhedral Formula does not apply.

Blue truncated icosahedrons. Geometric soccer ball or football shape. Archimedean solid. Regular polygon outline with pentagonal and hexagonal faces. Blue gradient polygonal figure | Eulers Polyhedra Math Magic | Euler's Polyhedron Formula — This is a truncated icosahedron, which is a convex polyhedra, so Euler’s Polyhedral Formula apply. Shutterstock Asset id: 2309812551 by Mima Subota.

A green Rhombicuboctahedron with 8 triangles and 18 squares as faces. — This Rhombicuboctahedron is an example of convex polyhedron, so Euler’s Polyhedral Formula applies. Cyp, CC BY 4.0 <https://creativecommons.org/licenses/by/4.0>, via Wikimedia Commons.

Icosidodecahedron with faces that are triangles and pentagons. — This Icosidodecahedron is another example of a convex polyhedron, so Euler’s Polyhedral Formula applies. Attribution must be given to Robert Webb’s Stella software as the creator of this image along with a link to the website: http://www.software3d.com/Stella.php. A complimentary copy of any book or poster using images from the Software would also be appreciated where feasible., Attribution, via Wikimedia Commons

This polyhedron is donut shaped | Eulers Polyhedra Math Magic | Euler's Polyhedron Formula — This hexagonal torus / toroidal polyhedron is not a convex polyhedra, so Euler’s Polyhedral Formula does not apply in this case. Tom Ruen, SVG version from de:User:Antonsusi, Public domain, via Wikimedia Commons