The goal of this blog is to create a list of super facts. Important facts that are true with very high certainty and yet surprising, misunderstood, or disputed by many. This blog aims to be challenging, educational, and fun, without it being clickbait. I determine veracity using evidence, data from reputable sources and longstanding scientific consensus. Prepare to be challenged (I am). Intentionally seek the truth not confirmation of your belief.
Category: Math & Logic
Important but surprising facts relating to Math or Logic
Super fact 113 : There are shapes that have a limited volume but an infinite surface area. Two examples are Gabriel’s Horn and the Menger Sponge.
3D illustration of Gabriel’s horn. RokerHRO, Public domain, via Wikimedia Commons.
You get Gabriel’s horn by rotating the curve y = 1/x around the x-axis in a coordinate system with the horn starting at x = 1, and y = 1, as in the picture below. Well, you can choose other start values too. A Gabriel’s horn with a beginning radius 1, or widest radius 1, (as in this example) will have a volume equal to pi. Pi is a popular constant. However, it will have an infinite surface area.
This seems like a paradox. The amount of paint you need to fill up Gabriel’s Horn (with widest radius = 1) is pi. However, the surface area of the outside and inside is infinite. So, wouldn’t you need an infinite amount of paint to paint the inside of Gabriel’s Horn? The amount of paint you need to fill up the horn should be more than if you just paint the inside, shouldn’t it? What’s going on? The solution to the paradox is to realize that the radius of Gabriel’s Horn will become increasingly small as it stretches out to the right, and for a coat of paint to take up volume it must have thickness. This is explained well here.
If you rotate this curve around the x-axis, you get a trumpet shape. That is Gabriel’s Horn. I was lazy and drew this using ChatGPT instead of drawing it myself.
To understand why it is possible for the surface area of Gabriel’s Horn to become infinite you can imagine two cylinders of equal volume, one short (and thick), and one long (and thin). The longer and thinner cylinder will have a larger surface area as shown in the picture below. As Gabriel’s Horn is stretched out and getting thinner and thinner you get an infinite surface area, as you go towards infinity, while the volume does not become infinite. This is analogous to the infinite series in my previous post where adding an infinite number of subsequently smaller addends results in a finite number (corresponding to the volume being finite in this case).
This picture (drawn by me) shows that when you elongate a cylinder but keep the volume the same the dark blue surface area gets larger.
The way you construct a Menger sponge or a Menger cube is by starting with a cube.
Then divide every face of the cube into nine squares in a similar manner to a Rubik’s Cube, dividing the cube into 27 smaller cubes.
Then remove the smaller cube in the middle of each face and remove the smaller cube in the center of the larger cube, leaving 20 smaller cubes. This is a level 1 Menger sponge.
Repeat steps two and three for each of the remaining smaller cubes and continue to iterate infinitely many times.
As you are repeating this process over and over the volume of the Menger sponge will decrease a little bit in every step whilst the area will grow towards infinity.
An illustration of the iterative construction of a Menger sponge up to M3, the third iteration.
The Jerusalem Cube is like the Menger sponge/cube but instead of removing cubes you remove cross or plus looking 3D shapes from the larger cubes.
Super fact 112 : Adding infinitely many numbers may result in a finite number. In addition, adding infinitely many numbers may result in an irrational important constant such as Pi. The same holds true for infinitely nested radicals (square roots).
The differently colored rectangles represent the fractions in the equation. Each subsequent addend is the half of the previous. Despite having an infinite number of addends, the total sum is just 1.
That you can add infinitely many numbers and get a finite number as the result is possible to understand if you imagine cutting a rectangle into smaller and smaller pieces and then adding them to get the rectangle back. If you start with half the rectangle and then you add the half of the remaining half and then the half of that remaining half, etc., you can keep doing that forever without exceeding the size of the rectangle. This is illustrated in the picture above. Note all these pictures are drawn by me.
If you’ve never seen an infinite series before this may come as a surprise. However, what is even more surprising is that you can add an infinite number of addends that are constructed from simple patterns and get all kinds of surprising results including irrational numbers with special meaning such as pi. You can easily find thousands of examples in mathematical handbooks and online. This reality is important in mathematics and our understanding of the world, as well as surprising, and therefore a super fact in my opinion.
Three fascinating examples of infinite series. Note that I indicate multiplication using a star *.
Infinitely Nested Radicals
In addition, to adding an infinite number of addends you multiply an infinite number of factors and end up with a non-infinite (finite) result. You can even have an infinite number of nested radicals. To explain what a radical is. A square is a number multiplied by itself. For example, the square of 5 is five times five, which is twenty five. A cube is a number multiplied by itself three times. The cube of five is five times five times five, which is one hundred and twenty five. The square is denoted by adding a superscript of 2 (5 with a superscript 2). The cube is denoted by adding a superscript of 3 (5 with a superscript 3).
The square root is the opposite of the square. The square root of twenty five is five. The cube root is the opposite of the cube. The cube root of one hundred and twenty five is five. The square root and the cube root are examples of radicals. Radicals are indicated by using a little house on top of the number as shown in the pictures below. For radicals that are not square roots you add a number indicating what type of radical you have. The cube root has the number three above the house. All the examples below are square roots and in those cases the number two is left out.
The three pictures below show one example of infinitely nested radicals (square root) using numbers n(n-1) repeatedly in the square roots. When n = 2 then n(n-1) is 2*1 = 2. When n = 3 then n(n-1) is 3*2 = 6, etc.
Infinitely nested square roots using n = 2 is the same as 2. Infinitely nested square roots using n = 3 is the same as 3, etc.Infinitely nested square root for n = 9,10,11,12
Infinite Series and Pi
The constant pi is a special mathematical number that tells you exactly how the distance around the edge of any circle compares to the distance straight across the middle (diameter). Pi is an irrational number, meaning it cannot be expressed as a fraction and when written as a decimal it has an infinite number of decimals that have no repeating patterns. Despite pi being irrational, it shows up as the result of a very large number of infinite series that follow surprisingly simple patterns.
The first 200 decimals of pi.Infinite multiplication and infinite number of addends.Infinite series and infinitely nested square roots (radicals) resulting in pi.
Super fact 111 : Russel’s Paradox is a logical contradiction discovered in 1901 that showed that the mathematical discipline of “Set Theory” was fundamentally flawed. Mathematicians had naively assumed that any definable property can be used to form a collection (or set) of items, but that is not true. An example of the Paradox is “A male barber shaves all men who do not shave themselves and only men who do not shave themselves. Does he shave himself?” Both “yes” and “no” are impossible answers. That is an example of an impossible set. Set theory needed an exclusion of such impossible sets.
Bearded client visiting barber shop. Shutterstock asset id: asset id: 1821348236 by Body Stock.
Russell’s paradox is a famous logical contradiction discovered by the philosopher and mathematician Bertrand Russell in 1901. To solve the contradiction, you need to remove the assumption that any property can form a set. In other words, not every set is possible. Basically, self-reference cannot be allowed.
To take the example above “A male barber who shaves all men who do not shave themselves and only men who do not shave themselves.” Is something that cannot exist. If the barber shaves himself then he is shaving someone who shaves himself, which was not allowed. If the barber does not shave himself, then he is not shaving all the men who do not shave themselves. Either way, it does not work. Such a barber cannot exist. In general, you cannot define a set anyway you like.
I consider this a super fact because it shows that contractions can be hidden even in mathematical disciplines, and it is important because you certainly don’t want contradictions hidden in a mathematical or scientific discipline. Contradictions lead to more contradictions and lots of problems.
Bertrand Russell portrait. Honourable Bertrand Russell.jpg: Photographer not identifiedderivative work: Conquistador, Public domain, via Wikimedia Commons
A Crazy Barber Story Involving Our Children
This happened soon after the September 11 attacks in 2001. In addition to planes crashing into buildings, there were attempts at biological warfare by spreading anthrax through the postal service. This is something we paid special attention to at my work because we were making postal sorting machines. It is also the reason I do not like people who write addresses in cursive.
Anyway, my wife called me at work, and she was very upset because our daughter’s hair was falling out. She touched her hair and it just fell off. She did not know what could be causing her hair to suddenly fall out, but she thought that it might have been biological warfare. I told her to call our doctor who had the good sense of suggesting that perhaps the kids had been playing barbershop. As it turned out they had. Our son confessed to cutting off our daughter’s hair. He had realized that this was bad, so he tried to put her hair back as well as he could. Afterwards, she was walking around with loose hair on top and that’s when my wife found her.
Our son is cutting his sisters hair. The picture is generated with the help of ChatGPT.
Super fact 80 : A conic section is a shape formed by slicing a cone with a plane. There are four such shapes, circle, ellipse, parabola, and hyperbola. The conic sections universally describe motion under gravity. The orbits of planets around their stars are circles or ellipses, comets fly around space in elliptical orbits, or parabolic or hyperbolic paths. Objects thrown up in the air follow parabolic paths. They are the basis for a huge amount of engineering applications.
Types of conic sections : circle , ellipse , parabola , hyperbola Shutterstock Asset id: 2377159367 by ProfDesigner
The four conic sections, circle, ellipse, parabola and hyperbola are fundamental and very useful shapes in mathematics, physics and engineering. Well, a circle is a special case of an ellipse, so it is really only three conic sections. The motion of the planets and other stellar objects are described by the conic shapes. Isaac Newton derived his law of gravitation from Kepler’s laws, which describe planetary orbits as ellipses.
The conic sections are all described by second degree equations (quadratic equations) and are in that sense the simplest shapes aside from points and lines. It is important to understand that there is an infinite amount of shapes that are almost conic sections and look like conic sections, but it is the exact mathematical properties of the four conic sections that make them so common in physics, mathematics, nature and engineering.
The black boundaries of the colored regions are conic sections. Not shown is the other half of the hyperbola, which is on the unshown other half of the double cone. by Magister Mathematicae, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=18556148
It may not come as a surprise that the circle is a fundamental and important shape, but I believe that the fact that the other conic sections are also fundamental in mathematics, physics and engineering come as a surprise to people outside of the STEM fields. It is a true and an important fact regarding how our world works.
Conic Sections
As mentioned, the conic sections are fundamental shapes that appear in a lot of places in STEM. Below are a few examples.
Parabola
Math function parabola. Shutterstock Asset id: 1628916337 by EleonoraDesigner
A parabola is formed when a plane cuts a cone, so the plane is parallel to a side of the cone. Parabolas are shapes that are roughly U-shaped and described by the equation y = x^2 or more generally by y = ax^2 + bx + c. Parabolas have a so called focus point. See the picture below. If you throw a ball, or any object, up in the air its trajectory will be a parabola (ignoring distortions caused by friction and wind). I should say the parabola you get in this case is upside down. The parabola is important when you design any kind of projectile.
Part of a parabola (blue), with various features (other colours). The complete parabola has no endpoints. In this orientation, it extends infinitely to the left, right, and upward. Picture is from Wikipedia Melikamp, CC BY-SA 3.0 https://creativecommons.org/licenses/by-sa/3.0, via Wikimedia Commons.
Antennas shaped like parabolas (in 3D) will direct incoming radiation and waves towards their focus point. If the surface is reflective a light located at the focus point will reflect to create a straight beam. Parabolas are used for radio telescopes, satellite dishes, car headlights, flashlights, solar cookers, solar power plants, water fountains, suspension bridges, business modelling and thousands of engineering applications. Parabolas like circles and the other conic sections shape our modern world (pun intended).
Würzburg-Riese radar built by Germany in WW2 had a 7.4 meter (24 foot) dish. From this page. Alan Wilson from Stilton, Peterborough, Cambs, UK, CC BY-SA 2.0 https://creativecommons.org/licenses/by-sa/2.0, via Wikimedia Commons
Ellipse and circle
As mentioned, a circle and an ellipse are conic sections formed by intersecting a plane with a cone. You get a circle when the cuts perpendicular to the cone’s axis (see pictures above) and an ellipse form when the plane intersects the cone at a slant but not slanted so much that it becomes a parabola or a hyperbola. An alternative for an ellipse is that the sum of the distances from any point on the curve to two fixed points (called the foci) is a constant. See the picture below. The two definitions are identical. For a circle the two foci are merged into one point at the center.
There are a lot of real world examples of ellipses. Planets orbit the Sun in elliptical paths. The sun is in one of the foci points. The orbits of other stellar objects and satellites are also elliptical. Charged particles follow elliptical paths within magnetic fields. Elliptical patterns are observed in the rotation of ocean currents, elliptical models and algorithms are used in medical imaging, computer science and encryption. Also whispering galleries.
Hyperbola
Comets and spacecraft that are not orbiting another body, in other words, they have enough speed to escape the gravitational pull and continue into deep space, will travel along a hyperbola. The boundary of a shockwave from a supersonic jet (a sonic boom) creates a hyperbolic curve on the ground as it moves. The intersection of two sets of concentric ripples in water makes a hyperbola. The light beam from a lamp or flashlight makes an ellipse or an hyperbola on a plane depending on the angle.
Newton’s Law of Gravitation
Johannes was an early 17th century German mathematician who derived three laws that describe how planetary bodies orbit the Sun using the observational data collected by the Danish astronomer Tycho Brahe. The three laws are the following:
Planets move in elliptical orbits with the Sun as a focus.
A planet covers the same area of space in the same amount of time no matter where it is in its orbit.
A planet’s orbital period is proportional to the size of its orbit (its semi-major axis).
Illustration of Kepler’s laws with two planetary orbits. The orbits are ellipses, with foci F1 and F2 for Planet 1, and F1 and F3 for Planet 2. The Sun is at F1. The shaded areas A1 and A2 are equal and are swept out in equal times by Planet 1’s orbit. The ratio of Planet 1’s orbit time to Planet 2’s is (a1/a2)^3/2 Hankwang, CC BY-SA 3.0 <http://creativecommons.org/licenses/by-sa/3.0/>, via Wikimedia Commons
Later Isaac Netwon would use Kepler’s three laws to derive his law of gravity. Newton showed that an inverse-square force (gravity) directed toward the sun was necessary to explain the orbits.
I started this blog, superfactful, in August of 2024. The goal of my blog is to create a list of facts that are important, not trivia, and that are known to be true yet surprising, shocking or disputed by large segments of the public. I determine what is true by evaluating the evidence I find in reliable reputable sources and if a longstanding scientific consensus is available that certainly helps. In some cases, I have expertise in the subject myself, which also helps. Whether a fact is important and surprising or disputed is a judgment call. In some cases, there are polls to help me determine how surprising or disputed the fact is. I am trying to avoid trivia and click bait, and I am only focusing on what is true, important and mindboggling.
It is a project I hope to learn a lot from, and I hope others will also learn something from reading it. We are all drowning in misinformation, false beliefs and unsubstantiated assumptions. We often know and understand less than we think. I have been bamboozled in the past and I am pretty sure you have too. If this blog can spread a little bit of light, I am happy.
In 2024 I posted 25 super facts and in 2025 I posted 53. I am hoping to one day to have collected 200 super facts. I have also made 64 other kinds of posts on this blog such as book reviews for educational books as well as other fact related posts. Below are ten selected super facts from 2025. To read the full post click the links.
That Earth is round was well known long before Columbus
Super fact 28: 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.
I’ve realized that this comes as a surprise to some. To read the post click here.
Columbus thought earth was smaller. He did not know about the Pacific Ocean. Earth Pacific Ocean view Stock Illustration ID: 1617553012 by Matis75
EV Cars Indeed Emit Less Carbon Pollution
Super fact 29: EV Cars emit less pollution than Internal Combustion Engine, even considering manufacturing, disposal and EV Cars being charged by dirty grids.
There is a lot of misinformation about EVs including that EVs are not better for the environment. To read the post click here.
Lifecycle greenhouse gas emissions comparison of average gasoline car and 300 mile range EV. Feedstock and fuel include the generation of electricity for EVs.
Scientists Agree that Global Warming is happening and that we are the Cause
Super fact 34: Climate Scientists agree that Global Warming or if you call it Climate Change is happening, and that it is caused by us primarily because of our burning of fossil fuels. There is a long-standing scientific consensus on these two facts because the evidence is conclusive. Typically, studies show an agreement of at least 97% or 98% among climate scientists.
Polls show that most American are unaware of the consensus among climate scientists. To read the post click here.
There is almost total agreement among climate scientists that global warming, or climate change, is happening and is caused by us. To understand why, you need to know a little bit about the impressive evidence, which for all practical purposes is conclusive. Take a look at this post “Global Warming is Happening and is Caused by us”
The green graph corresponds to “most scientists think global warming is happening (%).” The black graph corresponds to “there is a lot of disagreement among scientists (%)”. The yellow graph corresponds to “Most scientists think global warming is NOT happening (%)”. Graph taken from the Yale Program on Climate Change Communication.
Time Dilation Goes Both Ways
Super fact 38 : If two observers are moving compared to each other both will observe the other’s time as being slower. In other words, both observers will observe the other’s clocks as ticking slower. Time slowing down is referred to as Time Dilation.
Clocks slow down as you travel at high speeds. However, the person travelling think they are standing still. It is the other person who is travelling. This is confusing. To understand it click here.
In this picture Amy is traveling past Alan in a rocket. Both have a laser. Both measure the speed of both laser beams to be c = 299,792,458 meters per second. The speed of light is a universal constant.
Emissions of ozone-depleting gases have fallen by 99 Percent
Super fact 41 : Largely thanks to the Montreal Protocol in 1987 the emissions of ozone-depleting gases have fallen by more than 99%, 99.7% to be exact, according to Our World in Data. This has resulted in halting the expansion of the ozone holes and the reduction in emissions of ozone-depleting gases is saving millions of lives every year.
A gigantic victory for the environment that few are aware of. To read the post click here.
The phase out of six ozone depleting gases. Data source UN Environment Program (2023).
Sulfur dioxide pollution has fallen by 95 percent in the US
Super fact 44 : Sulfur dioxide pollution in the US has fallen by approximately 95% since the 1970s. This significant reduction is primarily due to regulations like the Clean Air Act. Global sulfur dioxide pollution has also fallen but not as much.
Another big victory for the environment that we seldom hear of. To read the post click here.
US sulfur dioxide pollution since 1800. Data Source: Hoesly et al (2024) – Community Emissions Data System (CEDS). This graph is taken this page in Our World In Data. US Emissions peaked in 1973.
I should mention that by clicking this link you can visit the graph above Our World in Data and select different countries and regions and play around with the settings.
We Exploded Thousands of Nuclear Bombs
Super fact 48 : Since 1945 we have set off more than 2,000 Nuclear Bombs corresponding to a yield of an estimated 42,000 times that of the Hiroshima Bomb.
That we have exploded these many nuclear bombs was a surprise to me and perhaps to you too. To read the post click here.
This is an illustration of the Tsar Bomba explosion by by mbafai Shutterstock Asset id: 2208486661. To see a photo of the actual Tsar Bomba explosion click here (it is copyrighted).
The Euler Number Math Magic
Super fact 53 : The Euler number denoted e, is an irrational number, which like the number pi is extremely important in mathematics. In addition, the relationship between the Euler number and pi; seemingly unrelated numbers, is quite amazing, especially if you throw the imaginary number: i = square root of -1 into the mix. Euler’s formula e^ix = cos(x) + isin(x), where x is degrees expressed in radians, is mind blowing to say the least. Radians means that 180 degrees is replaced by pi, and 90 degrees is replaced by pi/2, etc. A simpler special case, but equally amazing is Euler’s identity e^ix = -1, or e^ix + 1 =0. This is amazing math assuming you understand it.
One of the most amazing math facts explained. To read the post click here.
Euler’s formula in cyber space with grid 3d illustration, Asset id: 1636161301 by Giggle2000
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.
A surprise to the people who are convinced that there really is a mystery. To read the post click here.
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.
Evolution is a Fact
Super fact 63 : Evolution is both a fact and a scientific theory. It is a fact that life has changed over time. This is supported by overwhelming evidence, while the theory of evolution provides a comprehensive scientific explanation for these changes, using processes like natural selection.
Yes, there are scientific facts, and that evolution is happening is an observed scientific fact. To read the post click here.
The fossil record is a lot more solid and much less problematic than the creationist books I have read claimed. Shutter Stock Photo ID: 1323000239 by Alizada Studios
Super Facts 