Super fact – Super Facts

Shapes with Limited Volume and Infinite Area

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.

A trumpet looking shape with a decreasing thickness as you move right along the trumpet. It stretches out infinitely far. — 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.

A coordinate system with the curve y = 1/x beginning at x = 1 and y =1. — 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).

The Menger Sponge

The picture shows a Menger Cube with square holes in its surface. There are also square holes inside the cube. — An illustration of M4, the sponge after four iterations of the construction process. Niabot, CC BY 3.0 https://creativecommons.org/licenses/by/3.0, via Wikimedia Commons

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.

Four cubes, each representing a step in the creation, M0, M1, M2, M3 — 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.

Looks like Menger cube but with different shapes. — Third iteration Jerusalem cube. Affixidien, CC BY-SA 3.0 https://creativecommons.org/licenses/by-sa/3.0, via Wikimedia Commons

There are many other shapes with a finite volume and infinite surface area. However, there are no geometric shapes with an infinite volume and limited surface area.

Other Mathematics Superfacts

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Infinite Series Magic

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 picture features an infinite series and a rectangle divided into differently colored smaller and smaller rectangles. — 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 infinite series following very simple patterns. Two results in a simple fraction and the third yields the constant pi. — 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.

The picture displays the generic formula for this infinitely nested square root and three examples. — 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 = 5,6,7,8

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.

pi = 3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196… — The first 200 decimals of pi.

Two different infinite series, the first being a numerator with infinitely many factors multiplied and a denominator with infinitely many factors multiplied. The second an infinite number of fractions as addends. — Infinite multiplication and infinite number of addends.

Two infinite series and one infinitely nested radical. — Infinite series and infinitely nested square roots (radicals) resulting in pi.

Other Mathematics Superfacts

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Russels Paradox

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. Barber scissors and straight razor, barber shop, suit. Vintage barber shop, shaving. Portrait bearded man. Mustache men. Brutal guy, scissors, straight razor. | Russels Paradox — 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.

It is a black and white photo of the mathematician Bertrand Russel. — 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.

Was our son the barber who cut everyone’s hair and only those who did not cut their own hair? No because that barber can’t exist. Some sets can’t exist and you need to include that in the definition of what a set is, or in this case what kind of groups of barbers you can have. — Our son is cutting his sisters hair. The picture is generated with the help of ChatGPT.

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30-60 million Bison used to roam the plains

Super fact 110 : At the end of the 18^th century there were 30-60 million North American Bison (buffalo) roaming the plains. The mass destruction of the bison began in 1830 and was intentional and by the end of the 19^th century there were only a few hundred left. Since then, they have recovered and today there are 500,000 Bison including 30,000 wild Bison.

The picture shows a male plains bison standing on a plain and looking into the camera. — Scientists are helping users of American rangelands meet the challenge of managing multiple uses and sustainably. This picture is in the public domain because it contains materials that originally came from the Agricultural Research Service, the research agency of the United States Department of Agriculture. From Wikimedia commons author Jack Dykinga.

In the past there were at least 30-60 million bison, roaming North America. As mentioned by the end of the 19^th century there were only a few hundred left. They have recovered since then and now there are 500,000 Bison including 30,000 wild Bison. For comparison there are 86.2 million cattle in the US and another 11.8 million in Canada. This is just one of many reasons that the wild mammal biomass has substantially declined around the world. Wild mammals have declined by 85% since the rise of humans even as the total mammal biomass has nearly tripled.

The reason for the extremely sharp decline of the Bison in the 19^th century was because the U.S. government intentionally drove the bison to the brink of extinction. The American bison was a major resource for the traditional way of life of the native Americans and therefore the extermination of the Bison became an important tool in the efforts to subjugate Native Americans.

30-60 million Bison used to roam the plains | 1892: bison skulls await industrial processing at Michigan Carbon Works in Rougeville (a suburb of Detroit). Bones were processed to be used for glue, fertilizer, dye/tint/ink, or were burned to create "bone char" which was an important component for sugar refining. In the 16th century, North America contained 25–30 million buffalo. — 1892: bison skulls await industrial processing at Michigan Carbon Works in Rougeville (a suburb of Detroit). Bones were processed to be used for glue, fertilizer, dye/tint/ink, or were burned to create “bone char” which was an important component for sugar refining. In the 16th century, North America contained 25–30 million buffalo. This picture is in the public domain and taken from this Wikipedia.

I consider this a super fact because it is a shocking historical event that it seems many are unaware of. I was certainly surprised the first time I read about it.

How Did the Bison Population Recover ?

At the end of the 19^th century indigenous groups used the Flathead Indian Reservation to quietly protect small captive herds, keeping bison alive while they vanished from the wild. In addition, a few individuals as well as Native American families captured stray calves and started private breeding programs.

A herd of grazing bison on a field in Yellowstone. In the background are geysers and hills. — Herd of American bison grazing in a green meadow at Yellowstone National Park, with geysers and mountains in the background under a bright blue sky. Shutterstock asset id: asset id: 2688666937 by NicoleHFlores

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The Number of people in the World without Electricity has halved since 2000

Super fact 109 : The global number of people without electricity has halved since 2000, but it has increased in Sub-Saharan Africa. In the year 2000 1.35 billion people in the world was without electricity and in the year 2023 it was 675 million people according to Our World in Data. The dark aspect of the statistics is that the number of people without electricity in Sub-Saharan Africa has increased from 500 million people to more than 600 million people.

The graph shows the number of people without electricity in five regions of the world, Europe and Central Asia, Latin America and the Caribbean, East Asia and the Pacific, MENA, Afghanistan, and Pakistan and Sub-Saharan Africa. The graph shows that the number of people in the world without electricity has gone from 1.35 billion to 675 million, and that all regions except Sub-Saharan Africa show a dramatic improvement. As a result, Africa now has the vast majority of people in the world without electricity. | The Number of people in the World without Electricity has halved since 2000 — The global number of people without electricity has halved since 2000, but it has increased in Sub-Saharan Africa. Access to electricity is defined as having an electricity source that can provide very basic lightning and charge a phone or power a radio for 4 hours per day. Data source: Data compiled from multiple sources by the World Bank. CC BY. The graphs come from this page from Our World in Data.

This also means that the share of people without electricity living in Sub-Saharan Africa increased from 37% in the year 2000 to 80% now. However, it is also true that the share of people in Sub-Saharan Africa with electricity has doubled, rising from 26% to 53%. What is going on is that population growth has outpaced this expansion, meaning the number of people without electricity has still risen.

For the rest of the world, it is unequivocally good news. For example, in South Asia more than 500 million people lacked electricity in the year 2000, 414 million lacked electricity in 2010, and only 27 million people lacked electricity in 2023. Another way to look at the access to energy gap is by considering how long it is possible to run an air conditioner in different countries. Click here for data and analysis. In summary, the news for Sub-Saharan Africa is complicated but for the rest of the World, it is very good news. To read more about this topic click here.

I consider this a super fact because I believe it is an important but surprising fact in two parts. First the great news for the world and secondly the mixed news for Sub-Saharan Africa.

No Relief for the Heat Down in Africa

As mentioned, four of the five regions of the world, Europe and Central Asia, Latin America and the Caribbean, East Asia and the Pacific, MENA, Afghanistan, and Pakistan have made substantial progress. However, Sub-Saharan Africa is a complicated case. Note that MENA stand for Middle East and North Africa.

An alternative way of looking at access to electricity is to consider how much electricity is being used in each country and how that translates into the number of hours or minutes that an air conditioner could be running. Note it doesn’t mean that an air conditioner will be shut off after, let say 25 minutes, just that is much electricity one person use. What should be noted from the graph below is how dire the situation still looks like in Sub-Saharan countries. Sub-Saharan Africa is still very behind in this regard, and the fact that global warming is likely to hit Africa very hard that is not good news.