A Nation of Immigrants

If the United States is a nation of immigrants, then Jordan is a nation of refugees. While exact numbers are hard to pin down, about one third of the Jordanian population are refugees. This includes people of 44 different nationalities including large numbers of Palestinians, Iraqis, and Syrians. When I take a taxi, I often ask where the driver is from. While most answer Jordan, when asked about their background it is the rare driver who is "Jordanian Jordanian" as opposed to "Palestinian Jordanian". I play frisbee and soccer with young refugee girls from Iraq and Syria and of my six students last semester, two were refugees.

The refugee presence affects life here in both large and small ways. One of the best symbols of the influence was the bedouin tents in the Dana Biosphere Reserve. The Bedouin have lived in the region for generations and most live a traditional lifestyle. As the seasons change, they move their tents to take advantage of weather patterns. They sip sweet tea while herding goats and sheep and at night they sleep on the ground as they always have. But instead of being made of traditional goat hair which would need to be rewoven every year, many of the tents we saw had UNHCR (United Nations High Commission on Refugees) printed on them. I'm not sure how the Bedouin obtained these tents and certainly their living conditions are as much in need of help as many of the refugees but they are some of the few people who can claim to be Jordanian Jordanian - ones whose families were here long before the Hashemite Kingdom of Jordan came to be.

Many of my drivers have been struggling to make ends meet. They drive for Uber between classes or between their other part-time jobs as the constant influx of people has made it difficult to get jobs (even when many of the newcomers are forbidden from taking regular jobs). On the other hand, the people are overwhelmingly optimistic. They talk of their plans to start a business or of opportunities in other countries. Jordan is a country where people can escape from the horrors of war and earn an advanced degree.  I see strong parallels between Jordan and the United States as the land of opportunity.

I came to Jordan to teach algebra and work on my Arabic but have learned so much more. Every day I see the richness that immigrants add to daily life in Amman, whether it is in the smile of a young Iraqi girl catching a frisbee, the Palestinian style falafel that I ate for lunch, or a mathematical discussion with a Kuwaiti student.

Eratosthenes's Earth

Eratosthenes' measurement of the Earth. Illustration. Britannica ImageQuest, Encyclopædia Britannica

Thousands of years before Columbus headed west from the Canary Islands in search of new lands, Eratosthenes used the Egyptian landscape and the sun to calculate the circumference of the earth. He was the head librarian of the famous library of Alexandria in the third century BCE.

A friend of his lived in the town of Aswan (then called Syene) in upper Egypt. This friend mentioned that at noon on the summer solstice, the sun cast no shadow at all; in fact the sun would even illuminate the water in a very deep well. Intrigued by this fact, Eratosthenes went out at noon on the summer solstice and saw that in Alexandria the sun did cast a shadow and the rays hit the earth at an angle of 7.2 degrees. Of course, he did not have our modern units of degrees and instead measured this angle as \(\frac{1}{50}\) of a circle. Now instead of just noting this fun fact, Eratosthenes realized that he could use it to calculate the circumference of the earth.

For the method to work, Eratosthenes needed to make two assumptions which, fortunately, we now know are valid.

1. The earth is round. It should be clear that the question of the circumference of the earth does not even make sense if this is not true.
2. The sun is so large and far away that its rays are parallel to each other as they hit the earth. 

These two assumptions together mean that the curvature of the earth is responsible for the difference in the angle the sun’s rays hit the earth in Aswan and in Alexandria as seen in the diagram above.

Notice that the angle the sun makes in Alexandria is the same as the angle from the center of the earth, which is \(\frac{1}{50}\) of a circle. Therefore the arc between Aswan and Alexandria must be \(\frac{1}{50}\) of the circumference of the earth. In other words, the circumference of the earth is 50 times the distance from Aswan to Alexandria. This distance is 843km so the earth must have a circumference of 42,150 km. The actual circumference of the earth is estimated to be 40,075 which tells us that our answer is within 6% of the correct one.

As with the units for the angle, Eratosthenes did not use kilometers for this measurement. Instead he used stadia. The conversion between kilometers and stadia is unknown as it did not have a consistent definition in antiquity. Therefore, there is dispute about how accurate Eratosthenes’s result was. 

However, we can work backwards from Eratosthenes’s result of 252,000 stadia to obtain a conversion factor. The distance between Alexandria and Aswan must have been measured as 5040 stadia since it is \(\frac{1}{50}\) of the circumference of the earth. It is unlikely that his measurement for this distance was as precise as our current measurement of 843km but equating 5040 stadia with 843km gives us a conversion of \(\frac{843,000m}{5040s}\approx 167\) meters per stadion. This is well within the  accepted range of 157-209 meters and is evidence that his estimate for the circumference of the world was quite good.

Some questions for future explorations:

• Can you use or adapt this method if the sun is not directly overhead in either town?
• Does longitude or latitude affect the measurements?
• Do the measurements need to be taken at the solstice? In other words, does the tilt of the earth affect the results?
• How would you adapt this method to estimate the radius of the earth?

Further reading:

• E Gulbekian, The origin and value of the stadion unit used by Eratosthenes in the third century B.C, Arch. Hist. Exact Sci. 37 (4) (1987), 359-363.
• Rawlins, Dennis, The Eratosthenes-Strabo Nile Map. Is It the Earliest Surviving Instance of Spherical Cartography? Did It Supply the 5000 Stades Arc for Eratosthenes' Experiment?. Arch. Hist. Exact Sci. 26 (3) (1982), 211-219.  

Egyptian Fraction Infographic

This infographic explains the example in this post.

Fulbright Kid

I have a collection of children's movies in Egyptian Arabic from when I lived in Cairo. The DVD cover for Up with its colorful balloons and fun-looking characters always tempted my son and he used to beg me to watch it. He would not accept my explanation that it isn't in English and it eventually occurred to me that he had no idea what that meant. He had no exposure to foreign languages and thought I was just making excuses. One day, I acquiesced and popped it in the DVD player. He stared at the screen for 2 minutes with his mouth agape before asking me to turn it off.

My son is a loud and friendly kid. He will talk to anyone on the street and he insists on yelling across restaurants to talk with the waitstaff. After 6 months in Jordan, he is finally starting to ask people if they speak English before jabbering away at them and while he still doesn't ask this of other kids on the playground, he has learned to play with everyone whether they speak English or not. He is learning some Arabic in school and when he needs to calm down, we count to ten together in Arabic.

In addition to learning about different languages, he has learned quite a bit about Jordanian culture. He knows that the call to prayer means people think about God at least five times a day and that he can't play with his friends on Fridays since that is a day for families to spend together. He has also learned that the bidet is not a small toilet and that in Jordan wasting water with a long shower might mean we run out of water until our tank gets refilled on Sunday nights.

He has a culture class in school where he learns about traditions from around the world. He chose our Christmas destination of Athens because he had been learning about Greek gods and now that they are discussing Indian traditions and holidays, he wants to visit India for Diwali. A Chinese classmate has him excited about the prospect of visiting China and he can't wait for our upcoming trip to Egypt. When he looks at a map, he no longer sees far off places but he views the world as something to explore. And when he sees that a movie is in a foreign language he knows that while he can't understand it there are millions of people who can and that learning the language will not only allow him to understand the movie but will enable him to meet new people and explore new places.

Egyptian Fractions

Rhind Papyrus: © Trustees of the British Museum.

How would you divide 8 cupcakes equally between 5 people?

The answer we learned in elementary school is to give each person \(\frac85\) or \(1\frac35\) of a cupcake. But, practically, how do we cut a portion that is precisely \(\frac35\) of a cupcake?

One solution is to use an iterative process. To begin, you have more cupcakes than people, so you give each person 1 cupcake and then you’re left with 3 cupcakes to divide among 5 people. You can easily cut each in half, so you have 6 halves to distribute. Of course, this means that each person gets a half and then there is 1 piece leftover which can be further divided into 5 equal pieces (or eaten by the cake cutter when nobody is looking). These last pieces are a fifth of a half - or \(\frac1{10}\) of a cupcake.

So we have taken the 8 cupcakes and given each person 1 cupcake, then \(\frac12\) a cupcake, and finally \(\frac1{10}\) of a cupcake. Since everyone got the same amount of cake with no leftovers, we see that

\[\frac85=1+\frac12+\frac1{10}.\]

An infographic showing this process can be seen at this link.

This kind of decomposition is called an Egyptian Fraction Decomposition (EFD) and it was first seen on the Rhind Papyrus. This papyrus dates to approximately 1550 BC and was found near Luxor, Egypt in the 1800s AD. It contains tables of Egyptian Fraction Decompositions as well as methods for decomposing fractions and some problems about dividing a specified number of loaves among 10 men.


The method we used to divide the cupcakes resulted in smaller and smaller fractions with a 1 in the numerator and each subsequent denominator dividing the prior one. But why couldn't we have just said that \[\frac85=\frac15+\frac15+\frac15+\frac15+\frac15+\frac15+\frac15+\frac15?\]

Well, not only is this not the easiest way to divide up the cupcakes since it requires each to be split into 5 equal pieces, but it violates the rules of EFD. Mathematics is all about taking a set of rules and seeing what questions you can answer within that framework.

Okay, then lets list the rules for an Egyptian Fraction Decomposition.

• The terms of the decomposition must have a 1 in the numerator.
• Each term must be different from the other terms.
• Infinite sums (what mathematicians call a series) are not allowed, so there must be a finite number of fractions in the decomposition. 

Then we can see that \[1=\frac12+\frac12\] and  \[1=\frac12+\frac14+\frac18=\frac1{16}+\frac1{32}+\frac1{64}+\cdots\]  violate the rules but \[1=\frac12+\frac13+\frac16\] is a valid EFD.

Now that the rules have been clearly laid out, we can start asking questions about EFDs.

• Will we always eventually have just 1 leftover piece to divide into n equal parts?
• Will we always be able to find a finite list of fractions?
• Are there algorithms for finding an EFD which do not result in denominators dividing each other?
• Is it possible for a given fraction to have more than one EFD?
• What if you are only good at cutting in half so the fractions always have the denominator as a power of 2. Will you always be able to divide the cupcakes fairly?
• What other questions arise this way?

Abstract Algebra in Indonesia

With my abstract algebra students in Indonesia

Mathematics knows no races or geographic boundaries; for mathematics, the cultural world is one country. – David Hilbert

The Hilbert quote above embodies one of my favorite things about being a mathematician. My job has taken me around the world including conferences in Portugal, France, and India and collaborated with colleagues from Spain and Egypt.

Last November, I had the opportunity to teach an algebra course at the Universitas Bung Hatta in Padang, Indonesia.

My visit was coordinated through the Volunteer Lecturer Program of the International Mathematical Union. They bring mathematicians to universities in developing countries to teach courses. I learned a lot about math education in Indonesia and I think I made a positive impact on the students' education.

By spending three weeks living in Padang, shopping in the local market, and talking with students and faculty at the university I got a real sense of daily life in West Sumatra and a better understanding of life outside of the United States.