The Eye of Horus

After writing a lesson on fractions using Egyptian fractions ideas for our students in the Grady Math Program, I have been totally consumed with the mathematical contributions of the Egyptians, and in particular Egyptian fractions.

For some unknown reason, the Egyptians wrote proper fractions as the sum of distinct unit fractions (that is, fractions with 1 in the numerator), also called Egyptian fractions.  Hieroglyphic and hieretic tablets display characters for unit fractions only, with the only exception being 2/3.  And strangely, this representation of proper fractions was continued for nearly 4000 years into 17th century Europe!

Many questions regarding Egyptian fractions have intrigued modern mathematicians and historians.  While it is easy to show that every proper fraction has a distinct sum of unit fractions representation using the Fibonnaci greedy algorithm, it is unclear if the Egyptians has a systematic way of representing them as these representations are not necessarily unique.  What is the "best" representation for a unit fraction?  Is it the smallest number of terms?  Or the representation with the smallest denominator? 

Interestingly enough, the Eye of Horus (shown above) is a collection of hieretics for unit fractions 1/2, 1/4, 1/8, 1/16, 1/32 and 1/64.  This symbol represented the Old Kingdom number one as 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 = 63/64, which I suppose the Egyptians presumed was close enough to 1. 

An open conjecture of Paul Erdos is that every proper fraction of the form 4/n (n positive integer) can be written as a sum of three distinct unit fractions.  There are many other open questions on Egyptian fractions that extend to ideas in combinatorics, number theory and algebra.  But for now, how can you redraw an Eye of Horus so that the sum of distinct of unit fractions is actually 1?  C'mon Egyptians, you could do better!

See Mathemathematics in the Time of Pharaohs by Richard J. Gillings and Fractal Music, Hypercards and More by Martin Gardner for a brief history of Egyptian fractiosn, a heiretic tablet of unit fractions and more!

5 Degrees of Separation on Facebook

Here an interesting article I read on nytimes.com regarding degrees of separation on the social networking site Facebook.  In summary, "degrees of separation" is lower in the digital world.  This calculation is simply the diameter of a dynamic random graph!  How would Kevin Bacon feel about all of this?

Four Degrees of Separation

By ANAHAD O'CONNOR

Perhaps the saying should be four degrees of separation, rather than six?

Using data on the links among 721 million Facebook users, a team of scientists discovered that the average number of acquaintances separating any two people in the United States was 4.37, and that the number separating any two people in the world was 4.74. As John Markoff and Somini Sengupta report in today’s New York Times, the findings highlight the growing power of the emerging science of social networks:

The original “six degrees” finding, published in 1967 by the psychologist Stanley Milgram, was drawn from 296 volunteers who were asked to send a message by postcard, through friends and then friends of friends, to a specific person in a Boston suburb.

The new research used a slightly bigger cohort: 721 million Facebook users, more than one-tenth of the world’s population. The findings were posted on Facebook’s site Monday night.…

“When considering even the most distant Facebook user in the Siberian tundra or the Peruvian rain forest,” the company wrote on its blog, “a friend of your friend probably knows a friend of their friend.” The caveat there is “Facebook user” — like the Milgram study, the cohort was a self-selected group, in this case people with online access who use a particular Web site.

How many people around the world are you connected to through your Facebook account? And how many of your Facebook “friends” are actual friends, or simply “buddies”? To learn more about the research and its implications for social networks, read the full report, “Separating You and Me? 4.74 Degrees,” and then please join the discussion below..

Math Advocacy Group Volunteers at Grady High School in Atlanta

Now that we've been living in Atlanta for almost two years, we started looking for an opportunity to get involved in our community.  After correspondence with the organizers of the Math Advocacy Group at Grady High School, we volunteered to tutor for Super Saturday in preparation for the End of Course Tests (EOCT)  administered by the Georgia Department of Education for 9th and 10th grade students.

Grady High School is an urban school with a very diverse student body.  My husband and I volunteered to tutor optional Saturday morning sessions before the test date.  For the first hour, students would attempt to take a practice exam.  And then for the next two hours, we would present our solutions to various problems and try to get the students participation as much as possible.  It was quite a challenge as we had the full spectrum of students and behavioral problems.  But by the end of the session, we hope the students learned something from us.  We wish them all the best on their standarized testing.

It was clear that some students in the session did not understand the problems that were expected of them.  We wish we could spend more time with them to help them understand these problems.  Sadly, I wouldn't be surprised if a decent percentage of the students who attended Super Saturday will not pass the EOCT.  We hope that we can help such students pass next year by commiting ourselves to tutor them for a full year and monitoring their progress.  Other ways we would like to get involved with Grady include coaching a math competition team, teaching an SAT course, and making a presentation on our experiences in math related jobs. 

Despite the challenges, we are having a great time with these students and we are very excited to get more involved.  I admire the teachers at Grady that we worked with for their dedication.  They were all so wonderful and really passionate about what they do.  We were really touched by their warmth and trust in welcoming us to their team.  In fact, we got an invite to Prom and it made us feel at home... and like we were in High School again!