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! 

NSF PRISM Fellowship

It's official!  Next year I will be participating in the NSF PRISM (Problems and Research to Integrate Science and Mathematics program) in Atlanta Public Schools through Emory University's Center for Science Education.

As a PRISM fellow, I will work with a local High School Mathematics teacher to bring problem based learning lessons into the classroom curriculum.  I'm excited about this opportunity because rather than teaching Mathematics as a set of facts and formulas, I hope allow students to think critically and creatively while collaborating with their classmates on fun and interesting mathematical problems relevant to the curriculum.  In High School, students are actively considering choices for their future.  Perhaps with our efforts, they will consider studying math or a math-related field!

I have been assigned to work with a teacher at Decatur High School in Accelerated Math 2 for sophomores.  After a 2-week summer institute where we will write our cases together, during the school year we will execute them in the classroom.

Math Jokes at The Pi Mile Race

In belated honor of Pi Day, my husband and I ran the 39th Annual Pi Mile Race at Georgia Tech last Saturday.  Weeks before the race we talked of training and friendly competition but somehow never managed to get out for a single practice run.  The day before the race, there was a tornado alert in Atlanta and the city shut down in fear.  Although we were secretly hoping the race would be cancelled, no one who admit it.  On Saturday morning we woke up to bright and sunny skies and reluctantly headed to campus.  

We casually ran side by side and inevitably the pi jokes ensued: 

Why have you half-way finished the pi mile race when you only ran π/3?
cos(π/3)=1/2

Why did the sign read 1/2 way point after running only π/6?
'cause sin(π/6)=1/2

We made a full circle and finished the pi race where we started.

We had a splendid time at the Pi Race.  It turned out to be a fantastic and entertaining way to start a productive day!  I hope this gets us pumped for the Peachtree Road Race on July 4th.  Can you think of a joke involving trig functions and periodicity?

Combinatorics of Khadaffi

 Given Libya's current political turmoil, and as a student studying Combinatorics (the study of counting), I must ask:  In how many different ways can you spell the name of Libya's ruthless leader?  Perhaps there is no formal construction for such a thing, but the Library of Congress and various press have used the following spellings for him*:

1. Qaddafi, Muammar
2. Al-Gathafi, Muammar
3. Al-Qadhafi, Muammar
4. Al Qathafi, Mu'ammar
5. Al Qathafi, Muammar
6. El Gaddafi, Moamar
7. El Kadhafi, Moammar
8. El Kazzafi, Moamer
9. El Qathafi, Mu'Ammar
10. Gadafi, Muammar
11. Gaddafi, Moamar
12. Gadhafi, Mo'ammar
13. Gathafi, Muammar
14. Ghadafi, Muammar
15. Ghaddafi, Muammar
16. Ghaddafy, Muammar
17. Gheddafi, Muammar
18. Gheddafi, Muhammar
19. Kadaffi, Momar
20. Kad'afi, Mu`amar al- 20
21. Kaddafi, Muamar
22. Kaddafi, Muammar
23. Kadhafi, Moammar
24. Kadhafi, Mouammar
25. Kazzafi, Moammar
26. Khadafy, Moammar
27. Khaddafi, Muammar
28. Moamar al-Gaddafi
29. Moamar el Gaddafi
30. Moamar El Kadhafi
31. Moamar Gaddafi
32. Moamer El Kazzafi
33. Mo'ammar el-Gadhafi
34. Moammar El Kadhafi
35. Mo'ammar Gadhafi
36. Moammar Kadhafi
37. Moammar Khadafy
38. Moammar Qudhafi
39. Mu`amar al-Kad'afi
40. Mu'amar al-Kadafi
41. Muamar Al-Kaddafi
42. Muamar Kaddafi
43. Muamer Gadafi
44. Muammar Al-Gathafi
45. Muammar al-Khaddafi
46. Mu'ammar al-Qadafi
47. Mu'ammar al-Qaddafi
48. Muammar al-Qadhafi
49. Mu'ammar al-Qadhdhafi
50. Mu`ammar al-Qadhdhāfī 50
51. Mu'ammar Al Qathafi
52. Muammar Al Qathafi
53. Muammar Gadafi
54. Muammar Gaddafi
55. Muammar Ghadafi
56. Muammar Ghaddafi
57. Muammar Ghaddafy
58. Muammar Gheddafi
59. Muammar Kaddafi
60. Muammar Khaddafi
61. Mu'ammar Qadafi
62. Muammar Qaddafi
63. Muammar Qadhafi
64. Mu'ammar Qadhdhafi
65. Muammar Quathafi
66. Mulazim Awwal Mu'ammar Muhammad Abu Minyar al-Qadhafi
67. Qadafi, Mu'ammar
68. Qadhafi, Muammar
69. Qadhdhāfī, Mu`ammar
70. Qathafi, Mu'Ammar el 70
71. Quathafi, Muammar
72. Qudhafi, Moammar
73. Moamar AI Kadafi
74. Maummar Gaddafi
75. Moamar Gadhafi
76. Moamer Gaddafi
77. Moamer Kadhafi
78. Moamma Gaddafi
79. Moammar Gaddafi
80. Moammar Gadhafi
81. Moammar Ghadafi
82. Moammar Khadaffy
83. Moammar Khaddafi
84. Moammar el Gadhafi
85. Moammer Gaddafi
86. Mouammer al Gaddafi
87. Muamar Gaddafi
88. Muammar Al Ghaddafi
89. Muammar Al Qaddafi
90. Muammar Al Qaddafi
91. Muammar El Qaddafi
92. Muammar Gadaffi
93. Muammar Gadafy
94. Muammar Gaddhafi
95. Muammar Gadhafi
96. Muammar Ghadaffi
97. Muammar Qadthafi
98. Muammar al Gaddafi
99. Muammar el Gaddafy
100. Muammar el Gaddafi
101. Muammar el Qaddafi
102. Muammer Gadaffi
103. Muammer Gaddafi
104. Mummar Gaddafi
105. Omar Al Qathafi
106. Omar Mouammer Al Gaddafi
107. Omar Muammar Al Ghaddafi
108. Omar Muammar Al Qaddafi
109. Omar Muammar Al Qathafi
110. Omar Muammar Gaddafi
111. Omar Muammar Ghaddafi
112. Omar al Ghaddafi

I suppose the issue here is that there is no generally accepted methodology to romanize Arabic names.  Further, his name contains several sounds that may not have an exact equivalence in English. 

Can you think of any other ways to spell his name?  How about R-U-T-H-L-E-S-S?!

*List from http://www.abcnews.com/

Pi Day 2011

Guess what next Monday, 3/14 is?  Pi Day! 

March 14th is the official day that this irrational number representing the ratio of the circumference of a circle to its diameter is celebrated around the world.  Traditional celebrations include eating pie and contests as to who can recite the most digits of pi.  In 2009, National Pi Day was recognized by Congress and today, it continues to be celebrated in Math classrooms and by Math enthusiasts around the nation. 

How will you celebrate?  I'm planning to run the Pi Mile foot race hosted by Georgia Tech.  For more celebration ideas and events, please visit Pi Across America.  Don't forget to have your pi and eat it too!

Mathematics and Hollywood

My husband and I missed Oscar night this year.  My husband was fast asleep after waking up at 3:30am to watch the legendary 10+ hour World Cup cricket match that resulted in a tie between India and England, while I was busy studying.  In fact, I didn't even know I was missing anything until the next morning when my husband woke up, didn't know if it was day or night and read the award headlines on CNN.com. 

In honor of the 83rd Academy Awards ceremony, I was reminded of a few films awarded in the past that had a Mathematics theme:

• A Beautiful Mind, 2001 Best Picture winner
• The Matrix, 1999 Best Visual Effects (I was a little disappointed when I found out this movie was not about Linear Algebra)
• Goodwill Hunting, 1997 Best Original Screenplay

 Other Mathematics themed films that were not as fortunate to receive an award:

• Suspect X, 2008
• π, 1998
• 12 Monkeys, 1995

Who said Mathematics is not glamourous?  Now, off to see The King's Speech!

The World of Sporcle.com

Who doesn't need a "Mentally Stimulating Diversion" from time to time?  One of my officemates introduced me to a website called Sporcle.  This sticky site hosts thousands of trivia quizzes of the form, "How many X can you name in Y minutes?"  Varied and random topics range from history to geography, movies to sports, and world scrambles to crytograms.  Even math trivia topics such as Digits of Pi and A in Math! 

I recently took the Mathematician Names trivia quiz (along with hundreds of others!).  This trivia quiz lists 50 well known mathematicians by last name and gives you eight minutes to enter their first names.  While most of the mathematicians' last names looked familiar, I can't claim I was on first name basis with any of them.  This trivia quiz was a struggle for me despite my background.  I managed to cough out a measly 20/50 first names which surprisingly placed me into the 95th percentile!  But more importantly, I learned something.  Thank you Lejeune Dirichlet, Pafnuty Chebyshev and Gottfried Leibniz!

Sporcle.com is a great site if you are training to be on Jeopardy or for a Spelling Bee, need a short break or just looking for a friendly competition between friends.  New trivia quizzes are added to the lot on a daily basis.  You can also upload your own user-generated trivia quizzes.  Have fun, but keep an eye on the timer... and the clock!

IBM's Watson Wins on Jeopardy!

The three-day competition against contestants Watson, Ken Jennings and Brad Rutter on Jeopardy! was all the buzz this week. While Ken and Brad are familiar human Jeopardy! mega-champions, Watson is a supercomputer with 16 terabytes of memory, designed to compete on the quizshow as part of IBM's Greatest Challenge.  By earning $77,147 in total earnings, more that the the combined score of Jennings ($24,000) and Rutter ($21,600), Watson won $1M to which will be donated to charity Vision Source.

I was fortunate to meet Dr. Bill Murdock, a member of the DeepQA research team in IBM's Watson Research Center, at an Emory Computer Science Colloquium yesterday.  Bill Murdock helped Watson distinguish correct answers from wrong answers by building components that apply logic, learning, and analogy to the results of natural language processing.  He worked on this IBM Greatest Challenge since the project initiated in 2006 and developed many of the DeepQA components used in the Watson question answering system, particularly in the areas of typing answers and evaluating evidence from passages.

Although Watson's answers were sometimes not relevant and far from correct (think:  Toronto as a US city), more often then not, he answered with speed and confidence.  However, the answers he searched for were all facts known by some human on this Earth.  What if we could program a computer to think and create new ideas?   Will Watson's offspring prove century old conjectures on Mathematics?  What do you propose to be the next IBM Greatest Challenge? 

Congratulations, Watson!!

Mathematics of the Heart

With Valentine's Day approaching, why not discuss mathematics of the heart?  I am not talking about our sentimental deep love and understanding of mathematics.  I'm talking about the cardiovascular mathematics research led by Dr. Alessandro Veneziani at Emory University.  His research has been applied in medical practice to simulate the human heart during surgeries.  See video below to learn more about this amazing, life-saving application of mathematics:



For more information see Emory's eScienceCommons blog post or contact Dr. Veneziani directly.  Happy Valentine's Day! ♥♥♥

My Triangle by James Blunt

It is often the case that successful children's television programs cater to children and  parents alike.  This was confirmed when I recently watched a parody of James Blunt's "You're Beautiful" on Sesame Street with my best friend's one year-old daughter.  Turns out, brilliant television programs cater to children, parents AND mathematics enthusiasts.  See video and lyrics below and don't be shy to sing along!

  

This shape was brilliant.
This shape was brilliant.
This shape was pure.
I saw three angles,
of that I'm sure.

And I saw three pointed corners,
And then I saw three straight sides.
The top was very narrow and
The base was oh so wide.

A triangle.
My triangle.
Oh triangle, it's true.
I saw your shape in a crowded place.
Now I don't know what to do,
'cause you're gone and I'm so blue.

I searched low and high
Over earth and sky
But I can't find your triangle.
Tell me why.

And I miss your base,
And I miss your height,
And my dreams are triangular
Every night.

My triangle.
My triangle.
So beautiful, it's true.
It must be those angles
Put a smile on your face
Not to mention the hypotonuse.

But I need to know the truth.
Oh triangle, where are you?

Khan Academy and Calculus

This semester I am teaching Calculus II.  My class is small and intimate with a dozen students, half of whom were in my Calculus I course last semester.  The biggest challenge I have as an instructor of Calculus is the variety of knowledge students enter the course with.  Some students have taken Calculus in High School, while other students may not have seen the prerequisite material before.  As an instructor, how can I make Calculus refreshing and enjoyable for all students in the course?



Recently, I came across a great resource to supplement my Calculus lectures when I was searching the internet for ways to enhance tomorrow's lecture on L'Hospital's Rule:  Khan Academy.  Salman Khan, not to be confused with the Bollywood actor, has posted 1800+ self-narrated, self-produced online tutorials on various topics in math, science and humanities.  The collection of Calculus topics spans an introduction of limits to Green's theorem.  He quit his job as a hedge fund manager to fulfill his mission to provide a world-class education to anyone, anywhere.  Amazing!

Khan Academy's free mini-leactures are a wonderful resource for students who need to brush up on their Algrebra and Pre-Calculus, or those who would like to revise concepts discussed during lectures.  I highly recommend Khan's conversational approach to communicating the ideas of Calculus in a simple, understandable way.   See it for yourself at http://www.khanacademy.org/ and maybe you can find ways to contribute to this fantastic endeavor in making Mathematics and other subjects accessible to all.   

Adding and Counting by Ken Ono

Last Friday, I attended an Emory Public Lecture titled "Adding and Counting" by Ken Ono.  During this well attended lecture, Professor Ono beautifully revealed the new theory he and his colleagues have discovered:  that partition functions behave like fractals and a closed formula to count the number of partitions for any number n.  The results confirm an observation made by Ramanujan in 1919.



The partition function p(n) counts the number of ways a number n can be partitioned.  For instance, over addition, n=4 may be partition in five distinct ways:  1+1+1+1=1+1+2=1+3=2+2=4.  Thus, p(4)=5.  For a better idea of what it means to "behave like fractals", watch the video above and notice the same structure appears repeatedly upon zooming in.  A formula for p(n) has intrigued mathematicians for centuries.

Professor Ono joined the faculty of the Mathematics Department at Emory University last semester and I have had the pleasure of taking a Number Theory course with him last semester.  For more information on this new theory, please see the Emory eScienceCommons and AMS press release.  Congrats to Professor Ono and his team on this remarkable advancement of Mathematics!

Back to School

With 4-6 inches of snow and only 6 plows, icy Atlanta transformed to Hoth-lanta last week, resulting in a 5-day extended winter break for University students.  Although there was a delay in the start of the semester, there was no delay in discussing interesting topics in my Combinatorics course this week. 

A Steiner system with parameters l, m, n such that l<m<n is an n-element set S with an m-element subset of S (called blocks) such that any l-element subset of S is contained in exactly one block.  To simplify, let l=2 and m=3 to form a Steiner triple system S(2,3,n).  For what n does a Steiner triple system exist?
 

This question inspired Reverend Thomas Kirkman and he solved a special case of this problem in 1847.  His solution is widely known as the Kirkman Schoolgirl Problem.  So in the spirit of a fresh, new semester, here is the riddle for you:  15 girls walk to school each day in five groups of three.  How many days does it take for each pair of girls to walk together exactly once?  Note this is a Steiner triple system S(2,3,n) where n is the number of days.  Hint:  See picture.

Sir Cumference and the First Round Table by Cindy Neuschwander

I spent the first week of the New Year with my niece in New York City.  At age one, she has brilliantly mastered the art of counting from one to ten.  I tried to take it a step further by teaching her the even numbers.  When I said, "one," she replied "two."  To "three," she replied "four," and so on.  I was thrilled until I reintroduced the odd numbers and somehow her number system omitted "three."  I'm happy to report that this issue has since been fixed and she can now count to 15.  With these types of counting skills, perhaps she will be a Combinatorialist like her aunt!  And what would life without three be anyway?

I then strayed from counting and taught her some geometric shapes.  What's a better way than with Sir Cumference and the First Round Table by Cindy Neuschwander?  In this clever children's book, Sir Cumference is challenged to construct a table to accommodate King Arthur and his 11 other Knights to meet to discuss the invasion of the Circumscribers.  After trying a rectangle, square, parallelogram and octagon shaped table, with the help of his wife Lady Di of Ameter, his young son Radius and the carpenter Geo of Metry, it becomes clear to Sir Cumference that a circle shaped table is most suitable for the King’s long awaited discussion for peace.

There are several books in the Sir Cumference series.  Looking forward to the next one!

Happy New Year!

2011 is prime number since the only numbers divisible by 2011 are one and itself.  So 2011 will be a prime year!   My husband and I have come up with 10 goals for ourselves to ensure our precious time this year is used wisely.  Of course, weekly blog posts made my list! 

When was the last prime year?   When is the next prime year?  Were these years prime for you?  Just because a year may be prime, what would you like to achieve to make it prime?  Reach for the stars!!  A Happy New Year to you and yours!