Fractal Geometry

Benoît Mandelbrot, the father of fractal geometry, died last Thursday at the age of 85. We will learn about fractal geometry in class when we get to the modern era.

You can read more about it here.

Archimedes Palimpsest

Palimpsest comes from the Greek word palimpsestos meaning "scraped again". It is a manuscript that has another text written over it. Around 1200 A.D., a Christian monk turned Archimedes' book into a new prayer book. In October 1998, it was sold for $2 million at an auction. You can watch the documentary that we saw in class about the story again here: part 1 , part 2 and part 3.

Since 1998, a group of scientists have been working on the recovery of the writings of the greatest mathematician of the ancient era. You can find up to date information here.

As well, you can find a digital copy of the recovered Archamides Palimpsest after the prayers are removed on google books.

Dangerous Knowledge

“Dangerous Knowledge” is a fascinating documentary by the BBC about the story of four masterminds, Goerg Cantor, Ludwig Boltzmann, Kurt Gödel and Alan Turing. What do these four have in common?

Cantor created "set theory". Cantor’s Theorem says that there exist infinitely many infinite sets of different cardinalities. The theory was nonsense and shocking to some of the mathematicians of his time and it created a lot of opposition. Some philosophers saw his theory as a challenge to the uniqueness of “God”, the absolute infinity. Cantor is also known for the Continuum Hypothesis which states that there is no set with cardinality between the cardinality of natural numbers and of real numbers. (We will go over Cantor’s Theorem and the Continuum Hypothesis in detail in class). Cantor could neither prove nor disprove the Continuum Hypothesis. In 1940 and 1960 respectively, Kurt Gödel and Paul Cohen showed that, in fact, one cannot prove or disprove this hypothesis using the set theory axioms. Gödel is best known for his Incompleteness Theorem which states one cannot find a complete and consistent set of axioms for all mathematics. This gave a negative answer to Hilbert’s second problem. If you would like to know more about Cantor, Gödel, Boltzmann and Turing, watch this 90 minute documentary!

Numeral Systems

In the last few weeks we learned about various numeral systems from Babelonian to Hindu/Arabic numerals. In this interesting video you can watch a summary and more ...

Is there a meaning to Beckham's wearing jersey number 23?

If the title of the post interested you enough to find out, read the short essay by Marcus du Sautoy, professor of mathematics and a soccer fan. 23 seems to be the number of choice for other sport celebrities and world figures like Michael Jordan or Pope John XXIII. Nature also gave us 23 pairs of chromosomes. And let's not forget that according to historical records Julius Caesar was stabbed 23 times, which most likely was a pure coincidence and not a deliberate counting by the assassins.  So, what's so special about number 23?
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Beckham in his prime

See also an interactive resource to learn through games all about prime numbers.

Why do we need to study the history of mathematics?

If you answered instinctively -- Because I need the course credit for my major -- I recommend that you view this captivating lecture (you might also download the audio only) from the famous Gresham College in the U.K.  Professor Robin Wilson in his lecture Keep Taking the Tablets on the history of ancient Egyptian and Mesopotamian mathematics explains eloquently:

First of all, mathematics is part of our heritage – as central as literature and painting – and we need to study a culture’s mathematical artefacts for the same reasons as we study its cave paintings, its buildings or its literature: indeed, it’s been claimed that mathematics is one of the two oldest professions in the world, and so we certainly need to investigate its origins.

 Secondly, mathematics is not just the tidied-up God-given subject that we find in our textbooks. On the contrary, it reveals itself as a fundamentally human endeavour – whether we’re interested in the counting of sheep, the collection of taxes, or the measurement of time. On the other hand, if we’re primarily interested in the mathematics itself, then our understanding may be significantly increased if we can explore the context from which it developed – whether as a practical problem arising in technology or astronomy or the biological sciences, or whether it arose out of a desire to solve a mathematical puzzle or develop a mathematical idea.

And finally, going back to the original sources can provide new motivation – our reasons for studying a subject now may differ markedly from the original purpose in studying a problem. Perhaps bored schoolchildren would find quadratic equations more fun if they’re told that the methods originated in Iraq some four thousand years ago, and if they’re encouraged to decipher the solution themselves by studying the markings on a clay tablet.

You can download the video, audio, lecture notes and print the transcript
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Keep Taking the Tablets. Robin Wilson. 2004

Five Numbers

Here you can listen to Simon Singh at BBC Radio who talks about histories of five special numbers. He talks about 1, 2, 6, the number that defines the universe and the first Ramanujan number! LISTEN...