Trivium & quadrivium

"It is known that Church councils from the time of Charlemagne demanded that the clergy have a knowledge of music and be able to compute the date of Easter." 

This quote shows that during the early part of the Middle Ages, different schools of disciplines like math, music were used for the purpose of the Church. The knowledge gained in these disciplines were ultimately used to determine the exact day of Easter. We can see that the Church still had a very big influence in the academic world, if there was one during that time. It is hard to imagine people need to study math and astronomy to become a clergyman or a pastor. 

"He is deeply religious and finds mystic and scriptural significance in all numbers. The fact that God performed 22 works in creation is adequate explanation for Isidore as to why there are 22 generations from Adam to Jacob, 22 books of the Old Testament to Esther, and 22 sextarii in a bushel."

This quote shows that at that time numbers were viewed in a mystic way. But I am really impressed by people during that time to find some many connections between numbers and the scripture. Even though these connections may not be significant in terms of mathematics, it is clear that people were trying to listen to what God has to say to them through these numbers. We can also see that a lot of social and intellectual elements in the societies were still controlled by the Church. 

"With the rise of the universities, with their curricula based on the trivium and the quadrivium there came an opportunity for a more rapid spread of arithmetical knowledge."

This quote tells us that universities were the intellectual revolution during that time in the Middle Ages. It allowed spreading of knowledge. There was also the Hindu-Arabic influence which opened the doors to a lot of different areas in arithmetic. Things started to move away from the church. Arithmetic was not studied only by clergymen. University students were required to read books from Euclid. Operations were no longer done by fingers and abacus. Even though in today's standard these topics are taught in high schools, these were very tough topics for people of that time as they lacked computers, instruments and technologies. I feel that different eras in history require revolution (in this case universities) so that improvements can be made academically and socially. Other examples are the industrial revolution and the establishment of public education.

Alice Major on Mayan and other numbers

Different cultures interpret numbers differently. It is interesting that the author mentions quite a number if lucky/unlucky numbers in different cultures. Further to the Cantonese's 8 meaning fortune, 16 mean continuous so we always like to put 16 and 8 together. So 168 in Cantonese means "Fortune/prosperity comes continuously". In fact, a lot of us would give $168 as wedding gifts or whenever there is a significant occasion (We Chinese like to give money in a small red packet as gifts). A lot of the Chinese numbers are actually homophones of other Chinese words so people can play around with them endlessly. 

It seems like the Mayan's interpretation of different numbers has to do with their gods and deities. There are actually stories behind each number so we see that each number is represented by a different head glyph. The interpretation depends on the culture, history and the people of each civilization. When you mention a number to someone, it is likely it is not just a number to that person. There is always a story behind that number. 

One interesting experience I want to share: I play and watch soccer quite a lot. At the beginning of a soccer season, my teammates and I had to choose what number jersey we had to wear. Interestingly, the number a player chooses more or less reflects the personality of that person (it also reflects the position that you play in). For example, if you choose 10, it means you want to be the centre of attention. In soccer, the number 10 is the heart of the team. The person is the playmaker. Every single play needs to go through him. If you choose 9, that means you want to lead the team in attack. Number 9 or 10 are usually associated with flair and flamboyance. If you choose 5 or 6, you are the rock at the back in defence. These are the people who do the dirty work for the team. They do not want the attention and usually these players are the unsung heroes of the team. So we can see that in different contexts, the numbers can take on totally different meaning.

Maybe this is a good idea to share with our secondary school students. Since this ties to the history of mathematics, it allows students to know more about the history and stories behind these numbers. And this hopefully will allow students to appreciate them more. A good way to introduce this to them is to have them research meanings behind numbers in their own cultures. 

I myself play the goalkeeper in soccer. Usually goalies is assigned the number 1 jersey. So for me, the number 1 represents calmness, organizer (you need to organize your defence) and communication since these traits are very important for the goalie as the last line of defence. It also represents someone who can handle pressure well because if you see a goalie makes a mistakes, it usually results in a goal for the opponent. I think it is the hardest job out there on the field!

Dancing Euclidean Proofs!

 The first thing that came into my mind when I watched the video was I have never seen something like this before. Putting Euclidean geometry and dances together is not something that I would have thought of. However, I have to say that Sam and Azul did a very good job in presenting the proofs as dances. You know right away what they were trying to do when they drew on the sand using their bodies. This is especially true for dance number 1. To me, I did not even have to look at the Euclidean geometry proof itself to know that they were trying to draw an equilateral triangle. 

Another thing that stops me is the visual representation of the proofs. As mentioned in the article, if we are just reading the text, we are completely detached from its representation on the page. But with the dances, it is like making the proofs into a movie. That visual and sound of the music would make a deeper impression than just reading the Euclid text. Similarly, if we learn something by visualizing it, the memory will definitely last longer than just reading it on paper.

The third thing is the connection between the dances and the land. In the second dance, we saw that the students used a wooden stick and a rock as a line and point respectively. The drawing on the sand also leaves a trace of the circle and other lines. I think it is a very ingenious idea to have the land (the beach) play a part in the dance. So it is very nice to see the connection among the Math, dances and the land. It would be cool if we use the wave and water as an eraser. So when the waves come ashore, the water will remove the trace of the circle and then the students can draw on them again like a new whiteboard. But that will take too much time and effort to catch that on video. 

Euclid Poems

 The first thing that stands out when I look at "Euclid Alone Has Looked on Beauty Bare" by Edna St. Vincent Millay is that the word "Beauty" is capitalized. It seems like the poet wanted to refer "Beauty" as a person. The "her" in the last line of the poem also seems to suggest likewise. I think the poet was referring Euclid's work as the "Beauty" as Euclid's work is beautiful in the eyes of mathematicians. His book "The Element" is such a revolutionary piece that it is like a beautiful art piece in the eyes of the poet. The poem also seems to make connection between Euclid's work and the second coming of Christ. The line "O holy, terrible day" is a reference to the Bible in the book of Joel.  "The sun will be turned to darkness and the moon to blood before the coming of the great and dreadful day of the Lord." (Joel 2:31). This verse describes the second coming of Christ when it is great for people who believe in him but a terrible day for those who do not. The poet is putting Euclid in the place of Jesus Christ that he opened a new way of thinking for the entire world. Euclid is like the mathematical saviour of the world.

As I read the second poem, I think it echoes with the first one by saying that Euclid's work is something of a beauty. By saying "To tell that you have only seen her clothed?", the poet is saying there are still covered part of Euclid's work. Only a handful of people get to see the true beauty of it. At the end of the poem it says "Remove Once only, and then but far away, your sandal?" to suggest that the true beauty of Euclid's work should be uncovered for all to see. 

In both of these poems, the poets both suggest that Euclid's work is revolutionary and it provided people with a new mathematical way of thinking. There is a long lasting significance of his work up until today.