Explication and commentary on poems about Euclid

 

Euclid was a Greek mathematician from Alexandria, Egypt around 300 BCE. There is almost nothing known of his life and no first-hand descriptions of his physical appearance (The Story of Mathematics, n.d.). Euclid is often referred to as the “Father of Geometry” as he wrote the textbook, “Elements”, which represents the “peak” of mathematics up until that time in Greece. He also wrote on division of geometric figures into parts in ratios, mathematical theory on reflection, determining location of objects on the celestial sphere, and texts on optics and music (The Story of Mathematics, n.d.). The Elements was a compilation and explanation of all known mathematics containing 465 theorems and proofs; Euclid reworked math concepts of his predecessors and this field became known as Euclidean Geometry. His five general axioms where: 1) things which are equal to the same thing are equal to each other; 2) if equals are added to equals, the wholes (sums) are equal; 3) if equals are subtracted from equals, the remainders (differences) are equal; 4) things that coincide with one another are equal to one another; 5) the whole is greater than the part (The Story of Mathematics, n.d.). Among other things, Elements also contains information number theory; specifically, the Fundamental Theorem of Arithmetic and proof that there are infinitely many prime numbers.

In Millay’s poem, I think she refers to ‘Beauty bare’ as an actual entity, hence why ‘Beauty’ is capitalized. From lines 1 and 2, I think she references Euclid is the only one who looks at ‘Beauty’ because he is the “Father of Geometry” and tells others who talk foolishly about ‘Beauty’ to “shut up and stop talking.” Lines 4 and 5 refer to these people not being able to see the shapes Euclid sees.

In Kramer’s poem, he implies that ‘Beauty bare’ is embodied as a woman and questions if no one else has seen her naked (Lines 1 and 2) or heard her (Line 4). Kramer then critiques Millay’s poem by comparing the words coming out of her mouth as idiocrasy to Orpheus, a musician and poet in Greek mythology, who tried to bring his dead wife Eurydice back to life through music (Lines 9 and 10). Kramer then claims that Millay has not seen ‘Beauty’ herself (Line 11). It then appears that Kramer then asks the entity of ‘Beauty’ to then remove Euclid from the history of famous mathematicians (Lines 13-15). Kramer may be questioning the importance Euclid has in the history of mathematics; Euclid’s main contribution to mathematics was summarizing and compilating other work of mathematicians before him and being given credit as the “Father of Geometry” and founder of Euclidean geometry.

References:

The Story of Mathematics, n.d. Euclid of Alexandria – The Father of Geometry. Retrieved October 21, 2020 from https://www.storyofmathematics.com/hellenistic_euclid.html.

 

Eye of Horus and unit fractions in ancient Egypt

The Eye of Horus was also known as Wadjet, a symbol believed to provide protection, health, and rejuvenation (Ancient Origins, n.d.). Ancient Egyptians believed that Osiris was the king of Egypt and his brother Set murdered Osiris and became the new king. Osiris was temporarily brought back to life and impregnated his wife with a son named Horus. Horus eventually went on to avenge his father’s death but lost an eye. One version of the story was that his eye was ripped into 6 pieces. His eye was magically restored, and ancient Egyptians believed that it had healing properties.

The Eye of Horus was also a math symbol. Each of the six parts of the ripped eye was given a fraction as a unit of measurement; right eye is ½, pupil is ¼, eyebrow is 1/8, left side of eye is 1/16, curved tail is 1/32, and teardrop is 1/64 (Ancient Origins, n.d.). These fractions all add up to 63/64. These unit fractions are powers of two in their denominators, and used to represent fractions of hekat, which was the unit measure of capacity for grains.

Figure 1: Eye of Horus divided into 6 parts mathematically

One of the most interesting findings was that the Eye of Horus also has connections to medicine and neuroanatomy. Specifically, the Eye of Horus resembles the corpus callosum, metathalamus, olfactory tract, and brain stem located in the human brain.

Figure 2: Eye of Horus in the human brain (Refaey, Quinones, Clifton, et al., 2019)

Sports has been a big part of my life and I have worn many numbers. My favourite number is the number 9, which was worn by my favourite hockey player, Paul Kariya. Kariya was an amazing hockey player and often compared to as “the next Wayne Gretzky. He was also born in Vancouver, BC. When I played hockey, I couldn’t always get the number 9, so I settled on numbers that either had 9 in them (like 19, 39, 91) or multiples of 9 (like 18). Numbering in sports has always interested me and I was always curious to learn about the stories behind a number. Mathematically, there are so many interesting things with number 9. One of them being that for any natural number multiplied by 9, and the digits are repeatedly added until it is a single digit, the sum of the numbers will be 9. For example: 9 times 987654321 = 8,888,888,889. Then 8+8+8+8+8+8+8+8+8+9=81. Then 8+1=9.

Figure 3: Matt Yuen #18

 

References:

Ancient Origins, (n.d.) Eye of Horus: The True Meaning of an Ancient, Powerful, Symbol. Retrieved October 16, 2020 from https://www.ancient-origins.net/artifacts-other-artifacts/eye-horus-0011014.

Refaey K, Quinones G C, Clifton W, et al. (2019) The Eye of Horus: The Connection Between Art, Medicine, and Mythology in Ancient Egypt. Cureus 11(5): e4731. doi:10.7759/cureus.4731

Constructing a magic square

When I first attempted this problem, I chose to work with number 9 (as 9 is my favourite number) in the corner box. Here is one 3x3 square I got stuck on. I found that this did not work because I would need the diagonal for numbers less than 6; 9+6 does not allow for a number in the final box.

9	2	4
1	6
5

 

I ran into this same problem when 9 was in the middle. There were just too many “open boxes” around 9 that needed two numbers that added to 6 without repeating numbers.

	1	4
	9
2	5

 

Finally, I achieved a solution when 9 was not the middle or corner piece.

2	9	4
7	5	3
6	1	8

 

Was Pythagoras Chinese?

I think it can change students’ perspectives on mathematics if we acknowledge non-European sources. Throughout my learning of mathematics from high school to post-secondary, a majority (if not all of it) came from European and Greek sources. Acknowledging mathematics from non-European sources can be an opportunity to bring multicultural education into the classroom and learn mathematics from different cultural lens. This could be a sense of pride for students when they learn about a subject that has origins from their own cultural/ethnic backgrounds thereby forming a connection to the topic. As mentioned in the reading, both Chinese and Greek mathematicians were highly intelligent and made many similar discoveries and big accomplishments such as the approximation for π, Right Triangle Theory, and other contributions to number theory and geometry. Gustafson concludes that the accepted history of mathematics in the Western world largely ignores China’s contributions to mathematics due to their geographic isolation during ancient times (2012).  

I do not think it would make much of a difference to students’ learning if we were to rename the Pythagorean theorem as the Gougu theorem because the theory is essentially the same. However, it is worth noting that China still refers to the Pythagorean theorem as the Gougu theorem, therefore it might be confusing for Chinese students when they learn the same idea referred to by a different name and vice versa. I do feel like there is a sense of colonialism when it comes to naming of these mathematical theorems and concepts because its naming comes ultimately by those who tell its story. It is the sense of negligence or unknowingly claiming ownership over earlier civilizations or groups that discovered the idea and renaming it something else. It is like retelling a joke your friend told you in class but louder so the entire class can hear. It becomes something you become known for without acknowledging where it came from, which is why regardless of what the mathematical theorem or concept is called, other origins of the idea should be acknowledged when presented.

 

Visual proof of Gougou Theorem

Reading: Gustafson, R. 2012. Was Pythagoras Chinese-Revisiting an Old Debate. The Mathematics Enthusiast: Vol. 9: No. 1, Article 10.

Ancient Egyptian ‘algebra’: The method of ‘false position’ (estimate, check, adjust)

Problem: Matty has a basket of apples. If he were to double the quantity of apples he currently has while giving away its fifth, he would be left with 18. How many apples would he have to begin with?  

Modern method: let x be the number of apples.

Egyptian ‘false position’: Try x = 5.

So, we get f(5) = 9 when x = 5. We want f(x) = 18, but notice that f(5) = 9, which is half of f(x) = 18. We can try x=10:

We can conclude that x = 10 gives us the same solution using the modern method.

Therefore, we can conclude that Matty would have had 10 apples to begin with. 

Word problems as a genre in Mathematics Education

Gerofsky quotes Hoyrup who says, “Babylonian problems look like real-world problems at first; but as soon as you analyze the structure of the known vs unknown quantities, the complete artificiality of the problem is revealed” (2004). This chapter of Gerofsky’s book A Man left Albuquerque Heading East provided an interesting analysis on Babylonian word problems and posed several ideas I did not notice when reading and solving modern word problems. The main argument was that some Babylonian problems appeared to be typical problems seen in everyday life with realistic quantities, while the context of other problems also took place in everyday life but had far-fetched numbers or dimensions to the problem making them appear impractical and unrealistic. This reminds me of a simple word problem I saw plenty of times in grade school: “Jim bought 200 apples, 300 bananas, and 400 coconuts for $1000. If apples cost twice as much as bananas and three times as much as coconuts and bananas are twice the cost of coconuts, how much is a single apple?” (I just made this problem up to illustrate a point, however, feel free to try and solve it!). As this problem is “relevant” in the real-world with a shopper buying fruits at a grocery store, the quantities are unrealistic and impractical. Whiles these problems are unrealistic, they can still be solved quickly using contemporary algebra. Similarly, word problems like this can be generalized quite easily by changing quantities and ratio of fruits.

Pure mathematics refers to the study of mathematical theory while applied mathematics focuses on application of mathematics to the world in a very direct sense. I believe that word problems can come from a pure mathematics point of view or an applied mathematics point of view. Consider the 2020 AMC 10A Problems booklet. I would categorize the first three problems as pure maths questions while problem 4 is an applied maths question. All four however, can be solved using contemporary algebra methods. Despite being viewed as a pure maths or applied maths problem, these are still mathematical exercises where background information is presented using ordinary language rather than mathematical notation (Vershaffel et. al, 2000).

Near the end of the chapter, Gerofsky poses several questions to the reader; “Are word problems used primarily to train students in the use of methods without necessarily providing understanding of those methods? Are problems chosen to illustrate the methods at hand?” I think word problems can be solved using multiple methods, so it is possible that students may not understand explicitly what method is being used besides arriving at the correct solution. However, word problems can be created to test a student’s ability to determine what the word problem is being asked and carry out the appropriate method to solve the problem. From Gerofsky’s paper, Hoyrup says that Babylonians trained students in methods available at hand rather than understanding of methods (2004). As historical evidence has shown, the Babylonians were very good at computations as their methods were very systematic. I think these ideas of Babylonian word problems illustrate the fact that the Babylonians were also great writers and thinkers and were among the first civilizations to incorporate ordinary language in mathematical exercises.

 

Sources:

Gerofsky, S. 2004. A Man left Albuquerque Heading East. Word Problems as Genre in Mathematics Education.

AoPS Online. 2020 AMC 10A Problems. Retrieved on October 4, 2020 from https://artofproblemsolving.com/wiki/index.php/2020_AMC_10A.

Verschaffel, L., Greer, B., De Corte, E. 2000. Making Sense of Word Problems, Taylor & Francis.

Assignment #1 - Problem 1.2.3 Solutions and Reflection

 

From the textbook, we were given the drawing of a pyramid frustrum, which can be separated into a cuboid, four corner square pyramids, and four triangular prisms between each corner prism. Adding up the volumes of these shapes will give the volume of the frustum; a modern approach to solving this problem. 

As the Egyptians did not draw three-dimensional shapes, it is likely they derived the solution through similar triangles. 

 

 

Lastly, we discovered a generalization to the equation to extend our problem to frustums of n-sided regular polygon bases:

I really enjoyed this project as it was an interesting look at how ancient Egyptians likely solved mathematical problems. It demonstrates that there are different methods to solve a single problem, and limited by techniques developed at the time, we can see that it is still possible to solve this problem using the same method they solved in the past. I can see this activity be incorporated in a classroom learning about surface area and volume, which can teach students to think creatively “outside the box” when determining frustum volumes by separating the frustum into different three-dimensional shapes.

Link to our Power Point: 

https://docs.google.com/presentation/d/19k7213cygiKP1h1O11EwXUJ8k5JpRw3ubVyEAjUGMWg/edit?usp=sharing