The dancing Euclidean proof

 1. “embodying mathematical concepts and relationships through dance and movement”

Dance and movement can be done by person(s) with or without an object such as a stick or an umbrella in an environment in which the action is possible. These actions can represent a mathematical or physical process such as drawing a line, a circle, marking a length on the line. Even the speed of a moving particle can be represented by adjusting the speed of the dancer movement. I see a lot of opportunities not fixed to euclidean proofs. For example, we can have two concentric circles with different diameters. Two dancers move along the circles at two different speeds such that they complete one round trip at the same time. The audience can clearly see that the dancer on the larger circle moves faster (greater linear speed) to keep the same angular speed as the other dancer on the smaller circle. 

2. “potential to make the beauty of Euclid’s proofs accessible to mainstream audiences”

Dancing and moving make people feel good, happy, and engaged. By allowing learners to participate or even watch others doing the dance we create a happy learning environment. This is a good benefit of dancing and movement whenever possible in helping people learn through doing something that makes them happy.

3. “Help students understand and appreciate the beauty of Euclid’s proofs in new multisensory, experiential ways”

 

By dancing to construct a mathematical process students are part of the construction tools and objects (pencil, divider, ruler, etc.). Before the dance, students need to plan each move and entire process - that gives the exterior point of view. During the process students manage the movement, direction, magnitude and speed. Therefore they get the whole interior view of the process. 

Explication and commentary on a poem about Euclid

 Poem 1: Euclid Alone Has Looked on Beauty Bare by Edna St. Vincent Millay

Euclid alone has looked on Beauty bare.

Let all who prate of Beauty hold their peace,

And lay them prone upon the earth and cease

To ponder on themselves, the while they stare

At nothing, intricately drawn nowhere

In shapes of shifting lineage; let geese

Gabble and hiss, but heroes seek release

From dusty bondage into luminous air.

O blinding hour, O holy, terrible day,

When first the shaft into his vision shone

Of light anatomized! Euclid alone

Has looked on Beauty bare. Fortunate they

Who, though once only and then but far away,

Have heard her massive sandal set on stone.

Speculation

Euclid alone saw the beauty of geometry (or geometry is the beauty) clearly (in his time). Euclid displayed the beauty he saw to the rest of the people. That made people who talked foolishly about the ‘geometry’ going silent. Geometry made people come face to face with the world they live in and not to be selfish in the old way in which they learned nothing. After passing through many different generations, many people still did not understand geometry. But those who understood geometry escaped from the dark world and came to the world in which they used geometry to build civilization and beauty.

The old days before the geometry were primitive and terrible. At the first opportunity of the beauty of geometry in details, Euclid was alone who saw it clearly.

When Euclid first understood the concept of geometry, it was him alone who saw the beauty in details and clear. The event that Euclid managed to compile a complete beauty of geometry happened once, but far away from our time. It was fortunate that people re-discovered the beauty. We have heard the great importance of geometry as a massive sandal set on stone more than 2000 years ago! (metaphor, according to wikipedia Eda was a feminist)

Poem 2: The Euclidean Domain

…Euclid alone

Has looked on beauty bare. Fortunate they

Who, though once only and then but far away,

Have heard her massive sandal set on stone.

—Edna St. Vincent Millay, Sonnet

Euclid alone has looked on Beauty bare?

Has no one else of her seen hide or hair?

Nor heard her massive sandal set on stone?

Nor spoken with her on the telephone?

Proud poets, as you penned your paeans to Beauty,

Did you not think it was your bounden duty

(Though it were one that any might have loathed)

To tell that you have only seen her clothed?

And as you sang praise, Orpheus, of Eurydice,

Your mouth became the orifice of your idiocy!

For Beauty bare you never yet had seen,

’Twixt Hades’ depths and lofty Hippocrene.

O Beauty! Would you, for this mathematician,

Remove (if it would cause to give permission

To look on Beauty bare too great a scandal),

Once only, and then but far away, your sandal?

Speculation

Repeat the second part of the poem by Edna St. Vincent Millay.

Is it true Euclid alone saw the beauty of geometry bare (clearly, completely)?

Has no one else seen her (geometry) hiding or even just her hair? (metaphor)

Has no one heard of the importance of geometry? Has no one heard that geometry set her sandals on stone (and sent out the loud sound wave) more than 2000 years ago?  

Has no one worked with geometry? Has no one spoken to geometry on the phone?(Sarcasm)

You (Edna) were a proud poet. You wrote your poem of praise to geometry (beauty). Did you think that it was your burden (duty) to tell that you only saw the surface of the beauty of geometry? It looked like this is something anyone was unwilling to do.

(Edna might not be trained as mathematician according to wikipedia)

As you sang the praise (Edna’s poem) - at this point David used the comparison with Orpheus mourning, praising his late wife Eurydice (both from ancient Greek mythology) - your mouth became the pipe producing your idiocy (stupidity). Because, you have never seen the bare beauty of geometry. It was between the depths of the Hades (god of the dead) and self imposed inspiration.

O beauty (geometry), would you, for this mathematician (is it David?) remove the event that ‘Euclid or someone discovered you in whole’ (which is taken as the beauty set her sandal on stone with a loud sound) into our world more than 2000 years ago? Would you (beauty) please, for once, go back in time and did not set your sandals on stone? To look at that beauty could come with a great scandal.

Notes: According to wikipedia, Euclid presented the already discovered geometries into one single logically organized work. His contribution seems to be the organization including writing axioms, and putting proofs in the collection.

Eye of Horus and Unit Fractions

 The ancient Egyptian symbol “eye of Horus” (from the sky god Horus who was usually depicted as a falcon) was considered as a protective amulet to the Egyptian. It has been commonly painted on the bows of boats both protected the vessels and "saw" the way ahead. Moreover, the eye is constructed in six fractional parts, representing the shattering of the eye of Horus into six pieces. According to historical documents, these six parts also represent six senses of human. The inner corner of the eye indicates one half, the iris is one fourth, the eyebrow is one eighth, the outer corner of the eye is one sixteenth, and the decorations below the eye are one thirty-second and one sixty-fourth respectively. 

Note: The infinite geometric series with ratio less than one is:

The unit fraction numbers in the Eye of Horus are the first six terms of the geometric series:

Hence, it looks like Egyptian scribes would approximate one using the first six terms of this series because the sum: 1/2 + 1/4 +...+ 1/64 = 63/64.

In Milo Gardner paper (The Arithmetic used to Solve of an Ancient Horus-Eye Problem, 2006), explanations were given on how Egyptian scribes divide one (64/64) by number. In ancient Egypt hekat was a volume unit. One hekat can be considered 64/64 hekat. There are also other volume units such as hin, dja and ro where

1 hin = 1/10  hekat

1 dja = 1/64 hekat

1 ro = 1/320 hekat

Scribes would express the division of a hekat in terms of Horus Eye unit fractions and these smaller units of hekat. Here are some examples from Milo Gardner paper(2006).

In addition to Horus Eye unit fractions, Egyptians also used other unit fractions. All fractions can be represented as a sum of unit fractions (1/n, n is any natural number). According to (Katz, 2008, pg.5) the Egyptians expanded the fraction with non-unit numerators into the sum of unit fractions. Whenever a number of objects need to be equally divided into a number of receivers, this expansion method can be useful. For example, suppose we need to equally divide five loaves of bread to seven people. Then one can expand 5/7 into sum of unit fractions as follows: 

5/7  = 1/2 + 1/7 + 1/14 

This expansion also gives the plan of how to cut the bread. That is each person will get half of a bread loaf plus one seventh of a bread loaf plus one fourteenth of a bread loaf. So it is very interesting to know that some ancient peoples made good use of unit fractions in their daily business when they did not have the luxury of modern mathematics.

Many numbers have special properties, meaning or applications. Cultures are also connected to some numbers. In this Wichita State University website (http://www.math.wichita.edu/history/Topics/snumbers.html), there is some information on special numbers such as perfect numbers (a positive integers that is equal to sum of its divisors excluding itself such as 6 = 1+2+3). 

Amicable numbers are the pairs of numbers with the following properties:

1. Sum of divisors of first number (excluding the first number) equals the second number

2. Sum of divisors of second number (excluding the second number) equals the first number

An example is 220 and 284. Sum of divisors of 220 is:

1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284.

Sum of divisors of 284 is:

1 + 2 + 4 + 71 + 142 = 220.

There are also special numbers in cultures. Ancient Greeks are said to have believed in four elements (earth, water, air, and fire). Native American culture also talked about four directions (east, west, north, south) from where the wind came, and each of these winds had accompanying stories.

Constructing a magic square

 

Three by three magic square: Each square has a number from 1, 2, ..., 9 used once. Sum along any row, any column, and diagonal are all 15. What are the numbers in each square?

                                                                        1 + 2 + ... + 9 = 45

Write the square as 3 by 3 matrix.

The method of 'false position'

 The method of false position is suitable for solving an equation with one unknown. Usually the equation contains proportionality w.r.t. the unknown. Chinese dish problem discussed last week can be solved by this method.

Response on " Was Pythagoras Chinese?"

•  Does it make a difference to our students' learning if we acknowledge (or don't acknowledge) non-European sources of mathematics? Why, or how?

Since I was born and raised in China before coming to Canada at age of 12, the foundation of my mathematical knowledge was learned in China. This includes arithmetic operations, integer operations, fraction operations, solving algebraic equations and system of equations. Because of the difference in mathematical curriculums, I was ahead of my class for several years since grade 7. In Canada, I learned pre-calculus in high school and more advanced mathematics in university. Hence, I have the experience learning math from two different cultures.

In China, elementary school math is taught with a blend of Chinese and Western sources. I remember learning how to use an abacus and we have our own version of Pascal’s triangle we call it “Yang Hui triangle”. Also, as mentioned in Gustafson, we learned right triangle theory as “gou-gu” theorem. Nonetheless, we also use Arabic numerals, and we call our variables x, y, z. Even though we have Chinese names for sine, cosine, and tangent, we write using the Western notations in problem solving. As a matter of fact, using various sources of mathematics have little impact on student’s learning. What made a huge difference between Canadian school and Chinese school is the way of teaching the knowledge. As mentioned in Gustafson’s work, memorization serves as a ground for knowledge building in Chinese education. Students are required to memorize definitions, theorems, as well as ways to solve problems. Moreover, no calculator is ever allowed in exams. In my opinion, this way serves well on building a good foundation but when students become more mature, I’d like being taught by Western approach more. When in advance level, students should use tools to aid achieving the purpose of understanding. Time should not be spent on doing arithmetic by hand, yet knowing how to do arithmetic by hand is also important. 

•  What are your thoughts about the naming of the Pythagorean Theorem, and other named mathematical theorems and concepts (for example, Pascal's Triangle...check out its history.)

I would say, the way of naming mathematical theorems after the founder is sometimes bit misleading. Nowadays with the advancement in technologies, the development of knowledge are watched closely by academic society across the world. It would be easier to identify the source of a new idea. However, when communications cannot be done efficiently in the past, the validity on awarding one (or more, for co-founding) person for the contribution is questionable. Different people can independently come up with breakthroughs (e.g. Newton and Leibniz). In my opinion, the naming should be referring the knowledge instead of referring the person who come up with it (e.g. intermediate value theorem). Of course the acknowledging of the inventor is also important, but that can be done separately. Another benefit for not naming after a person is to remove the culture bias and barrier in knowledge. For example, when we look at the theorems in math courses, we get the Eurocentric feeling in mathematics because most of the theorems are named after European mathematicians. However, mathematics has been developed across different countries and cultures for quite a long time. Instead of fighting for who did it first, people should pay more attention on the evolvement of the knowledge. 

Babylonian word problems

 Word problems can be both practical and imaginary (pure math). For example, the first problem in our reading is “I have added seven times the side of my square to eleven times its area, and it is 6;15”. If we think from plane geometry point of view with assumption that the numbers ‘seven’ and ‘eleven’ do not have dimension (unit-less), then we would say there is no practical value.

7(side of square) + 11(Area of the square) = 6;15

It is because the equation representing the statement does not satisfy dimensional equality to be practical. The left hand side is the sum of two objects with the first object in dimension of [length] and the second object in the dimension of [area] or square of [length]. Some people would treat it as a puzzle. In fact, the term ‘square’ is not necessary. ‘Square’ and ‘side’ can be replaced by ‘circle’ and ‘radius’ or, more generally, ‘f(x)’ and ‘x’. Thus, if we use symbolic algebra, it can be written as:

7x + 11 f(x) = 6;15

In this statement the function f can be an abstraction such as a measure of an n-dimensional structure built from a component object of value x. We can further generalize it by replacing the constants with other objects (words, parameters) such as:

ax + b f(x) = c

On the other hand, we can assign the length dimension (7 meter, 7 feet, etc.) to the number seven; assign no dimension (unit-less) to the number 11. Then the problem becomes “the area of the rectangle made of seven meter side and the side of my square added to eleven times my square is 6;15”. 

The problem now has a practical value. Therefore, when we talk about a word problem or a set of word problems such as writings on clay tablets found in Babylon archaeological extraction from the so-called lens of ‘practicality’, ‘generality’, or ‘abstraction’ while sitting at our time, we would see that they are connected or we can build the connections. In other words, in the language of contemporary (symbolic) algebra in which we have a store of not only natural language but also mathematical symbols, we see the connection. We can see them separate, and we see them together because we can specialize, generalize, and replace some objects with abstract objects. Any applied problem can be generalized, then add abstraction to it. When we studied ‘vector space’ (abstract idea) in linear algebra, we could also transform in opposite direction.

 However, if we can transport ourselves to the time of Babylonia, what would be the case? We would not have symbolic algebra. We could start from ‘generality’. Generalization is possible using words. The above same example can be reworded into: “I have added seven times the size of a side of my object to the eleven times the size of my object. It is 6;15”. Here the word ‘object’ can be square, circle, cube, etc.

For abstraction without symbols, it is possible but it would be limited by the skills in using words and sentences, and experiences either real or imaginary. In above rewritten word problem, we could build the ‘object’ as an abstract object. In “Babylonian algebra (Crest of the Peacock)” we learned that Babylonians did power of four and eight even though they could only have power of two (area) and three (volume) in applications. Therefore, it is safe to assume that limited abstraction is possible without using symbolic algebra.

By this discussion, we find that by learning the history of mathematics (Babylonian and Egyptian), we are studying applied and pure mathematics of those days, and we could learn how generalization and abstraction can be possible and to what extent without using symbolic algebra.