Monthly Archives: March 2015

Exploring Up-Arrow Notation

One of the greatest things about math is that there are so many ways to do the same thing. We can express the same equation in multiple fashions and still keep the meaning the same. If you stop and think for a moment, you too will realize how awesome this is! How many different ways can you write the word “the” in English? Take a moment to mull this one over; it’s a tough one. Now if you came up with more than one way, you and I need to have a sit down conversation because you are most likely a genius. Anyways I digress, the point is the fact that math allows us so much flexibility in the ways we represent things is beyond amazing.

Take a look at the following exponential expression: , if you are familiar with how exponents work, you’ll recognize that this equals 8. Unfortunately, if you are unfamiliar with this particular area of mathematics, this expression is basically meaningless to you. Luckily for us, math allows us to put this into a different form. So now we can take and represent it instead as . This shows us that 2 raised to the power of 3 is really just 2 multiplied by 2 multiplied by 2. Still not simple enough for you? Well hallelujah, we can express it in another fashion. We can change to the simpler version of . So now we have 2 plus 2 plus 2 plus 2 for the grand total of 8. At the beginning we only had one form to look at this expression but by the end we have 3. There are so many ways to represent things in math that people began to push the limits of that fact. Much like exponential or scientific notation, other mathematicians came up with their own notations (ways of representing expressions in math). The one that I am going to focus on today is Knuth’s Up-Arrow notation.

Here we have ourselves a very fun way of representing math. The closest thing that it can be compared to is exponential notation. Looking at the above example, we can draw a couple of conclusions about the mathematical expressions we used. First, multiplication is just a series of addition operations. What I mean by that is, the expression is really just a short way of writing 2 plus 2 plus 2 plus 2 (a series of addition operations) Can you imagine how awful it would be if every time you had to double something, you had to write out every addition operation it took? It would be pure anarchy! Along the same vein, exponential notation is just a way of expressing a series of multiplication operations. So in this instance, would be the short way of writing . So now enter Knuth’s Up-Arrow notation, this notation gives us a way to represent a series of exponential operations.

The whole idea of multiple exponential operations can be a little daunting so let’s go over an example. Lets take a regular exponential operation that we are used to seeing, like . So this is easy enough to understand, lets convert it into up-arrow notation. As you can guess from the name, up-arrow notation uses an “up-arrow” as its symbol, so 32 turns into 3↑2. And there you have it, up-arrow notation! Just kidding, we have barely scratched the surface that is the awesomeness of up-arrow notation. That was an easy example; so let’s take it one step further. Let’s say that you wanted to write out 3 raised to the power of 3 raised to the power of 3 (333), that wouldn’t be too bad right? What if you wanted to take it out one more step? How about two more steps? Eventually you are going to hit your limit of how small you are actually able to write. But don’t you fret your little mathematician head; up-arrow notation is here to save the day. 3 raised to the power of 3 raised to the power of 3 (333) becomes the nice and simple expression 3↑↑3. Whew that was a whole lot easier and shorter to write out. This could continue until you were blue in the face. For example, if we take 3↑↑4, this doesn’t translate to 3 raised to the power of 3 raised to the power of 4 (334), this actually is equivalent to 3 raised to the power of 3 raised to the power of 3 raised to the power of 3, or 3333. So as you can see, these numbers begin to get bigger very quickly!

Moving on to something even more complex (yay for complexity!!! Wait….), let’s look at when we add a third arrow into the mix. Basically when you add an arrow, you create a series of up arrow operations. So if you have “n” arrows, you can expand it out into a series of (n-1) arrow operators. So looking at the example 3↑↑↑2, let’s expand this out into a series. The problem would then look like 3↑↑↑2 = 3↑↑3 = 3↑3↑3 = 7625597484987. So we had n=3, so when we expanded it out, it became a series of two arrow operations, and then we expanded that out to a series of one arrow operations. So now when we change this to 3↑↑↑3, we can again do this expansion. This time we get 3↑↑↑3 = 3↑↑(3↑↑3) = 3↑↑(3↑3↑3). If we look back at our double arrow example, we know that we will have 3 raised to the power of 3 raised to the power of 3 and so on 7625597484987 times! So as you can see, the triple arrow form grows unbelievably faster than even the double arrow form.

I hope that you are beginning to see the usefulness of this notation. You can take unnecessarily complicated expressions and shorten them to something much easier to read and much easier to write. Up-Arrow notation even goes as far as a quadruple arrow notation, but we will save that for another time. In the future, I hope you’ll use what you’ve learned here to confuse a friend or show off at a classy math party. Until then, just enjoy this fun little notation.

My Top Secret Messages to Maryam Mirzakhani

If you’re anything like me, you need to send TOP SECRET messages all the time.

Just the other day, I was working on a really hard problem set for my History of Math class, so I decided to ask my good friend Maryam Mirzakhani to do it for me. This, of course, went against my University’s cheating policy, so I needed to be sure that my message was encrypted securely enough that my resourceful and mathematically gifted professor Evelyn Lamb couldn’t read my message and fail me for cheating.


Luckily, by the grace of modular arithmetic, I was able to have a quick exchange with Maryam just in time to hand in my assignment undetected. Below I’ll discuss the rad encryption algorithm Maryam and I used to exchange messages, and the clever but unfortunately unsuccessful algorithms my suspicious professor tried to discover our ploy.

We decided to encrypt with RSA and pay homage to the best public-key cryptosystem around. RSA is an asymmetric algorithm, which means that the keys of the sender and the receiver are completely independent. Maryam and I needed to independently complete the steps below to exchange encrypted messages.

1) I chose 2 extremely large prime numbers p and q.
I went with my favorite primes, 61 and 11.
2) Set my modulus to be n = p * q, and held on to a value I’ll call ϕ(n) = (p-1)*(q-1)
So for me, n = 11*61 = 671, and ϕ(n) = 10*60 = 160.
3) Chose the exponent “e” for my public key
The number e just needs to be coprime with ϕ(n), a common choice is 216 + 1 = 65,537 but 3 is sometimes just as good a choice.
I chose e = 7, just because I happen to like 7.
4) Found my private key exponent, “d” as the multiplicative inverse of e mod ϕ(n).
That is, find d such that d*e = 1 (mod ϕ(n)).
Normally, you can do this using the extended Euclidean Algorithm.
But I instead used the coveted Wolfram-Alpha algorithm, and found that d = 23.

After these steps Maryam and I each had a public and private key- you can think of these as keys that interchangeably lock and unlock the message. Anyone listening in (like Professor Lamb) can see each of our public keys- this is what allows strangers on the internet to securely exchange messages.

The public key consists of n and e, and the private key is d. My public key was (n = 187, e = 7) and my private key was d = 23 (but don’t tell Professor Lamb!) Maryam broadcast her public key, which was (n = 779167, e = 17).

I want to encrypt my message:
Hi Mimi! How great is the weather in California? Hey, I have a favor to ask…

First I converted the letters in my message into numbers by some publicly known agreed upon encoding, and broke my message into chunks so that the value of each chunk was less than Maryam’s public key value n, again with a publicly agreed upon scheme:
720 010 500 077 001 050 010 900 105 000 330 007 200 111…

I then encoded each chunk into the cypher-text c using Maryam’s public key (n = 779167, e = 17) as: c = me (mod n)
So specifically, c1 = 72017 (mod 779167)
c2 = 1017 (mod 779167) and so on.

I sent these encoded cypher-text chunks to Maryam, who then used her private key d to decode them into the message that I wrote:
m = cd (mod n)

This is because I encoded the cypher-text as c = me (mod n), so when Maryam computed cd, she had actually computed (me)d (mod n) = med (mod n). Recall that Maryam very carefully chose e and d so that e*d = 1 (mod ϕ(n)). This means, thanks to Fermat’s Little Theorem, that med (mod n) is the same as m1 (mod n). Excellent news, this is just my original message! Thanks, modular arithmetic!

We could now securely exchange messages, and for even more security I even left a signature in my message so that Maryam could be sure the message actually came from me.

But not so fast! Professor Lamb noticed that Maryam and I were exchanging mysterious messages, so she took a stab at decoding them.

Pollard’s p-1 Algorithm
RSA is a secure algorithm because it is very difficult to factor large numbers.

Recall that when I sent Maryam a message, I encoded the message m into cypher-text c using her public key (n and e) as:
c = me (mod n)
and she decoded the message using her private key d as:
cd = med (mod n) = m (mod n)
This is secure because, if you remember back to how Maryam chose e and d,
e*d = 1 (mod ϕ(n))

This means that for Professor Lamb to decode the message that I sent to Maryam, she needed to find d. To find d, she needed to know what ϕ(n) = (p-1)*(q-1) was, because you need to know the modulus before you can find the inverse of an element, and to find ϕ(n) she needed to figure out p and q. Therefore, the only thing standing between me and expulsion for cheating is the fact that it’s very hard to factor very large numbers. Notice, however, that all the other information is publicly available- c, e and n can be viewed by everyone.

Professor Lamb decided to try Pollard’s p-1 algorithm to factor Maryam’s public key modulus, n = 779167. She first decided to try the algorithm on a smaller, more manageable number, so she tried n = 5917. Here’s what she did:

1. She chose a positive number B.
Professor Lamb liked the number 5, so she set B = 5.

2. Computed m as the least-common multiple of the positive integers less than B.
m = lcm(1, 2, 3, 4, 5) = 60

3. Set a = 2.
Easiest step ever.

4. Found x = am – 1 (mod N) and g = gcd(x, N)
x = 260 – 1 (mod 5917) = 3417 (mod 5917)
g = gcd(3417, 5917) = 61

5. If g isn’t equal to 1 or N, then you’re done!
Professor Lamb found that 61 was a prime factor of 5917! Slick!

6. Otherwise, add 1 to a and try again. If you’ve already tried 10 times, just give up.
Luckily she didn’t need to use this step, but for a lot of different n’s she probably would have.

Feeling triumphant and confident in Pollard’s p-1 algorithm, Professor Lamb turned to Maryam’s public key modulus, n = 779167. The first 3 steps were the exact same as before, and for step 4 she found:
x = 260 -1 (mod 779167) = 710980
g = gcd(710980, 779167) = 1

Drat! Professor Lamb then had to proceed to step 6, increased a to 3 and try again:
x = 360 -1 (mod 779167) = 592846
g = gcd(592846, 779167) = 1

Double drat! Professor Lamb continued this for approximately 10 steps, and then gave up. (Really I should just be glad that she didn’t try to factor my public key modulus n = 187. Our encryption would have been much more secure if I had chosen much larger primes!)

Luckily for me, Maryam and I chose a secure encryption algorithm. RSA is set up so that to decode the message, you need to know the prime factors p and q of the modulus n. You need p and q so that you can find the inverse of the public key mod (p-1)(q-1), and these public and private key exponents work to encode and decode the message because of Fermat’s Little Theorem.

Professor Lamb tried to decode our secret messages by factoring Maryam’s public key modulus with Pollard’s p-1 algorithm, but unfortunately it did not yield a prime factor. Because finding large prime factors is such a difficult problem, Professor Lamb wasn’t able to read our secret messages, and I got an A on my homework.

Obvious disclaimers!

– I obviously didn’t ask Maryam Mirzakani to do my Math History homework. She’s an incredibly intelligent lady, working on much, much more difficult things, and apparently getting awesome results.

– I obviously don’t endorse cheating and Professor Lamb’s homework is not too difficult. It is just difficult enough 🙂

– Even though I motivated the need for privacy in my silly article with my desire to keep my professor from finding out I was cheating, privacy is obviously very important for a wide range of reasons(possible hyperlink?), and is equally important to protect people who don’t have anything to hide.

– The ascii art image of Maryam Mirzakani is obvious very cool! It was made by my very talented friend Tobin Yehle, who wrote a neat program to translate photos into ascii art.

Circle Limit III

The angles of a triangle must add up to 180°. This is a simple fact that you were probably taught fairly early in your math career. It’s been known for millennia and is pretty simple to prove: for a right triangle, assume we have two parallel lines, one line perpendicular to them, and a fourth line between one of the intersections and an arbitrary non-intersection point on the opposite line as shown below.
This makes a triangle with one right angle, C, and two acute angles, A and B. We also need to consider angle D, the complementary angle to A. We know that A+D has to be 90° since they sum together to make a right angle, so the measure of angle D must be 90° – A. Since D and B are alternate interior angles with respect to the parallel lines and the red transverse line (remember all those awful congruence theorems you learned in your high school geometry class?) they have to be congruent angles. This means that the measure of angle B has to be 90°-A as well. So if we sum up the angles inside the triangle, A + B + C = A + 90° – A + 90° = 180° + A – A = 180°. The proofs for acute and obtuse triangles are similar, but a bit more complicated so we won’t go through them. The point is, we proved it! Triangles have to have 180°, right? Wrong.
The proof we used—and indeed all proofs that triangles must have 180° inside them—relies in some way on an infamous postulate used by Euclid around 300 BCE that says (more or less) that given a line and a point not on that line there is exactly one line through the point that does not intersect the original line. This postulate, though reasonable sounding, foiled mathematicians for thousands of years. Despite attempt after attempt to prove this postulate, no one was ever able to succeed. In fact, it was eventually proven that there IS no proof of this persnickety postulate. The angry mathematicians, having been foiled by this simple-yet-unprovable statement, began to consider what would happen if, indeed, it were not true. What would happen if, for example, there were an infinite number of lines through the point that didn’t intersect the original line? This line of questioning led to the discovery of hyperbolic geometry: a world where there are infinitely many parallels to a line through a given point off the line.
One of the many interesting aspects of hyperbolic geometry is that triangles don’t have to have 180°—In fact, they must have less than 180° (otherwise they could be a triangle in spherical or euclidean geometry). These triangles can still tessellate a plane though! In one particular representation of hyperbolic space, called a Poincaré disk, this tessellation would look like the image below.
The Poincaré disk is a way to show the hyperbolic plane on a circle. The idea is that straight lines are represented as curves from one side of the circle to another with the intention of preserving angles without necessarily preserving lengths. These curves must be circles that intersect the boundary of, or must be diameters of, the disk. The result is that each triangle in the picture above is the same size! From the large-looking central triangles to the itsy bitsy ones on the edge, each triangle would have exactly the same area in a hyperbolic space.
M.C. Escher was a Dutch artist whose graphics are widely known for their otherworldly bizarre mathematics. Stairs that led up to themselves and water that flowed in a ring are just two examples of his pieces, enacted with an almost formulaic mathematical exactness. He is well known in scientific communities for the diagramesque works of art.
You may be asking what this little Dutch artist has to do with our discussion of “curved” triangles. Well, Escher had become somewhat famous for using tessellations in his work. Creating shapes, especially in the shape of animals, which would tessellate all the way across the pieces, forming a lattice of cells that had only to be filled with a clever image. In the early 1950s, he became curious about finding different ways to “draw” infinity on a page. A letter from a friend came to him with some of these Poincaré tilings in the hyperbolic plane and became enamored with them. The images in the letter were a type of tiling denoted by {p,q} that was a tiling of p-gons with q of them meeting at each vertex. These images of hyperbolic tilings inspired Escher to create his Circle Limit series in 1959 and 1960. Circle Limit III was inspired in particular by the {8,3} tiling—4 octagons meeting at every vertex, and is a beautiful reimagining of the tiling with fish in place of the triangles.

Circle Limit III by M.C. Escher. His other work, including the other Circle Limits, can be found at


Circle Limit III with the {8,3} tiling overlaid on it. Image by Doug Dunham.

Escher’s works seem to represent the very nature of the hyperbolic plane that we have talked about. After all, in a world where there are an infinite number of parallel lines, why couldn’t I draw infinite fishes on a page?
Anyone wanting to know more can Google hyperbolic geometry, parallel postulate, M.C. Escher, or triangle group.

Continued Fractions and the Pell Equation

Recently in class we lightly touched on the subject of continued fractions, and it brought back memories of a class I took last semester on Number Theory. I thoroughly enjoyed the class and a section I enjoyed more than any other was on the Pell Equation. This equation, of the form x2-Dy2=1, is a Diophantine equation, a polynomial equation with more than one variable, named after Diophantus of Alexandria. The equation was first studied by the Indian mathematician Brahmagupta, although he never gave a general solution, rather he used specific examples. The first person to provide a general solution was Lord Brouncker; however, Euler attributed the solution to John Pell, most likely because he confused Pell with Brouncker. The Pell Equation can be used as an approximation of the square root of non-square numbers. When D=11, a solution of that particular Pell equation (in this case, the first integer solution) is (10,3). 10/3=3.333 repeating and √11 is just over 3.1.

The D=2 Pell Equation graphed. Where the function intersects a point where x and y are integers represents an integer solution. Image: David Eppstein, via Wikimedia Commons.

The method I use to solve the Pell Equation relies completely on continued fractions, specifically by writing √D as a continued fraction; this was primarily how Lord Brouncker found a solution. A continued fraction is the sum of a fraction within a fraction within a fraction. What is interesting is that all numbers can be written as continuous fractions. Rational fractions always terminate, while irrational fractions can continue forever or enter a periodic cycle. An example of a continued fraction is:


Our fraction will have the form:


and we will shortly define all of the α that appear. The fraction above also has a floor function. The floor is defined as the highest integer component of a number. A couple examples of this are floor(3.5)=3 and floor(√2)=1.

Continued fraction of √2, which has period p=1. Image: Zahnradzacken, via Wikimedia Commons.

Continued fraction of √2, which has period p=1. Image: Zahnradzacken, via Wikimedia Commons.

To tackle the Pell Equation it is necessary to define α and β. The first is α=√D+floor(√D)=, and the second is β=α-floor(α)=√D-floor(√D), which is equivalent to α=floor(α)+β . The method using continued fractions is recursive such that αn+1=(1/βn) and βnn-1. The reason why D must be a non-square number is that the square root of a non-square integer number will have a non-terminating periodic continued fraction. If D were a square number, β=√(square)-floor(√square)=0 as it becomes a whole number minus the floor of that same whole number. The periodic fraction will need to be truncated at a certain point, and that point is when the bottom denominator is equal to the original α.

Now the calculations can begin. For simplicity I am selecting D=3. α=√3+1 and

β=√3-floor(√3)=√3-1. The first step is to calculate the period of our fraction, that is find all αn.

α1=1/β=1/(√3-1) then multiply the top and bottom by the conjugate √3+1 which will give:

α1=(√3+1)/2=/=α. (floor(α1)=1) Then calculate β11-1=(√3-1)/2

α2=1/β1=2/(√3-1)=√3+1=α, thus D=3 has period p=2.

The continued fraction is:


When working backwards to calculate √3 as a fraction with only one denominator, the fraction comes out the be (2√3+3)/(1√3+2). Then it can be claimed that this can be written as:

D=(a√D+Dc)/(c√D+a) and we can compose a 2×2 matrix [a,Dc][c,a] and the determinant is:

a2-Dc2 (which looks a lot like the original equation). If you take a=2 and c=1, and substitute them in for x and y, you get 22-3*12=4-3*1=4-3=1 Thus (x,y)=(2,1) is the first solution to x2-3y2=1

One more interesting thing we can do with periodic continued fractions is, given such a fraction, we can find D and then find the solution to the equation. An example would be

α=6+1/(3+1/(6+1/…))) where 3 and 6 alternate. Because of the periodic nature of the fraction, it can be rewritten as α=6+1/(3+1/α)). Solving for alpha, the equation becomes

2+α=19α+6, then 3α2-18α=6, and if we divide by 3 and add 9 to both sides to complete the square we get (α-3)2=11, thus this is the continued fraction of √11. Because the continued fraction was given at the beginning, α=√11+3 can be plugged into it and use the same process as D=3 to get the first solution to the equation x2-11y2=1, which happens to be (x,y)=(10,3).

There are a few little quirks that really interest me about the Pell Equation. What might be considered intuitive is that if D=n2-1, then x=n and y=1. Another quirk is just how large first solutions can become. When D=61, the smallest solution (x,y)=(17663190049,226135980), yet when D=63, (x,y)=(8,1). Another quirk solution is (x,y)=(1,0), as it is true for any D.

The Pell Equation is, at least to me, an equation that is beautiful because it looks so simple and yet has some surprising methods to solve it and has a wide range of solutions. It relies upon periodic continued fractions and numerical methods to solve and has so far been my favorite problems to work on.

Sources: (Used for generating equation images)

The Bridge of Asses

Image: Jenny Mealing, via Wikimedia Commons.

I don’t mean to be crude or inappropriate with my title.  After all a donkey used to be called an ass.  I don’t know what brought about taboo on the word, but in fact, it is by many considered a bad word.  I only used this title because it seemed a good attention grabber.  The phrase did, grab my attention after all.  But, what is “the Bridge of Asses,” and how does it have anything to do with Euclid or mathematics?  Why am I writing about “the bridge of Asses?” It may be because, “Ass” is the first swear word I ever used.  I was in a Shakespeare play called Much Ado about Nothing, and one of my character’s lines was, “You Are an ASS!”  Of course it was not so hard for me to accept the fact that I was swearing, as it was to accept the fact that my parents seemed to think it was funny and OK.  I was 12, but I digress. Let me tell you the real reason I am discussing, “The bridge of Asses.”

Sometimes we want to be able to tell if someone is really interested in something, or even if they are able to quickly grasp a concept.  Euclid’s fifth proposition in the first book of his elements was used to do just that.  Now before I proceed, lest I be accused of shaming people who have a hard time with math, I must say that I struggle very much with math and while reading about Euclid’s fifth proposition often felt like the “ass.” Don’t mock me! We all have our strengths and weaknesses. I am just trying to tell you about a something which I find interesting. Let’s talk about some history.

Around 1250 a man named Roger Bacon gave an alternate name to Euclid’s fifth proposition in the first book of his elements, which I will from here on out refer to as “the Bridge of Asses” or the fifth proposition.  The name he gave it was Elefuga, another word I will use freely to refer to the fifth proposition.  Elefuga, derived from Greek, means, “escape from misery.”  Medieval boys were presented with the Elefuga shortly before their “escape from misery.”  That is to say most medieval young men’s experience in geometry ended shortly after they encountered the fifth element, because it proved they simply did not want to go on or their mentor felt they should not.  They, like a donkey fears crossing a bridge, had a hard time grasping the fifth proposition or refused to grasp it. I personally believe they refused to try to grasp it or the mentor did not want to walk them through it well enough. This is because I think with time and patience people can overcome most barriers, but again I am digressing.

To better explain this, “the Bridge of Asses,” also known as the isosceles triangle theorem, is Proposition 5 of Book 1 of Euclid’s Elements.  But, also, pons asinorum, the Latin translation of “the Bridge of Asses,” became a metaphorical statement for a problem that will separate the confident from the unconfident. In other words it is a critical test, of the ability and understanding, of an individual. You see things like this all the time in movies. Usually someone has a sensei or master and they are trying to prove themselves. Eventually they come to the test that decides if they will continue with their training or not. For Bruce Wayne in Batman Begins it is, possibly, when he brings the flower to the League of Shadows high up in the mountain so that he can begin training with them.  Now we want to pass “the Bridge of Asses” for math, or proposition 5. Let’s see if you and I can manage to cross the bridge of elements together.

First, what is proposition 5?  Straight from Euclid’s elements, it is that, “In isosceles triangles the angles at the base equal one another, and if the equal straight lines are produced farther, then the angles under the base equal one another. Now, just hearing it makes sense, but to cross “the Bridge of asses” we must also prove proposition 5 and most importantly understand the proof.

Now I have read many blogs and articles proving the fifth proposition so I feel that I must make it clear that I am deriving this proof from an article, “the Bridge of Asses,” from [1].  Also to make the proof more clear, I am going to list our proof in steps.

  1. We need to draw an isosceles triangle. It will have points ABC. For review, because I had forgotten, we must recognize that isosceles means that the sides AB and AC are equal.
  2. Now we want to extend past AB and AC indefinitely.

Step 2

  1. Now we want to add two more points D and E. The line AD will pass through the point B. The line AE will pass through the point C. AD and AE will be equal.

Step 3

  1. Now that we are past this point we must notice that the angle at DAC and the angle at EAB are equal. This is a simple to believe since they are the same angle.
  2. From step four we say that the triangle DAC and the triangle EAB have equal angles when all the corresponding side’s angles are compared. We can use the side-angle-side theorem to prove this.  It says that two triangles are equal when the triangles have two sides of the same length and the angle of those two sides is the same.

Step 4-5

  1. From step five we can conclude that the angles ADC and AEB are the same as well as that the lines DC and EB are equal.
  2. Now if we subtract AB from AD and AC from AE we can show that BD = CE.
  3. IT now holds by side-angle-side theorem that the triangles DBC and CEB are equal. If they are equal then so are the angles DBC and ECB
  4. We have now proven that the angels ABC and ACB are equal because the angle ABC = 180 degrees – the angle DBC and the angle ACB = 180 degrees minus the angle ECB when the two angles being subtracted are equal.

I hope that you found the proof I presented sufficient.  I don’t claim it as my own since I had to get help to cross this bridge. Hopefully I was able to help you across also, if you even needed help.  If you are still unsure, I suggest a pen and paper.  After all, that is really how it came to make sense to me.

Well now that we have crossed “the bridge of asses” together we are ready to further our careers in mathematics.  Really though I think the concept of “the Bridge of Asses” has a significant meaning. We will continually come across bridges in our education and careers. Sometimes we will feel that the bridge we are presented with is scary and hard to cross.  When I first saw the proof of proposition five that is what I thought. But if we take the time, think about it, and cross the bridge we will be that much better. Just like you and I crossed this bridge we can cross others. Don’t hold back, break a problem into steps, study it, think about it, and together we will cross “the Bridge of Asses.”





The Pythagorean Theorem


Have you ever pondered where mathematical equations come from or how they were derived? If the answer is yes, I want you to think if you have ever wondered where the Pythagorean theorem came from. Whether you’re in geometry, trigonometry, algebra or calculus you have to admit that we see this theorem often in each one of these math classes. Wow, it seems like it’s stalking us! Well we can thank a few cultures for that!

The History

The Babylonians are known for their discovery of Pythagorean triples. How do Pythagorean triples relate to the Pythagorean theorem, you ask? Well, they actually do play a big role and I’ll explain why later. Similarly, Pythagorean triples were also discovered by the Chinese during the Han Dynasty. However, the Chinese mentioned one thing in their proof that the Babylonians left out. This was the relationship between Pythagorean triples and a right triangle. This Discovery, made by the Chinese, is very similar to the Pythagorean theorem that we know and use today.

There was also a gentleman by the name of Pythagoras who was made famous for his discovery of the Pythagorean theorem. Though he had a prior knowledge about Pythagorean triples he was still able to find a relationship between the Pythagorean triples and right triangles. The Pythagorean theorem is mostly attributed to Pythagoras because authors like Plutarch and Cicero gave him the credit. So, I want to provide you readers with a little bit of background information on Pythagoras.

Pythagoras is from Samos Island. From a young age he was very well educated but, at the time, his passion was poetry not mathematics. However, later on Pythagoras stated to become much more interested in math and science because of the influence of Thales. Pythagoras even traveled to Egypt and he attended math related lectures there. As he gained more interest in mathematics he decided to move to the island of Croton fulltime, and specialize in Geometry. It wasn’t until later in his career that he derived the Pythagorean Theorem.

What is the Pythagorean theorem?

The cool thing about the Pythagorean theorem is that it is known to be one of the earliest geometry related theorems! The theorem states that in right triangles the square of the hypotenuse equals the sum of the squares of the other two sides. This may be a little bit confusing written out in word so I have provided a picture below!

In this particular picture, c2 = a2+b2. Hopefully that makes more sense! Now lets break down the Pythagorean theorem just a little bit more. Imagine that you have two square of two different sizes and you used them to construct multiple right triangles. Now I know that this sounds a bit confusing and you may be thinking how can I get multiple triangles from just two squares? Well, What if we put the smaller square in the center of the larger square, but we rotated the small square slightly so that it resembled a diamond. It should look something like this!

Image: Yuval Madar, via Wikimedia Commons.

Now you are able to divide the drawing up in different lengths by using different variables. From the drawing you can see that the letter “c” labels each side of the diamond or the Hypotenuse (the longest side) of the triangle.   The letter “a” labels the shortest side of the triangle and “b” labels the medium size leg of the triangle. Also notice that side “a” and side “b” both create a right angle within the triangle. I’m sure that this is making sense visually but not mathematically. Well then, I will explain in mathematical terms how these two squares and this picture relates to the Pythagorean theorem.


  • The area of a square can be written like this: (a + b)^2 = a^2 + b^2 + 2ab
  • The area of a square can also be written in term of the four triangles that we created, with the variable “c”, in the diagram above: c^2 + 4(ab/2)
    • So this means that (a + b)^2 +2ab = c^2 + 4(ab/2)
  • Now all we have to do is simplify the equation!
    • If we subtract 2ab from the left side and we are able completely cancel it out.
      • So we end up with (a + b)^2 = c^2
    • Lastly if we distribute the ^2 (on the left side of the equation) to both the “a” and “b” variables we end up with: a^2 + b^2 = c^2 and that’s the how we derive the Pythagorean theorem!!!

What can we do with the Pythagorean theorem?

Like I said before the Pythagorean theorem is used on right triangles. More particularly we use the theorem when we know the value of two sides of the triangle and we want to find the value of the remaining side of that particular triangle. We can also find the distance between points with this theorem. The Pythagorean Theorem is often used in higher-level math classes like calculus. For example in calculus three, we use this theorem to find the distance between two points on a plane, finding the surface area and volume of different shapes and etc.


Thanks to Pythagoras, the Babylonians and the Chinese we have the Pythagorean theorem. The famous theorem is a^2 + b^2 = c^2. We are able to derive this formula by taking the area of a square. And lastly the theorem is used a lot in finding the side lengths of a triangle and is also helpful in higher-level math courses.


The Path of Chinese Astronomy

“Second star to the right and straight on ‘til morning.” I’m pretty sure that if most people now took Peter Pan’s directions to Neverland, they’d never get there. In these modern times, we rely significantly less on the sky to navigate and it’s only because of the knowledge accumulated and developed over time. Finding new information usually isn’t an easy task without using prior knowledge of another subject or idea as a starting point. For example, astronomical observations used known mathematics to become more pertinent to modern-day science. Such was the case for China, as time was measured and kept constant with the usage of the cycles of the sun and moon as well as intercalation– the insertion of days and months to make the lunisolar calendar follow the moon phases. The study of the night sky flourished for the Chinese during the Han dynasty (206 BCE – 220 CE), continuing through to the modern day.

Mathematical proof for the Pythagorean Theorem from the Zhou Bi Suan Jing. Image: Chinese Pythagorean theorem, from page 22 of Joseph Needham’s Science and Civilization in China: Volume 3, Mathematics and the Sciences of the Heavens and the Earth, published in 1986 by Cave Books Ltd., based in Taipei, via Wikimedia Commons

The lunisolar calendar was used to mark the passing of the seasons and special occasions. The Chinese used advanced algebra for this purpose. It was mainly equatorial based, which focused on circumpolar stars and ecliptic frameworks that stemmed from Western science. The Zhou Bi Suan Jing (The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven), one of the oldest and most famous Chinese mathematical texts dating back to the Zhou dynasty (1046 BCE – 256 BCE), used the Pythagorean Theorem on astronomical calculations as well as for multiple equatorial based problems. To go into detail, one of the 246 problems in the compilation was to find the height of the sun from the earth, as well as the diameter of the sun. One person was to wait until the shadow of a 264 cm gnomon (the part of a sundial that casts the shadow) was 198 cm so that a large 3-4-5 right-angle triangle could be formed. This larger triangle would be from the sun straight to the ground, along the ground to the gnomon (forming the right-angle), and from the gnomon to the sun (angle of elevation). The smaller triangle, consisting of just the gnomon and its shadow, was used to find the equivalent measurements of the larger triangle so that the Pythagorean Theorem could be applied. As a result to this problem, the base of the triangle would be 24,900,000 m, the height of the sun 33,200,000 m, and the hypotenuse going toward the sun 41,500,000 m.

Su Sung’s diagram for the Cosmic Engine. Image: Page 451 of Joseph Needham’s book Science and Civilization in China: Volume 4, Part 2, Mechanical Engineering, via Wikimedia Commons

Moving through to the Tang dynasty (618 CE – 907 CE), Yi Xing was a well-known monk, engineer, and astronomer who had used the knowledge of the previous dynasty to work on an astronomical clock, which displayed the relative positions of the sun, moon, zodiacal constellations and major planets occasionally. The improvements on the function of an astronomical clock would later be succeeded by Su Sung during the Song dynasty (960 CE – 1279 CE), when he created a water-driven astronomical clock for his clocktower, and designed  and constructed a Cosmic Engine that operated as an astronomical hydromechanical clock tower. Su Sung had worked off of the achievements of Zhang Heng, an astronomer, inventor, and guru of mechanical gears who lived from 78 CE – 139 CE. Along with that, Su Sung was among the first during the dynasty to work on empirical science and technology.

Caption: A Ming dynasty (1368 – 1644) mariner’s compass diagram developed from Shen Kuo’s studies.. Image: Unknown, via Wikimedia Commons

Another genius during this time was Shen Kuo, who was most known for finding the concept of the geographic north pole (true north) and the magnetic declination (angle on the horizontal plane between magnetic north and true north) towards the north pole by using a more precise measurement of what’s known as the astronomical meridian (a large circle that passes through the celestial poles, the nadir (vertical direction pointing in the direction of the force of gravity), and the zenith (vertical direction opposite to the force of gravity, opposite of the nadir) for a given location). He also used advanced math to calculate the position of the pole star that had moved over many centuries, which made sea navigation more accurate using a magnetic needle compass. Shen Kuo also theorized that the sun and moon were both spherical and used cosmological hypotheses to predict planetary motion. He worked with his colleague, Wei Pu, to record and plot the moon’s orbital path for a duration of five years. However, much of their efforts were wasted thanks to political rivals who only used part of the corrected plots calculated by Shen Kuo and Wei Pu for planetary orbital paths and speeds.

The Song dynasty was followed closely by the People’s Republic of China from 1912 to modern-day, during the time of rapid development in science and technology. They moved away from the study of celestial objects and focused more on the application of past astronomical studies on mechanical technology and modern science. So where the use of calculation, measurement, and logic was previously aimed toward the shapes and motions of celestial objects, it was now applied to military technology, arsenals, shipyards, steamships, and artillery. In short, the Chinese did not reduce observations of nature to mathematical laws until much later, since for a short period after the Song dynasty the focus was mainly on literature, arts, and public administration. Chinese mathematics had shifted more towards Western mathematics in terms of being used for technology and modern science rather than for astronomical studies. Despite this, China and many other Asian cultures still use the lunisolar calendar, remade each year using the same mathematical calculations from the Han dynasty.