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Differential Geometry: A Visual Appeal

[cochleoidcatacaustic]

An Unusual Relationship

While it is common to see the trigonometric functions used on a day to day basis, most students seem to forget (or ignore) the existence of the hyperbolic functions. These functions have multiple analogies to the trigonometric functions, and yet they rarely see use in areas of mathematics outside of differential equations and complex analysis. The hyperbolic functions have very basic definitions, as will be shown below. \( \sinh (u) \), pronounced as either "sinch" or "shine" is the hyperbolic sine function. \( \cosh (u) \), pronounced "kosh" is the hyperbolic cosine function. The four other hyperbolic functions, hyperbolic tangent \( \tanh (u) \), hyperbolic cotangent \( \coth (u) \), hyperbolic secant \( \operatorname{sech}(u) \), and hyperbolic cosecant \( \operatorname{csch}(u) \), are all defined in relation to these first two hyperbolic functions.

$$ \sinh (u) = \frac{e^u - e^{-u}}{2} $$

$$ \cosh (u) = \frac{e^u + e^{-u}}{2} $$

$$ \tanh (u) = \frac{\sinh (u)}{\cosh (u)} = \frac{e^u - e^{-u}}{e^u + e^{-u}} $$

$$ \coth (u) = \frac{\cosh (u)}{\sinh (u)} = \frac{e^u + e^{-u}}{e^u - e^{-u}} $$

$$ \operatorname{sech}(u) = \frac1{\cosh(u)} = \frac{2}{e^u + e^{-u}} $$

$$ \operatorname{csch}(u) = \frac1{\sinh(u)} = \frac{2}{e^u - e^{-u}} $$

Additional definitions are given at the Wikipedia page on Hyperbolic Functions.

[Images taken from Wikipedia entry on Hyperbolic function]

These images above show the hyperbolic sine and cosine functions, and how their definitions are related to the exponential function \( e^x \): the hyperbolic cosine function is the average of \( e^x \) and \( e^{-x} \), while the hyperbolic sine function is half the difference of \( e^x \) and \( e^{-x} \).

Initially, despite having similar names, it would seem as if these functions have no relation to the trigonometric functions; the trigonometric functions are used for finding angles and calculating components of vectors, and it seems like there isn't any relationship between the trigonometric functions and the exponential function...so what's the catch?

Well, it turns out the hyperbolic functions are just as good as the trigonometric functions, but they have their own uses. While the trigonometric functions can tell you a circular angle at a given point, the hyperbolic functions can give you a hyperbolic angle at that point. This hyperbolic angle also happens to be the area of a hyperbolic sector up to that point, as seen in the image below.

[Image taken from Wikipedia entry on Hyperbolic function]

The image depicts the unit hyperbola, where the hyperbolic angle \( a\) reaches up to the point \( (\cosh (a), \sinh (a) ) \). This is similar to how a circular angle, \( \theta \), will produce a point \( (\cos(\theta), \sin(\theta)) \) on the unit circle. Clearly, the trigonometric functions and the hyperbolic functions have more in common than we originally thought. But wait, there's more!

Discovered around 1740 and published to the mathematical world in 1748, a famous equation was figured out by Leonhard Euler that provided an astonishing amount of groundwork for complex analysis. This equation is known today as Euler's formula. No, not the one relating the vertices, sides, and edges of a polyhedron.

$$ e^{i \theta} = \cos(\theta) + i \sin (\theta) $$

You'll probably recognize it for a certain value of \( \theta = \pi \):

$$ e^{i \pi} = \cos(\pi) + i \sin(\pi) = -1 + i (0) = -1 $$

Yes, this formula has found its way across the internet, scaring teenagers with its daunting, yet elegant, complexity. Most stumble over the \(i = \sqrt{-1}\) and the \(\pi \) being multiplied together and then exponentiated, and they say something along the lines of "How does this even produce a number? Exponentiating an imaginary number has no actual meaning!" Euler's formula says it does have a meaning: a rotation in the complex plane by an angle of \( \theta = \pi \), or 180 degrees (a half-turn). This is what makes this formula so amazing and eloquent, despite its simplicity.

The uses for Euler's formula doesn't stop there, however! It shows us a relationship between the exponential function and the trigonometric functions! Using Euler's formula, we can define the trigonometric functions in terms of the exponential function in the same way as the definitions of the hyperbolic functions. First, we realize that

$$ e^{-i \theta} = \cos(-\theta) + i \sin(-\theta) = \cos(\theta) - i \sin(\theta) $$

since the cosine function is even and the sine function is odd. Finding the sum of this result with Euler's formula, we can remove the imaginary part of these equations to obtain a formula for cosine:

$$ e^{i \theta} + e^{-i \theta} = 2\cos(\theta) $$

$$ \cos(\theta) = \frac{e^{i \theta} + e^{-i \theta}}{2} $$

We repeat the same process in order to obtain a formula for sine, but instead compute \( e^{i \theta} - e^{-i \theta} \) in order to remove the real part of the equations:

$$ e^{i \theta} - e^{-i \theta} = 2i \sin(\theta) $$

$$ \sin(\theta) = \frac{e^{i \theta} - e^{-i \theta}}{2i} $$

These formulas definitely look very similar to the formulas for \( \cosh(u) \) and \( \sinh(u) \). In fact, we can define each function in terms of the other to truly show how closely they're related:

$$ \cos(x) = \cosh(i x) \qquad \cosh(x) = \cos(i x) $$

$$ \sin(x) = -i \sinh(i x) \qquad \sinh(x) = -i \sin(i x) $$

This astounding result only makes us question the one remaining question about this surprising relation between trigonometric functions and hyperbolic functions: Why aren't the hyperbolic functions known or used as often as the trigonometric functions?

Well, that's beyond the scope of this post. ;)

[cochleoidcatacaustic]

Inversive Geometry

Inversive Geometry is a small subset of the mathematical subject known as geometry that we were taught at some point in our lives. This topic of geometry isn't usually talked about until higher levels of learning, but the fundamentals can be taught with only prior knowledge of basic geometry.

Inversive geometry works by having a starting point, usually called the inversion center, and an inversion circle around the inversion center with a radius \( k \) called the inversion radius. From these geometric constructions, we can find anything from inverse point to an inverse curve which is a locus of inverse points. It's interesting to note that the inverse curve of an inverse curve is the original curve--very similar to how an inverse function or a reciprocal would work. Many inverse curves produced from well-known curves also produce other well-known curves.  For example, the inverse curve of a parabola is a cardioid. An example of this will be shown at the end.

The most primitive example of inversive geometry is inverting a single point. Let's call this point P. To start off, we create the inversion center O and inversion circle \( O \). Then, a length is drawn from the inversion center O to the point P. A circle with diameter OP is drawn which will intersect the inversion circle \( O \) twice. The intersections of these two circles will be called N and N'. (A special property of these two points is that the line formed between P and either N or N' will always be perpendicular to the line between O and either N or N'.) A line between these two points N and N' is constructed. The inverse point P' is the result of the intersection between this line and the original line between O and P. This can be seen in the image below for clarity.

[Image taken from Wikipedia Entry on Inversive Geometry]

As a result of this procedure, points that are on the inversion circle map to the exact same points when inverted.

If this procedure is repeated for all the points within the domain of a plane curve, then we end up with the inverse curve as the locus of all those points. For an example of this, we look towards the parabola. It is easiest to visualize this in polar coordinates due to a very simple formula for finding the inverse curve of a polar equation with respect to the unit circle:

$$ r_{inv} = \frac{k^2}{r_{orig}} $$

For this formula, \( k \) represents the radius of the inversion circle. The polar equation for a parabola is \( r = \frac{l}{1- \cos (\theta)} \) where \( l \) is half of the latus rectum. To find the inverse curve of this parabola with respect to the unit circle ( \( k = 1 \) ), we use the formula and find that the equation for the inverse curve is \( r = \frac{1}{l}(1 - \cos (\theta)) \), which is the polar equation for a cardioid scaled by a factor of \( l \). This can be seen in the image below. Notice how the points on the inversion circle are where the two inverse curves intersect each other.

[Image taken from Wikimedia Commons]

Inversive geometry has a lot of great applications in not only applied math but also more analytical math such as complex analysis and other subjects. I hope this post sparked some interest towards inversive geometry as there is so much more within this area of mathematics!

[cochleoidcatacaustic]

Complex Analysis and Visualization

About a year ago a lot of my mathematical studies had been focused on complex analysis, a topic not very commonly heard about outside of the realm of math. I am an avid user of the online graphing calculator known as desmos and was thinking about how difficult it is for us to visualize complex functions. Essentially, complex functions are "normally" viewed in four dimensions due to the complex term in both the input and output functions. Think of how a real-valued function works by having input values on the x-axis and output values on the y-axis, and now contrast that with how a complex-valued function works by having input values on a plane, and therefore also needs a plane for its output values. We can describe a complex-valued in two ways, either using complex variables \(w = f(z) \) or with real variables split into real and complex parts \(u+iv = f(x+iy) \).

Back to desmos, I was very interested in how complex functions could be visualized without having to look into dimensions beyond what we typically see when graphing real-valued functions, i.e. a 2 dimensional coordinate plane. After seeing other examples of ways that complex functions could be interpreted as a mapping of one graph to another, I decided to experiment a little and create a visual model of a square being mapped based on a given complex-valued function.

Graphed above with the desmos graphing calculator is the function \( w = { \sin^{-1} z} \), also known as the inverse sine function. As you can see, there is a blue square with lines going through it--this represents the input of the complex function. It is grid shaped so that you can understand and visualize how the function maps the grid to the wavy red and orange maps in the background, which correspond to the output of the complex function. While it might not immediately make sense as to what's happening in this graph, you can understand it by first looking at the x-axis, where the positive real numbers exist. We know that \( \sin^{-1} (0) = 0 \), and \( \sin^{-1} (1) = \frac{\pi}2 \approx 1.5708 \), and this continues all the way down the x-axis until \( \frac32 \). The corresponding output for these values can be seen on the x-axis as well but in red because the results of the function are also real. The wavy lines of the graph are therefore representations of how the function skews based on the imaginary values also added together for the inputs and outputs. However, what makes this graph really interesting is when you see how it shifts just by a simple rotation of the original blue graph...

As you can see, the once-centered blue box has been rotated about \( 60 \) degrees clockwise. But the graph of the output in red has drastically changed. It's these small but very interesting results that amaze me and really show how unique complex analysis can be, despite being such a foreign concept.

[cochleoidcatacaustic]

Base Infinity

I started working on this a few months ago, thinking a lot about prime factorization of numbers. Specifically, knowing that each whole number greater than 1 had an unique prime factorization (if order doesn't matter). I had also just taken the AMC10B 2013, and if I recall correctly, one of the questions on it (that I couldn't figure out) had to do with representing numbers in certain bases. Anyway, I thought, 'why don't I take the prime factorization of each number, and write it out by writing the exponents of each prime in descending order?' If this seems confusing, let me explain with an example.

Take \(756\). Its prime factorization is \(756 = 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot 7 = 2^2 \cdot 3^3 \cdot 7^1\). Including zeroth powers for primes that are not factors of \(756\), we have \(756 = 2^\textbf{2} \cdot 3^\textbf{3} \cdot 5^\textbf{0} \cdot 7^\textbf{1} \cdot 11^\textbf{0} \cdots\). Since any prime greater than \(7\) is going to have \(0\) as its exponent, start at \(7\) and write the exponents in order. So, \(756\) would be represented in this form by \(\textbf{1032}\).

Now, remember how I mentioned bases before? Right, so take the smallest possible base that the number can be represented in, and that's its base. So, back to \(756\), it was represented by \(1032\), whose highest "digit" is a \(3\), so the smallest base it can be represented in is base \(4\), also known as quaternary. So, finally, one would represent \(756\) as \(1032_4\). Hopefully that wasn't too confusing.

Anyway, I started making a long list of all the numbers and their respective "Base Infinity" representations. I called it base infinity because there won't be a highest base (there isn't a highest power of two, and the power of two are the first numbers to bring in a new base, see below). For a little preview, here's the first thirty four base infinity representations of the positive integers (I decided to include \(1\) as \(0_1\) because it technically has no factors that are primes...)

$$ \begin{array}{cc} \begin{array}{r|l} n & b_{\infty} \\\\ \hline \\\\ 1 & 0_1 \\\\ 2 & 1_2 \\\\ 3 & 10_2 \\\\ 4 & 2_3 \\\\ 5 & 100_2 \\\\ 6 & 11_2 \\\\ 7 & 1000_2 \\\\ 8 & 3_4 \\\\ 9 & 20_3 \\\\ 10 & 101_2 \\\\ 11 & 10000_2 \\\\ 12 & 12_3 \\\\ 13 & 100000_2 \\\\ 14 & 1001_2 \\\\ 15 & 110_2 \\\\ 16 & 4_5 \\\\ 17 & 1000000_2 \\\\ \end{array} \begin{array}{r|l} n & b_{\infty} \\\\ \hline \\\\ 18 & 21_3 \\\\ 19 & 10000000_2 \\\\ 20 & 102_3 \\\\ 21 & 1010_2 \\\\ 22 & 10001_2 \\\\ 23 & 100000000_2 \\\\ 24 & 13_4 \\\\ 25 & 200_3 \\\\ 26 & 10001_2 \\\\ 27 & 30_4 \\\\ 28 & 1002_3 \\\\ 29 & 1000000000_2 \\\\ 30 & 111_2 \\\\ 31 & 10000000000_2 \\\\ 32 & 5_6 \\\\ 33 & 10010_2 \\\\ 34 & 1000001_2 \\\\ \end{array} \end{array}$$

If you noticed any patterns, we're on the same page! First of all, the bases have an interesting pattern. Going in sequence, the highest power so far, no matter where in the sequence you are, will always be a power of two. This is a direct consequence from two being the smallest prime. The sequence of bases also happens to be the number of elements in the largest set of divisors which are in geometric progression for each number. Then I started taking certain parts of the sequence and seeing what pattern the original numbers formed. First I took all the representations that started with one and had any number of zeros after (i.e., \(1_2, 10_2, 100_2, 1000_2, \ldots \)). If you look back at the list, they form the prime numbers, which makes sense by definition of the representation. Then I looked at representations that started with two and had any number of zeros after, which is the square primes, as in \(4, 9, 25, 49, 121, \ldots \). I considered the other representations such as starting with three and any number of zeros after, starting with four and any numbers of zeros after, and they all represent the n-th power primes. I thought that was pretty interesting, how it came to be represented like this.

Then I looked at any representations with a two in them at all. This would include representations such as \(12, 21, 102, 22, 23,\) and \(52\), just to name a few. I looked at the original numbers, not quite sure what the sequence \(4, 9, 12, 18, 20, 25, 28, 36, \ldots \) was, but found in OEIS that it was numbers that are divisible exactly by the square of a prime. I also looked at representations with a three in them at all, which would end up being numbers with at least one \(3\) in their prime signature. Then I looked at representations with a one in them at all, which are somehow called the weak numbers. This could be done with \(4, 5, 6, \ldots \) as well.

Looking back at the sequence of bases, \(1, 2, 2, 3, 2, 2, 2, 4, 3, 2, 2, 3, \ldots \), I was thinking a bit about probability. Especially with the common occurrence of the number \(2\) in this sequence, I started writing down the probability of each number after successive terms were added. Here's a little table showing the probabilities for the first 10 terms of the sequence.

$$ \begin{array}{rccccc} \text{Seq} & \to & P(1) & P(2) & P(3) & P(4) \\\\ 1 & \to & P(1) = 1 & & & \\\\ 1,2 & \to & P(1) = \frac 12 & P(2) = \frac 12 & & \\\\ 1,2,2 & \to & P(1) = \frac 13 & P(2) = \frac 23 & & \\\\ \hline \\\\ 1,2,2,3 & \to & P(1) = \frac 14 & P(2) = \frac 24 & P(3) = \frac 14 & \\\\ 1,2,2,3,2 & \to & P(1) = \frac 15 & P(2) = \frac 35 & P(3) = \frac 15 & \\\\ 1,2,2,3,2,2 & \to & P(1) = \frac 16 & P(2) = \frac 46 & P(3) = \frac 16 & \\\\ 1,2,2,3,2,2,2 & \to & P(1) = \frac 17 & P(2) = \frac 57 & P(3) = \frac 17 & \\\\ 1,2,2,3,2,2,2,4 & \to & P(1) = \frac 18 & P(2) = \frac 58 & P(3) = \frac 18 & P(4) = \frac 18 \\\\ 1,2,2,3,2,2,2,4,3 & \to & P(1) = \frac 19 & P(2) = \frac 59 & P(3) = \frac 29 & P(4) = \frac 19 \\\\ 1,2,2,3,2,2,2,4,3,2 & \to & P(1) = \frac 1{10} & P(2) = \frac 6{10} & P(3) = \frac 2{10} & P(4) = \frac 1{10} \\\\ \end{array} $$

What I noticed was that, after the horizontal line, the probability of one was the same as the probability for the highest number so far. I knew one would only appear once in this sequence because no other numbers have no factors for their prime factorization, which would be base one, or unary. The probability of two also is higher than any of the other probabilities after the first two terms.

I haven't thought of anything new about this interesting representation for numbers, but it is full of many mysteries and would love to see if anything else pops up that I hadn't known about!