Posts Tagged ‘math’

Arithmetic Derivative Graph and Thoughts

April 11, 2011

A recent thread on r/math brought the arithmetic derivative to my attention. It’s a very interesting idea. As I tried to rap my head around it, I wished I had a graph of its actions on some of the natural numbers. So I made one:

The Arithmetic Derivative as a graph on natural numbers

(more…)

Unicode For Mathematical Typesetting

March 29, 2011

In the last month, the way I typeset math has changed dramatically. It’s a result of a combination of several tools that I’ve discovered. (more…)

Rethinking Grade School Algebra

March 28, 2011

There’s a question I like to ask random people: where is the flaw in the argument that -1 = 1 because 0*(-1)=0*1? I very rarely get a satisfactory response. Usually the answer is that “you’re not allowed to multiply both sides by zero.” But we can come up with a slightly subtler argument: -1 = 1 because (-1)^2 = 1^2. Some just don’t answer, other will insist that its not allowed… To me it suggests something is deeply wrong with how most people understand algebra.

They don’t know mathematics, they know voodoo-mathematics, a series of mysterious steps that result in their test being returned with a checkmark beside the question.

Now it may seem that I’m being a pedant. After all, they know it isn’t true; what does it matter if they can’t tell me why? But even if we set aside the fact that it simply feels wrong to not understand why the math works, it has practical implications because there are cases where the mistake won’t be as overt as above. And then these people won’t see the mistake.

So I’d like to use this essay to go over grade school algebra from a different perspective. (more…)

Mandelbrot Set on the Riemann Sphere

March 5, 2011
Mandelbrot on the projection of the bottom of the Rienman sphere straight down. The mandelbrot st on the projection of the top half of the Riemman sphere straight down.

The mappings are B_1(0) \to \hat{\mathbb{C}}:

\phi^- (z) = \frac{z}{2-|z|}

\phi^+(z) = \frac{1}{z(2-|z|)}

For most practical purposes you’ll want to (as seen in above picture) restrict the domain of \phi^+ to not include some neighborhood of 0 to make it a bounded function on \mathbb{C}.

Function Approximation by Geometric Derivatives

December 9, 2010

If the idea behind a normal derivative might be describe as the amount one needs to ad to move forward. If so, then the idea behind a geometric derivative is the amount one needs to multiply by.

The definition of a geometric derivative is D^* f(x) = \lim _{h\to 0} \left(\frac{f(x+h}{f(x)}\right)^\frac{1}{h}

One thing one might wonder is whether we can use them to approximate functions, like we can use derivatives to approximate functions with Taylor series. A quick Google search didn’t reveal anything, but it isn’t too hard to figure out.

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Multi-Colour 3D Printing by Filament Swapping

December 8, 2010

Multi-Color Diffusion Eq (c=1,x real, u(x,0)=H(x)) black red pink by Christopher olah

The above picture is solution to the diffusion equation u_t = u_xx; ~~ t \in [0,\infty),~ x \in \mathbb{R} with the initial conditions u(x,0) = \{ 0 ~ x>0, 1 ~ x\geq 0. (WordPress doesn’t seem to like the array environment…)

English Translation: If you imagine a metal bar heated up on one side, as time progresses the temperature will even out. This is a plot of the temperature with one side being the length of the rod and the other being time.

But that’s probably not too interesting to most people reading this post. The interesting things is how I got the multi-colour object.

It was made by feeding one short piece of filament into the printer after another, during the print job. It was surprising to see how nicely one colour faded into the next.

Unfortunately, this broke the hacklab.to Break-R-Bot Maker Bot. The problem was that there was a sharp point on one filament that deflected the next one to the side. It was easy enough to fix (thanks to Rob for helping me!), but it seems like a bad idea to test it again oh the hacklab printer. I’m building my own, so the experiments should continue in a few weeks, anyway.

It seems like the problem should be possible to avoid as long as one makes sure that the filaments have flat ends. I am also planning to experiment with using a hot air gun to fuse pieces of filament.

(Thanks to Stefan for taking the picture of the models for me!)

3D printing of Mathematical Objects!

August 21, 2010

 

 

 

Back in the fall I did some work on getting hacklab.to‘s 3d-printer to print mathematical objects created in sage. Unfortunately, shortly after I got it to work and printed a test sphere, the 3d printer broke. Thus began a long succession of the makerbot — nicknamed the break-r-bot — being fixed and broken… spending most of its time broken.

But recently it was fixed and I decided to dig out my old code and get to work on it.

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Separation Axiom Visualisations

August 14, 2010

A couple days a go, I saw some nice visualisations of separation axioms on Wikipedia. Unfortunately, it wasn’t a full set. Well, here is a full set (well, T0, T1, T2, T2 1/2, T3, T4, T5):

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Variations of the Mandelbrot Set

April 23, 2010

Here’s some variations of the Mandelbrot Set I’ve been looking at. They aren’t terribly interesting from a mathematical perspective, but their pretty!

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The Mandelbrot Set: Compact?

March 28, 2010

Several weeks ago, I read something on Wikipedia that shocked me: “The Mandelbrot set is a compact set.

At first I didn’t believe it. How could the Mandelbrot set, in its infinite complexity, be compact? (more…)

Why are there mini sets in the Mandelbrot Set?

March 10, 2010

One of the most interesting properties of the Mandelbrot Set is that we can find what appear to be miniature Julia Sets and mini Mandelbrot Sets.

It’s easier to look at a different question first: why is it that the Julia sets associated with a point on a Mandelbrot set tend to be similar to that region of the Mandelbrot set?

Consider the functions that create them:

Julia: \lim_{n\to\infty}(z\to z^2+k)^n(x)

Mandelbrot: \lim_{n\to\infty}(z\to z^2+x)^n(x)

And the question is: why is the region of the Mandelbrot set around a point x similar to a the Julia set where k=x?

It’s obvious that at that point, the two functions are the same, but, given that we’re studying chaos it seems odd that the slight variations in the region around it wouldn’t produce a totally different appearance.

There are a couple ways to approach this problem. The first is to note that this isn’t, at the vary least, necessarily true as demonstrated by the following experiment. I constructed a sequence of fractals formed by the function \lim_{n\to\infty} (z\to z^2 +0.3 +x/m)^n(x) where I varied m from one to a hundred along the integers:

(You may need to click to see the animation!)

(Images created with gnofract4d and animated with imagemaick.)

That only goes to dividing by a hundred, but the influence is pretty negligible at the end. I could divide by more until the influence was arbitrarily negligible.

The second way of looking at this is that Julia sets of similar k values tend to be similar and therefore the associated region of the Mandelbrot set is similar.

Finally, we can look at the fact that the Mandelbrot set is T_2 and thus, provided a point is not right on the edge of a sudden change, we can construct a set containing it and no the change (ie. select an arbitrarily small neighbourhood such that the change is negligible, as said previously).

The “mini-Julia sets” form when we get a miniature, often deformed, plane (let’s dub it a microplane, because making up terms is fun!). The microplane will have a root of the Mandelbrot set in its center.

If the microplane is small enough, it may be contained in one of the previously described neighbourhoods in which the iterated function is essentially a Julia function and thus form what appears to be a Julia set, though it is different in some respects (eg. connected, joined to the Mandelbrot, et cetera).

But the really odd things are the “mni Mandelbrot Sets”. What is going on with them?

I don’t really have a satisfactory answer. I do have a few observations, however:

Firstly, these `mini Mandelbrot Sets’ are quite different from the Mandelbrot Set. They seem to form on roots. They seem to have bulb-like Julia sets around them…

Here’s a visualisation of the formation of one:

It’s completely different from the formation of the Mandelbrot Set!

Finally, I’d like to put forward a theory: there is a natural propensity for an iterated function to form self-similarity.

Complex Powers

March 9, 2010

Exponents + Complex Numbers = Cool Stuff

I’ve been messing around with these sort of things and discovering all sorts of neat stuff. I haven’t got far enough in Needham to really understand all of it, but here are some things I have found.

The plots of x^k where k is a point on the complex unit circle are fascinating. Here’s an animation I made:

Complex Plots of x^k where k is a point on the complex unit circle (Animation -- gif)You can also watch a high-quality video on youtube.

At k=i argument and magnitude of the output seem to be switched. They swirl into these circles and then swirl out such that the high magnitude and low magnitude ends are switched.

Another interesting thing is that i^x and e^{ix} are related. In fact: i^{-2ix/\pi}=e^xi^{-2i/\pi}=e

Update: The reason why this happens is fairly straight forward. Consider x^{e^{\pi i n}} = (re^{i\theta})^{\cos(\pi n)+i\sin(\pi n)}=e^{\ln(r)\cos(\pi n)}*e^{\ln(r)i\sin(\pi n)}*e^{i\theta\cos(\pi n)}*e^{-\theta\sin(\pi n)}=e^{\ln(r)\cos(\pi n)-\theta\sin(\pi n)}*e^{i\theta\cos(\pi n)+\ln(r)i\sin(\pi n)} and thus magnitude and argument switch as n increases.

Compactness Graph

March 9, 2010

Here’s the first revision of a graph of the implications of topological properties that I made:

(Click on it to see a better version!)

It’s mostly based off the stuff in Counterexamples In Topology (great book, BTW) but I did add some stuff (like Baire!) and merged/reorganised it. Diagram was made by Graphviz.

Most of the implications are trivial, but there are a few I haven’t prooved yet (most of the ones involving seperation axioms).

Gamma Fractals

February 12, 2010

There doesn’t seem to have been much done in the way of studying fractals of iterated functions involving gamma. At least nothing that I could find.

So I started playing around with it. It’s a bit difficult to work with as sage (4.3.2) keeps crashing when I try to plot it (try to run complex_plot(gamma(gamma(x)), [-10,10], [-10,10]) or complex_plot(gamma(gamma(gamma(x))), [-5,5], [-5,5]) ), but I still got some pictures…

Identity

Gamma Function

\Gamma^2 (Gamma of Gamma)

Fractals with non-Integer powers

February 11, 2010

I mess around with fractals a lot, and sometimes stumble on some interesting things. Recently I came across some odd jagged/discontinuous fractals:

Julia Set of z \to z^{\frac{3}{2}}+0.4+0.1i

This tends to happen in fractals as soon as I involved non-Integer powers.

Why is this happening? To understand this, we need to look at some basic complex analysis.

Recall that a complex number can be interpreted as a vector like (real, imaginary). But we can also think if a vector in terms of direction and magnitude. For our purposes, we will think of vectors as a magnitude and an angle from the positive portion of the real number line (sometimes called the `argument,’ we will just call it the angle).

Complex numbers have the interesting property that when we multiply them their magnitudes multiply like normal numbers but their angles add. For example, -1 has an angle of \pi so when we multiply two of them we end up with a magnitude of 1 and an angle of 2\pi and thus 1.

When we raise a value to a real power, the angle is multiplied by that value. We can visualise this as:

z \to z^2
\implies

It’s fairly clear that for the inverse, for any value, there are two valid answers.

\implies
z^2\to z

We can also look at it with an alternative visualisation I just cooked up:

Notice that the angle of the ray of discontinuity is arbitrary, but its existence is inevitable.

(For more on complex analysis, I recommend Visual Complex Analysis by Tristan Needham (Google Books, Amazon). It’s awesome!)

(Sage users: Here is a generalised version of the function I used to make the above image. It takes a value n and returns a visualisation of the nth root:


def root_visualize(n):
 l=[]
 for j in range(n*8):
 l.append(complex_plot(lambda x: (x*exp(-sqrt(-1)*j*pi/4))^(1/n)*exp(sqrt(-1)*j*pi/n/4),[-5,5],[-5,5]))
 print j
 return animate(l)

Be warned: it is slow)

Now, notice that z^{1.5} is the same as z\sqrt{z}. Thus, it has the same discontinuity. That seems like a reasonable answer to “why is this so jagged?”

But it leads to some interesting questions. The intuitive reaction is (for me at least): part of the fractal is missing. You’re cutting apart the nice, smooth mutlifunction and thus are only seeing part of the fractal. Well, at n iterations there 2^n possible ways to have cut this hypothetical super-fractal (as I will refer to it).

Let’s look at the Julia super-fractal of z\to \pm z^{1.5}+0.3 for four iterations…

If you stare at all the possibilities of the forming set you will notice some patterns and what look like reciprocals and other patterns. One may intuitively wish to add them to extract a single fractal, but this isn’t possible since they’d cancel… Perhaps adding the absolute values?

(For more on the formation of fractals, see my previous post.)

Of course, one really should look at a larger number of iterations than I am. There is the small problem of this being O(n*2^n), but it should be easy to parallelize…

So that’s the extent that I’ve explored this to so far. I’d be thrilled to get some feedback. Do you have any thoughts about this? Know of similar things that have been studied? (I’ve never formally studied Chaos Theory and suspect I have lots of holes in my knowledge. Reminder to self: take/audit course on chaos theory next year…)


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