Category Archives: Mathematics

Mathematics, Programming

Mandelbrot Set and the Fractals of Life

I've been playing around with Rust for a few weekends just to see what all the hacker hype is about. I haven't done a math post in a minute, so I wanted to use this small project as an exploration of mathematical visual arts.

Tangentially, there's an amazing PBS Nova documentary about Fractals - Hunting the Hidden Dimension. If you haven't watched it already, do it NOW! (WARNING: May induce wook-like philosophies)

https://en.wikipedia.org/wiki/Fractal_landscape

I love fractals. They are the most intriguing shapes which can represent very natural phenomena. Every snowflake is a unique fractal. Mountain ranges are landscape fractals with infinite triangles. Rainforests -> trees -> tree roots -> branches -> stems -> leafs represent a natural fractal ecosystem. Extending this analogy to space, a planet -> lunar system -> solar system -> globular cluster -> galaxy -> cosmic web form a fractal universe. Watch this incredible Youtube video visualize our entire human experience as an isomorphic fractal structure to the universe and the quarks that make up our known existence.

In mathematical terms, fractals are shapes with non-integer dimensions, infinite length arounds its edges, and self-similar. This means one can infinitely zoom in or out of a fractal, and the fractal will still retain the exact same pattern.

Fractals were largely ignored in the mathematical community UNTIL they could be visualized by the power of computer graphics (shoutout Benoit Mandelbrot). Today, we see fractals in a plethora of digital art, design, and visual performance tools: see Electric Sheep or go attend any electronic music show with a projector.

Mandelbrot Set Fractal Properties

On March 1, 1980, Mandelbrot created the first visualization of the Mandelbrot Set on an IBM computer. I wanted to recreate this moment using modern computer languages.

A Mandelbrot Set is a set of complex numbers C that are bound and do not diverge from the recursive function f(z+1) = z^2 + C when iterated from z = 0.

To accomplish this, I wrote a concurrent image output Rust program which leveraged all 4 of my CPU cores by splitting the rendering output work among 8 threads by pixel row. I then iterated through the set of Complex numbers and determined which numbers did not recursively diverge from the function listed above.

Here are my initial thoughts of working with Rust:

  1. The language itself is like a combination of C++, JavaScript, and OCaml. I really enjoyed working with the success and error handlers of the language.
  2. The syntax is also NOT human readable at first glance - there is an unmistakable learning curve to this language.
  3. The cargo packaging system for Rust is simply amazing. I was easily able to add Complex numbers and concurrency to my program with a few import statements (I secretly wish Golang had a halfway decent package management system).
  4. Sublime Text really struggles with displaying the compiler errors in the code editor. This is the exact reason why I switched to bindings-packed VSCode.
  5. Documenting and writing test cases using /// and #[test] are extremely intuitive and forces you to write quality code.
  6. Rust is unbelievably fast.

Anyway, here's my super detailed 8000x6000 grayscale Mandelbrot Set. If you look close enough, you'll see even more miniature Mandelbrot Sets :).

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Mathematics, Sports

Contrarian Positions in Horse Racing

Currently reading: Exotic Betting [1].

I find in life there are a lot of contrarian positions you can take in which you would be surprised how many times you are correct. It would be mundane as all hell for things to go the way you planned... sprinkle in some volatility for a good time :).

I've been particularly interested in horse racing as of recent. WARNING: Gambling is a losing deal in the long run, but the idea of contrarian (exotic) positions is to catch that rare (and more common than not) spurt of insane upside, resulting in an illustrious sense of accomplishment, or.. just another day at the track.

The house always takes its portion (~15%) during every race. The longer you bet over many races, the more likely you are to lose money. This is rule number one of gambling: the house always wins in the long run. No one can escape the law of large numbers. No one. That's why it's a law.

Keep this pool chart as we progress through some examples:

Exactas / Trifecta / Superfecta Tips & Strategies

  1. NEVER BOX YOUR BETS / NEVER USE ALL. There is no point making these exotic bets unless you have an opinion. Otherwise, you are diluting the pool of bets (and your win bets) trying to get lucky. We are trying to beat the pool payouts, remember?
  2. If you're lost on any bets, trifecta is a terrible idea. Trifectas are all about eliminating horses from contention.
  3. Betting a favorite in any of these exotics, severely depresses the payoffs in that single race, since that's where the pool's money is.
  4. Exactas to increase win odds without increasing risk. Put your hero (4-1) first, and run the favorite and 2 long shots that you like in second. This avoids the win pool of the favorite, multiplying your odds. Let's avoid following the crowd.
  5. Exactas to bet contrarian against the 1-1 favorite. Run a variety of exactas (different weights based on odds) with your hero and 2 long shots that you like. When the favorite misses the board, there is a phenomenon where the exact pays more generously than expected.
  6. Exacta (Contrarian) Situation: Skeptical of favorites but too confused to construct an aggressive play. Key 2 wing-and-a-prayer longshots (2nd to worst odds and 5th to worst) with 4 or 5 others on the field, leaving the favorites out all together.
  7. Trifecta small boxes: Usually players will play trifecta if they like 3 horses. It's far more likely for 2 to run their usual race and for the third to drop out. To accommodate this contrarian position, require 2 of your horses to hit the board and be loose on the third. Bet 1-2-3, 1-2-4, 1-2-5, and 1-2-6 boxes.
  8. Trifecta small part-wheels: Focus on 1 horse to place and then wheel the third slot. 1/3,5,7/3,4,5,6,7 and 3,5,7/1/3,4,5,6,7. We sacrifice the third place and gain additional horses.
  9. Trifecta aggressive punches: Do these every now and then to win big and when you're confident. $1 box 1-2-3, $1 box 1-3-4, $3 part-wheel 1/3/2,4 and $3 part-wheel 1/2,4/3.
  10. Superfecta: Key one to win and then bet a trifecta underneath.
  11. Superfecta variation: Key 2 with 4. 1,2/1,2/3,4,5,6/3,4,5,6 - 24 combinations. 1,2/3,4,5,6/1,2/3,4,5,6 and 1,2/3,4,5,6/3,4,5,6/1,2 for 72 total combinations.

Lastly before you make your exotic bets, make sure to bet superfectas in $0.10 amounts and trifectas / exactas in $1 amounts. This is due to another house, the IRS.

IRS rules require that winnings at odds of 300-1 or higher be reported to the government, if they exceed $600 and that the payoffs are subject to withholding if they total $5000 or more. [1]

Don't worry though, you can always repeat the bet. Think of 2 $333 winning tickets and ducking the $600 tax ;).

Best of luck contrarians.

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Mathematics, Programming

Math and Computer Science

I'm currently working on my final semester at UC Berkeley, and although I'm an EECS major, I'm taking all math classes to finish out my minor.. Abstract Algebra, Real Analysis, and Number Theory. These classes are all proof based: something that I'm used to but not proficient at.

Having been in class for only 4 weeks, I can thoroughly say that the Mathematics and Computer Science mindset take the same kind of genius. If you can do one, then you can do the other. Although learning the language of Abstract Algebra vs. learning the syntax for NodeJS is drastically different, the algorithm and problem solving mindset remains the same.. The only difference is the syntax.

Coming from a computer science standpoint, mathematics has a lot more definitions such as isomorphism, groups, abelian, etc.. But at the end of the day, these definitions define your working groups for proofs -- your bread and butter if you will. They're isomorphic to the bread and butter of computer science terminology: linked list, array list, arrays, hashes, etc..

I'm having a hard time memorizing these definitions with all the things that are going on in my life, but this has been one of the most academically interesting semesters at my time at Cal.. More updates on math in the coming days!

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