A Few Mathematical Concepts That Blow My Mind (Part 1.5)

Gabriel’s Horn

I know I said in my first Math Post that I would cover Euler’s Identity next, but I just came across this beauty, so I had to share it. This one’s called Gabriel’s Horn - a solid cone that has infinite surface area, but finite volume. An interesting little (or infinitely big) occurrence straight from the world of improper integrals. Let’s go ahead and dive into the math of it all:

• First we start by letting R be the region bounded by the graphs y = 1/x and y = 0 for x > 0
• then we get the area by computing the integral:
• 

  
  

• this can be interpreted as the area being infinite or increasing without bound. So we’ve got infinite area, let’s see what happens when we rotate this function along the x-axis and get a 3D solid
• 

  
  

So interestingly enough, we have a solid with infinite surface area, but finite volume of pi
it looks a litle like this:

 (as a side note it looks like I can't just paste an equation I made from a google doc in here, so I wound up having to export the google doc as a pdf, take a screen clipping to make it an image and upload it here. It didn't come out as pretty as I'd like, but at least the equations are a little more presentable than in the first math post.) Once again, thank you all for reading.
-Carlos

My Thoughts on the Progression of Technology and its Relationship with the Environment

    For a long time, I've faced an inner problem: I love technology. Technology is amazing; it's essential to maintain the standard of living to anyone who has the ability to read this in that we've all benefited from rapid transportation like cars, advances in medical technology like vaccines and modern dental practices, and let's not forget our ability to harvest energy resources, which drives our modern society. That being said, I really love nature too. I think one of the most beautiful things I witnessed firsthand was the view from the top of a small mountain I hiked in New Hampshire. Nature is beautiful to look at, but it's also beautiful in that it's so effective for how "simple" it is. (Note, I don't believe that nature is "simple" or has simple processes and ecosystems, I'm just saying it's simple in the way that it requires a lot less knowledge an expertise to grow a blueberry bush than to design and produce a CPU or something like that) And, not only is it beautiful, but it's essential for everyone's survival (currently thanking the trees around coral gables for providing me with some air)

     And so for a long time we've been presented this theme that as technology rises, nature declines; as we continue to build faster cars and smarter computers, nature pays the ultimate price. For a long time as well as still today, this idea is pretty accurate - we have to burn things like oil and coal to produce energy, which pollutes the air surrounding, which slowly kills the environment, we have nuclear power plants which can sometimes suffer meltdowns and spill radiation into the surrounding area. And that's all just byproducts from "good" technology. "Bad" technology gave us things like the atomic bomb, which did unspeakable things to the environment surrounding such as contaminating the soil and water supplies. In our quest to become a better, more advanced society, we've made some pretty big mistakes that have damaging impacts on the environment.

    In my opinion, our human society grows and matures similarly to how an individual grows and matures. When we were very young and starting to get a grasp on the way things worked in the world around us, we broke a few things. And let's be honest, sometimes we just broke things for fun. I personally can't even remember the number of things I've broken in my search to figure out how something works. But, as I matured, I learned how to operate and do certain things without potentially damaging other people's belongings. I also learned that some things are not worth messing with unless I really know what I'm doing (a moment of silence for all the computers I broke/messed up in my earlier youth). And so what I'm trying to say is much like how I've grown and matured, I think the human race is starting to reach this certain benchmark of age and maturity in that we're realizing the fact that what we do has an effect on the environment. I think a more important fact we're beginning to learn is that nature and the environment is just as necessary for a healthy life as modern technology.

     Now that we understand that we have to figure out ways for our technology and nature to play nice, imagine the possibilities. Imagine a society where technology is based on nature and is made to work alongside nature rather than against it. Imagine a society that understands a simple truth that nature and technology do not have to be mortal enemies. I see the relationship between humans (and technology) and nature as one of those things that have to get worse before it gets better. I really do believe that the worst is behind us, and we will begin to get better. There are so many interesting and innovative ways to cleanly harvest renewable energy resources and things like solar panels are becoming more and more efficient where in the words of theoretical physicist and futurist Michio Kaku, eventually solar energy will become more efficient and less expensive to the point where it will meet the rising costs of oil and gas and cause people to switch over. This is only one (but a very big) example of working with nature instead of against it, but I believe as well as hope that this mindset will begin to catch on as we continue to progress. This idea of binding technology with nature is something that I find to be really interesting, so if anyone would like to discuss it further with me, either post a comment or send me a message so we can continue the discussion. Once again, Thank you all for reading.
-Carlos

A Few Mathematical Concepts that Blow My Mind (Part One)

As I said in my first post, the first real thing I discuss will have something to do with math. I was thinking about what specifically I wanted to do, so I decided to search the web as well as some of my old math notebooks for inspiration. While flipping around, I discovered the remnants of me showing a friend of mine the concept that 0.99999.... is equal to 1. Every time I see that proof, I still can't help but think "whoa!" and that's when I decided to make this post about an assortment of mathematical things that just blow me away. First, Let's start with the proof that 0.9999.... = 1:

• lets start with x = 0.99999....
• multiply by ten to get 10x = 9.999999......
• subtract by x on both sides 10x - x = 9.99999999.... - 0.99999....
• this gives you 9x = 9
• divide both sides by 9 to give you x = 1, which means 0.9999....=1

One of the things that gets me about this proof explained this way is that the technique used to perform this proof is vastly simple, yet we get a pretty profound conclusion. I find it to be quite beautiful, actually. to borrow from mathematician Steve Strogatz, "Many people don't believe it could be true. It's also beautifully balanced. The left side represents the beginning of mathematics; the right side represents the mysteries of infinity." I find this symbolic interpretation to be beautiful as well. I think this concept is really cool if you apply it generally to all things as well. It's like the simplest of things that we thought we understood fully and thoroughly can still surprise us with hidden characteristics of vast complexity and deep profoundness. to illustrate, I'll pull up a quote of Gandalf from Lord of The Rings (isn't it only fitting that a wizard would be a good example to explain math?) "My dear Frodo. Hobbits really are amazing creatures. You can learn all there is to know about their ways in a month, and yet after a hundred years they can still surprise you." If I've lost you, All I'm trying to say is that Gandalf's quick understanding of  "all there is to know about Hobbits" is like our quick understanding of the concept of 1. both are simple and don't require much thought to wrap your head around. But, both of these things have another side more profound; in Hobbits, this is explained through Gandalf still being surprised about some of their actions, and in math, this is explained through .9999.... stretching on to infinity and giving us a much harder time understanding. This proof is great, but it doesn't really tell us much about how in the world .999.... and 1 represent the same real number. Let's take a different approach and prove this concept with the help of infinite series and sequences (and Wikipedia!).

Alright, so first let's think of the number 0.9999..... as an infinite series; that is to say that:

• 0.9999..... = b0.b1b2b3b4..... = b0 + b1(1/10 ) + b2(1/10)^2 + b3(1/10)^3 + b4(1/10)^4 + ....
•  now, with the convergence theorem, we can say that if  |r| < 1 then
  ar + ar^2 + ar^3 + ... = (ar)/(1-r)
•  now simply applying this theorem to our series, we get:
  0.999... = 9(1/10) + 9(1/10)^2 + 9(1/10)^3 + .... = (9(1/10))/(1-(1/10)) = 1
  (I know this isn't the most visually appealing way to display this equation, but the blogger word processor doesn't seem to have a function that lets me display fractions in a more readable way. Next time, I'll try writing this all in a real google doc and see what happens when i try to paste it in here)

Pretty much what's happening here is that if we keep adding up all those 9's infinitely we're eventually going to get 1. This points to another rule in math that says the sum of an infinite series is defined as the limit of what that series approaches and this concept can be applied to any kind of infinitely repeating number like 0.999... They are two equally valid ways to express the same real number (but for all our sakes, just refer to it as 1 in conversation with someone else). Well, This post got longer than I had imagined it to be, so I've decided to give all the mathematical concepts I came up with their own section, and I'll post about them little by little along the way. I won't post them one right after the other so that I can have a little more diversity and so that I don't sound like a text book :P.  If anyone wants to discuss this or if I've made some sort of error somewhere, feel free to say something about it in the comments. thank you all for being awesome and taking the time to listen to me ramble on about math! For those of you who found this post interesting, as a sneak peak, I'll be discussing Euler's identity in part 2, although I don't know yet when I'll begin writing about it.
-Carlos

First Post - An Introduction

First off, Welcome! I'm starting this blog as a way for me to visualize all of my ideas, beliefs, aspirations and all of that sort as well as provide people with a way to explore my mind and exchange/share alternate ideas with me. One of the things I love most about human interaction is the concept of 2 individuals being able to truly open their minds to one another without any sort of inhibition and discover together what makes us "tick". The things I'll mostly be posting about will probably fall in the categories of futurism, various topics in science (mostly things dealing with computers), mathematics, religion (as well as the lack thereof), and some other random topics and ramblings including the occasional poem. These are some things of the things that interest me the most, but feel free to discuss anything with me - I'm always down to have new favorite interests. For those of you wondering "just who the hell is this guy?", allow me to introduce myself. My name is Carlos, I'm currently studying computer science at the University of Miami, and I guess you could call me an Agnostic, but i prefer to just be seen as a budding scientist. I like to steer clear of religious labels because like Neil deGrasse Tyson said, "when you claim to be of a certain religion, all of this baggage comes with it, and people wind up assuming they know everything about you. I'd rather we just talk and discuss each others beliefs." Well, I think that's it for now, but expect more posts to come at a pretty regular rate; I plan on doing a few posts per week, but we'll see how it goes. As a sneak peak, my next post will have something to do with mathematics. I can't wait to get started and start posting about some interesting topics! see ya for now,
-Carlos