There is a puzzle night going on at the Institute at 549 North 12th Street, Philadelphia, PA this Saturday, February 7th. February is not my favorite month, really. It cold and grey and it's "holidays" celebrate a fuzzy rodent that DOESN'T give out candy, and a holiday that excludes single people. Oh, and then there is that third one, President's Day, that hasn't been fun since I used to get the day of from school and is really just an excuse to sell cars. What I'm saying is, the excuse to get a group of people together to do puzzles is a needed change in short nightmare of a month.
I decided to share some of the puzzles here on the blog since I wanted to share the fun with rest of you. Here is the first puzzle:
By adding mathematical symbols, can you make each statement true? You
cannot add any extra numbers, for example 4 * 1 + 1 + 1 is not a legal
move, and cube roots don't work, but square roots are fair game. 6 is
done to give you an idea.
0 0 0 = 6
1 1 1 = 6
2 2 2 = 6
3 3 3 = 6
4 4 4 = 6
5 5 5 = 6
6 + 6 - 6 = 6
7 7 7 = 6
8 8 8 = 6
9 9 9 = 6
If you're interested in some more, check out the Math Counts Meetup page . Or stick around this week, the rest of the puzzles will go up. If you live in the area, come out and bring friends! There will be more puzzles not listed on the website that will be there.
Monday, February 2, 2015
Monday, January 19, 2015
Mushroom Life
Here is a fun game to keep you a little occupied.
Maybe if you are one of the ten people who follow my facebook page, or google+, or any of my social media accounts, you might come to the conclusion that I'm obsessed with this game this morning. Which is very true, and it's one of the few math games I can get away with playing at work and claim that I'm being productive.
John Conway's Game of Life is a zero player game where the initial state is chosen by a person and then AI is allowed to take over and create many patterns. Someone modified the game a bit, so that pins would become mushrooms in a couple generations. I really want to find the person who made this game, and failing in that I want to steal this code. Is it still wrong if your honest about your intentions to do wrong?
I introduced a coworker to the original game this morning, and he came back to me with this version. The both of us being mushroom dorks, this was a fun version to use. At first it was disappointing because it was too small and can't handle a small glider gun. As I sat and watch the patterns evolve across the screen, I started to recognize some of the patterns. This is what happens when you spend three years watching mushrooms grow. Mushroom farming is a zero player game in which the outcome is controlled by the initial state.
It's still a complex system, and the rules are different. But if the super complexity is ignored for a moment, stuff like nutrition in the soil and the presence of predators, then outcome of setting up the original box is something amazing to watch. The health of the mycelicum, where water collects in a bed, all of that and more create the patterns that occur.
 |
| A mushroom spaceship, or "a weekender" |
John Conway's Game of Life is a zero player game where the initial state is chosen by a person and then AI is allowed to take over and create many patterns. Someone modified the game a bit, so that pins would become mushrooms in a couple generations. I really want to find the person who made this game, and failing in that I want to steal this code. Is it still wrong if your honest about your intentions to do wrong?
 | |
| The final stages of an exploder |
It's still a complex system, and the rules are different. But if the super complexity is ignored for a moment, stuff like nutrition in the soil and the presence of predators, then outcome of setting up the original box is something amazing to watch. The health of the mycelicum, where water collects in a bed, all of that and more create the patterns that occur.
Thursday, January 8, 2015
ISWG a day late, and horns with no volume
A day late for ISWG, so I'll give you two posts in one! The first one is my insecurities, and my second one is being too smart for my own good.New years resolutions, amiright? I stopped setting them a while ago, but I still try to improve myself. I stopped setting them because of that god awful trap that I tended to fall into where I miss the goal and I think "Oh, damn it I failed! No I'll never be able to do it!" and then I would give up until next year.
Now, I've got stuff I do and I feel about when I can't do it as well as I can. It's a journey. Write when I can, try to improve my organizational skill, get better faster harder stronger. You know the whole drill.
How are you guys doing on your resolutions?
This month the Minions of Math are going to start talking about infinity. Not to ruin the surprise, but we'll talk more about infinity next month too. This is a video month, and I watched this video by Numberphile this morning before running 2 miles in 3o Fahrenheit weather. That combination has proven time and time again to kick my mind into overdrive, so if figured I would take the time to address one of the paradoxes here: Gabriel's Horn.
There is a soft spot in my heart for Numberphile and Vihart because those two along with Gödel,Esher, Bach and the great Lewis Carrol did more for my interest in mathematics than 13 years of schooling. With that said, I've noticed that the deeper and deeper I get into math the Numberphile videos are a bit harder to watch since a majority of them seem to be aimed at I Fucking Love Science crowd. Instead of complaining about this, here are some more thoughts about Gabriel's Horn.
He points out that what makes it a paradox is the fact that it's a horn that tapers into infinity, and as such it has infinite surface area, BUT it has a finite volume. It seems paradoxical, until calculus is involved.
OK not true. The truth is 0.999999 repeating is so close to 1 that it's 1. That answer always struck me as unsatisfying when I was younger, but if we except that axiom we can do something more amazing with Gabriel's horn. More amazing than what the guy in video says with the whole "You couldn't paint the surface, but you could fill it with a finite amount of paint."
If there is a number that is so close to 1 that it's 1, then a number like 0.000 . . . 001 can be so close to zero that it's 0. Why Gabriel's Horn works is it's an infinite series, which can be best explained by a joke:
There's a math conference in town, and all the mathematicians walk into a bar. The first mathematician orders a beer. The second orders half a beer. The third orders a quarter. The forth orders an eight. The bartender stops them at this point and hands them two beers.It works for 2: 1 + 1/2 + 1/4 + 1/8 + 1/16 . . . to infinity. It works for 1: 1/2 + 1/4 + 1/8 + 1/16 . . . to infinity. In fact, it works with zero if ou work with small enough numbers. In a sense, the volume kind of works the same way for the horn, it's an infinite series. Less cool when you know how it works. What I've been working up to this whole time is that IF this is the case, AND a the sum of an infinite series can equal zero, THEN a Gabriel's trumpet exists that not only has infinite surface area, but also has a volume equal to zero. I'll leave it as an exercise to the reader.
Wednesday, December 31, 2014
Update day: the year ahead and what I've been up to
Hi guys, long time no see. I haven't posted on update days in a while, probably 3 months without looking it up. So what have been up to? Lots of stuff, many of it posted here. Animal wise though? One stray cat.
This cat has been hanging around my apartment since the summer at least. I started keeping him fed because I noticed squirrel damage was less with him around. It started getting cold, and we, roommate and I, were able to get him a place to stay. I thought.
He's back in the neighborhood, and a limp we noticed before may be infected. I don't know since I just got a message from my roommate on facebook about this and haven't seen this first hand. I'm just gonna dump right here because whatever, that's what this is for.
I feel like I haven't done what I should or need to do for this cat. I can't have animals in my apartment, if I could this dude would be living in my place. I can't have animals, so I become more concerned with cats in the area because I've always had pets growing up. Now I work to keep myself alive and think "Man, I wish I had enough so I could move to a place to get an animal, and also have enough to keep said animal supported". Basically at this point my goal is this cat. I'm going home, finding this cat, and keeping it inside while I figure out a way to keep him well. Fun times for me.
This cat has been hanging around my apartment since the summer at least. I started keeping him fed because I noticed squirrel damage was less with him around. It started getting cold, and we, roommate and I, were able to get him a place to stay. I thought.
He's back in the neighborhood, and a limp we noticed before may be infected. I don't know since I just got a message from my roommate on facebook about this and haven't seen this first hand. I'm just gonna dump right here because whatever, that's what this is for.
I feel like I haven't done what I should or need to do for this cat. I can't have animals in my apartment, if I could this dude would be living in my place. I can't have animals, so I become more concerned with cats in the area because I've always had pets growing up. Now I work to keep myself alive and think "Man, I wish I had enough so I could move to a place to get an animal, and also have enough to keep said animal supported". Basically at this point my goal is this cat. I'm going home, finding this cat, and keeping it inside while I figure out a way to keep him well. Fun times for me.
Monday, December 29, 2014
Drawing spheres on the surface of a hypercube
I could write another description of what the higher dimensions are, but personally I "A Wrinkle in Time" to describe that to me when I was 8 years old. Whatever that book left out was filled in by years of video games and movies to fill in the gaps. You know what the first dimension is, and the second and the third. You can't picture anything higher than that, but that's not a problem because the best of us can't. I'm here to talk about a much crazier, much wilder idea.
The problem with describing anything higher than 3 dimensions is that the task of imagining what a 4 dimensional cube or sphere looks like and this is impossible. Physically adding an extra dimension for the sake of modeling the 4th dimension is like blaming penguins for me being bad at analogies. This doesn't have to be the case though, because it's easier to imagine what a pyramid looks like on the surface of a hyper-sphere or a hyper-cube. This is because the surface of a 4th dimensional object has 3 dimensions. Allow to me to explain.
A cube is a 6 sided object with 12 equal edges. What makes it a cube is the fact that it has height, length, and width. Each side, or face, has only length and width thereby making it a 2 dimensional plane. A cube is a solid object bounded by a 6 equal squares. A cube is an easy place to start because it can be given to a child along with a Sharpie and they can draw all manner of shapes on its surface. Then Euclidean geometry can be explained to the child because they need to grow up sometime and understand how basic shapes can be modeled by numbers. All the shapes will fall under the basic rules: squares will have angles that add up to 360o and areas equal to their length times their height, triangles have angles that add up to 180o and have areas equal to its height times its base.
Anyone went the extra step for geometry knows that the properties of these shapes change when drawn along a curved surface, like a sphere. A sphere is a solid object bounded by equal circles, which means that a line is no longer straight, a line becomes curved. All lines are great circles and all great circles intersect. A great circle is the equator for simple reference. These lines can still draw shapes on a sphere it's just that the properties of the shapes change. A triangle angles are equal to 180o plus the area of the triangle. This makes transferring shapes from the surface of a globe to a flat map hard since the shapes are not the same.
A hyper-sphere is a catchall term referring to a sphere with more than 3 dimensions. A 3-sphere is a 4 dimensional sphere, bounded by 3 dimensional circles. All great circles on a 3-sphere are great spheres. Much like we can draw squares, circles and triangles on the surface of a ball, we could draw cubes, spheres and pyramids on the surface of a 3-sphere. A cube on the surface of hyper-dimensional object would have 12 equal curved edges, or better yet it would be the intersection of six spheres touching. As long as the distance of each edge remains the same, then distortion of the cube would be the same no matter where it is along the surface of the hyper-sphere. Along a hyper-hyperbola however, its proportions would change depending on its location along the 3 dimensional curve. Sometimes it would have finite volume, other times it would have infinite volume.
I'm stopping here for today. I've thought too hard about this for too long which feels like prolonged exposure to hallucinogenics. My advice is to picture distorted boxes, because as soon as you try to imagine yourself standing on the surface of a 4D sphere it brings up a lot of weird questions. Does a 4th dimensional light source create a 3rd dimensional shadow? Am I a 3D shadow of my 4th dimensional self? Don't think about it, it's too much.
The problem with describing anything higher than 3 dimensions is that the task of imagining what a 4 dimensional cube or sphere looks like and this is impossible. Physically adding an extra dimension for the sake of modeling the 4th dimension is like blaming penguins for me being bad at analogies. This doesn't have to be the case though, because it's easier to imagine what a pyramid looks like on the surface of a hyper-sphere or a hyper-cube. This is because the surface of a 4th dimensional object has 3 dimensions. Allow to me to explain.
A cube is a 6 sided object with 12 equal edges. What makes it a cube is the fact that it has height, length, and width. Each side, or face, has only length and width thereby making it a 2 dimensional plane. A cube is a solid object bounded by a 6 equal squares. A cube is an easy place to start because it can be given to a child along with a Sharpie and they can draw all manner of shapes on its surface. Then Euclidean geometry can be explained to the child because they need to grow up sometime and understand how basic shapes can be modeled by numbers. All the shapes will fall under the basic rules: squares will have angles that add up to 360o and areas equal to their length times their height, triangles have angles that add up to 180o and have areas equal to its height times its base.
Anyone went the extra step for geometry knows that the properties of these shapes change when drawn along a curved surface, like a sphere. A sphere is a solid object bounded by equal circles, which means that a line is no longer straight, a line becomes curved. All lines are great circles and all great circles intersect. A great circle is the equator for simple reference. These lines can still draw shapes on a sphere it's just that the properties of the shapes change. A triangle angles are equal to 180o plus the area of the triangle. This makes transferring shapes from the surface of a globe to a flat map hard since the shapes are not the same.
A hyper-sphere is a catchall term referring to a sphere with more than 3 dimensions. A 3-sphere is a 4 dimensional sphere, bounded by 3 dimensional circles. All great circles on a 3-sphere are great spheres. Much like we can draw squares, circles and triangles on the surface of a ball, we could draw cubes, spheres and pyramids on the surface of a 3-sphere. A cube on the surface of hyper-dimensional object would have 12 equal curved edges, or better yet it would be the intersection of six spheres touching. As long as the distance of each edge remains the same, then distortion of the cube would be the same no matter where it is along the surface of the hyper-sphere. Along a hyper-hyperbola however, its proportions would change depending on its location along the 3 dimensional curve. Sometimes it would have finite volume, other times it would have infinite volume.
I'm stopping here for today. I've thought too hard about this for too long which feels like prolonged exposure to hallucinogenics. My advice is to picture distorted boxes, because as soon as you try to imagine yourself standing on the surface of a 4D sphere it brings up a lot of weird questions. Does a 4th dimensional light source create a 3rd dimensional shadow? Am I a 3D shadow of my 4th dimensional self? Don't think about it, it's too much.
Monday, December 22, 2014
Monday morning tin foil
I want to be ahead of the curve here, so I'm going to say that The Interview was actually really bad and in danger of losing money, so the NSA and Sony got together and hacked the company just so they could blame it on North Korea. Never mind that it put hundreds of people financial security at risk, it was all just act so they could have an excuse to blame our enemies and possibly start a new war, I guess? Does this sound like a good conspiracy theory yet? Oh, wait!
Why would they do this? Well, isn't it convenient that it happens to be going on right as protests are going on in Washington D.C. with a new civil rights movement? Or that it just happens to coincide with legalization of medical marijuana, which is like the government just giving up on the war on drugs? The hack was planned to distract us from what's really going on, much like how the Kardashian's were built in a bunker to keep us docile and stupid.
Open your eyes, sheeple! Doesn't it seem too easy, or too convenient that a country we've been enemies with for years just happened to hack us?
Why would they do this? Well, isn't it convenient that it happens to be going on right as protests are going on in Washington D.C. with a new civil rights movement? Or that it just happens to coincide with legalization of medical marijuana, which is like the government just giving up on the war on drugs? The hack was planned to distract us from what's really going on, much like how the Kardashian's were built in a bunker to keep us docile and stupid.
Open your eyes, sheeple! Doesn't it seem too easy, or too convenient that a country we've been enemies with for years just happened to hack us?
Monday, December 15, 2014
Post later!
I just finished a new post. It's really long and math heavy, like usual, so I'm gonna find some pictures to make it easier to read. Big walls of unformatted text is hard to read as well as my biggest weakness as a blogger.
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