Monday, September 29, 2014

Infinity, Part 1 to 3: Until the End of Time and the Universe Explodes; Limits and Asymptotes, Visualizing Infinity; Going Halfway, Travelling from A to B Forever (The Language of Mathematics, Series IVb, #154 to #156) - UPDATE 2

As a follow-up to one of the first videos that I produced for The Language of Mathematics, the 2007 teaser video regarding Zero and Infinity, below you will find three additional videos discussing the concept of infinity in more detail. The original teaser video is also embedded.

My Two Infinities, Part 1: Until the End of Time and the Universe Explodes (Language of Math #154)

My Two Infinities, Part 2: Limits and Asymptotes, Visualizing Infinity on a Graph (Language of Math #155)

My Two Infinities, Part 3: Going Halfway, Travelling from A to B until Reaching the Planck Length (Language of Math #156)

The Language of Mathematics (5): Zero and Infinity

Friday, February 14, 2014

A Blast from the Past: The Language of Mathematics on Google Maps Street View

Winter is nearly at an end, spring is in the air, and it’s almost time to do some math. I’ve been gearing up to start producing another full season of math videos for The Language of Mathematics and Math in Real Life. There is a lot planned for 2014 so it should be a nice full year of some math goodness.

Once the weather warms up and the consecutive days of rain cease, we’ll be heading out with the camcorder and the tripod to find some nice walls where we can continue working on Series IV as well as producing some videos on a few random and not-so-random topics. A lot more on that later. For now, I’d like to share a little blast from the past.

A few weeks ago, while a friend was surfing Street View of Vancouver on Google Maps she came across one of my favorite walls in the city, and it so happens to have captured one of my handiworks from the math videos.

The site has some sentimental value for me. It’s the first wall I used when I began this project back in 2007, and shows the location of the first math video I loaded on YouTube. For those that have been around long enough, if you recall, it was the wall that I used to create the videos where we discussed the Real Number Set (Part 1 and Part 2) in Series I.

The image that my friend found shows my handiwork from Series IVa on the wall. It’s from the video where we discussed systems, SI units, and where we created a conversion table for colored squares and triangles. You’ll find the video we created below, and if you are so inclined, you can use Google’s Street View to check out the area further if you want to have a feel for why this wall is the only location that I have come back to for every series since starting this project - a wall that we will most likely visit for many years to come.

View Larger Map

2011: Series IV - Systems, SI Units, and a Conversion Table for Colored Squares and Triangles (Math Series IVa #146)

As for the rest of the videos that were produced in this area, you will find them below. For multi-part sets, the links are provided below the videos that have been embedded. I hope you enjoy watching them as much as I enjoyed creating them. Peace.

2007: Series I - The Language of Mathematics (3): The Real Number Set - Part 1
Part 2: The Language of Mathematics (4)

2008: Series II - Language of Mathematics II (42): Exponents and Radicals (1 of 3)
Part 2 and Part 3. I also put together a set of these videos linking the concept of exponents and radicals with the real number set: Part 1 and Part 2

2009: Series IIIa - Language of Mathematics III (63): Black holes and elementary particles (Part 2)
Part 1: The Language of Mathematics III (62)

2010: Series IIIb - Why Do We Factor? Introduction to Factoring Polynomials: The Language of Mathematics IIIb (105)

Monday, January 6, 2014

An Exercise for the Mind: a 10 by 10 Math Puzzle: a Pattern Recognition Game: Meditation on an Open Maze

A few years ago one of my students introduced me to a simple and intriguing game. He didn’t know where the game originated from but said that it was popular in Turkey, and that he and his friends loved playing it.

It is probably best categorised as a mathematical puzzle that requires no arithmetic calculations, just placements of numbers based on two rules; hence I prefer to refer to this puzzle as a pattern recognition game, or simply, an exercise for the mind using an open maze – a meditation of sorts.

The Purpose of the Game
    To take a 10 by 10 grid, equaling 100 squares, and completely filling in every square based on two types of movements.

The Rules are Simple
    1) Create a 10 by 10 grid (graph paper or a spread sheet will come in very handy).

    2) Fill one of the squares with the number 1, then, based on the following two movement types, use consecutive numbers to fill in the rest of the grid:
      Movement Type I: If the next number in the sequence is going to be placed vertically or horizontally, then it must be placed exactly three squares away from the previous number - there must be a two square gap between the numbers.

      Movement Type II: If the next number in the sequence is going to be placed diagonally, then it must be placed exactly two squares away from the previous number - there must be a one square gap between the numbers.

    3) Numbers can only be placed in empty squares - once a square has been used it can not be used again. All placements must occur within the grid.

    4) Game is complete when the number 100 is reached - when all empty squares in the grid are filled. If you cannot continue placing numbers based on the two movement types above, and empty squares remain in the grid, then the puzzle is not complete and the game lost.

Two Incomplete Examples and a Solution
    Example 1: The first example is one where I tried to get the worst possible result. I was able to end the game in 6 moves.

    Example 2: The second example is my best result so far. In this example I was able to reach 99 – one move short of finishing the game.

    A Solution: The final example is a solution completed by my student who introduced me to the game. He had been playing this game for approximately three years, and this was the only solution that he had been able to obtain during this period. This means that we know there will be at least four solutions to this game – the one presented below and its three mirrors. It would be interesting to find out how many solutions actually do exist - any mathematicians or programmers out there interested in tackling this problem?

    Aside from creating a weighted scoring system or using symbols instead of numbers to fill in the squares, two variants could be: 1) give a starting point and an end point for the game – this will dramatically increase the difficulty of the puzzle; 2) create different types of grids but maintain the same placement rules – an example is presented below.

For anyone who wants to try this out, below you will find a sheet with 12 grids (click to enlarge). Ideally, a simple program based on the parameters above loaded on a device with a touchscreen would be the best way to play this game. If you are into such things, I hope the game entertains you, and if you find anymore solutions, please send them my way. Thanks in advance.

Saturday, November 16, 2013

Bill Nye, Brian Greene, Neil deGrasse Tyson, and Lawrence Krauss have a brilliant little discussion on the limitations of mathematics, and its importance and relevance to humanity

Math lovers and aficionados will find the following discourse both entertaining and informative.

Below you will find the video and partial transcript of Arizona State University’s Origins Project’s Q&A segment from their ‘The Storytelling of Science’ panel discussion, featuring “well-known science educator Bill Nye, astrophysicist Neil deGrasse Tyson, evolutionary biologist Richard Dawkins, theoretical physicist Brian Greene, Science Friday’s Ira Flatow, popular science fiction writer Neal Stephenson, executive director of the World Science Festival Tracy Day, and Origins Project director Lawrence Krauss.”

The first question asked of the panel was:
Q: “If you could give us all a one word piece of advice for our own science storytelling, what would it be?”
Bill Nye was the first to reply with, “Algebra, learn algebra.” Neil deGrasse Tyson follows with, ‘Ambition’. Lawrence Krauss with, ‘Passion’. Neal Stephenson with, ‘Empathize’. Richard Dawkins states that since empathize has already been taken, he will choose ‘Poetry’. While Ira Flatow states that ”you should be able to tell it so that your mother can understand it.”

The second question asked by the audience is what kicks off the fireworks:
Q: “I’ve always wanted to be an astronautical engineer, but I am horrible at math, but I’ve got lots of passion. Can this dream ever be a reality and where do I start?”
The dialogue of the panelists was as follows:
Lawrence Krauss: “As Bill said, math is the language of science, and I think you have to be able to be adept at it.”

Neil deGrasse Tyson: [interrupting Krauss] “Math is the language of the universe.”

Lawrence Krauss: “Yes, You’re right, I agree with you there.”

…audience applauds and cheers…

Lawrence Krauss: “I agree, but let me just finish. Too many people think that you have to be a mathematical wizard… you don’t have to be the best mathematician in your class, you don’t have to be a wiz. It takes all type to do science, and any stereotype just doesn’t work. If you’re interested, do it.”…

Bill Nye: “But the other thing, I would say, you say you’re bad at math, I bet you’re not that bad. And I just want to remind you that when it comes to math there is no substitute for practice. It sucked for me, it sucks for everyone. You just have to practice. So when you come to me and say ‘I’m bad at math’, I am open minded of course but skeptical. I bet you can do it whoever you are.”

Lawrence Krauss: “You know, that’s an important point. We were talking about it last night too… I like science museums because they show science as fun but science is hard work like anything, like music, like anything else to do it well, and it takes a lot of work. And if you don’t enjoy it you can’t do the work, but just enjoyment alone isn’t enough, you really got to be willing to work at it.”

Neil deGrasse Tyson: “I think what’s going on here is, people presume that if the math is not coming easy that therefore you’ll never learn it. And I meant it literally that math is the language of the universe, and it’s like any other language, especially a language that does not share the Roman alphabet. So, for example, if you wanted to study Chinese, it looks completely intractable at first… and you can ask the question, ‘how long does it take one to become fluent in Chinese, if you’re not Chinese yourself?’ …it can take… almost 10 years, if you never go to China. If you go to China, maybe 5 years of intensive exposure - and you’ve never done that with math - imagine that level of exposure to math, what kind of fluency you would have at the other end of that pipeline. So at least give yourself the opportunity that any person learning a foreign language would give themselves before you turn around and say you’re not good at math.”

…audience applauds and cheers…

Brian Greene: [addressing Neil deGrasse Tyson] “The question that comes to mind for me is, how do you know that math is the language of the universe?”

Neil deGrasse Tyson: “The universe told me.”…

Bill Nye: “It’s a first approximation”…

Brian Greene: “I was wondering, I have a question about this, could you imagine that one day far into the future we encounter some alien civilization and they say, ‘hey, show us what you’ve done to understand the universe’, and we bring out our math books with all our theorems in physics and they turn to us and say, ‘Math! We tried that, it takes you just so far! And the real way to do it is like this!’

Neil deGrasse Tyson: “I would say, that whatever that real way is it’s not manifest to us at this moment, and until that day happens where an alien tells us how backwards we are, all I can say is that the math that we did invent out of our human brain - as you [pointing to Brian Greene] surely know Eugene Wigner said the unreasonable effectiveness of mathematics in describing the universe - the fact that it works at all is sufficient enough for me.”

…a little chaos ensues…

Lawrence Krauss: “I want to go on record, and this is a momentous occasion, I want to go on record as agreeing with Brian. In a sense that it is fascinating if you’re a theoretical physicist to wonder when you find something fascinating - some mathematical formula that’s fascinating - whether it’s a property of our brains or whether it’s a property of the universe, and we just don’t know I think is the answer….”

Brian Greene: “Right, but let me answer your question. I find it slightly confusing because, Neil, you describe math as something that we create, so why is it the thing that we create is somehow intrinsic to the universe?...”

Neil deGrasse Tyson: “I don’t lose sleep over that, I celebrate it.”

Brian Greene: “It’s a good thing, I celebrate it too.”

Lawrence Krauss: “But it is the question, there may be limitations of our understanding of the universe because of the way our brains work…”

Neil deGrasse Tyson: “That’s surely the case. That’s surely the case.”…

Lawrence Krauss: “But seriously that’s an interesting question and we really have to wonder about that, and again, as some of us are on the forefront of physics, you wonder at some point when it’s going to end.”…

Bill Nye: “But, to the questioner’s question, I wouldn’t worry about the possibility that mathematics is going to turn out to be ineffective in describing the universe and use that as a reason to not keep practicing. Press on.”

…audience applauds and cheers…
The above dialogue and more takes place in the first few minutes of the following video embedded below.

Q&A Segment - The Great Debate: THE STORYTELLING OF SCIENCE (Part 2/2)

Part one of ‘The Storytelling of Science’ follows and is well worth the watch as well.

The Great Debate: THE STORYTELLING OF SCIENCE (Part 1/2)