Monday, April 16, 2007
Happy Birthday, Leonhard Euler
Yesterday was Leonhard Euler's 300th birthday. Happy birthday! to one of the most prominent and prolific mathematicians this world has ever seen. In my mind, one of the most unique characteristics about Euler was that he was how normal he was outside of his genius. Most great mathematicians seemed burdened by their genius in how it affected their lives outside of their profession - indeed, many of them had psychological or social difficulties (ever see the movie "A Beautiful Mind"?). But Euler seemed to escape all of that; his remarkable contributions were all the more unique in how unremarkable the rest of his life was.
Monday, April 2, 2007
From Two to Ten Dimensions
In real life, we know that we live for the moment in three dimensions - we can call them height, width, and depth. (Mathematicians cleverly call them "x, y, and "z", or some other combination of letters.) We can also view time as a dimension as well - so we can say that we live in four dimensions - the three spatial dimensions and time.
In a future post, I'd like to talk about 'time'; it's a topic I've thought about a lot recently. But, to set the stage, we need to delve into the "dimensionality" of our current reality.
Our reality has these three spatial dimensions. We can construct three straight lines, meeting at a single point, which are all ninety degrees away from each other - the x, y, and z axes. Even though it is possible mathematically, we cannot imagine a way to create a new line, intersecting at the same point, that is ninety degrees away from each of the other three lines. If we could do that, then we would have a fourth spatial dimension.
Although four dimensions are impossible to imagine, we can gain some insight by going in the other direction; that is, we can imagine a reality which has less dimensions than the one in our reality. This is the pretext for the book Flatland by Edwin Abbott Abbott. This rather dated book, originally published in 1899, describes what life could be like if lived in only two spatial dimensions. For those interested, this book is in the public domain, and can be downloaded in its entirety here.
After reading this text, we then wonder if there really is a fourth spatial dimension, and if so, how would we living in three dimensions interact with it? Fourth dimensional beings (and perhaps we ourselves have a fourth dimensional component?) would view our existence quite differently than we who are trapped in it do, much like we would view the world of the flatlanders.
Interestingly, today's physicists and mathematicians now theorize that there are indeed more than three dimensions plus time in reality; the current thought is that there is ten dimensions. Ten may seem like a rather arbitrary number, but you can visit the site Imagining the Tenth Dimension by Rob Brayton to see an interesting flash production which attempts to explain the current thinking.
I personally would not be surprised if there are indeed more than 10 dimensions. In fact, I expect that there are infinitely many dimensions (there certainly are mathematically). But, what is reality?
In a future post, I'd like to talk about 'time'; it's a topic I've thought about a lot recently. But, to set the stage, we need to delve into the "dimensionality" of our current reality.
Our reality has these three spatial dimensions. We can construct three straight lines, meeting at a single point, which are all ninety degrees away from each other - the x, y, and z axes. Even though it is possible mathematically, we cannot imagine a way to create a new line, intersecting at the same point, that is ninety degrees away from each of the other three lines. If we could do that, then we would have a fourth spatial dimension.
Although four dimensions are impossible to imagine, we can gain some insight by going in the other direction; that is, we can imagine a reality which has less dimensions than the one in our reality. This is the pretext for the book Flatland by Edwin Abbott Abbott. This rather dated book, originally published in 1899, describes what life could be like if lived in only two spatial dimensions. For those interested, this book is in the public domain, and can be downloaded in its entirety here.
After reading this text, we then wonder if there really is a fourth spatial dimension, and if so, how would we living in three dimensions interact with it? Fourth dimensional beings (and perhaps we ourselves have a fourth dimensional component?) would view our existence quite differently than we who are trapped in it do, much like we would view the world of the flatlanders.
Interestingly, today's physicists and mathematicians now theorize that there are indeed more than three dimensions plus time in reality; the current thought is that there is ten dimensions. Ten may seem like a rather arbitrary number, but you can visit the site Imagining the Tenth Dimension by Rob Brayton to see an interesting flash production which attempts to explain the current thinking.
I personally would not be surprised if there are indeed more than 10 dimensions. In fact, I expect that there are infinitely many dimensions (there certainly are mathematically). But, what is reality?
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