There are many mathematical facts (results) which proved to be mind-boggling. While math may be dense and difficult at times, the results it can prove are sometimes beautiful, mind-blowing, or just plain unexpected.
I'd like to share these three mathematical results which I found astonishing till my school life.
1.) Brouwer’s Fixed Point Theorem
Let’s say we have a picture (for example, the World Map) and we take a copy of it. We can then do whatever we want to this copy—make it bigger, make it smaller, rotate it, crumple it up, anything. Brouwer’s Fixed Point Theorem says that if we put this copy overtop of our original picture, there has to be at least one point on the copy that is exactly overtop the same point on the original. (It could be part of India, Africa, or might be a point in the Ocean) but it has to exist.)
This theorem comes from a branch of math known as Topology, and was discovered by Luitzen Brouwer.
This also works in three dimensions: Imagine we have a glass of water, and we take a spoon and stir it up as much as we want. By Brouwer’s theorem, there will be at least one water molecule that is in the exact same place as it was before we started stirring. (Intresting no!)
2.) The 4-Color Theorem
In mathematics, this theorem, or the four color map theorem, states that, given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colors are required to color the regions of the map so that no two adjacent regions have the same color. (Amazing!) . Adjacent means that two regions share a common boundary curve segment, not merely a corner where three or more regions meet.
This theorem was first discovered in 1852 by a man named Francis Guthrie, who at the time was trying to color in a map of all the counties of England (this was before the internet was invented, there wasn’t a lot to do).
In 1976 (over a century later), this problem was finally solved by Kenneth Appel and Wolfgang Haken.
3.) Fermat’s Last Theorem (cool name!)
Remember Pythagoras’ theorem from school? It has to do with right-angled triangles, and says that the sum of the squares of the two shortest sides are equal to the square of the longest side (x squared + y squared = z squared). Pierre de Fermat’s most famous theorem is that this same equation is not true if you replace the squared with any number greater than 2 (you could not say x cubed +y cubed = z cubed, for example), as long as x, y, and z are positive whole numbers.
While Fermat posed this problem in 1637, it went unproven for quite a while. And by a while, I mean it was proven in 1995 (358 years later) by a man named Andrew Wiles.
So these were the three results of which I was staggered when read for the first time.
(I’ll surely add more facts and astonishing results soon! )
Hope you enjoyed it.
Thanks for scrolling!
(Image source - Google Images)