Part Six: FRONTIERS
Chapter 25. The Loneliest Numbers
Prime numbers, solitary and inscrutable, space themselves apart in mysterious ways.
This final part of the book goes into a little more of number theory, one of the most complex areas in mathematics. It starts with this chapter that talks about the nature of numbers, and which ones are the loneliest. The loneliest is one, and the second loneliest is two. We arrive at this conclusion by first seeing how prime numbers behave, and more specifically twin primes. These (primes and twin primes) are less and less when the population of numbers increases, so they become more solitary. But 1 is not considered a prime, because mathematicians have excluded it solely for convenience. If it were taken into account, it would mess the whole theorem of prime numbers. This is an example of how we can create definitions and work until some point with them using certain rules, but these are not rigid and we can come back and modify them in order to keep progressing with our discoveries.
Chapter 26. Group Think
Group theory, one of the most versatile parts of math, bridges art and science.
“As these examples suggest (flipping and rotating mattresses, the choreography of square dancing, the fundamental laws of particle physics, the mosaics of the Alhambra and their chaotic counterparts), group theory bridges the arts and sciences. It addresses something the two cultures share – an abiding fascination with symmetry. Yet because it encompasses such a wide range of phenomena, group theory is necessarily abstract. It distills symmetry to its essence.” (page 212)
The important aspect of group theory is symmetry, and this allows us to see the hidden unity of things that would otherwise seem unrelated. Other examples are the symmetry of water molecules to the logic of a pair of electrical switches.
“If something’s bothering you, just sleep on it.”
Chapter 27. Twist and Shout
Playing with Möbius strips and music boxes, and a better way to cut a bagel.
Möbius, suiböM. The Möbius strip is quite a thing. It’s a one sided strip and a good introduction to topology and other non-Euclidean geometry.
“So what is topology? It’s a vibrant branch of modern math, an offshoot of geometry, but much more loosey-goosey. In topology, two shapes are regarded as the same if you can bend, twist, stretch, or otherwise deform one into the other continuously – that is, without any ripping or puncturing. Unlike rigid objects of geometry, the objects of topology behave as if they were infinitely elastic, as if they were made of an ideal kind of rubber or Silly Putty.” (page 220)
Important resources mentioned in this chapter are Vi Hart’s videos on Youtube (Möbius Story: Wind and Mr. Ug; and Möbius music box) and also from her father, George, the Möbius bagel (you’ll maximize the surface for spreading cream cheese!). Also, architecture inspired by Möbius like the National Library of Kazakhstan, other sculptors like Max Bill and Keizo Ushio, and paintings and drawings like the ones from Escher.
Chapter 28. Think Globally
Differential geometry reveals the shortest route between two points on a globe or any other curved surface.
This chapter focuses on differential geometry. This is the study of the effects of small local differences on various kinds of shapes. He explains this by making the example of why planes don’t travel in an apparent straight line but are seen to go up a little before going down. Another figure that is explained in this chapter is the geodesic and the two-holed torus. All these analysis serve us to find the shortest path to do something.
“Sometimes when people say the shortest distance between two points is a straight line, they mean it figuratively, as a way of ridiculing nuance and affirming common sense. In other words, keep it simple. But battling obstacles can give rise to great beauty – so much so that in art, and in math, it’s often more fruitful to impose constraints on ourselves. Think of haiku, or sonnets, or telling the story of your life in six words. The same is true of all the math that’s been created to help you find the shortest way from here to there when you can’t take the easy way out.
Two points. Many paths. Mathematical bliss.” (page 236)
I wonder how this can be applied metaphorically to entrepreneurship, the line of success, and the common conception that life is linear.
Chapter 29. Analyze This!
Why calculus, once so smug and cocky, had to put itself on the couch.
This chapter puts a little more trouble in the way of math. It explains how some things are not subject to our general laws of mathematics and if you apply them in certain order (different, but following the same rules) we would arrive at different answers. One example is the operation 1 – 1 + 1 – 1 + … What’s the answer of this? 1? 0? ½? This has lead to new discoveries such as the Fourier series and the Gibbs phenomenon. What it’s interesting is that when we analyze these things, we can get more understanding of things in our daily life like digital photos.
Chapter 30. The Hilbert Hotel
An exploration of infinity as this book, not being infinite, comes to an end.
This last chapter is about infinites. In here, we see the nature of infinites and their different categories, like Cantor’s theory of infinites state. He says there are three levels of infinites and this means that some infinites will have the same correspondence with other infinites, others would be greater, and others smaller. This has revolutionized the whole concept of infinites. Then, he explains the Hilbert Hotel example, a creation by David Hilbert in order to defend Cantor’s theory. This hotel is one with infinite number of rooms and infinite guests and the conclusion is that the Hilbert Hotel can’t accommodate all the real numbers. That’s because they are too many of them, an infinity beyond infinity. This illustrates how one infinity can be greater than other.
So that’s it. A long history of mathematics explained in a very fun and precise way, with many illustrations and practical examples we can find in real life. Although it may seem like a book written for children, it’s very insightful and you can take a lot of advantage in it.
Prime numbers, solitary and inscrutable, space themselves apart in mysterious ways.
This final part of the book goes into a little more of number theory, one of the most complex areas in mathematics. It starts with this chapter that talks about the nature of numbers, and which ones are the loneliest. The loneliest is one, and the second loneliest is two. We arrive at this conclusion by first seeing how prime numbers behave, and more specifically twin primes. These (primes and twin primes) are less and less when the population of numbers increases, so they become more solitary. But 1 is not considered a prime, because mathematicians have excluded it solely for convenience. If it were taken into account, it would mess the whole theorem of prime numbers. This is an example of how we can create definitions and work until some point with them using certain rules, but these are not rigid and we can come back and modify them in order to keep progressing with our discoveries.
Chapter 26. Group Think
Group theory, one of the most versatile parts of math, bridges art and science.
“As these examples suggest (flipping and rotating mattresses, the choreography of square dancing, the fundamental laws of particle physics, the mosaics of the Alhambra and their chaotic counterparts), group theory bridges the arts and sciences. It addresses something the two cultures share – an abiding fascination with symmetry. Yet because it encompasses such a wide range of phenomena, group theory is necessarily abstract. It distills symmetry to its essence.” (page 212)
The important aspect of group theory is symmetry, and this allows us to see the hidden unity of things that would otherwise seem unrelated. Other examples are the symmetry of water molecules to the logic of a pair of electrical switches.
“If something’s bothering you, just sleep on it.”
Chapter 27. Twist and Shout
Playing with Möbius strips and music boxes, and a better way to cut a bagel.
Möbius, suiböM. The Möbius strip is quite a thing. It’s a one sided strip and a good introduction to topology and other non-Euclidean geometry.
“So what is topology? It’s a vibrant branch of modern math, an offshoot of geometry, but much more loosey-goosey. In topology, two shapes are regarded as the same if you can bend, twist, stretch, or otherwise deform one into the other continuously – that is, without any ripping or puncturing. Unlike rigid objects of geometry, the objects of topology behave as if they were infinitely elastic, as if they were made of an ideal kind of rubber or Silly Putty.” (page 220)
Important resources mentioned in this chapter are Vi Hart’s videos on Youtube (Möbius Story: Wind and Mr. Ug; and Möbius music box) and also from her father, George, the Möbius bagel (you’ll maximize the surface for spreading cream cheese!). Also, architecture inspired by Möbius like the National Library of Kazakhstan, other sculptors like Max Bill and Keizo Ushio, and paintings and drawings like the ones from Escher.
Chapter 28. Think Globally
Differential geometry reveals the shortest route between two points on a globe or any other curved surface.
This chapter focuses on differential geometry. This is the study of the effects of small local differences on various kinds of shapes. He explains this by making the example of why planes don’t travel in an apparent straight line but are seen to go up a little before going down. Another figure that is explained in this chapter is the geodesic and the two-holed torus. All these analysis serve us to find the shortest path to do something.
“Sometimes when people say the shortest distance between two points is a straight line, they mean it figuratively, as a way of ridiculing nuance and affirming common sense. In other words, keep it simple. But battling obstacles can give rise to great beauty – so much so that in art, and in math, it’s often more fruitful to impose constraints on ourselves. Think of haiku, or sonnets, or telling the story of your life in six words. The same is true of all the math that’s been created to help you find the shortest way from here to there when you can’t take the easy way out.
Two points. Many paths. Mathematical bliss.” (page 236)
I wonder how this can be applied metaphorically to entrepreneurship, the line of success, and the common conception that life is linear.
Chapter 29. Analyze This!
Why calculus, once so smug and cocky, had to put itself on the couch.
This chapter puts a little more trouble in the way of math. It explains how some things are not subject to our general laws of mathematics and if you apply them in certain order (different, but following the same rules) we would arrive at different answers. One example is the operation 1 – 1 + 1 – 1 + … What’s the answer of this? 1? 0? ½? This has lead to new discoveries such as the Fourier series and the Gibbs phenomenon. What it’s interesting is that when we analyze these things, we can get more understanding of things in our daily life like digital photos.
Chapter 30. The Hilbert Hotel
An exploration of infinity as this book, not being infinite, comes to an end.
This last chapter is about infinites. In here, we see the nature of infinites and their different categories, like Cantor’s theory of infinites state. He says there are three levels of infinites and this means that some infinites will have the same correspondence with other infinites, others would be greater, and others smaller. This has revolutionized the whole concept of infinites. Then, he explains the Hilbert Hotel example, a creation by David Hilbert in order to defend Cantor’s theory. This hotel is one with infinite number of rooms and infinite guests and the conclusion is that the Hilbert Hotel can’t accommodate all the real numbers. That’s because they are too many of them, an infinity beyond infinity. This illustrates how one infinity can be greater than other.
So that’s it. A long history of mathematics explained in a very fun and precise way, with many illustrations and practical examples we can find in real life. Although it may seem like a book written for children, it’s very insightful and you can take a lot of advantage in it.