Part Three: SHAPES
Chapter 12. Square Dancing
Geometry, intuition, and the long road from Pythagoras to Einstein.
This chapter is an introduction to geometry, and gives a couple of reasons why geometry has become one of the favorite subjects to study. Basically, he says that it’s because it marries logic with intuition. For one part, we have the formal argument, the proof, that involves strict steps to achieve an end; that is the logic part. But when one has to encounter a geometrical problem, one has to apply intuition into the start of the solution. In this chapter, he illustrates this point by explaining the Pythagorean theorem in two ways, one more geometrical and the other more algebraic. Then, he says that the geometrical is more convincing since it illuminates, and that’s what makes it more elegant, rather than the other one which still arrives at the same answer but with not the same level of concreteness and understanding. Finally, the Pythagorean theorem matters because it reveals a fundamental truth about the nature of space, although later Einstein, Bernhard Riemann, and other non-Euclidean geometers modified it in order to be applied to curved spaces.
Chapter 13. Something from Nothing
Like any other creative act, constructing a proof begins with inspiration.
In this chapter, the implications of geometry in other branches of science and reasoning is explored. Since Euclid’s Elements, many have used the same logical procedure of proofs. Some of those are Isaac Newton (The Mathematical Principles of Natural Philosophy), Spinoza (Ethics Demonstrated in Geometrical Order), and Thomas Jefferson.
“Still, what’s missing in all this worship of Euclidean rationality is an appreciation of geometry’s more intuitive aspects. Without inspiration, there’d be no proofs – or theorems to prove in the first place. Like composing music or writing poetry, geometry requires making something from nothing.”
So, geometry is composed of two main things; logic and intuition. Intuition is what gives us something from nothing, like building an equilateral triangle, and it guides us to certain point where logic must follow some steps to complete the proof.
“And who knows? If we highlight this other side of geometry – its playful, intuitive side, where a spark of imagination can be quickly fanned into a proof – maybe someday all students will remember geometry as the class where they learned to be logical and creative.”
Chapter 14. The Conic Conspiracy
The uncanny similarities between parabolas and ellipses suggest hidden forces at work.
After going through some of the origins of geometry, now we encounter the conic sections. He starts by explaining what a focus is in an ellipse and gives some examples on how it is applied in the world. For example, bulbs, telescopes, TV reception, radio telescopes, etc. Then, the parabola is explained along with what a focus is in one. Here, the same principle of the focus applies; that a focus would have the same distance from any point in the graph to another outside of it (in the case of an ellipse, this distance is the sum of the foci to a point in the ellipse described). Now, going a little further, we see that these functions come by cutting a cone. The circle is taken by a parallel cut to the base; the ellipse is taken by a not so steep cut; the parabola is taken by a parallel cut to the side of the cone; and further, a hyperbola is taken by slicing the cone very steeply, creating two hyperbolas. Also, planets are said to move in elliptical orbits.
Chapter 15. Sine Qua Non
Sine waves everywhere, from Ferris wheels to zebra stripes.
Trigonometry is introduced in this chapter, and focus primarily on the sine function, explaining some real world examples and application where we can find this function.
“Yet trigonometry, belying its much too modest name, now goes far beyond the measurement of triangles. By quantifying circles as well, it has paved the way for the analysis of anything that repeats, from ocean waves to brain waves. It’s the key to the mathematics of cycles.”
In a later explanation, Strogatz shows how this application of trigonometry in cycles and circles works, by explaining how a sine function is based on 90 quadrants, summing the total of the 360 degrees of a circle (in similar angles, the value of sine is the same in the circle).
“Sine waves are the atoms of structure. They’re nature’s building blocks. Without them there’d be nothing, giving new meaning to the phrase “sine qua non””.
This last point can be seen also in quantum mechanics, cosmological scales, astronomy, the Big Bang, Stars, Galaxies, and the Ferris wheels.
Chapter 16. Take It to the Limit
Archimedes recognized the power of the infinite and in the process laid the groundwork for calculus.
This chapter explains the concept of limits and infinites, giving an introduction to calculus. It all started (mainly) with Archimedes. He “realized the power of the infinite”, and solved for the areas of circles by inscribing a 96-sided polygon and separating it into triangles. Another solution is to arrange the pieces of a circle (triangles) almost infinitely so it ultimately becomes a rectangle. Then, by knowing the value of pi and its radius, we derive the formula Area = πr2. “Pi” is the ratio of the circumference and the diameter (this would apply to all circles since all circles are similar).
So there it is, Archimedes contributing to the groundwork for calculus nearly 2,000 years before its invention by Newton and independently Leibniz.
Geometry, intuition, and the long road from Pythagoras to Einstein.
This chapter is an introduction to geometry, and gives a couple of reasons why geometry has become one of the favorite subjects to study. Basically, he says that it’s because it marries logic with intuition. For one part, we have the formal argument, the proof, that involves strict steps to achieve an end; that is the logic part. But when one has to encounter a geometrical problem, one has to apply intuition into the start of the solution. In this chapter, he illustrates this point by explaining the Pythagorean theorem in two ways, one more geometrical and the other more algebraic. Then, he says that the geometrical is more convincing since it illuminates, and that’s what makes it more elegant, rather than the other one which still arrives at the same answer but with not the same level of concreteness and understanding. Finally, the Pythagorean theorem matters because it reveals a fundamental truth about the nature of space, although later Einstein, Bernhard Riemann, and other non-Euclidean geometers modified it in order to be applied to curved spaces.
Chapter 13. Something from Nothing
Like any other creative act, constructing a proof begins with inspiration.
In this chapter, the implications of geometry in other branches of science and reasoning is explored. Since Euclid’s Elements, many have used the same logical procedure of proofs. Some of those are Isaac Newton (The Mathematical Principles of Natural Philosophy), Spinoza (Ethics Demonstrated in Geometrical Order), and Thomas Jefferson.
“Still, what’s missing in all this worship of Euclidean rationality is an appreciation of geometry’s more intuitive aspects. Without inspiration, there’d be no proofs – or theorems to prove in the first place. Like composing music or writing poetry, geometry requires making something from nothing.”
So, geometry is composed of two main things; logic and intuition. Intuition is what gives us something from nothing, like building an equilateral triangle, and it guides us to certain point where logic must follow some steps to complete the proof.
“And who knows? If we highlight this other side of geometry – its playful, intuitive side, where a spark of imagination can be quickly fanned into a proof – maybe someday all students will remember geometry as the class where they learned to be logical and creative.”
Chapter 14. The Conic Conspiracy
The uncanny similarities between parabolas and ellipses suggest hidden forces at work.
After going through some of the origins of geometry, now we encounter the conic sections. He starts by explaining what a focus is in an ellipse and gives some examples on how it is applied in the world. For example, bulbs, telescopes, TV reception, radio telescopes, etc. Then, the parabola is explained along with what a focus is in one. Here, the same principle of the focus applies; that a focus would have the same distance from any point in the graph to another outside of it (in the case of an ellipse, this distance is the sum of the foci to a point in the ellipse described). Now, going a little further, we see that these functions come by cutting a cone. The circle is taken by a parallel cut to the base; the ellipse is taken by a not so steep cut; the parabola is taken by a parallel cut to the side of the cone; and further, a hyperbola is taken by slicing the cone very steeply, creating two hyperbolas. Also, planets are said to move in elliptical orbits.
Chapter 15. Sine Qua Non
Sine waves everywhere, from Ferris wheels to zebra stripes.
Trigonometry is introduced in this chapter, and focus primarily on the sine function, explaining some real world examples and application where we can find this function.
“Yet trigonometry, belying its much too modest name, now goes far beyond the measurement of triangles. By quantifying circles as well, it has paved the way for the analysis of anything that repeats, from ocean waves to brain waves. It’s the key to the mathematics of cycles.”
In a later explanation, Strogatz shows how this application of trigonometry in cycles and circles works, by explaining how a sine function is based on 90 quadrants, summing the total of the 360 degrees of a circle (in similar angles, the value of sine is the same in the circle).
“Sine waves are the atoms of structure. They’re nature’s building blocks. Without them there’d be nothing, giving new meaning to the phrase “sine qua non””.
This last point can be seen also in quantum mechanics, cosmological scales, astronomy, the Big Bang, Stars, Galaxies, and the Ferris wheels.
Chapter 16. Take It to the Limit
Archimedes recognized the power of the infinite and in the process laid the groundwork for calculus.
This chapter explains the concept of limits and infinites, giving an introduction to calculus. It all started (mainly) with Archimedes. He “realized the power of the infinite”, and solved for the areas of circles by inscribing a 96-sided polygon and separating it into triangles. Another solution is to arrange the pieces of a circle (triangles) almost infinitely so it ultimately becomes a rectangle. Then, by knowing the value of pi and its radius, we derive the formula Area = πr2. “Pi” is the ratio of the circumference and the diameter (this would apply to all circles since all circles are similar).
So there it is, Archimedes contributing to the groundwork for calculus nearly 2,000 years before its invention by Newton and independently Leibniz.