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Chapter 2: An ancient theorem and a modern question: Difference between revisions
Chapter 2: An ancient theorem and a modern question (edit)
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<math> 2^a \cdot 2^b = 2^{a+b} </math> | <math> 2^a \cdot 2^b = 2^{a+b} </math> | ||
Now, you may notice that this doesn't help if we are interested in numbers like <math> 2^{\frac{1}{2}}</math> or <math>2^{-1}</math>. These cases are covered in the | Now, you may notice that this doesn't help if we are interested in numbers like <math>2^{\frac{1}{2}}</math> or <math>2^{-1}</math>. These cases are covered in the recommended section if you are interested but are not strictly necessary for understanding this chapter. | ||
=== Pythagorean Theorem <math> a^2 + b^2 = c^2 </math>=== | === Pythagorean Theorem <math>a^2 + b^2 = c^2</math> === | ||
''For any right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.'' | ''For any right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.'' | ||
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This is the fancy name for the basic geometry we are familiar with where parallel lines do not intersect. The rules or "postulates" of Euclidian geometry are as follows. | This is the fancy name for the basic geometry we are familiar with where parallel lines do not intersect. The rules or "postulates" of Euclidian geometry are as follows. | ||
[[File: | [[File:Screenshot from 2020-06-14 18-55-40.png|thumb|Newton (1795–1805) 460 x 600 mm. Collection Tate Britain. Euclidean geometry is the study of mathematical objects that can be constructed by a straight edge and compass.]] | ||
==== Euclidian Postulates ==== | ==== Euclidian Postulates ==== | ||
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Euclid's fifth postulate cannot be proven as a theorem (by assuming only the first four), although this was attempted by many people. Euclid himself used only the first four postulates ("absolute geometry") for the first 28 propositions of the Elements, but was forced to invoke the parallel postulate on the 29th. In 1823, Janos Bolyai and Nicolai Lobachevsky independently realized that entirely self-consistent "non-Euclidean geometries" could be created in which the parallel postulate did not hold. (Gauss had also discovered but suppressed the existence of non-Euclidean geometries.) | Euclid's fifth postulate cannot be proven as a theorem (by assuming only the first four), although this was attempted by many people. Euclid himself used only the first four postulates ("absolute geometry") for the first 28 propositions of the Elements, but was forced to invoke the parallel postulate on the 29th. In 1823, Janos Bolyai and Nicolai Lobachevsky independently realized that entirely self-consistent "non-Euclidean geometries" could be created in which the parallel postulate did not hold. (Gauss had also discovered but suppressed the existence of non-Euclidean geometries.) | ||
=== Radians and <math> \pi </math> === | [[File:Euclid-woodcut-1584.jpg|thumb|Euclid, coloured woodcut, 1584.]] | ||
=== Radians and <math>\pi</math> === | |||
<math> \pi </math> is introduced in the books as the sum of all angles of a triangles, which is <math> 180^\circ</math>. This might be confusing to those who know that <math> \pi = 3.14 \cdots </math>. | <math>\pi</math> is introduced in the books as the sum of all angles of a triangles, which is <math>180^\circ</math>. This might be confusing to those who know that <math>\pi = 3.14 \cdots</math>. | ||
The explanation for this is simple. <math> \pi </math> is simply used as a shorthand for <math> \pi R </math> where <math> R </math> stands for radian. An arc of a circle with the same length as the radius of that circle subtends an '''angle of 1 radian''' (roughly 57.29). Adding three radians together brings you almost '''180 degrees''' around. <math> \pi </math> radians brings you ''exactly'' 180 degrees around. The circumference subtends an angle of <math> 2\pi </math>. To summarize: | The explanation for this is simple. <math>\pi</math> is simply used as a shorthand for <math>\pi R</math> where <math>R</math> stands for radian. An arc of a circle with the same length as the radius of that circle subtends an '''angle of 1 radian''' (roughly 57.29). Adding three radians together brings you almost '''180 degrees''' around. <math>\pi</math> radians brings you ''exactly'' 180 degrees around. The circumference subtends an angle of <math>2\pi</math>. To summarize: | ||
<math> 1 Radian = 1R = 57.29^\circ </math>: | <math> 1 Radian = 1R = 57.29^\circ </math>: | ||
<math> \pi \cdot 57.29 = \pi r = 180^\circ </math> | <math> \pi \cdot 57.29 = \pi r = 180^\circ </math> | ||
So just remember, <math> \pi = 180^\circ </math>. Further explanations are given in the [[Preliminaries| preliminaries]] section. | So just remember, <math>\pi = 180^\circ</math>. Further explanations are given in the [[Preliminaries| preliminaries]] section. | ||
[[File:S-c45c4ef6993dba6ec59e8dbdaf35b55822acac41.gif|thumb|A radian of 1 is the angle which subtends an arc of length 1 on a unit circle, or equivalently, an arc length of r on a circle with radius r.]] | [[File:S-c45c4ef6993dba6ec59e8dbdaf35b55822acac41.gif|thumb|A radian of 1 is the angle which subtends an arc of length 1 on a unit circle, or equivalently, an arc length of r on a circle with radius r.]] | ||
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==== Geodesic ==== | ==== Geodesic ==== | ||
A [https://en.wikipedia.org/wiki/Geodesic geodesic] is a curve representing the shortest path between two points in a space. It is a generalization of the notion of a "straight line". | A [https://en.wikipedia.org/wiki/Geodesic geodesic] is a curve representing the shortest path between two points in a space. It is a generalization of the notion of a "straight line". In a "flat" space, the straight line is indeed the shortest distance between two points, but in a curved space, this no longer holds true; the shortest distance between two points inherits some of the curvature from the space in which it exists. Geodesics are well explained in the videos pertaining the hyperbolic geometry in the essential section. | ||
=== Hyperbolic Geometry === | === Hyperbolic Geometry === | ||
A type of geometry which can emerge when the fifth postulate is no longer taken to be true. Objects like triangles obey different rules in this type of geometry. For instance, [https://en.wikipedia.org/wiki/Hyperbolic_triangle hyperbolic triangles] have angles which sum to '''less''' than <math> \pi </math> radians. In fact, we have we have a triangle with an area represented by <math> \triangle </math> and three angles represented by <math> \alpha, \beta, \gamma </math> then by the ''Johann Heinrich Lambert formula'': | A type of geometry which can emerge when the fifth postulate is no longer taken to be true. Objects like triangles obey different rules in this type of geometry. For instance, [https://en.wikipedia.org/wiki/Hyperbolic_triangle hyperbolic triangles] have angles which sum to '''less''' than <math>\pi</math> radians. In fact, we have we have a triangle with an area represented by <math>\triangle</math> and three angles represented by <math>\alpha, \beta, \gamma</math> then by the ''Johann Heinrich Lambert formula'': | ||
<math> \pi - (\alpha + \beta + \gamma) = C \triangle </math> | <math> \pi - (\alpha + \beta + \gamma) = C \triangle </math> | ||
where <math> C </math> is just some constant determined by the ''units'' by which we measure a give length or area. The ''units'' we use can always be chosen such that <math> C=1</math>. | where <math>C</math> is just some constant determined by the ''units'' by which we measure a give length or area. The ''units'' we use can always be chosen such that <math>C=1</math>. | ||
In contrast to euclidean geometry where the angels of a triangle alone don’t tell you anything about its size - in hyperbolic geometry if you know the sum of the angels of a triangle, you can calculate its area using the formula above. | In contrast to euclidean geometry where the angels of a triangle alone don’t tell you anything about its size - in hyperbolic geometry if you know the sum of the angels of a triangle, you can calculate its area using the formula above. | ||
[[File:Aeef84ebc9b315a863ff5dbd293254b4.gif|thumb|Hyperogue is a video game that takes place on the hyperbolic plane.]] | |||
== Preliminaries == | == Preliminaries == | ||
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* Understanding fractional and negative powers | * Understanding fractional and negative powers | ||
** [https://betterexplained.com/articles/understanding-exponents-why-does-00-1/ Understanding Exponents (Why does <math>0^0</math> | ** [https://betterexplained.com/articles/understanding-exponents-why-does-00-1/ Understanding Exponents (Why does <math>0^0=1)</math>?] | ||
** [https://medium.com/i-math/what-do-fractional-exponents-mean-1bb9bd2fa9a8 What Do Fractional Exponents Mean?] | ** [https://medium.com/i-math/what-do-fractional-exponents-mean-1bb9bd2fa9a8 What Do Fractional Exponents Mean?] | ||
** [https://medium.com/i-math/negative-exponents-reciprocals-and-the-decimal-system-revisited-f4f08894e285 Netaive Exponents and the Decimal System] | ** [https://medium.com/i-math/negative-exponents-reciprocals-and-the-decimal-system-revisited-f4f08894e285 Netaive Exponents and the Decimal System] | ||
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** [https://mathblog.com/linear-algebra Linear Algebra Done Right] by Sheldon Axler | ** [https://mathblog.com/linear-algebra Linear Algebra Done Right] by Sheldon Axler | ||
== Art == | == Art == | ||
[[File:Spiral6-4-large.jpg|thumb| Conformal mappings on the Poincare disk by Paul Nylander]] | |||
* Gorgeous hyperbolic art | * Gorgeous hyperbolic art | ||
** [http://bugman123.com/Hyperbolic/index.html Hyperbolic Geometry Artwork] | ** [http://bugman123.com/Hyperbolic/index.html Hyperbolic Geometry Artwork] | ||
[[Category:Graph, Wall, Tome]] | |||
[[Category:The Road to Reality]] | |||