Chapter 2: An ancient theorem and a modern question
Description goes here.
Preliminaries
- Know how to visually represent addition, subtraction, multiplication, and powers
- Know what squares (powers of two) and square roots are
- Know what logarithms are
- Know what an equation and the solution of an equation is (note that an equation can have more than one solution!)
- Now tie it all together
- And quick a introduction to radians
Community Explanations
Translation In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure or a space by the same distance in a given direction.
Exponents2 Exponentiation can be thought of as repeated multiplication. meaning: 23= 222 and 25= 22222 adding them together we also see that 2325= 22222222
The additive property of exponentiation tells us that we can also write it this way 23+5=2+25
Now, you may notice that this doesn’t help if we’re asking about 212or 2-1 How do you repeatedly multiply something 212or 2-1times? so for numbers other than the counting numbers, we need a different clear explanation (To be Expanded)
Better explained: understanding exponents
Exponents on Khan academy
Pythagorean Theorem | A2 + B2 = C2 “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” test your or learn more in Khan academy Pythagorean Theorem on better explain 6 animated proofs
Euclidean geometry This is the fancy name for the basic geometry we’re familiar with. to be expanded.
Euclidean Postulates 1. A straight line segment can be drawn joining any two points. 2. Any straight line segment can be extended indefinitely in a straight line. 3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center. 4. All right angles are congruent. 5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. This postulate is equivalent to what is known as the parallel postulate.
Euclid's fifth postulate cannot be proven as a theorem, 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.)
video explanation
Radians and pi pi is introduced in the book as the sum of all angles of a triangle, which is 180. this might be confusing to those who know that = 3.14... The explanation for this is simple, in this case is simply used as a shorthand for R - Where R 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. radians bring you exactly 180 degrees around. The circumference subtends an angle of 2π radians. To summarize: 1 redian = 1r = 57.29
57.29 = r =180
As a shorthand = 180
Hyperbolic Geometry
One class of models of Euclidean geometry minus the parallel postulate. Hyperbolic triangles have less than 180° (while spherical triangles have more than 180° - these are not talked about for now). to be expanded
Johann Heinrich Lambert formula for calculating the area of a hyperbolic triangle
In this formula: pi = 180 = the area of the triangle C is some constant. This constant depends on the ‘units’ that are chosen in which lengths and areas are to be measured. We can always scale things so that C = 1. 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 with the above formula. Wikipedia (couldn’t find exercises for this on khan academy, if someone can find exercises anywhere please share them)
For an animated experience of hyperbolic geometry you can play the game Hyperrogue available for android and desktop (free version or paid version)
Representational Models
Conformal Disk (Beltrami-Poincare): shown on the left
Projective (Beltrami-Klein) shown in the middle
Hemispherical (Beltrami)
Hyperboloid (Mindowski-Lorenz) shown on the right
Geodesic
a geodesic is a curve representing the shortest path between two points in a surface (a curved one, for example). It is a generalization of the notion of a "straight line" to a more general setting.
Stereographic Projection Video (1:07)
Logarithms
(becomes important in ch. 5, and stays important!)
If you just came across the first instance of “log” in the book (section 2.4), continue until the end of the section (one page later), and then come back here.
Logarithms are introduced for the first time in this expression that describes the distance between two points in hyperbolic space. the logarithm used here is called the natural logarithm, see next section.
Better explained: using logarithms in the real world Logarithms in khan academy
Natural logarithm e The natural logarithm of a number is its logarithm to the base of the mathematical constant e. e is equal to 2.718… this is the logarithm used in the expression above. better explained: demystifying the natural logarithm
Mathematical proofs To be expanded
On Proof And Progress In Mathematics - William P. Thurston
To learn some basic proving-things skills, you can read the book “how to prove it” by Daniel J. Velleman. You can read/download it here
Proof by contradiction
Curvature
3-Dimensional Hyperbolic Geometry (Hyperbolic Space) Hyperbolic and other geometrical visualizations
Essential
- An additcting puzzle game where you do Euclidian constructions
- An interactive animated version of a proof of the Pythagorean Theorem
- Pythagorean Theorem Proof by Community Contributor @TimAlex
Recommended
- Understanding fractional powers
- A more in-depth description of the logarithms and exponents with applications
- Hyperbolic geometry
Further Exploration
- To understand what geometry really is
- The Four Pillars of Geometry by John Stillwell
- A guide through Euclid's Elements
Books
The Four Pillars of Geometry by John Stillwell
Linear Algebra Done Right by Sheldon Axler
See More
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