MTH 10 FINALS

• Plato once said "Let no one enter, who is ignorant of Geometry"
• Geometry


  The word comes from two Greek words geo and metron which mean earth and measurement, respectively
• Geometry was developed independently in a number of early civilizations as a practical way of dealing with areas, lengths and volumes
• Sections of this module

  • Origins of geometry before Euclid
  • Euclidean geometry
  • Non-Euclidean geometries
  • Projective geometry
  • Topology
• In the 1850s, 400 clay tablets were unearthed containing Babylonian Mathematics written in cuneiform script

  
• Babylonians


  • Gave us the formulas for finding areas and volumes of some geometric figures like circles and cylinders
  • Approximated the value of π to 3
• There are no sources indicating that Babylonians were aware of the Pythagorean Theorem, since the records they left behind suggest that they already had example cases of Pythagorean Theorem known to us these days as the Pythagorean triples
• The Egyptian civilization also gave us formulas in finding areas and volumes, which were essential in the construction of their famous pyramids and the determination of food supply at that time
• Eratosthenes was known to be the first mathematician to calculate the circumference of the earth
• Pythagoras gave us the concept that given a right triangle with lengths of sides a and b; and, length of side opposite the right angle c, then a^2+b^2=c^2, now known as the Pythagorean Theorem

  
• Archimedes of Syracuse contributed a handful geometric concepts like the volumes of irregular shapes and derived an accurate approximation of the value of π using the method of exhaustion developed by Eudoxus of Cnidus
• The Greeks discovered irrational numbers like (square root of 2), pi and the famous golden ratio, also known as the divine proportion, phi
• How to get the golden ratio

  1. Divide any line segment (say one with unit length 1) into two parts

  2. The ratio of the whole (1) to the longer part (X) equals the ratio of the longer to the shorter part

  3. This gives the quadratic equation x^2 + x - 1 = 0, which has the solution x = (-1 + √5)/2, approximately 0.618
• Rectangles having the ratio of length and its width to be phi are called golden rectangles, and these are considered by many as a rectangle with the most pleasing proportions
• Euclid of Alexandria was responsible for a mathematical system that students all over the globe study in Mathematics
• In his textbook The Elements, Euclid described a mathematical system known as Euclidean geometry

  
• Geometry deals with points and set of points, but points, lines and planes are undefined terms in geometry
• Euclid's axioms


  • Things that are equal to the same thing are equal
  • If equals are added to equals, then the whole are equal
  • If equals are subtracted from equal, then the remainders are equal
  • Things that coincide with one another are equal to one another
  • The whole is greater than the part
• Euclidean postulates


  • A straight line can be drawn from any point to any point
  • A finite straight line can be produced continuously in a straight line
  • A circle may be drawn with any point as center and any distance as radius
  • All right angles are equal to one another
  • If a transversal falls on two lines in such a way that the interior angle on one side of the transversal are less than two right angles, then the lines meet on that side on which the angles are less than two right angles

  
• The first four postulates of Euclidean geometry seemed obvious and noncontroversial, but the fifth one had a long and complicated statement
• The 5th postulate is also called the Parallel Postulate as it characterizes what happens to a pair of parallel lines
• Many believed that Euclid made a mistake in including the 5th postulate, as they thought it could be derived or proven from the other postulates
• Over the years, the 5th postulate was restated by several mathematicians in equivalent forms
• Playfair's Axiom


  Through a point P not on line l, there exists exactly one line passing through point P parallel to l
• In Euclidean geometry, the sum of the interior angles of a triangle is 180 degrees
• Congruent triangles


  • Have the same size and shape
  • Satisfy the congruence criteria: SSS, SAS, ASA
• Similar triangles


  • Have the same shape but can be different sizes
  • Have corresponding angles equal
• Attempts to prove Euclid's 5th postulate instead gave rise to other geometries
• Hyperbolic geometry


  A geometry obtained by negating Euclid's 5th postulate, where there can be more than one line through a point parallel to another line
• Hyperbolic geometry was developed independently by Lobachevsky, Bolyai and Gauss in the early 19th century
• Nikolai Ivanovich Lobachevsky


  Russian mathematician and geometer
• Janos Bolyai


  Hungarian mathematician
• Johann Carl Friedrich Gauss

  German mathematician known to contribute in many fields of Mathematics
• Gauss was the first of the three but since his work was not published, Lobachevsky and Bolyai took the credit
• Lobachevsky published his work on this geometry first in an article entitled "Geometrische Untersuchungen zur Theorie der Parellellinien (Geometrical Researches on the Theory of Parallels)"


  1829

  
• Bolyai published his work on this geometry


  1832

  
• Hyperbolic geometry


  A new geometry obtained by assuming the negation of Euclid's 5th Postulate

  
• Poincaré disk model


  A nice model for hyperbolic geometry by considering a circular disk
• Hyperbolic geometry


  • Through a point P not on a line l, there exists at least two lines passing through point P parallel to line l
  • The sum of the angles of a (hyperbolic) triangle is less than 180
  • Similar triangles are congruent
• Elliptic geometry


  A non-euclidean geometry where there are no parallel lines, where all lines intersect