Euclidean Geometry http www youtube comwatch vKUGLOi Zy
Euclidean Geometry http: //www. youtube. com/watch? v=_KUGLOi. Zy. K 8
Pythagorean Theorem Suppose a right triangle ∆ABC has a right angle at C, hypotenuse c, and sides a and b. Then
Proof of Pythagorean Theorem What assumptions are made? Other proof: http: //www. youtube. com/watch? v=CAk. MUde. B 06 o Pythagorean Rap Video
Euclid’s Elements • Dates back to 300 BC • Euclid’s Elements as translated by Billingsley appeared in 1570 • Ranks second only to the Bible as the most published book in history
Euclid’s First 4 Postulates 1. We can draw a unique line segment between any two points. 2. Any line segment can be continued indefinitely. 3. A circle of any radius and any center can be drawn. 4. Any two right angles are congruent.
Euclid’s 5 th Postulate (Parallel Postulate) 5. Given a line l and a point P not on l, there exists a unique line l’ through P which does not intersect l.
Distance Let d(P, Q) be a function which assigns a positive real number to any pair of points in the plane. Then d is a distance function (or metric) if it satisfies the following three properties for any three points in the plane: 1. d(P, Q) = d(Q, P) 2. d(P, Q) ≥ 0 with equality if and only if P=Q 3. d(P, R) ≤ d(P, Q) + d(Q, R) (triangle inequality) (often write |PQ| for distance)
Euclidean Distance •
Taxi-cab Metric A different distance, called the taxi-cab metric, is given by |PQ| = |a-c| + |b-d|
Circles The circle CP(r) centered at P with radius r is the set CP(r)={Q : |PQ| = r}
Isometries and Congruence An isometry is a map that preserves distances. Thus f is an isometry if and only if |f(P)f(Q)| = |PQ| Two sets of points (which define a triangle, or some other figure) are congruent if there exists an isometry which maps one set to the other
More Axioms to Guarantee Existence of Isometries 6. Given any points P and Q, there exists an isometry f so that f(P) = Q (translations) 7. Given a point P and any two points Q and R which are equidistant from P, there exists an isometry f such that f(P) = P and f(Q) = R (rotations) 8. Given any line l, there exists an isometry f such that f(P)=P if P is on l and f(P) ≠ P if P is not on l (reflections)
Congruent Triangles: SSS Theorem: If the corresponding sides of two triangles ∆ABC and ∆A’B’C’ have equal lengths, then the two triangles are congruent.
Categories of Isometries An isometry is a direct (proper) isometry if it preserves the orientation of every triangle. Otherwise the isometry is indirect (improper). Important: It suffices to check what the isometry does for just one triangle. If an isometry f is such that there is a point P with f(P) = P, then P is called a fixed point of the isometry.
Transformations 1. An isometry f is a translation if it is direct and is either the identity or has no fixed points. 2. An isometry f is a rotation if it is a direct isometry and is either the identity or there exists exactly one fixed point P (the center of rotation). 3. An isometry f is a reflection through the line l if f(P) = P for every point P on l and f(P) ≠ P for every point P not on l.
Pictures of Transformations
Sample Geometry Proof Prove that if the isometry f is a reflection, then f is not a direct isometry.
What happens if… • You do a reflection followed by another reflection? • You do a reflection followed by the same reflection?
Parallel Lines Euclid stated his fifth postulate in this form: Suppose a line meets two other lines so that the sum of the angles on one side is less that two right angles. Then the other two lines meet at a point on that side.
Angles and Parallel Lines Which angles are equal?
Sum of Angles in Triangle The interior angles in a triangle add up to 180°
What about quadrilaterals?
More generalizing • What about polygons with n sides? • What about regular polygons (where all sides have the same lengths and all angles are equal)?
Exterior Angles of Polygons
Another Geometry Proof Theorem (Pons Asinorum): The base angles of an isosceles triangle are equal.
Symmetries of the Square A symmetry of a figure is an isometry of the plane that leaves the figure fixed. What are the symmetries of the square?
The Group of Symmetries of the Square The set {a, b, c, d, e, f, g, h} together with the operation of composition (combining elements) forms a group. This is a very important mathematical structure that possesses the following: 1. Closed under the operation 2. The operation is associative (brackets don’t matter) 3. There is an identity element 4. Every element has an inverse
Frieze Groups • A frieze group is the symmetry group of a repeated pattern on a strip which is invariant under a translation along the strip • Here are four possibilities. Are there any more?
Frieze Groups
Wallpaper Groups • Symmetry groups in the plane • Show up in decorative art from cultures around the world • Involve rotations, translations, reflections and glide reflections • How many are there?
Similar Triangles AB/DE = AC/DF = BC/EF
Pentagon Exercise Which triangles are congruent? Isosceles? Similar?
The Golden Ratio The golden ratio is defined to be the number Φ defined by Φ = (1 + √ 5)/2 ≈ 1. 618
Golden Pentagon
What is the ratio of your height to the length from the floor to your belly button?
The Golden Rectangle
The Golden Ratio A B Ratio of A to B is the golden ratio A = 1. 618 B
Where can we find the Golden Ratio?
The Parthenon
The Great Pyramid of Giza 2560 BC -Side lengths approximately 230 m -Base covers 53 000 m^2 -Sides angled at 51. 5 degrees. 1^2 + (√φ)^2 = φ^2 √φ φ 1 2 1+ 1. 618 = 2. 618
CN Tower Base to observation deck 342 m Base to spire 553. 33 m 553. 33/342 = 1. 618 = φ
Moving on… 1, 1, 2, 3, 5, 8, 13, … What is the pattern?
Fibonacci Numbers • Each number is the sum of the two before Fn= Fn-1+Fn-2 • Fibonacci Numbers in Nature • Youtube video on Fibonacci
Ratios of Fibonacci Numbers • • • F 2/F 1=1/1=1 F 3/F 2=2/1=2 F 4/F 3=3/2=1. 5 F 5/F 4=5/3=1. 67 F 6/F 5=8/5=1. 6 • • • F 7/F 6=13/8=1. 625 F 8/F 7=21/13=1. 6154 F 9/F 8=34/21=1. 619 F 10/F 9=55/34=1. 6176 F 11/F 10=89/55=1. 6182
Connection • So the Fibonacci numbers and the golden ratio are connected • More about the Fibonacci Sequence and The Golden Ratio
Pascal’s Triangle
More on Pascal’s Triangle • All You Ever Wanted to Know About Pascal's Triangle and more • Pascal's Triangle and the Fibonacci Series
Connection • So the Fibonacci numbers and Pascal’s triangle are also connected!
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