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On this date

On this date. 10/21/1805 Nelson defeats French and Spanish in the Battle of Trafalgar 27 to 33. 10/21/1914 Birth of Martin Gardner. 10/21/1969 Death of Sierpinski . On this date. 10/21/1805 Nelson defeats French and Spanish in the Battle of Trafalgar 27 to 33. On this date.

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On this date

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  1. On this date • 10/21/1805 Nelson defeats French and Spanish in the Battle of Trafalgar 27 to 33. • 10/21/1914 Birth of Martin Gardner. • 10/21/1969 Death of Sierpinski.

  2. On this date • 10/21/1805 Nelson defeats French and Spanish in the Battle of Trafalgar 27 to 33.

  3. On this date • 10/21/1914 Birth of Martin Gardner

  4. On this date • 10/21/1969 Death of Sierpinski “investigator of infinity”

  5. What did the acorn say when it grew up?

  6. n-gon • Sum of Interior Angles = (n-2) × 180°

  7. n-gon • Sum of Interior Angles = (n-2) × 180°

  8. n-gon • Sum of Interior Angles = (n-2) × 180° • Number of diagonals =

  9. n-gon • Sum of Interior Angles = (n-2) × 180° • Number of diagonals = n(n-3)/2

  10. n-gon • Sum of Interior Angles = (n-2) × 180° • Number of diagonals = n(n-3)/2 • Sum of Exterior Angles =

  11. n-gon • Sum of interior angles = (n-2) × 180° • Number of diagonals = n(n-3)/2 • Sum of exterior angles = n × 180° - (n-2) × 180°

  12. n-gon • Sum of interior angles = (n-2) × 180° • Number of diagonals = n(n-3)/2 • Sum of exterior angles = n × 180° - (n-2) × 180° = 360°

  13. 22=4 times 23 =8 times

  14. 22=4 times 23 =8 times

  15. How to make Sierpinski’s Triangle

  16. How to make Sierpinski’s Triangle • Doubling the length→ Triples the size, 3=2d so dimension: d=log23=1.5849

  17. How to make Sierpinski’s Triangle • Doubling the length→ Triples the size : 3=2d d=log23=1.5849

  18. Types of Geometry • Synthetic (Euclidean) – plane & solid • Coordinates (Descartes) • Vectors • Complex Plane (Argand-Gauss) • Inversion • n-dimensional • Non-Euclidean • Hyperbolic (Lobachevsky-Bolyai-Gauss) • Convex • Combinatorial • Differential • Fractal

  19. Euclid’s Five Postulates • 1. Between 2 points is a line segment. • 2. Any line segment can be extended. • 3. Given any line segment, a circle can be drawn centered at one end and through the other. • 4. All right angles are congruent. • 5. Given a point P and a line ℓ, there is only one line through P “parallel” to ℓ.

  20. Lobachevsky-Bolyai-GaussGeometry • Saddle (Hyperbolic Paraboloid)

  21. Hyperbolic Disk |z|<1

  22. Computer Generated

  23. Pythagoras • Golden Ratio 2cos36 =(1+sqrt(5))/2

  24. Rectangles are 2x1, Pentagram diameter is ¾ BC:AB=CD:BC

  25. Flags • Britain 1:2 (UK, Aus, Can, Ire, Bah) with the little correction of 10:19 for US • France/Spain 2:3 (France, Spain, Italy, Algeria, Latin America) • Germany 3:5 • Belgium 13:15 • Denmark 37:28

  26. British flag theorem • PA2+PC2=PB2+PD2

  27. British flag theorem • PA2+PC2=PB2+PD2

  28. (a+b)2= c2 +4(½ab)

  29. Law of Cosines

  30. Law of Sines • sinA=a/2R • sinB=b/2R • sinC=c/2R Therefore, a/sinA=b/sinB=c/sinC

  31. Descartes • Cogito ergo sum

  32. Descarte’sCartesian Coordinate System

  33. The Distance Formula The distance d between any two points with the coordinates (x1,y1) and (x2,y2) is given by A (x1,y1) B (x2,y2)

  34. A Poem by Rachel’s Dad (Dwight Paine)

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