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Interesting Math

Interesting Math. Discrete Math Mr. Altschuler. What Interests You?. Write on a small piece of paper, a subject of endeavor that interests you. I will try to structure a lesson around each of the subjects that you submit sometime during the semester. Examples: Cooking Sports Art

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Interesting Math

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  1. Interesting Math Discrete Math Mr. Altschuler

  2. What Interests You? • Write on a small piece of paper, a subject of endeavor that interests you. • I will try to structure a lesson around each of the subjects that you submit sometime during the semester • Examples: • Cooking • Sports • Art • Music • Reading • Exercising • Travel • Building • Or anything else

  3. Lets work with Art Today • All students should turn toward their “artistic” side. • We will take a survey. On the other small piece of paper, draw a rectangle that is “pleasing” to your eye. There is no correct or incorrect answer here. • The rectangle may be a: • Square, Oblong, not Oblong, Vertical, or Horizontal • On the paper please write the words vertical or horizontal to denote the orientation of your “pleasing” rectangle • Take one minute; then we will collect our results to see the distribution of the class’s “taste”

  4. The Greeks’ Idea of “Perfection” • The Greeks felt the “Golden Rectangle” should be horizontal, because vertical appeared less stable (it seemed to want to fall over). • It’s Aspect Ratio (length : height) was their “Golden Ratio” (depicted by the Greek letter phi - j): length height

  5. A Little Math

  6. A Little More Math

  7. A property of the Greek’s Golden Rectangle The blue rectangle has a golden ratio. Removing the yellow square (h-by-h) from that shape leaves the green rectangle, which also has a golden ratio. h l h h l-h h

  8. Back to The Greeks’ Idea of “Perfection” • The Golden Rectangle: • The Parthenon was built with the same aspect ratio:

  9. Irrationality of j • Every rational number can be expressed as one integer divided by another. An irrational number cannot expressed that way. • j is the “most” irrational number because the expression used to describe it converges the slowest

  10. Convergence of e versus j

  11. Places in nature where j arises • Nautilus Shell • width of adjacent rings follows j ratio http://www.natures-word.com/sacred-geometry/phi-the-golden-proportion/phi-the-golden-proportion-in-nature

  12. Places in nature where j arises • Plants – many plants sprout leaves at a 360o/j angle off the stalk to reduce shadowing from the leaves above (a byproduct of the “most irrational number) http://www.natures-word.com/sacred-geometry/phi-the-golden-proportion/phi-the-golden-proportion-in-nature

  13. InterestingMathematical Properties of j

  14. Golden Triangles • A golden triangle is isosceles • The Equal sides are tall, and ratio of their length to that of the base is the Golden Ratio. • A Regular Pentagram is the shape of the five-pointed stars on the US Flag. • The triangle near each point is a golden triangle.

  15. InterestingMathematical Properties of j • Choose ANY two numbers • Find the sum • Add to the sum the last number to find a new sum • Repeat the last step a few times. • The ratio of the final total to the final number always approaches j (will get closer with the more steps repeated).

  16. Of Local Interest • Major Pierre L’Enfant was commissioned by President Washington to layout the District of Columbia. He used Pentagrams varying in Size by j to create many of the major avenues http://scienceforums.com/topic/1842-vesica-piscis-real-sacred-geometry/page__st__105

  17. Math is Interesting • At least to me (hopefully to you too!). • We will explore many more interesting concepts in Discrete Math.

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