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 -  ): lengthheight

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 l-h h 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  Every rational number can be expressed as one integer divided by another. An irrational number cannot expressed that way.  is the “most” irrational number because the expression used to describe it converges the slowest

10. Convergence of e versus 

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

12. Places in nature where  arises Plants – many plants sprout leaves at a 360 o /  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. Interesting Mathematical Properties of 

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. Interesting Mathematical Properties of  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  (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  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.