Interesting Math

Discrete MathMr. 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– Music– Reading– Exercising– Travel– Building– Or anything else

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”

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

j

hl

lhl

A Little Math

jj

jj

j

j

j

1 : l/h since

1

1

1

:1/hby r demoninato &numerator gmultiplyin

hlhl

lhl

h

h

hl

lhl

A Little More Math

number irrationalan ...618.12

51

:answer positive only thefor 2

5112

11411

01

1

2

2

j

j

j

jj

jj

j

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

hh

h

Back to The Greeks’ Idea of “Perfection”

• The Golden Rectangle:

• The Parthenon was built with the same aspect ratio:

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

...111

11

11

j

Convergence of e versus jabsolute value of ratio of errors

No. of Terms Value Error Value Error j/e1 1 -1.71828 2 0.381966 0.222 2 -0.71828 1.5 -0.11803 0.163 2.5 -0.21828 1.666667 0.048633 0.224 2.666667 -0.05162 1.6 -0.01803 0.355 2.708333 -0.00995 1.625 0.006966 0.706 2.716667 -0.00162 1.615385 -0.00265 1.647 2.718056 -0.00023 1.619048 0.001014 4.488 2.718254 -2.8E-05 1.617647 -0.00039 13.899 2.718279 -3.1E-06 1.618182 0.000148 48.3310 2.718282 -3E-07 1.617978 -5.6E-05 186.4111 2.718282 -2.7E-08 1.618056 2.16E-05 789.6312 2.718282 -2.3E-09 1.618026 -8.2E-06 3644.1013 2.718282 -1.7E-10 1.618037 3.15E-06 18201.03

2.718282 <<actual 1.618034 <<actual

value of e value of j

j is the slowest converging irrational number

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

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

InterestingMathematical Properties of j

jj

jj

jj

j

1...382.01

1...618.011...618.2

...618.1

2

2

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.

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).

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

Math is Interesting

• At least to me (hopefully to you too!).• We will explore many more interesting

concepts in Discrete Math.