interesting math
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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 - PowerPoint PPT PresentationTRANSCRIPT
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.