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The Golden Ratio
Given a line segment AB, the point C on the line such that theratio of the length of the line AB to the length AC is the same asthe ratio of the length of line segment AC to the length CB.
A C B
This ratio is known as the golden ratio and is denoted by the greekletter φ. AB
AC = ACCB = φ
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The Golden Ratio: ABAC = AC
CB = φ
A C B
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Golden Ratio: Solutions to quadratic equations
An equation of the form:
ax2 + bx + c = 0
has solutions
x =−b ±
√b2 − 4ac
2a
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The value of the Golden Ratio
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The value of the Golden Ratio
φ =1 +√
5
2≈ 1.618
( φ is the only positive solution to the equation x2 − x − 1 = 0. )
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Construction 1: The Golden Cut
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Golden Rectangle
A Golden rectangle is a rectangle that has side lengths that are ingolden proportion.
b
a= φ
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Construction 2: Golden Rectangle given shorter side
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Construction 3: Golden Spriral
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Golden Triangles
Golden Acute Triangle Golden Obtuse Triangle
b
a
b
b
a a
b
a= φ
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Is this a Golden Acute Triangle?
72◦ 72◦
36◦
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Construction 4: Acute Golden Triangle over a given base
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Construction 5: Golden Triangles over a given basealternate construction
Suppose that a golden cut was done previously, and so you hadlengths as in the line below.
CA B
Obtuse Acute
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Construction 6: Subdividing an Obtuse Golden Triangle
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Regular Polygons
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Construction 7: Regular Pentagon over a given side
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Fibonacci Numbers: The Classic Problem (page 25)
A certain man puts a pair of rabbits in a place surrounded on allsides by a wall. How many pairs of rabbits can be produced fromthat pair in a year if it is supposed that every month each pairbegets a new pair, which from the second month becomeproductive?
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Fibonacci rabbits, the first few months
1st month
2nd month
3rd month
4th month
5th month
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Fibonacci Numbers
Let fn be the number of pairs of rabbits after n months.
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Fibonacci Numbers
The Fibonacci Numbers are the numbers in the sequence definedby
f1 = 1 f2 = 1
fn = fn−1 + fn−2
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Binet’s formula for Fibonacci numbers
fn =(1 +
√5)n − (1−
√5)n
2n√
5
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Given that f19 = 4181 and f16 = 987,what are f17 and f18?
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Ratios of Fibonacci Numbers
f2f1
=f6f5
=
f3f2
=f7f6
=
f4f3
=f8f7
=
f5f4
=f9f8
=
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Fibonacci Spiral
2
1
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Fibonacci Flowers