The Golden Ladder
Transcript of The Golden Ladder
The Golden LadderBy Weixin Wu
Advisor Professor ShablinskyMay 2nd 2015
What is the golden ratio?
Two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities.
Exact value of phi(ϕ) and its properties
ϕ = (√5 + 1) / 2 ≈ 1.618
φ = 1 / ϕ or ϕ * φ = 1
φ = (√5 - 1) / 2 ≈ 0.618
ϕ 2 = 1 + ϕ ( x2 – x – 1 = 0, the positive root is ϕ)
Some famous Euclidean geometric shapes
Apply Golden Ratio to tangent circles
d = ϕ
d = 1
d = φ
d = φ2
d = φ3
Flip all triangles into the previous triangle
d = ϕ
d = 1d = φ2
d = φ
Recursive property one
All triangles are similar.The ratio between the corresponding sides of any triangle and the next smaller triangle is ϕ.
Proof of recursive property one
S1
S2
S3S3
S4
S5
Proof of recursive property one
S1
S2
S3
S3
S4
S5
ϕ ?
Proof of recursive property one
S1 = (ϕ + 1)/2
S2 = (ϕ + φ)/2
S3 = (1 + φ)/2
S4 = (1 + φ2)/2
S5 = (φ + φ2)/2…..
Sn= (φm + φm+1)/2
Sn+1= (φm + φm+2)/2
Proof of recursive property one
Sn = (φm + φm+1)/2
Sn+1 = (φm + φm+2)/2
Sn+2 = (φm+1 + φm+2)/2
Sn+3 = (φm+1 + φm+3)/2
Proof of recursive property one
Case one:
Proof of recursive property one
Case two:
Proof of recursive property one
Sn is the length of the nth side
Recursive property two
The area ratio between any triangle and its next smaller triangle is ϕ 2.
A1
A2
S1
S2
S3
S3
S4
S5
?
Proof of recursive property two
By using Heron’s formula:
Proof of recursive property two
√ 𝑃 1√ 𝑃 2=√
(𝑆1+𝑆2+𝑆3) /2(𝑆3+𝑆4+𝑆5)/2
=√ (ϕ∗𝑆3+ϕ∗𝑆 4+ϕ∗𝑆5)(𝑆3+𝑆4+𝑆5)
=√ϕ (𝑆3+𝑆4+𝑆5)𝑆3+𝑆4+𝑆5 =√ ϕ
√ (𝑃 1−𝑆1)√ (𝑃 2−𝑆3)
=√ (𝑆2+𝑆3)/2(𝑆4+𝑆5)/2
=√ (ϕ∗𝑆 4+ϕ∗𝑆5)(𝑆 4+𝑆5)
=√ ϕ (𝑆4+𝑆5−𝑆3)𝑆4+𝑆5−𝑆3 =√ ϕ
Proof of recursive property two
Proof of recursive property two
An is the area of the nth triangle
An An +1 =ϕ 2
Recursive property three
The ratio between the area of any triangle and the total area of all embedded triangles is .
Proof of recursive property three
A1
A2
A3A4
An
∑i=n+1
∞
A i=ϕ ?
Proof of recursive property three
𝐴𝑛
𝐴𝑛+1+𝐴𝑛+2+…+ 𝐴𝑚𝑎𝑠𝑚𝑔𝑜𝑒𝑠𝑡𝑜𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑦
An+An +1+A n+2+…+Am𝑎𝑠𝑚𝑔𝑜𝑒𝑠 𝑡𝑜𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑦
𝐴𝑛An +1=ϕ 2
Proof of recursive property three
An+An /𝜑 2+A n /𝜑4+…+An /𝜑2𝑚𝑎𝑠𝑚𝑔𝑜𝑒𝑠 𝑡𝑜𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑦
𝐴𝑛∗1
1− 1𝜑 2
By using geometric series, the above expression becomes:
Proof of recursive property three
𝐴𝑛
𝐴𝑛+1+𝐴𝑛+2+…+ 𝐴𝑚=
𝐴𝑛
𝐴𝑛∗1
1− 1𝜑2
−𝐴𝑛
𝐴𝑛
𝐴𝑛∗1
1− 1𝜑2
−𝐴𝑛
=𝜑2−1=𝜑
Proof of recursive property three
An is the area of the nth triangle
An
∑i=n+1
∞
A i=ϕ
More proofs in the future
The Golden Ladder in 3D
Thank you!