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!