The Golden Ratio: Beginning the Geometry of

Here is the previous post concerning the Golden Ratio. For a collection of all the posts concerning the Golden Ratio, click #GoldenRatio below.

Now that we know what the golden ratio is, and why people like it, let's look at it in a geometrical sense, as the greeks would have. One definition of the golden ratio is the one that started this series, based on the golden rectangle. Another definition is as follows:

Divide a line segment into two parts. The Golden Ratio is the ratio of the larger part to the whole, when that ratio is also equal to the ratio of the smaller part to the larger part.

Basically, imagine the line segment pictured above. When we set the two proportions equal to each other we get

a+ba=ab=ϕ\frac{a+b}{a} = \frac{a}{b} = \phi

Solving this like we did in the first post will indeed yield 1+52\frac{1+\sqrt{5}}{2}.

Since this post is kind of short too, I'll add in a teaser. If someone can effectively answer my question before this evening (in the next ten hours), I'll put up the tomorrow's post tonight.

Here's the question: What are the lengths of the diagonals of a regular unit pentagon? Prove it.

See the full proof in here

Note by Bob Krueger
6 years, 1 month ago

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1 vote

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Click here for the image

ABCDE is a regular pentagon with side length 1

AD,AC & EB is the diagonals of the pentagon

AD intersect EB at point F,AC intersect EB at point G

AF=EF=y, so AEF\bigtriangleup AEF is a isosceles triangle

DEF\bigtriangleup DEF is a isoceles triangle, so ED=FD=1 ED=FD=1

So,the diagonals=1+y = 1+y

FAE=AEF=EDF=36\angle FAE=\angle AEF=\angle EDF=36^\circ

AED=AFE=108\angle AED=\angle AFE=108^\circ

AFEADE \bigtriangleup AFE \sim \bigtriangleup ADE So that, 1+y1=1y \frac{1+y}{1}=\frac{1}{y}

a+ba=ab=ϕ \frac{a+b}{a}=\frac{a}{b}=\phi

By substituting a=1,b=ya=1, b=y

We get, 1+y1=1y=ϕ \frac{1+y}{1}=\frac{1}{y}=\phi

The length of the diagonals is ϕ1.618 \phi \approx 1.618

Lim Zi Heng - 6 years, 1 month ago

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Is there another way?

Bob Krueger - 6 years, 1 month ago

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An alternate way is to apply cosine rule in one of the triangles.

If a is the length of the diagonal, use cosine rule to get a2=2.618 a^2 = 2.618approximately equal to 1+ϕ1 + \phi and from here recall that 1+ϕ=ϕ21 + \phi = \phi^{2}it is easy to get a=ϕa = \phi .

Harman Kahlon - 6 years ago

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@Harman Kahlon Right. This is the way I first discovered it.

Bob Krueger - 6 years ago

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this is the way I think

Lim Zi Heng - 6 years, 1 month ago

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I will attach an image after I know how to attach an image

Lim Zi Heng - 6 years, 1 month ago

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I attach the image already

Lim Zi Heng - 6 years ago

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it is very interesting to know that the equivalent resistance of thiscircuit where all the resistances are 1 ohm is GOLDEN RATIO. I think this is because of the existence of the ratio in the radical form

ayush chowdhury - 6 years ago

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Very interesting indeed.

Bob Krueger - 6 years ago

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