golden ratio variety perspective view golden ratio greek letter phi context dynamic proportion golden rectangle constructed construct simple square draw line midpoint square opposite corner use line radiu draw arc height rectangle complete golden rectangle golden rectangle golden ratio length longer rectangle divided shorted simple application pythagorean derived observe you cut out perfect square height golden rectangle end rectangle remaining subrectangle golden rectangle ratio length longer shorter algebraically equality fact quadratic formula learned secondard school see equation being another interesting fact limit ratio consecutive term fibonacci sequence namely sequence term suces term sum term sequence ratio consecutive term give sequence numerically get closer closer example decimal form already quite close many person claimed golden ratio figure design number ancient architectural work great pyramid egypt parthenon greece work leonardo vinci parthenon bce annunciation 1472leonardo vinci generally been met criticism regarding their scientific rigour event golden ratio dynamic proportion special statu regard pictorial decomposition idea many artist least been exposed whether they consciously otherwise incorporate into their own artwork alleged example artist utilized dyamic proportion american painter george bellow maxfield parrish dempsey firpo 1924george bellow notion dynamic proportion root rectangle being taught illustration course very day

There are a variety of perspectives from which to view the
Golden Ratio $\varphi$ (Greek letter "phi").
In the context of dynamic proportions, $\varphi$ corresponds to the Golden Rectangle
constructed as follows
[3]:

Construct a simple square.

Draw a line from the midpoint of one side of the square to an opposite corner.

Use that line as the radius to draw an arc that defines the height of the rectangle.

Complete the golden rectangle.

The Golden Rectangle

The Golden Ratio $\varphi$ is equal to the length of the longer side of this rectangle
divided by that of the shorted side.
By a simple application of the Pythagorean theorem,
it may be derived that:
\[
\varphi = \frac{a + b}{a} = \frac{1 + \sqrt{5}}{2}
\approx 1.6180339887\ldots
\]

UGED 1533 - Mathematics in Visual Art Chapter Slide 2

Observe that if you cut out a perfect square with the same height as the
Golden Rectangle from one end of the rectangle,
the remaining subrectangle is also a Golden Rectangle, in that the ratio of length of the longer side to that of the shorter side is equal to $\varphi$.
Algebraically, this corresponds to the equality:
\[
\frac{\varphi}{1} = \frac{1 + \varphi}{\varphi}
\]
In fact, applying the quadratic formula that we learned in secondard school,
we see that $\phi$ is one of two solutions to the equation:
\[
\frac{x}{1} = \frac{1 + x}{x},
\]
the other solution being: $\frac{1 - \sqrt{5}}{2} = -\frac{1}{\varphi}$.

UGED 1533 - Mathematics in Visual Art Chapter Slide 3

Another interesting fact about $\varphi$ is that it is the "limit"
of the ratio of consecutive terms in the Fibonacci sequence, namely
the sequence where the first two terms is 1, and each sucess term is equal to the sum of the previous two terms. Hence, the sequence is:
\[ 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, \ldots \]
The ratio consecutive terms give the sequence:
\[
\frac{1}{1}, \frac{2}{1}, \frac{3}{2}, \frac{5}{3}, \frac{8}{5}, \ldots
\]
which numerically gets closer to closer to $\varphi$.
For example, $\frac{8}{5}$ is decimal form is $1.6$,
which is already quite close to $\varphi \approx 1.6180339887\ldots$.

UGED 1533 - Mathematics in Visual Art Chapter Slide 4

Many people have claimed that the Golden Ratio figures
in the designs of a number of ancient architectural works such as the Great Pyramid
in Egypt and the Parthenon in Greece, and in the works of Leonardo da Vinci
[4].

The Parthenon, 438 BCE

The Annunciation
(c. 1472) Leonardo da Vinci

UGED 1533 - Mathematics in Visual Art Chapter Slide 5

Such claims have generally been met with criticisms regarding their scientific rigour
[5].
In any event, that the Golden Ratio
or other dynamic proportions have a special status with regard to pictorial
decomposition is an idea which many artists have at least been exposed to,
whether or not they (consciously or otherwise) incorporate it into their own artwork.
Two alleged examples of artists who utilized dyamic proportions are
American painters George Bellows (1882 - 1925) and Maxfield Parrish (1870 - 1966).

Dempsey and Firpo
(1924) George Bellows

Notions in dynamic proportions such as root rectangles
are being taught in illustration courses to this very day.