The Golden Ratio in Sonnets

Since the Golden Mean is 1.618 (which can be represented by the ratio 8:5), it is clear that on the surface there is a connection between sonnets and the Golden Ratio.  But this is not always the case.  Firstly, sonnets are usually comprised of 14 lines; if the volta or break occurs at line 8, one line in the poem goes unaccounted for.  Because of this formal convention and the mathematics behind it, the true Golden Ratio in sonnets would be 8.5:5.5 to account for the remaining line.  This ratio, of course, is not quite applicable to Shakespearean sonnets which usually have their volta or break after line 12, immediately preceding the poem's couplet ending.

Whether you are looking at Petrarchan or Shakespearean sonnets, or even more contemporary ones, it is important to remember the cultural and historical origins of the form.  For classical culture and thought, mathematics embodied a divine spirit that could be honored through numerical iterations.  If one interprets Romanticism as a revolt against scientific rationalism and bears in mind the context of this blog post, he or she can clearly view sonnets as early methods for promoting creationism and rejecting scientific study.

The link to the original blog post can be found here

Golden Mean: Post 3

Golden Mean, Golden Rectangle, Golden Spiral:

Golden Rectangle

Photo by Reilly B.

Original Post by Sophia Darby: "Similar to the Parthenon in Greece, the architecture of the Wheaton Library is an expression of the golden ratio. The perimeter of the rectangle is measured from the horizontal line at the top of the stairs and vertically up past the length of a column to the top of the building. A ratio of 1 to 1.618 represent the height and width of the rectangle. In addition, Fibonacci numbers and the golden mean are often found together. For example, there are 8 columns, and the stairs are split into 3 sections (both numbers found in the Fibonacci sequence)."

While it is difficult to determine the measurements because of the sheer size of this building, the measurements certainly add up when scaled. There are actually 6 columns, not 8 (the framing device of the two extremities of the entrance mirrors their appearance, but they do not stand alone like the others and are not cylindrical). The larger rectangle Sophia refers to is separated into five more digestible rectangles, which does, in fact, align with the Golden Mean because 5 is a Fibonacci number. The width of the larger rectangle is equivalent to the width of three of the smaller rectangles mentioned earlier, and 3 is also a Fibonacci number. 5/3 = 1.666; this is certainly close enough to phi to declare this rectangle a Golden Rectangle.

Original post from Math 125 Blog can be found here

Golden Mean: Post 4

Golden Mean, Golden Rectangle, Golden Spiral

Golden Rectangle & Golden Spiral

Photo by Sophia Darby

"The musical instrument in this picture, like many other instruments, contains the golden ratio in a variety of places within its structure. In the violin, one example of the golden ratio is the proportion of the top of the violin to the bottom of the neck and the measurement from the neck out to the side of the upper bout. This also creates a golden rectangle if you extend the lines into space." -Kelsey Goodwin

Cellos present the golden ratio between their neck and body, thus demonstrating divine proportions, which is a exalted term for the appearance of the number 1.618033, represented by the Greek letter phi (ϕ).

Violins, violas, and cellos were all eventually designed using the golden ratio in order to improve the acoustics of the instruments.

I've overlaid the successive golden rectangles and their corresponding segments of the golden spiral onto Sophia's image in order to show this (skewed to a small extent due to the slight angle of the image).

This ratio is also evident in the lengths of the cello's different segments - The length of the neck plus the length of its body divided by the length of its body is equal to all of the following: 

the length of the body / the length of the neck

the length from the top of the body to the second 'tip' on its side / the length from the bottom the peg box to the bottom of the neck

the length from the top of the body to the second 'tip' on its side / the length from the top of the body to the first 'tip' on its side

the length from the top of the body to the first 'tip' on its side / the distance between the two 'tips' on its side

To view the original blog post click here

Golden Mean Post 1

Golden Mean, Golden Rectangle, Golden Spiral

Golden Spiral

photo taken by Emerald B. edited by Sophia D.

"The golden ratio, which was used by the Greeks, is a “formula” for beauty. It states that the most beautiful object to the human eye is a ratio of 1.618. For example to get this ratio you could have a rectangle with a length of 34 and a width of 21. When you divide these terms you get 34/21, which equals 1.618. These terms are not random however, 21,34 are consecutive Fibonacci numbers. Any two consecutive fibonacci numbers when divided will equal the golden ratio, demonstrating the ubiquitous nature of both the golden mean and the fibonacci sequence. 

A golden spiral, like the one you see above, is not a golden rectangle. Instead, the golden spiral is created by drawing a spiral that fits perfectly inside a golden rectangle. Therefore, it uses a rectangle and the 1.618 ratio to create the most pleasing of spirals to the human eye. For example, draw with your mind's eye a rectangle around the largest centrally located spiral of the 7 spirals on the railing. Begin with a bottom right corner where the spiral meets the "bud" of the flower shape, draw up to the top of the spiral, then left to the edge, down, and right again. The rectangle that you create will have a length and width that when divided will equal the 1.618.

Another method to determine if a spiral is golden or not is to put it inside a golden rectangle and begin to eliminate squares. If you are continuously able to cut off squares and shrink the golden rectangle with the spiral remaining inside, then it adheres to the principles of the golden mean. This method is made possible by the underlying principle that when you chop a square off the end of a golden rectangle, you are left with another golden rectangle. If you continue to do this, your rectangle will get smaller and smaller but remain golden. In other words the golden ratio can be perpetuated in a fractal pattern." 

description by Kyle McNicoll edited by Sophia D.

To view the original blog post click here

Golden Mean: Post 2

Golden Mean, Golden Rectangle, Golden Spiral: 

Golden Rectangle

Original Photograph by Kaitlin M.

The following description was written by Angela M: "Our IDs as a whole resemble a golden rectangle. You can see that the area with the red background creates another golden rectangle and the rest is cut off into a square."

The measurements are as follows: 

ID card: L - 8.5 cm, W - 5.4 cm

Red rectangle: L - 5.4 cm, 3 cm

It can be argued that the ID card itself is a golden ratio. 8.5 / 5.4 = 1.574, which is close enough to phi (1.618) to convince one that a golden ratio was the goal. The red rectangle, however, does not have this same justification. Not only is it way too large for it to be considered a golden rectangle itself, but the part at which it was cut off is also not close enough to phi to justify calling it a golden mean. The Wheaton portion of the ID has a length of 5.5 cm, while the red rectangle has a width of 3. 5.5 / 3 = 1.83, which is way too large for phi. 

It was a rather clever choice, and we were initially convinced that it was divided according to the principles of the golden mean. While it's sad that it doesn't quite add up, it definitely got us to think a lot more about these everyday objects we tend to take for granted.

Original post from the Math 125 blog can be found here.