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Golden Ratio
The golden ratio is an irrational number (like $ \pi $) represented by the lowercase Greek letter $ \phi $.
It is also the solution to the quadratic equation $ x^2 - x - 1 = 0 $: $ \phi = \frac{1 + \sqrt{5}}{2} $ (approximately 1.618)
The reciprocal of the golden ratio ($ \frac{1}{\phi} $) is represented by the uppercase Greek letter $ \Phi $.
Local Pentagonal Symmetry:
The geometry of the golden ratio is this: in a line segment, the ratio of the long piece of the line segment to the shorter piece of the line segment must equal the ratio of both pieces summed together to one of the constituent pieces of the line segment.
$ \frac {a+b}{a} = \frac {a}{b} {\stackrel {\text{def}}{=}} \phi $
Here is an excellent proof taken (and slightly modified) from this site that proves Penrose tiling are non-periodic:
- Consider any tiling expanding an infinite plane. We can deflate this tiling as many times as we like. Consider the ratio of the total number of kites to the total number of darts after each inflation. At any stage in the inflation process let k represent the total number of kites and d represent the total number of darts. Since each kite is decomposed into two kites and one dart, and each dart is decomposed into one kite and one dart, as the value of $ \frac{k}{d} $ converges we must have
- Rearranging gives
- Applying the quadratic formula:
...which is the golden ratio. So the ratio of the total number of kites to the total number of darts in a tiling covering the infinite plane is equal to $ \phi $, the golden ratio. This is an irrational number, so the tiling must be non-periodic.
Robinson triangles are an important property of pentagons and pentagram symmetry. In order for P2 & P3 Penrose tiling to have pentagonal symmetry, the ratio of the area of the Robinson triangles must be $ \phi:1 $ or $ 1:\phi $, depending on if the larger triangle is on the top of the ratio or the bottom.
Real-World Applications:
The golden ratio isn’t just used in the field of mathematics; it is often used in history and in modern contexts, sometimes unconsciously, and can be found in front of us almost every day.
Many of our everyday designs, ranging from credit cards to television screens, to historical designs like medieval manuscripts, follow the golden ratio and form golden rectangles.
Some of the Egyptian pyramids closely follow current mathematical pyramids, the latter of which can be taken apart to show the relationship between the sides of the square base, the height, and $ \phi $.
In nature, flowers often follow the golden spiral with the arrangement of their leaves (see Introduction for picture). This is also why seeds have to planted in a certain way (in a line) so they have space to grow in a spiral[1].
Many 20th century artists and architects proportioned their works to the golden ratio, often using the golden rectangle when positioning the features of their works. This was named divine proportion in the 16th century, and it led to artists tying the golden ratio to religious works (such as Salvador Dali with his The Sacrament of the Last Supper). One such alleged example is the Mona Lisa, which shows the proportions of the lady’s eyes and face matching the Fibonacci sequence’s numbers.
Further Readings:
- http://jwilson.coe.uga.edu/EMT668/EMAT6680.2000/Obara/Emat6690/Golden_Ratio/golden.html --> This is a very good source for exploring the real-world applications of golden ratio.
- Robinson triangles are not the only shape that is related to the golden ratio. Here are some sites that explain golden rectangles, golden spirals, and golden stars (pentagram) and where they show up in our everyday lives:
