Archimedean Spiral

The Archimedean Spiral (aka arithmetic spiral), as the name suggests, was discovered by the famous Greek mathematician Archimedes in the third century B.C. Archimedes had a very well rounded education and as a result was able to take his strong math skills and apply them to physics and engineering. He has been credited with inventing many practical mechanisms that laid the foundation for some of the greatest advances of that time period. Much of these engineering innovations were the result of applying the Archimedean Spiral because it can easily be computed and works very well in real world applications. Unlike the Golden Spiral and other logarithmic spirals, the Archimedean Spiral increases at a constant speed and angular velocity with respect to a fixed point. These properties make it much more appealing and friendly for mathematicians to work with as opposed to the complex spirals found in nature. However, the Archimedean Spiral and other arithmetic spirals do in fact exist very rarely in nature. The shape of the sun’s magnetic field forms an arithmetic spiral because their solar winds increase at a uniform rate as they extend outward. In certain instances, the propulsion of fireworks traces out variations of the spiral. Ultimately, the Archimedean Spiral is a man made tool that functions well for its intended purposes. It has been used exclusively in the development of scroll compressors, gramophone records, watch springs, and screw mechanisms capable of raising water for irrigation. Without the brilliant inventions of Archimedes, many of the devices that enable us to efficiently perform tasks would not exist.

The Golden Spiral

Two of the most common ways of visualizing the Golden Ratio (Φ) are the Golden Spiral and the Golden Rectangle. Both of these geometric figures are drawn by constructing rectangles whose sides are 1 and Φ. These proportions can be formed over and over again by adding or removing squares from the original rectangle and repeating the process. Constructing an arc passing through opposite corners of the rectangles forms the Golden Spiral shown below. The Golden Spiral is a special type of logarithmic spiral that grows continuously by a factor of Φ each quarter turn. Different variations of logarithmic spirals can be found in countless examples throughout nature. The Milky Way Galaxy, hurricanes/cyclones, sea shells, and even some fingerprints. In rare examples such as the Nautilus sea shell, a growth pattern closely related to Φ yields a fairly accurate Golden Ratio. Like a perfect circle, the perfect Golden Ratio is nowhere to be found in nature. However, similar spirals can be seen all over if you look closely. The swooping flight of a hawk toward its prey traces out a similar spiral. The reason for this is because the hawk’s line of vision is sharpest when equal to the pitch of the spiral. Another similar example is the circling of some insects toward a light source. Many more cases can be made for the approximations of the ratio existing in everything from DNA to snowflakes. However, when one is looking for a pattern they will find it in everything. Therefore, many cases of the ratio in nature are recorded due to people wanting to find it and making the necessary “stretches” to argue its existence. Next post I will write about another well known spiral, the Archimedean Spiral.

The Fibonacci Sequence

Of all the mathematical patterns discovered throughout history, none have had the profound impact as the Fibonacci Sequence. Originally discovered in ancient India, the sequence was not formally known in Europe until Leonardo of Pisa (aka Fibonacci) published his ground breaking book Liber Abaci in 1202. By studying the reproduction of rabbits and analyzing the growth of the population over time, Fibonacci was able to observe a breeding pattern unrealistic in most other species. The unique characteristics of the sequence set it apart from all other recursive mathematical series. The first two numbers of the sequence are 0 and 1. All numbers following are calculated by taking the sum of the two previous numbers. Thus the first fifteen numbers of this infinite sequence are the following: {0,1,1,2,3,5,8,13,21,34,55,89,144,233,377}.
How the sequence corresponds to the Golden Ratio is seen by dividing any number in the sequence by the previous number. By doing this you obtain numbers very close to one another and that oscillate around Golden Ratio. In fact after thirteen iterations the numbers becomes fixed (≈1.618) and converge to more and more precise approximations of the Golden Ratio. Next post I will touch on some mind-blowing examples of this pattern in nature.

The Golden Ratio

Dating back to over 2,400 years ago, famous mathematicians including Pythagoras and Euclid have dedicated countless hours into the study of what is now referred to as the Golden Ratio. This perfect ratio has been a topic of interest for mathematicians, philosophers, architects, and even artist. Ancient Greeks used mathematics to develop a relationship between beauty and truth. Aristotle believed in the existence of an ideal median that divided two extremes of a single entity. This perfect balance between excess and deficiency must satisfy properties such as symmetry, proportionality, and harmony. The Golden Ratio encompasses all of these characteristics and has been the cornerstone for architects and artists due to its aesthetically pleasing visual representations. The Parthenon is believed to have been designed using approximations of the ratio. Renowned artists such as Leonardo DaVinci and Salvador Dali are believed to have incorporated the ratio into some of their most famous artwork. The Golden Ratio, represented by the Greek letter Φ, is approximately 1.61803...or [ (1+√5)/2 ]. This is an irrational constant, meaning that it can’t be plotted on a number line because its decimal places go on infinitely, never converging to a finite value. By many, the Golden Ratio is thought to be a preexistent model for the natural balance of equilibrium that is a part of any changing life form. Below are several geometric representations of the ratio, and how it is found in the human body. Next post I am going to touch on how this ratio is found in countless mathematical sets including the Fibonacci Sequence. Because some of these pictures are difficult to understand I have included the verbal proportions of the body as well.
*Note all of the following ratios are equivalent, but the perfect ratio can only be approximated based on how close your body is to having the ideal proportions.
Length of face: width of face
Distance between the lips and where the eyebrows meet: length of nose
Length of face: distance between tip of jaw and where the eyebrows meet
Length of mouth: width of nose
Width of nose: distance between nostrils
Distance between pupils: distance between eyebrows

Human Evolution

I have recently read a blog post by Susan Hawks. I found the blog to be very interesting although she does makes some rather big stretches in her argument. Based on Susan’s profile, one can assume she is a very free thinking person. She has a unique philosophy of spirituality and governments' role in society. I agree with most of her ideas and liked her connection between the metamorphosis of a caterpillar into a butterfly and our current state in human existence. I agree that in order for human evolution to continue and new institutions to emerge; older systems must die off to allow for these new organizations to take hold. She believes that a radical rebirth of society is necessary and inevitable. Susan views government as an immune system that’s role is to protect the organism (the human race). However, during times of such rapid growth the government does not recognize this process as necessary, and fights the transformation to the point of actually self-destructing. Susan is quoted as saying, “When the caterpillar is in a chrysalis stage, its physical body goes through such a rapid transformation that its own immune system recognizes the change as an outside attack. It then begins to attack its own organism because it doesn’t recognize that the process it is undergoing is natural and necessary.” If interested one can read her entire argument and other posts by going to . Also, here is a short clip from NOVA summarizing its series on fractal geometry. I would recommend anyone interested to check out the site because it offers very good videos on math and science.

Complex Numbers Imaginary or Infinite

To understand a fractal one must understand complex numbers. A complex number is a number that cannot be graphed on a number line. There are many complex numbers that exist in nature for whatever reason, and allow us to make sense of this chaotic universe. One of the most famous complex numbers, the imaginary number i, is crucial to the understanding of the Mandelbrot Set. The actual function of the Mandelbrot Set is [z -> z^2 + c]. All this function does is input a given constant c and filter the new number back through the same function infinitely many times. The Mandelbrot Set requires that z be grounded at zero to start, but many other fractals can be generated by experimenting with different starting values. The reason complex numbers are essential to fractal geometry is due to their mathematical properties that create infinite sets that are also bounded by parameters. Some of these concepts require a basic understanding of abstract math and being able to visualize the imaginary coordinate system. Regardless, these complex numbers do in fact exist and make possible many of the technological advances being seen in all fields of science. Harmonic motion requires the use of complex numbers in order to model certain frequencies that otherwise could not be explained with trigonometric functions. Consequently, the idea of infinity that has yet to be really comprehended is the central theme to appreciating the beauty of fractal geometry. The set generated by this function continues infinitely, plotting more and more precise points, and when graphed produces the Mandelbrot Set. The mind-blowing aspect of this visual image is the fact that one can zoom in infinitely and continue to see self similarity. As one zooms in on the intricate spirals they see previous patterns unfolding in a dynamically changing number set. These mysterious patterns are now being found in everything ranging from the arteries in your arm to the cyclical patterns of the economy. They are replacing many primitive systems as the blueprint for trying to understand complex dynamic systems. Fractal antennas are imbedded in tens of millions of communication devices that allow for any two people to simultaneously communicate from any two points on a globe. The evolution of computer animation uses fractals to create the movies and video games that are constantly pushing the limit to virtual reality. More importantly than how humans are manipulating fractals to their advantage, is the pre-existent nature to which they are being discovered all around us. Next post I will go into detail about how fractals are being used to model natural phenomenon.

Intro to Fractal Geometry

Fractals are possibly the most fascinating discovery in the field of geometry in our lifetime. Fractal geometry is being used to model some of the most complex dynamic systems in all of life. They are being integrated in fields such as communication, computer animation, engineering, economics, and even medicine. Unlike Euclidean geometry in which everything is reduced to nice regular shapes, fractal geometry is able to describe the intricate patterns in nature that have baffled even the greatest mathematicians throughout history. Fractal geometry was first discovered in 1970 by the French mathematician Benoit Mandelbrot. Unfortunately at that time the math community was not open to such an innovative leap in geometry, and Mandelbrot was an outcast among his colleagues. Mandelbrot looked down upon Euclidean geometry because it was only useful for studying the world that we as humans created based on a number of pre-existent constants that enabled us to make sense of a seemingly chaotic world. Mandelbrot was quoted in his book The Fractal Geometry of Nature (1983), as saying, "Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line." Once again the limitations of mathematics prevented humans from seeing a much deeper form of order that exist all around us. The math behind fractal geometry is surprisingly simple. It uses a basic algorithm that feeds a number through it and then loops the new number over and over again. This process, known as iteration, is very tedious and sometimes impossible to compute by hand. Fortunately, Mandelbrot was offered a job at IBM that gave him access to supercomputers that could do the iterations for him. This opened Pandora’s Box of possibilities in terms of what we could do with fractals. Mandelbrot plotted these millions of points and arrived at the famous Mandelbrot Set which can be visualized by applying a continuous color scheme to convey the necessary details that are fundamental to fractal geometry.



It didn’t take long for people to see that fractals were much more than a means of creating cool images. Next blog I will talk about the properties of fractals, their first implications, and how they are spreading like wild fire across all fields of science.