Many people are aware of my abhorrent disdain for circles. I am not a fan. Yet, we see circles showing up over and over again, in architecture, in art, and in product design. I am a person who lives in a world of squares and rectangles. Clearly, they are more efficient and this form makes my heart sing. Circles create angst and frustration. They mock common sense – at least for me.
Recently, the circle design has been seeing the ascension of an even nastier variant, the surge of designs that are using the spiral. The rise of the Fibonacci. OMG.
Squares and Rectangles
These shapes represent stability. In fact, the rectangle is the most used area shape in design. The reason for its popularity is because it is a trusted familiar shape that represents honesty, solidity, and stability. As squares and rectangles have straight lines and right angles they have a very mathematical, balanced feel. These shapes scream rational, practical, and conformity. As far as shapes go, these are neither flashy, nor attention seekers – some may even venture to say that they are boring, however clever designers may twist or turn them to add interest to a design.
So, it is easy to appreciate squares and rectangles and perhaps it is obvious to all why I find them so comforting.
Circles
Some argue that these rounded shapes tend to send a positive emotional message of harmony and protection. The circle is often used in a design to represent unity, commitment, love, or community. Curves in general when used in shapes tend to be viewed as feminine in nature while straight lined shapes are more masculine.
Circles have no beginning or end, they represent life and the lifecycle. The circle along with the oval is readily found in nature with the sun, moon and earth, not to mention fruit and flowers.
Circles have a free sense of movement – wheels, balls, merry go rounds. Their movement may also represent power and energy.
Due to their curved lines, ovals and circle are graceful and complete. They give a sense of integrity and perfection.
They are not used as much in design for spacial reasons, but this means that when they are used, that they attract more attention than their right angled counter parts.
Therefore, I am sure that you all see why I am so reticent to like circles in designs. They are the exact opposite to squares and rectangles.
Spirals
Now, another hot trend is gaining momentum in design and engineering. It is a design feature that is even worse than circles, it is the horrid spiral. Spirals are shapes that are most often found in nature, from shells and snails to stars in the galaxy, water draining from the tub, or dirt being whisked up by the wind. Spirals represent the notion of growth and evolution, the circles of life, seasons or time. Spirals represent transformation, fertility, life, and death. They are creative and due to their curved nature also have a feminine feel. Spirals can move in clockwise or anticlockwise directions and can take you on a journey. They are free flowing, boundless and open. The spiral is a shape that can go on for eternity.
How awful. I am sure that you agree that spirals cannot hold a candle to squares and rectangles. To me, spirals are denigrating and harmful to designs. I cannot see the attraction, can you?
Fibonacci Reborn
All of sudden, we are seeing a keen interest in Fibonacci and his golden maths. These numbers are popping up everywhere. But, what is Fibonacci and where did it come from anyway?
In mathematics, the Fibonacci numbers, commonly denoted F_{n} form a sequence, called the Fibonacci sequence, such that each number is the sum of the two preceding ones, starting from 0 and 1. That is:
 F 0 = 0 , F 1 = 1 ,
and
 F n = F n − 1 + F n − 2 ,
 for n > 1.
One has F_{2} = 1. In some books, and particularly in old ones, F_{0}, the “0” is omitted, and the Fibonacci sequence starts with F_{1} = F_{2} = 1. The beginning of the sequence is thus:
 ( 0 , ) 1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 , 34 , 55 , 89 , 144 , …
Fibonacci numbers are strongly related to the golden ratio: Binet’s formula expresses the nth Fibonacci number in terms of n and the golden ratio, and imply that the ratio of two consecutive Fibonacci numbers approximates the golden ratio asymptotically as n increases.
Fibonacci numbers are named after Italian mathematician Leonardo of Pisa, later known as Fibonacci. They appear to have first arisen as early as 200 BC in work by Pingala on enumerating possible patterns of poetry formed from syllables of two lengths. In his 1202 book Liber Abaci, Fibonacci introduced the sequence to Western European mathematics, although the sequence had been described earlier in Indian mathematics.^{}
Did you know that Fibonacci also introduced numbers to the western world, they replaced Roman numerals which were the rage at the time. In the Liber Abaci (1202), Fibonacci introduced the socalled modus Indorum (method of the Indians), today known as the Hindu–Arabic numeral system. The book advocated numeration with the digits 0–9 and place value. The book showed the practical use and value of the new HinduArabic numeral system by applying the numerals to commercial bookkeeping, converting weights and measures, calculation of interest, moneychanging, and other applications. The book was wellreceived throughout educated Europe and had a profound impact on European thought. That action alone makes him a hero in my book.
Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is an entire journal dedicated to their study, the Fibonacci Quarterly. Applications of Fibonacci numbers include computer algorithms such as the Fibonacci search technique and the Fibonacci heap data structure, and graphs called Fibonacci cubes used for interconnecting parallel and distributed systems. They also appear in biological settings, such as branching in trees, the arrangement of leaves on a stem, the fruit sprouts of a pineapple, the flowering of an artichoke, an uncurling fern and the arrangement of a pine cone’s bracts.
Fibonacci numbers are also closely related to Lucas numbers L n in that they form a complementary pair of Lucas sequences U n ( 1 , − 1 ) = F n and V n ( 1 , − 1 ) = L n. Lucas numbers are also intimately connected with the golden ratio.
Art and Nature
As was stated, the Fibonacci Sequence shows up in nature all the time. The same is true in art and photography.
Some argue that the absolute human definition of beauty is clearly defined by the Fibonacci Sequence.
Mona Lisa (also known as La Gioconda or La Joconde) is a 16th century portrait painted in oil on a poplar panel by Leonardo da Vinci during the Italian Renaissance. The work is owned by the Government of France and is on the wall in the Louvre in Paris, France with the title Portrait of Lisa Gherardini, wife of Francesco del Giocondo. It is perhaps the most famous and iconic painting in the world.
Prevalent in the major works of Leonardo Da Vinci and underlying many of his design compositions, is the phi relationship (also known as the Golden Ratio or the Golden Mean), a ratio of approximately 1:1.618, found in nature and creation, and inherent in the Fibonacci sequence. The Golden Rectangle, the Golden Triangle, and the Golden Pyramid, all based on the Golden Ratio are all appear prominent in the work of Leonardo Da Vinci. He referred to the Golden Ratio as the “divine proportion”.
Science and Space
But this is what’s really cool: Certain solar system orbital periods and relative planetary distances are also related to phi. Some scientists say that the shape of the universe itself is a dodecahedron based on Phi.
The dimensions of the Earth and Moon are in Phi relationship, forming a Triangle based on 1.618.
The illustration shows the relative sizes of the Earth and the Moon to scale.
 Draw a radius of the Earth (1).
 Draw a line from the center point of the Earth to the center point of the Moon (square root of Phi).
 Draw a line to connect the two lines to form a Golden Triangle (Phi).
Using dimensions from geometry’s classic Pythagorean Theorem, this is expressed mathematically as follows:
Dimension (km) 
Proportion (Earth=1) 
Mathematical Expression 

Radius of Earth  6,378.10  1.000  A 
Radius of Moon  1,735.97  0.272  
Earth + Moon  8,114.07  1.272  B 
Hypotenuse  10,320.77  1.618 (Φ)  C 
Hypotenuse / (Earth Radius + Moon Radius) 
1.618 (Φ)  A²+B²=C² 
Another way of looking at the relationship is to take 10320.77² / 8114.07², which is 106,518,293.39 / 65,838,131.96, which is 1.618.
Great Pyramid of Egypt
This triangle is known as a Kepler triangle. This geometric construction is the same as that which appears to have been used in the construction of the Great Pyramid of Egypt.
The phi ratio is found in the architecture of the Great Pyramid in the triangle formed by the height, halfbase, and the apothem, or diagonal. In other words, the basic cross – section of the structure demonstrates the Golden Section.
If the half – base is given a value of 1, this gives the value of phi for the apothem, and the square root of phi for the height. The Golden Section shows up again and again in Giza and in much more baffling and tedious ways.
Conclusions
While I personally find great comfort in my squares and triangles, I cannot ignore the circles and spirals exist and others embrace them. Nor, can I pretend that the Fibonacci Sequence is not evident in all aspects of life. It seems to be everywhere.
In the 1800s, after French mathematician Edouard Lucas officially named Fibonacci’s Sequence as such, scientists started noticing occurrences of it in the actual natural world: in the spirals made by the growth of sunflower seeds, or leaves around a stem. Even the genealogy of male bees fits the pattern.
Which immediately tempted scientists to make grandiose assumptions about the greater meaning of the sequence. The ancient GrecoRoman notion of the golden ratio connected to Fibonacci’s work; scientists and artists alike began to see perfect, universal math in everything from snail shells to tsunamis — even when it was a stretch.
Art history is lousy with traces of Fibonacci’s influence — some founded, some not. His fan following speaks to our desire to find answers in nature, and to involve ourselves with those.
Do we like Fibonacci sequences because they occur in nature, or do we like the idea that Fibonacci sequences occur in nature because we like Fibonacci sequences?
The truth is likely somewhere within the process, the math. Building on itself, onward and outward.
Now, whatever you do – do not get me started on triangles! They are actually worse than circles. No spacial efficacy whatsoever. Not even suitable for a good burial. So inefficient. Where do you even stand up? So bad, just the worst. Hey, wait a minute, if you are buried in one, maybe standing up is not an option? Hmmm.
References:
Dee, E. (2018). Mona Lisa. Pinterest. Retrieved on December 29, 2018 from, https://www.pinterest.ca/pin/295196950547774267/?lp=true
Meisner, G. (2012). Phi and the Solar System. PhiPoint Solution, LLC. Retrieved on December 29, 2018 from, https://www.goldennumber.net/solarsystem/
Stanford, M, R. (2013). The Fibonacci Sequence: Life Imitates Art Imitates Math. Walker Art Center. Retrieved on December 29, 2018 from, https://walkerart.org/magazine/thefibonaccisequencelifeimitatesartimitatesmath
Zepeda, R. (2014). Egypt Pyramids – Secret Information, Knowledge, and Wisdom. HubPages. Retrieved on December 29, 2018 from, https://hubpages.com/education/EgyptPyramidSecretInformation
WRD. The Meaning of Shapes in Design. The Creative Fringe. Retrieved on December 29, 2018 from, http://www.whiteriverdesign.com/meaningshapesdesign/
About the Author:
Michael Martin has more than 35 years of experience in systems design for broadband networks, optical fibre, wireless and digital communications technologies.
He is a Senior Executive with IBM Canada’s Office of the CTO, Global Services. Over the past 14 years with IBM, he has worked in the GBS Global Center of Competency for Energy and Utilities and the GTS Global Center of Excellence for Energy and Utilities. He was previously a founding partner and President of MICAN Communications and before that was President of Comlink Systems Limited and Ensat Broadcast Services, Inc., both divisions of Cygnal Technologies Corporation (CYN: TSX).
Martin currently serves on the Board of Directors for TeraGo Inc (TGO: TSX) and previously served on the Board of Directors for Avante Logixx Inc. (XX: TSX.V).
He serves as a Member, SCC ISOIEC JTC 1/SC41 – Internet of Things and related technologies, ISO – International Organization for Standardization, and as a member of the NIST SP 500325 Fog Computing Conceptual Model, National Institute of Standards and Technology.
He served on the Board of Governors of the University of Ontario Institute of Technology (UOIT) and on the Board of Advisers of five different Colleges in Ontario. For 16 years he served on the Board of the Society of Motion Picture and Television Engineers (SMPTE), Toronto Section.
He holds three master’s degrees, in business (MBA), communication (MA), and education (MEd). As well, he has diplomas and certifications in business, computer programming, internetworking, project management, media, photography, and communication technology.
The aesthetics of Geometry!
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