with the grain. vol 2

The Perfect Measuring System

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 I've had this idea floating around in my head for some time now but before I get into the deep dark secrets of measuring systems, I'd like to tell you a story. It all started earlier this summer while I was helping to renovate an old, outdoor cedar deck. During the demolition phase I kept noticing these holes in the railing and on some of the deck boards. The holes were random, but so perfectly round and even that I assumed someone had drilled them. It wasn't until I heard a buzzing sound coming from inside one of the boards that I caught the culprit making the holes: the humble carpenter bee.

 I was stunned by the consistency of these holes (it really doesn't take much to impress me). Each one was made by a different bee, yet they looked machined and exactly the same. Beeing the ever curious thinker that I am, and getting bored with all of this demolition (sorry I'm just abuzz with puns right now!), I gathered a few boards that had holes in them and measured each one. 3/8" exactly. Every. Single. One. Perfectly round, and within 1/64" of eachother - better tolerances than most machines. It was then that the idea hit me. Is it a coincidence that our system of linear measurement (the Imperial System) perfectly describes this act of nature? Are all measurement systems so well in tune with the world around us? Is there a perfect system of measurement?

The OG woodworker....the mighty carpenter bee.

The OG woodworker....the mighty carpenter bee.

 

The need to explain the world around us

Just to be clear, during this article when I reference a system of measurement I'm talking about linear measurement only (inches, feet, centimeters, kilometers, etc.). As is common with humans, we have a need to understand things. We analyze, we categorize, we systematize...pretty much anything ending with "ize" we do. It's this innate desire to uncover the secrets of the world around us that drives us to develop consistent patterns. In this case, the patterns became systems of measurement.  

 We've been doing this since ancient times and more often than not, the basis for our measurement systems has been something that we're all familiar with: our body. The Bible mentions several measurements used by the ancient Hebrews and many of those (the fingerbreadth, handbreadth, span, and cubit) were based on the proportions of the human body. The Greek Daktylos ("finger") was about 3/4", while the Pous ("foot") was surprisingly close to our own 12" ft. Speaking of feet (not usually a great conversation starter), even as "recent" as the middle ages we were still using the body as the basis for many of our measurements. At one point a yard was determined by the size of a man's waist...yikes.

Many animals like horses and donkeys are still measured using a "hand" roughly equal to 4 inches.

Many animals like horses and donkeys are still measured using a "hand" roughly equal to 4 inches.

 As time went on, however, our societies grew more complex. Commerce, travel, and science progressed and we required more exacting and universally accepted standards of measurement. It was during this period that we started looking to nature for those all too recognizable patterns. So we went with what we knew best...crops used to make beer. "Fourteenth-century statutes recorded a yard...of 3 feet, each foot containing 12 inches, each inch equaling the length of three barleycorns", states the Encyclopedia Britannica.  Proposals like these continued for centuries, but time after time they were found lacking. The need arose for a system that could be easily replicated anywhere in the world, used proficiently in business and the sciences, and could be based on universal standards that everyone could agree on. It was this driving need for standardization on a global scale that brought about the Metric System. 

 

The French did what?!

 They gave us the croissant, the Eiffel Tower, and in 1799 the Metric System.  The defining characteristics of this system are the use of the now familiar base 10 or decimal system (.001, .01, .1, 1, 10, 100, 1000, etc.), prefixes that denote those multiples (deka, kilo, centi, milli), and most importantly a basis of measurement that was unchanging and universally accepted: the meter. Defined as " one ten millionth of the distance between the North Pole and the Equator through Paris", the meter could be measured from anywhere in the world and found to be exactly the same . They had done it. The French had found a system of measurement that would bring us out of the dark ages and into the future! They had found the perfect measurement system...or had they?

 

What's the deal here?

 Before you send the confetti flying and let loose the tears of joy, let's consider some of the drawbacks of the Metric System. While revolutionizing the sciences and engineering through its clear, concise, and easily repeatable decimal system, its major flaw lies in the very base it's rooted in - 10. More specifically, the limits it places on performing fractional arithmetic, and visually representing ratios. You're scratching your head aren't you? It will all make sense, I promise! For example, imagine a circle. Now divide it into two equal sections. Each piece represents 1/2, or as a decimal, 0.5. Now take that same circle and divide it into 3 equal pieces. In this case each piece represents 1/3, but in decimal form this would be 0.333333333333....(the three's go on infinitely). What happened here? Well, because of the base 10 only numbers that end in multiples of 2 or 5 (the only divisors of 10) will have a finite representation in decimal form. Let's repeat the previous example but this time substituting the circle for Imperial measurements (linearly speaking, it's base 12 not base 10) and see what the results are. Take a yard and divide it in half. You get 18 inches. Take the metric equivalent, the meter, and do the same thing. You get 50 cm. Again, this works because the divisor, 2, is a multiple of both 10 and 12 so you get a rational number. Now try dividing by three. The yard becomes 12 inches. The meter however, becomes 33.3333333....cm. Due to the base 12, which is divisible be 2, 3, 4, and 6 the ability to represent fractional ratios easily and precisely is greatly increased. This is beneficial because being able to rationally visualize whole number ratios has been the basis of our design for millennia! Check out my previous blog on the The Golden Ratio for a little more on that topic.

 However, where the Imperial System seems to come out a winner in this regard, it's found wanting in its complete lack of consistency. In Metric you have 10 mm per cm. There are 100 cm per meter, and 1000 m per kilometer. Pretty basic, and incredibly intuitive. The Imperial System divides an inch into 1/64, 1/32, 1/16, 1/8, 1/4, and 1/2. Definitely not intuitive. Then, there are 12 inches per foot, 3 feet per yard, and...1,760 yards in a mile (I had to Google that one).  So what's the point of all of this? Is there a perfect system of measurement or not?? Well, to be honest...no. Remember the story of the perfect hole made by the carpenter bee? I was stunned that it matched the Imperial measurement of 3/8" so precisely. Truth be told, it also matches the Metric measurement of 10 mm just as closely. Each system here has its merits as well as its drawbacks. What's important is what you're measuring, how it can most clearly be represented, and what you're most comfortable using. For me, growing up in the United States, I tend to gravitate towards the Imperial System for linear measurement. Although, lately I find myself often abandoning all traditional measurements and using simple ratios combined with visual instinct to help aesthetically balance my work because let's face it, the only "perfect" measurement is the one that looks right to your eyes.

-Steve

Just a few more goodies for you math nerds!

 

 

with the grain. vol 1

The Golden Ratio

 Somewhere between watching a late night art documentary on Netflix and falling asleep in math class, many of us have probably heard of "The Golden Ratio" (not to be confused with "The Golden Rule"...totally different). Usually, not to far behind is the "Fibonacci Sequence" (FEE-BOW-NACHEE) which loves to pop up in math-related movies and plenty of sci-fi/mystery novels. Aside from making you sound educated and artistic in casual conversation, what's the big deal with these two? How are they related to each other, and how can they be used to improve or refine your designs? Let's start with the basics: What is the Golden Ratio?

 From zero to infinity

 To understand where the Golden Ratio comes from, we must first understand the Fibonacci Sequence. At its core, the sequence is simply "a set of numbers that starts with a one or a zero, followed by a one, and proceeds based on the rule that each number (called a Fibonacci number) is equal to the sum of the preceding two numbers". Well, that sounded fancy. Just bear with me here, it goes like this: 

0 + 1 = 1

1 + 1 = 2

1 + 2 = 3

2 + 3 = 5......etc.

 This pattern goes on infinitely, and if you omit the sums and just make note of the resulting numbers you end up with this list: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144...etc, thus creating the "Fibonacci Sequence"! Now this is where it starts to get interesting ("starts to??!"...I know, I know, I've had you on the edge of your seat this whole time!). If we take this sequence and begin to divide each number by it's preceding number we start to notice a pattern:

2 ÷ 1 = .5                                    21 ÷ 13 = 1.615

3 ÷ 2 = 1.5                                 34 ÷ 21 = 1.619

5 ÷ 3 = 1.67                               55 ÷ 34 = 1.618

8 ÷ 5 = 1.6                                89 ÷ 55 = 1.618

13 ÷ 8 = 1.625                           144 ÷ 89 = 1.618

 As the numbers get larger, the answers all start to approximate the same number: 1.618. This is true no matter how high up the sequence you go as along as the two numbers are consecutive. It is this very number, the proportion that describes any two consecutive numbers in the Fibonacci Sequence, that we commonly refer to as "The Golden Ratio".

The Divine Proportion

 Taking this ratio one step further, you can derive a rectangle where side x is equal to 1 and side x + y is equal to 1.618, known as a "Golden Rectangle":

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You'll notice that in the above picture, both the larger rectangle (the pink square + the green rectangle), and the green rectangle alone can be considered Golden Rectangles as they share the same 1:1.618 ratio. If we continue to subdivide the smaller rectangle into further Golden Rectangles we get something that looks like this:

 Now if we create a spiral that crosses through each of these rectangles we end up with...you guessed it, the "Golden Spiral"! 

As the spiral grows, it gets wider every 1/4 turn by a factor of 1.618 commonly denoted by the Greek letter Phi (φ).

As the spiral grows, it gets wider every 1/4 turn by a factor of 1.618 commonly denoted by the Greek letter Phi (φ).

 So why do these shapes matter? Well, it wasn't long after these proportions were discovered that people started to see them occurring naturally everywhere! We can witness this ourselves by taking a close look at the growth pattern of many plants and ferns, the incredibly efficient arrangement of sunflower seeds, the incremental growth of the nautilus shell, the proportions of the human body, the swirling vortex of a hurricane, and even the spiral shapes of galaxies. In fact, we subconsciously attribute greater beauty to those whose faces more closely exhibit these proportions. This ratio seemed to be so deeply ingrained in the natural world that it quickly became known as the "Divine Proportion".

The precision with which these examples of naturally occurring Golden Ratios stick to the same proportion is astounding. It's no wonder many people attributed magical powers to this number.

The precision with which these examples of naturally occurring Golden Ratios stick to the same proportion is astounding. It's no wonder many people attributed magical powers to this number.

...You may even find the Golden Ratio in some unexpected places!

...You may even find the Golden Ratio in some unexpected places!

 It wasn't long before people realized that not only are we surrounded by the Golden Ratio, but that it was aesthetically pleasing, well balanced, and incredibly strong. Throughout history humans have used this seemingly magical number to produce some of the most stunning and memorable works of architecture, art, and even music.

The Greeks were well known for incorporating Phi into much of their architecture. Not only is it visually balanced, but clearly it has withstood the test of time.

The Greeks were well known for incorporating Phi into much of their architecture. Not only is it visually balanced, but clearly it has withstood the test of time.

Many famous artists were well aware of the spacial laws of nature and, as a result, were able to create more realistic paintings.

Many famous artists were well aware of the spacial laws of nature and, as a result, were able to create more realistic paintings.

Even in the world of music, using the numbers from the Fibonacci Sequence will yield favorable sounding proportions (the basis for musical scales and chords) that have been in use since monastic choral chanting all the way to today's pop music.

Even in the world of music, using the numbers from the Fibonacci Sequence will yield favorable sounding proportions (the basis for musical scales and chords) that have been in use since monastic choral chanting all the way to today's pop music.

The use of Phi today

 It's no wonder that the Golden Ratio continues to be used prolifically throughout modern society. Much of our technology, architecture, product design/ergonomics, and even the very web page you're reading this on have been heavily influenced by Phi. 

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If applied well, you won't notice the Golden Ratio because it simply "looks right" to our eyes. However, when misapplied or executed poorly, our brain will recognize it!

If applied well, you won't notice the Golden Ratio because it simply "looks right" to our eyes. However, when misapplied or executed poorly, our brain will recognize it!

 Clearly, there is much to be gained by incorporating these proportions into your own designs, but how do you go about it?

Finding a balance

 When most people first learn about the Golden Ratio they have a tendency to use it in every aspect of a design. Every dimension, every line, every intersection has to be a reflection of Phi. Unfortunately, this inevitably leads to forced and often cluttered looking designs. While it is a useful tool, the Golden Ratio is not a panacea for bad design. Like everything else in life, there are often a multitude of variables that force us to establish certain limits on our work. Our use of φ can be best applied to filling in the aesthetic gaps left by these limiting factors. 

 In terms of furniture design, we often start with one or more fixed dimensions. For example, in designing a hall table the height is commonly based on average human proportion so it will most likely be 30-32" tall. This is a fixed dimension that won't change. However, by applying the Golden Ratio I can come up with a proportionally balanced length. If my height is 30", the length would be derived as 30" x 1.618 = 48.54" which I would round to 48". Similarly, Phi can be used to give a more natural feel to graduated proportions such as drawers. Many people will graduate them using a fixed number, but I find that φ provides a balanced sense of motion to the piece.

While the drawers and doors are governed by φ, you'll notice that the overall height, width, and possibly depth may be limited by other factors such as available space or personal preference

While the drawers and doors are governed by φ, you'll notice that the overall height, width, and possibly depth may be limited by other factors such as available space or personal preference

 These are just a few examples of incorporating the Golden Ratio into your designs. No matter the application- furniture, graphic design, art, photography- the key here is not to use Phi as a crutch. It's best to consider the limiting variables when planning a design so as to let it evolve naturally until you strike a good balance between form and function. Once you feel you have a solid foundation you can then go back and see where Phi can tweak some of the aesthetic proportions to help transform your design into something that is truly pleasing to the eye and balanced.

Dig Deeper

 I hope this has been an informative introduction to the Golden Ratio and the Fibonacci Sequence. If you are truly interested in applying this ancient proportion to your work, I strongly suggest that you not only do further research on the subject, but put it into practice! Start to observe the objects that surround you, both natural and man-made. Practice breaking them down into their most basic lines and shapes and, more often than not, you'll find that φ is at the root of them. The building of this mental "Phi library" will help you to naturally incorporate it into your designs and art in such a way that your results will seem natural and elegant.

 So that's it for now. As always, feel free to leave your comments and questions. I leave you now with some links to interesting tidbits to feed your ravenous curiosity. Until next time!

-Steve