Aim For the Stars!
a fun post about math
MATH / FUN / BEAUTY
Editor’s Note
Starting in December (my next post), I am shifting my posts to release at 6 pm Central. While I have a number of these posts written, I am also going to focus on making them not exceed 5-7 minutes of reading time. Anything longer and I will edit them down or split them in two.
The Inspiration
I eat a lot of fresh fruits and vegetables. On a recent evening, I had purchased a fresh pineapple which I really enjoy. That in turn reminded me of something special about pineapples and became the inspiration for this evening’s post.
The Setup
Tonight I will “aim for the stars”. I try to consider when I pick a topic whether it is boring or too dense. In either of those cases, I shy away from the topic. I am going to venture into a few more topics as I have a few more guideposts on the length and even perhaps more everyday news. I hope that anyone else who reads my posts will give me the benefit of the doubt if I have a clunker or two. I thank you for taking the time to read what I have to say.
Tonight’s topic is about math and how it influences our natural world in a surprising way. I think even if you are not “all-in” for math, you will enjoy the videos I have linked in the post. I think they are entertaining and might even be a fun set of ideas to play through with a child, nephew, or grandchild at some point when it comes to the flowers, pineapples, and pinecones.
Math is Amazing
In an earlier post, I droned on and on about calendars. Part of that discussion included Ockham’s Razor and the Babylonian contribution to our early understanding of the world. That is where our story this evening picks up. I have included the same diagram for Ockham’s Razor for the unfamiliar. I am a firm adherent to Ockham’s Razor. I believe that the best place to look for answers is the simplest solution. Seeking complex explanations often leads to a lot of wasted time.
Mathematics is referred to as a language but it is not a natural language like English. Mathematics as a language is interesting in that it is UNIVERSAL and focused on explaining ABSTRACT things. While some things are complex, MOST are not. The simplest explanation is often the best and formulas provide that concise simplification.
Babylon Rocks
Babylon, beyond its role in early calendars, was also the trailblazer for mathematics. An adjacent culture, the Sumerians were trading partners with the Babylonians. It appears that association led directly to a breakthrough in mathematics even before it had a name. The Babylonians developed a base 60 counting system. Why did that happen? We don’t know for sure but here is a theory:
The Babylonians counted in units of five, undoubtedly inspired by the five digits on their hands.
The Sumerians counted in units of twelve. So why is that? It turns out that counting emerged in some areas of the world by twelves. If you look at your hands and just count the obvious bone segments in your hand, there will be three bones in each of your fingers (ignore your thumb).
If you try it out, you will see that is an efficient way to hand count stuff and faster than counting by fives. I tried it out and compared counting to sixty on my hand. It was much faster doing the twelves.
Trading with a partner means payment. When people count differently, it turns out the best way to make that work is to multiple the bases together. Five times twelve is sixty.
Our modern world is full of things we measure by sixties and multiples. It all started in Babylon. We have 360 degrees in a circle, 60 seconds in a minute, 60 minutes in an hour. Pretty cool eh!
People may not have considered it but 60 is a pretty convenient number for SO MANY THINGS. That may be because it is divisible evenly by 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30. That makes a lot of things easier to figure out and split evenly!
The Babylonians developed a shorthand for multiplication which was ingenious. They figured out the squares of numbers and kept them close at hand. They would allow them to do multiplication without learning how to do it. I will not present it here but if you are interested, leave a comment and I will share the formula they figured out that made “multiplication” easier without much training.
The cool word for all this that we all MIGHT HAVE BEEN taught one day in ancient history class is that the system of counting was called sexagesimal. Now that is the big word for this post!
Here is a good link about Babylonian math. Referring to this region of the world, modern-day Iraq as the cradle of civilization is not too much of an exaggeration. It was a great climate without a lot of seasonal variation. There were also a lot of wild grasses that could eventually be cultivated. Botanists believe this is the place where people first began harvesting wild wheat to make flour and bread. Flour lasts a long time and with it people can FINALLY stay in one place and start working on stuff like math! It was also located adjacent to a land bridge out of Africa so migration in that direction would have been natural. As usual, if we stick with Ockham’s Razor, this is not a bad theory at all. So how did THIS TOPIC come to be in my mind?
Thanks to the Babylonians, generations have been adding to our understanding of our world through mathematics. I think I have posted already a bit about some of the greats in the field including Newton and Gauss for example. Tonight we will talk about a mathematician with the most debonnaire of names, Fibonacci. However, here is a song inspired by Babylon just because.
Babylonia brought the world a first significant attempt at the language of mathematics. The Babylonians were making these breakthroughs long before the very first elements of the Bible were written! The Babylonians even recognized a lot of what comes next, they just did not name it. It turns out the numbers we discuss next appear in nature all the time.
Numbers and Nature
Now I am going to jump ahead to a special number that is credited as being first observed by Euclid, a Greek mathematician and widely credited with early geometric principles. Fibonacci numbers are a very interesting and simple sequence. Every number is the sum of the previous two. Here is the start of the sequence:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, …
At this point, this seems like a weird and not very interesting number sequence. How can this be significant?
Now here is where it gets interesting. Take one of the numbers above and divide it by the previous number. We will start with 3/2 and then 8/5
3/2 = 1.5 ; 8/5 = 1.6 ; 13/8 = 1.625 ; 21/13 = 1.615
I do not want to make this boring but the ratios CONVERGE on a special number. Now it can get MORE AMAZING. It turns out we can start out with ANY TWO numbers and repeat the procedure described. After a few numbers the ratio will start getting close to 1.6. Euclid is credited with recognizing the prevalance of this ratio (~1.618) and it is now called “The Golden Ratio”. This turns out to be a property of the NATURAL WORLD. The picture of the pinecone and how its spirals look turn out to follow the Fibonacci sequence.
The picture below shows these numbers to be strange in many ways. This may be a bit more difficult to follow but is really worthwhile when you see the result. If we make some squares and draw them with the lengths from the original sequence above (1, 2, 3, 5, 8, 13, 21, 34, …) and then trace along the edges of the square, the picture is going to get REAL WEIRD.
Try to remember the shape of that simple spiral. You may recognize something like that in nature. Think of a snail shell for example if you turn it upside down. As you can see, the picture above was a PRETTY COOL way to pack those simple squares together. No wasted space is a pretty good principle to know something about. It looks neat and it is an EFFICIENT use of space. Nature LOVES simplicity and efficiency. We are going to lend a few references to some fun videos now. The ratio of the number approaches a number around 1.618 if we keep repeating the pattern described previously and that number is called “the Golden ratio”.
Here is the optimistic version of “the Golden Ratio” and Fibonacci numbers. There are two videos because the first is a bit much and probably overstates the importance of these numbers. The second video is a bit more pessimistic. Both are fun and might fit into your next walk on a treadmill.
Now for the sake of balance, here is the more pessimistic version. The great thing about the 2nd video is that it questions some of the 1st. Nevertheless, it is still VERY ENTERTAINING. Because this topic can get very complex quickly, I STRONGLY recommend the videos. I think the result will be clear and amazing for many of you. Even if you HATE math, the leaf pattern explanation is simply AMAZING! Who would have guessed that the orientation of leaves on a petal has something to do with mathematics! This sort of thing brings a smile to my face. This is not the only pattern of nature but the one I thought it would be fun to talk about this evening. If you want to understand why I provided a picture of a pinecone at the top, you have to watch the video(s). Sorry about that.
Beyond being “beautiful”, these patterns that emerge in nature are worth emulating because their efficiency has emerged over a long time as the VERY BEST WAY to do certain things. Over the next few generations, designers will find ways to emulate these patterns in the things we build. Emulating nature will be an exciting new phase in design. Many believe these patterns describe the shapes of hurricanes, spiral galaxies, and clouds. All of this is right under our noses. Music tonight starts and ends with the number one since that had to be the start of mathematics. Let me know what you thought of the post especially if you took the time for the videos.
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