The beauty in repetition
Music: Make Room by Jonathan McReynolds
The alarm on the phone starts ringing. I try to reach it with my eyes closed, moving my hand over my head where my bedside table is located. I feel something like plastic thinking it's the phone but something falls to the ground: the glasses again. At this point, the hand becomes a spider and crawls on the bedside table until it reaches the phone. I have to put it almost touching my eyes so I can possibly read anything, after all the glasses are too far now. 'Snooze', that's the button I want to press. The weird thing is that I have 3 more alarms so it doesn't really matter if I press snooze or not but I do it anyway.
What really wakes me up is the time: 7:10am? Too late already. Up to this point everything is done so quickly that it's hard to believe a second ago I was practically a zombie. I go to work, come back home tired, try to do something productive, fail at it and go to bed...
The next day in the morning the alarm on the phone starts ringing again. I try to reach it with my eyes closed again, and everything that happened the day before repeats.
Repetition is sometimes referred as monotony which is defined as lack of variety and interest; tedious repetition and routine. Or my favorite: tedious sameness. And I agree, a monotonous life can make you feel trapped, stressed and that your life is not going anywhere. Yet we all get excited by repetition in many places of our life and it can indeed be beautiful, mesmerizing and unintuitive.
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Popular songs tend to maintain the tempo through out. When the tempo starts slowing down it's usually because the song is ending and so the repetition stops. The chorus is a repeated section that contains the primary musical and lyrical motifs of a song. Usually is the part with more energy and the one where everyone starts singing. Why we love the chorus? That's the most repeated part and yet the one that transmits more to us, it's the part that's most familiar from all songs and the one that it can actually makes us remember old songs that we forgot.
When I hear a song for the first time that I love, I usually listen it in a loop so many times. I can't help it, I love the repetition and could spent a whole day like that. It's like repetition can make things last more.
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Not long ago, a very loved one left us and in the ceremony, a piano and 4 singers sang a gospel. One of the singers was the daughter. The song that was played is called Make Room by Jonathan McReynolds. As the tittle says, the song talks about making room for the things you treasure. Of course being a gospel the song refers to God but we all knew in that room, in that moment, Make Room wasn't about God, but about the recent lost. We will make room for her, in our hearths, in our memories and for the rest of our lives.
The second half of that song, you can hear the words: You can move that over, You can move that over again and again. Usually this is repeated in the background while the main melody does some variations but in that room, at least to me, that felt like the main part. It's repeated 30 times: You can move that over, You can move that over and with each repetition the emotion was stronger. Like cries of help coming from the bottom of the heart. That repetition had a strong effect and it was what it was needed in that moment, we needed that moment to take longer than usual and repetition made exactly that: You can move that over and You can move that over and You can move that over. If you want to hear the song I've put it on the top of this post, it's not the original because I feel this version better captures that moment in the ceremony.
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Repetition can also be quite unintuitive sometimes. One of the weirdest and beautiful facts of repetition comes from mathematics. I want to talk about fractals but before we go into the repetition part, I want to clarify some misconceptions.
A fractal is not a self-similar object by definition, those are just a small part of the whole thing. We will mainly talk about self-similar fractals because that's the whole topic of this post but I don't want you to leave with that impression. One of the most famous examples of fractals which is in general not self-similar but quasi-self-similar is the Mandelbrot Set which can be seen below.
But that's just an example, fractals don't need to be quasi-self-similar and their invention was to model nature and capture its roughness. But it turns out self-similar fractals gives a good model for regularity in roughness. What I mean by regularity in roughness can be seen by zooming into Koch snowflake, which is the following fractal:
When you zoom in, you can see that it never gets smooth:
This is a self-similar fractal. What's very nice about self-similar fractals is that we can make sense of the idea of fractal dimension quite intuitively. But before jumping into that, let's talk about weird characteristics of it. What's the perimeter of a Koch snowflake? To answer that I think it's best to see how we can build it. It basically consist of 3 parts, so if we can know the perimeter of one of its sides we can just multiply by 3 that amount to get the rest. To see that, let's construct the Koch snowflake from scratch.
To get the perimeter we first focus on one side and the multiply by 3 to get the whole perimeter. To do that, of course, we will have to repeat the same process infinitely many times. When repeating the process, patterns emerge and identifying those saves us some time. We don't have to do a Koch snowflake, which is effectively impossible due to it having infinite detail, to know its properties. Let's show how to compute the perimeter of the Koch snowflake.
As we can see the length of the perimeter ends up being infinite: As we get closer and closer the details never smooth out and the line keeps twisting more and more. But you can already see that the area inside this infinite line is not infinite but finite since it fits in your screen. If you want to know the details the area it encloses you can take a look at this:
That's really unusual, when a circle grows and has an infinite perimeter, its area also grows to infinity but that's not the case here. That's not the only odd property either. With fractals we can also define some type of fractal dimension which is quite intuitive in the case of self-similar fractals. Lets quickly talk about dimension first. We can make sense of dimension by deforming objects with different dimensionality and see how they change with the same transformations. Then we can apply those transformation to a fractal like the Sierpiński triangle:
where it can be build by repeating:
Let's see if we can talk about dimensionality in the context of fractals. First we transform a line, a square and a cube doubling a side on each figure and then computing its length, area and volum respectively. After that, we do the same for the Sierpiński triangle:
We can see for the Sierpiński triangle the dimensional is around 1.58. That means we have a figure that it's not a line nor a plane but something in-between. This fractional dimensionality goes beyond what we have seen and in general for non self-similar fractals we can have varying dimensionality depending on the level of 'zoom' on it. Maybe if we revisit fractals in the future we can dive more deeply on this topic :)
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Sometimes it is said that "Insanity is doing the same thing over and over again and expecting different results", but it turns out that if you repeat it enough times, infinitely many times, you actually get to new results.
Infinite repetition made entirely new objects emmerge: we have seen objects with infinite perimeter but finite area which ended up challenging our concept of dimensionality itself. Math and science in general allows us to explore old concepts and see them in a totally new perspective which gives us new ways to see and think about the world.
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The next post is titled: Video games and the next big step on humankind. I hope to you see you there too! :)
Thanks for reading,
Rob.
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