Math and me

One of the elements I hated most in school was mathematics. In the early grades I was really good at it. I always excelled at doing calculations in my head. Even the more complicated ones. Until this day I am pretty good at it.

But after the forth grade school slowly but steadily turned into hell for me. The early eighties and nineties were not a time when a person like me could go to school without being harassed. Had I known earlier that I am autistic and had I received even some basic levels of support, my whole school time would have been much easier.

To be fair and honest, a big deal of my problems when it came to understand the curriculum was selfmade. I can be pretty stubborn. I detest learning things I don’t have a connection to. And I had the typical arrogance of a schoolboy / teenager who is convinced that school is stupid. Which didn’t help much either.

So it was as it was. Nowadays there is a support system for kids that have special needs. These support systems may have and create problems of their own, but in general they are really helpful.

In math I quickly fell behind after the simplicity of numbers turned into the mess of formulas. Some of the teachers did a hell of a job trying to teach us how to deduce the formulas. But the explanations didn’t stick with me. And back in the days I didn’t know why. And I didn’t care.

For instance take the formula for calculating the area of a circle, which is ‘A = π * r^2’. I know that π is the relationship between diameter and circumference of a circle. And the circumference and the diameter are directly linked to the area. The formula for calculating the area of a square with a baseline b is ‘b^2’, the circumference is ‘b * 4’, therefore the area can be expressed as ‘(circumference / 4)^2’.

But why do you multiply the square radius of a circle with Pi to get the area of a circle? Most people are happy with knowing the formula, but can’t explain why it works. But for me, the ‘why it works’ is essential. Without it I will never be able to use this formula. I need to understand its nature, the relationship between this formula and reality. Once understood the formula itself becomes irrelevant for me since from that moment on I can use its intrinsics.

The underlying reality of a formula is what I would call realities logic or mechanics. And as soon as I understand this logic, this mechanic, I can create the formula on the fly if I have to. So I actually don’t need to memorize it. But since I can’t feel the relationship between π and the area of a circle I will have to look the formula up again the next time I need it.

Why do I write all this? The devslopes course is now at the beginning of week 6 for me. And the week starts with a coding challenge. The previous week ended with me having to write a medium article, so I decided to write about this coding challenge.

The challenge was: Write code for the regular paperfolding sequence. Described here: https://en.wikipedia.org/wiki/Regular_paperfolding_sequence.

And as soon as I read the wikipedia entry I started hating this challenge. Basically for the first time during the course I really hated something. Not because I hate coding and not because the challenge itself is a particular bad one, but because all these feelings and suppressed emotions, this feeling of being dumb and stupid, started crawling back into my mind. Because the door in my mind started closing after reading the mathematical formulas and explanations.

I read the wikipedia article over and over again. Read the explanation. Read other explanations. But all dwelled around some primitive explanation followed by some useless formulas.

I decided to dig into this on my own and to solve this riddle myself. Therefore I took a piece of paper and started folding it. And as soon as I was able to fold it in my mind and to feel it, I got an understanding of what the solution is.

The folding is pretty straight forward. Take a piece of paper. Fold it exactly in half by bending the upper border to the right until it lays exactly on the lower border. Hold the two borders and without releasing them create a fold in the middle of the paper as a sharp line with your fingernail or something else. This shard fold is exactly in the middle of the paper and runs in parallel to the upper and lower border.
Fold the remaining paper again in half, this time bending the first fold to the right until it lays exactly on the upper and lower border. Without releasing the two borders and the first fold you can now create a second fold in the middle as a sharp line. This second fold again runs perfectly parallel to the upper and lower border and to the first fold.
Fold the remaining paper a third time, bending the second fold to the right until it lays exactly on the upper and lower border and the first and second fold. Create the third sharp line, which runs exactly parallel to the two borders and the two previous folds.

Now don’t look at the face of the paper. Only pay attention to what once was the right edge of the paper. I’ll call this ‘the line’. All the folds run in 90° angles to this line. Each fold creates a corner on the line. Ignore the folds and only pay attention to the line with its corners.

Imagine you walk the line. Each time you arrive at one of the corners you have to turn left or right.

On an unfolded paper, this line has a beginning at the lower border of the paper and runs without corners to the upper border of the paper.

The structure of the line could be describe as
B —E
where ‘B’ stands for the beginning of the line and ‘E’ for its end.

When you fold the paper for the first time in the middle and close this folded paper, this line now has a corner with a perfect 180° angle exactly in its middle. When you follow this line from its beginning and reach the 180° corner, you have to take a right turn to follow the line again to its end.

Now the structure of the line could be describe as:
B — 1R — E
‘1R’ means: first level, right turn.

After the second folding the line looks like:
B — 2R — 1R — 2L — E
‘2R’ means: second level, right turn
‘2L’ means, second level, left turn

When you travel the path from B over to 2R and reach 1R you all of a sudden find yourself at the corner 2L. This one is the first left hand corner. Here the paper bends inwards, creating the need for the left hand turn.

And on level 3 this looks like:
B — 3R — 2R — 3L — 1R — 3R — 2L — 3L — E

Now two patterns emerge:
1: Each new fold creates a new corner between all the previous corners including beginning and end. The third level created new corners between the beginning and 2R, between 2R and 1R, between 1R and 2L and between 2L and E.
2: These newly introduced corners alternate between right and left, starting with right. The first new corner (3R) is a right turn, the second new corner (3L) a left turn, the third new corner (3R) again a right turn, the fourth (3L) a left turn.

And with these two patterns I finally understood the solution to the coding challenge as well as the wikipedia article. But for understanding it, I had to feel the solution, I had to create the solution for my own. Until then, an article like this particular wikipedia entry remains gibberish to me.

The final solution I posted is:

Instead of ‘R’ and ‘L’ the official solution uses ‘1’ and ‘0’, where ‘1’ means right and ‘0’ means ‘L’.