Posted in Science & Nature

## Folding Paper

Take any piece of paper and fold it in half. Then fold it in half again. Chances are, you will not be able to fold the paper more than seven times. Try it. No matter how thin the piece of paper is, it is extremely difficult to fold a piece of paper in half more than seven times. The reason? Mathematics.

A standard sheet of office paper is less than 0.1mm thick. By folding it in half, the thickness doubles and becomes 0.2mm. Another fold increases it to 0.4mm. Already, the problem can be seen. Folding a paper in half doubles the thickness, meaning every fold increases the thickness exponentially (2ⁿ). By seven folds, the thickness is 2 x 2 x 2 x 2 x 2 x 2 x 2 = 128 times the original thickness. This makes the piece of paper so thick that it is “unfoldable”.

Another limitation is that folding the paper using the traditional method means the area also halves, decreasing exponentially. With a standard piece of paper, the area of the paper is so small after seven folds that it is mechanically impossible to fold it. Furthermore, the distortion caused by the folds is too great for you to apply enough leverage for folding the paper.

Could these limitations be overcome by using a larger piece of paper? Sadly, no matter how large the piece of paper, it is impossible (or at least extremely difficult) to fold a piece of paper over seven times. This has been a mathematical conundrum for ages, until it was solved in 2002 by a high school student named Britney Gallivan. Gallivan demonstrated that using maths, she could fold a piece of paper 12 times. The solution was not simple though. To fold the paper 12 times, she had to use a special, single piece of toilet paper 1200m in length. She calculated that instead of folding in half every other direction (the traditional way), the least volume of paper to get 12 folds would be to fold in the same direction using a very long sheet of paper.

Mathematics, along with science, is what makes something that seems so simple, impossible.

Posted in Science & Nature

## Dimensions: Flatland

As we live in a three-dimensional world, it is difficult to imagine that there are higher dimensions. To illustrate this, the thought experiment of the hypothetical “Flatland” can be considered. Let us assume that there is a two-dimensional world called Flatland. Here, the concept of depth does not exist. Only forwards, backwards, left and right exist; there is no up and down. Everything that happens here would look like it was drawn on paper.

Now let us interact with Flatworld. If we were to touch Flatworld with our finger, it would be like poking your finger through a newspaper. The inhabitants of Flatworld would see a circle suddenly appear out of nowhere that grows larger and larger. A person would appear as if they were being seen through a CT scanner – in sections. The concept that things can be above or below would sound crazy to a Flatlander, even though to us it appears as a simple concept.

Let us take an ant walking along a piece of paper as an example of a “2D object”. If the ant wishes to go from one edge of the paper to the opposite edge, it must walk along the 2D plane. However, with our 3D powers, we can fold the paper into a cylinder; now the ant can walk to the other point in an instant (across the fold). To another ant on the other side, the ant would look as if it teleported and suddenly appeared out of nowhere.

In another experiment, we make a Mobius strip (a ribbon is twisted once then its two sides are joined) and make an ant walk along it. Although the ant would think that it was walking in a straight line along a two-dimensional surface, it would have walked on both sides of the strip – a three-dimensional concept. If the Mobius strip concept is confusing, think of a garden hose instead: an ant walking along a straight garden hose is walking in: 1D (straight line), 2D (hose is actually a flat surface) and 3D (the ant can walk in a corkscrew pattern along the hose).

If we were to tell that ant that it had just travelled in a higher dimension, that ant would either scoff at us or be genuinely terrified of the experience. To it, we (or the giant pink circle that it sees our finger as) would look like some omnipotent being that can see everything going on in its world and teleport from one place to another. And although the concept of depth would initially intimidate the ant, it would bring the level of the ant’s understanding of the world up one dimension. For if we see what we only know, then how can anyone see anything new? The only way to truly learn and understand new things would be to jump out of the box and see everything from the outside – just like an ant seeing the piece of paper it was on from a higher ground.

Although we may laugh at the foolishness of the Flatlanders (and the ant), to a being of the 4th dimension, we would appear just as stupid and naive. By applying what we learned from the world of Flatland to our three-dimensional world, we can expand our horizon of knowledge and understand what the fourth-dimension is.

(This post is part of a series exploring the concepts of dimensions. Read all of them here: https://jineralknowledge.com/tag/dimensions/?order=asc)

Posted in Science & Nature

## Dimensions: Exploring The Dimensions

In the 21st century, films and television have evolved to show 3D images. However, most people only have a crude understanding of what dimensions actually mean. This is a guide that will explore the incredible journey from a zero-dimensional point to a tenth-dimensional point and all the wonderful lines and folds that lie in between.

A point in space has no area – this is the zeroth dimension.
When two points are connected, it forms a line – the first dimension. This line allows one to travel from one 0D point to another, introducing the concept of length.
Another line is drawn branching off this line in a different direction – we have entered the second dimension. Now we have the concept of width.
By adding another concept – depth – we ascend to the third dimension. Now it is possible to go from one point on a 2D surface to another as we have “folded” a branch in the second dimension to meet the other branch. A simple explanation would be lifting your finger off one point and placing it on another point.

To simplify our journey to the third dimension we have:

• Assumed a “point” in space as a dot (.)(0D)
• Joined” two dots to form a line (|)(1D)
• Branched” the line to create two ends (Y)(2D)
• Folded” the branches together to make the two ends meet (P)(3D)

These four concepts of point, joining, branching and folding are crucial in understanding how the different dimensions interact.

An interesting thought regarding the concept of dimensions is perception. How would inhabitants of each dimension view different dimensions? This is easy for lower dimensions (2D and below) because we can see them as a dot, line, square and cube (our dimension). Ergo, we can easily understand all the concepts of the lower dimensions (e.g. width). However, the opposite would not be possible (e.g. a 2D being trying to understand depth) as the concept does not exist in their dimension.

To further explore this thought, we must explore the world of Flatland.

(This post is part of a series exploring the concepts of dimensions. Read all of them here: https://jineralknowledge.com/tag/dimensions/?order=asc)