If we were to imagine a fictional universe where there is no concept of up or down, where only right and left, front and back exists, we wouldn’t be the first to do so. On a first glance, this idea of 2 dimensions might seem trivial simply because, we have an additional dimension at our disposal to explain things more accurately., but it is bit intimidating when we define 2-d as “a 3-d space that simply misses one dimension”. Note that, we didn’t say “ignore”, or “negligibly reduced” but simply “has no place in existence”.
WHY EXPLORE 2D?
The world of two dimensions is no different than the three dimension. One of the few common properties they share are homogeneity and relativity (not to be confused with Einstein’s relativity). By this property, the lower dimensions of space are interchangeable and can are defined relative to each other. Interchanging the names given to each dimension lengths and breadth them wouldn’t change the shape or violate any laws of 2-d space, as length and breadth are agreed-upon frames of reference. No one of the dimensions has a higher priority over the other and therefore share a kind of symbiotic relationship. The direction x and y are interdependent.
“Flatland”, the famous fictional novel by Edwin Abbott Abbott creates a fictional world of 2d objects and how they interact with each other in 2-d space. The book is essentially a comment on Victorian culture, it is equally appealing to physicists and explorers of dimensions.
If one were to explain this world in a simple sentence, it would be the following– a world without height. Meaning, the sense of height, with respect to the reference plane, is absent. It is easy to confuse, the complete absence of height with an extremely small one (like that of paper). Every real-world object we encounter, however small, is actually 3-d whether it’s a thin sheet of paper we are talking about or the layer of ink sitting on top of it.
Any objects having its third dimensional length, even in orders of the smallest measurable length that holds any meaning (Planck’s length – 1.6 x 10-35m), might seem to vanish even under the most powerful microscope. Despite this, it can still be measured and thus fails to convey the idea of two dimensions, accurately. The first and second dimension, in relation to the third is therefore just a concept – a mathematical tool used to trivialize complex ideas that serves as a cornerstone in the study of higher dimensions, not limited to 3d. True 2-d shapes, on a purely intuitive and visual level, have no place in our world.
Before jumping into flatland objects, let us take some time to understand what a line segment really means. In our elementary/nursery years, we are typically used to representing a line as a diagram, because we weren’t too concerned with its dimensional accuracy or what it meaning it could hold while exploring higher dimensions. The fact of the matter is that there’s no way to “draw” a visible line without contributing towards it breadth in 2-d space, and that means violating the very definition of a single dimension. A line, therefore, is not a shape, it is an idea, a concept to explain the shapes that we encounter on a daily basis. It cannot be represented visually paper, nor through any media, but can only be imagined as a limited length in a specific direction or defined in the form of a equation. Alternately, it is the distance between two points in space. The same holds true for points, which is nothing more than a position that can be represented through numbers.
When we use the word “true” to define an object, we are essentially saying that the object in question, is not projection of a higher dimensional object. For e.g., a circle, as a projection of sphere is not “true “in the sense, its existence depends on that of the sphere. Other examples of “false” 2-d objects are shadows, surface of a paper. This is because shadow, cannot be considered an object because it is nothing more than the surface of the object it is falling on. Moreover, it cannot be isolated from the surface it falls on. Such is the case with surface of a paper as it depends on the paper’s existence.
So what exactly, is a 3-d object in relation to a 2-d object? What is the relation of a cube with respect to a square?
A cube is an infinite number of squares, stacked one upon the other in a direction perpendicular to both the x and y directions.
So far, we have understood what 2d objects really are. But what is a 3-d object? What is it relation with a 2d object? A naïve, but fair way of defining this would be – an infinite number of 2d objects stacked one upon the other in a direction that cannot be defined by the directions of the 2-d objects plane. The concept of vectors, therefore, play an important role in understanding the definition.
HOW 2-D CREATURES PERCEIVE EACH OTHER
Imagine, two entities of flatland A- a circle and B- a square, interacting with each other. How will they detect each others presence? How will they tell each other apart?
To answer the above question, let us understand try and explain this with the help of an example. Let us consider viewing a coin placed on a table top. Its top-down view projection, would be a circular shape of the face of the coin. This is its 2-d shape.
If we were to view the same coin, our eyes placed at the table’s edge, we will see this as a small, thin rectangle, with its length larger than its width. The length here is actually the diameter of the coin, and its width, its height. If this coin was perfectly 2-d dimension, it would have no height. Therefore, the rectangular projection in the table-edge view would be seen as nothing more than a line. This is the perspective with which “flatlanders” (for the lack of a better word) see objects in the world – as lines of varying length and effect of light on it. The above theory holds true for all objects that lacks height/depth.
The whole point of discussing such a seemingly trivial information is to illustrates the loss of visual information.
B in this world, is not therefore the shape we are familiar with. The concept of angle is foreign here, and hence the sharpest point near the viewer (in case of a square), is just a point that shines brighter than the other points in the line.
Say while, viewing one corner of a square B in front of A’ eyes, it will notice light affecting the corners abruptly (due to its sharp nature) more than it is affecting the edges moving away on either side. The circle A on the other hand will be affected similarly, except that the brightest point nearest to the source of light will fade gradually to the farther points due to the fact that its corner is smooth. This property of how light affects each point in the perimeter of a 2d object, can be termed as a say, shader. (similar to shader in computer graphics, which is defined as the manner in which light affects a texture of an object). This shader, is a function of the position and rotation of object relative to the source of light, colour, smoothness etc. (basically, all the properties that affects the distribution of light on its surface).
Despite this, the above solution fails to accurate separate closely related geometric objects such as a square from a rectangle, or a circle from oval. So how do you distinguish between them?
Well, the answer to that would-be rotation. Through a complete rotation of 360 degrees, the viewer (another flatland being, in this case) will note the change in light with the and be able to tell a square from a rectangle. Note that this change in brightness throughout a line, needs to be perceived with utmost precision, due to the fact that unlike the novel, we’re ignoring all other senses viz touch, smell and concerned with only the visual/dimensional senses for interacting with a flatland being. The following pictures are far from being accurate, but helps to understand the abrupt nature of light falling on sharp edges much better.
Another important factor is the sense of time interval between appearance of two consecutive edges.
The square has four equal sides, hence the time interval between the appearance (or disappearance) of any two consecutive corners must be equal given that a square rotates with an uniform angular velocity w(omega).
Whereas in the case of a rectangle, the corner comprising the smaller length, appears faster as compared to the greater length.
Circle on the other hand will be homogeneous from all directions, whereas the light on an oval object will vary less near the minor axes, and more near the major axes.
The whole point of elaborating the effect of light on flatland objects, and how they are perceived by each other is to illustrate the loss of information and complexity in understanding and explaining a flatland object in terms of rotational shader, and not 2d geometry. It is important to understand that the space in itself is 2d, but a flatland being’s brain is limited in its ability to perceive 1d cross-sections of every 2d object it encounters and later combine the multiple views in the form of 2d, just like we in our 3d space perceive another object as a continuous and infinite number of tessellations of 2d images. The above point begs the question, whether at all we are 3-d or a mere projection from the 4th dimension limited by our inability to comprehend what lies beyond the three axes of space.
This analogy and logical relation between 2-d and 3-d helps us analyze the relation between 3-d and what is beyond – the fourth dimension, in the following article!