There has been much debate among mathematicians, physicists and philosophers about whether a true two-dimensional world can exist. On one hand, the concept of a 2D world seems intuitively straightforward – we can easily imagine a world occupied by shapes with no thickness, like a drawing on a piece of paper. However, when examined more closely, there are good reasons to think that a purely 2D universe with 2D objects is an impossibility. In this article, we will explore some of the arguments against the existence of a 2-dimensional world and try to understand why the world we inhabit must have at least three spatial dimensions.

## What is meant by a 2D world?

First, it is important to clarify what we mean when talking about a hypothetical 2D world. A 2-dimensional or “flat” world can be imagined as a universe where there is only length and width, but no depth or height. Objects in this world would have an area but no volume. For example, a 2D person would essentially look like a shadow or silhouette – flat, without any thickness. They would have to slide past each other to get around, as it would be impossible to move over or under anything. Similarly, a 2D sphere would be a flat circle, and any other 3D shapes would be reduced to straightened out, flattened forms in a 2D plane. The world itself would exist on a flat 2D surface, somewhat like a sheet of paper, rather than having the added 3D space we experience.

## Theoretical arguments against 2D

There are several theoretical reasons why many scientists and philosophers consider a purely 2D universe to be an impossibility:

### Atoms and molecules could not exist

In our 3D world, complex structures are built out of atoms, which are composed of nuclei and electrons. These components exist in 3D space and interact through electrostatic forces that operate in all directions. But in 2D space, there is no “up” or “down” – only left-right and forward-backward. Atoms simply could not hold together structurally if electrons were restricted to orbiting a nucleus in just two directions. So in a truly 2D universe, matter as we know it could likely not exist.

### No complex structures or functioning systems

Similarly, there seems to be fundamental limitations in how complex functioning systems could develop in a 2D world. Our biology, for example, relies on complex protein molecules that fold in very specific 3D configurations. Cell membranes, organs, muscles and tissues rely on 3D shapes and structures. It is difficult to imagine how such complex 3D systems could evolve in a 2D environment. Higher cognitive functions like reasoning and consciousness also seem to require 3D connectivity in the brain. So many scientists argue there are insurmountable barriers to complex life in 2D.

### No gravity or motion physics

In a 2D world, there would be no gravitational forces as we know them. Gravity depends on the inverse square law, which requires a quantity (mass) to spread over the surface of a sphere. With no third dimension, this fundamental force would not operate. In the absence of gravity, basic physics principles like acceleration and orbital motion would also break down. So even basic physical interactions seem to require 3D space.

### No conservation of angular momentum

Conservation of angular momentum is another essential concept in physics that requires 3D space. When an object rotates in 3D space, it has an angular momentum that depends on its mass, shape and velocity of rotation. When it interacts with other objects, its original angular momentum is conserved and transferred elsewhere. In 2D, there is no way for rotating objects to conserve angular momentum, severely limiting how physics could work.

## Thought experiments making the case against 2D

In addition to these theoretical limitations, some influential thought experiments have been proposed that aim to show the logical impossibility of certain 2D scenarios:

### Maxwell’s demon

Proposed by physicist James Clerk Maxwell in the 19th century, Maxwell’s demon is a hypothetical intelligent being that can sort hot and cold molecules in a closed 2D box, apparently violating the second law of thermodynamics. The thought experiment aims to show that the demon would necessarily have to detect molecules approaching from above or below, requiring 3D space. This suggests certain physical processes cannot be confined to 2D.

### The enlarged chessboard problem

Imagined by mathematician Henri Poincaré, this thought experiment considers an infinite 2D chessboard. When a larger flat chessboard is placed on top of the original one, there should still be a one-to-one correspondence with the squares below. But Poincaré realized this correspondence breaks down at the boundaries, where the extra squares on the larger chessboard have no counterparts below. This highlights logical contradictions if space can be stretched without adding a third dimension.

### Movement in a 2D world

Simple thought experiments can also show the problems of movement in a 2D world. Imagine two 2D people approaching each other from opposite directions on a 2D plane. With no third dimension, they could not pass around each other. Similarly, any object rotated in a certain direction would inevitably collide with itself as it comes full circle. Movement and navigation seem extremely limited without access to “up” and “down”.

## Evidence from modern physics

Beyond these theoretical arguments, modern physics also provides more empirical evidence against the possibility of a 2D universe:

### Holographic principle

According to the holographic principle proposed by physicists like Gerard ‘t Hooft and Leonard Susskind, all of the information about a volume of space can be encoded on a bounding surface. If our 3D universe is a holographic projection of 2D information on its boundary surface, then a truly 2D universe with no boundary wouldn’t contain enough information to exist.

### String theory

In string theory, fundamental particles are modeled as 1-dimensional strings. The behavior and interactions of strings appear to require them to vibrate and oscillate across dimensions – at the very least, three spatial dimensions. So string theory assumes 3D space is a minimum requirement.

### Quantum field theory

Modern quantum field theories, which form the basis of particle physics, are defined in terms of fields permeating 3D space. Attempts to formulate quantum field theory in 2D have run into major obstacles. So our current physical frameworks practically assume 3D space in their foundations.

## Some proposed workarounds

While 2D worlds face many conceptual barriers, there have been some creative attempts to imagine possible workarounds:

### Holographic principle

Some have suggested that a lower-dimensional world could exist as a holographic projection of a higher-dimensional space, as described above. Though not “truly” 2D, this kind of holographic world could potentially mimic some properties of a flat plane.

### Artificial lattice structures

Engineered lattice materials with tightly controlled geometric properties could potentially simulate a 2D world by restricting objects and fields to specific planar pathways. However, these would still ultimately exist within our larger 3D universe.

### Flatland scenario

Proposed by mathematician Edwin Abbott in 1884, Flatland imagines a rigidly structured 2D world occupied by geometric shapes who are unaware of any third dimension. While ingenious, this scenario dodges many of the physical issues at hand. And it requires a 3D external observer.

### Computing artificial 2D environments

Purely computational worlds, simulations or virtual realities could potentially model 2D environments that sidestep some physical constraints. But they would still require operating within a 3D-based computing system.

## Conclusion

While we can easily conceive of a two-dimensional world in our imaginations and calculations, upon closer scrutiny there seem to be immense difficulties in actualizing a completely flat universe with no access to a third dimension. Fundamental physics breaks down, atoms and complex structures cannot form, and movement becomes severely limited. Creative workarounds have been proposed involving holography, lattices and virtual reality, but none seem capable of truly eliminating the need for a three-dimensional substrate. Until someone finds a loophole in these theoretical barriers, most scientists remain highly skeptical of the possibility that a purely 2D world could exist. So for now, the evidence firmly suggests that two dimensions are not enough, and our observable reality requires at least three.