# To 3D and Beyond

How many spatial dimensions do we live in? How many can we directly perceive? Are the two answers the same? If they are different, what effect, if any, do any higher dimensions have on us?

How many spatial dimensions do we live in? How many can we directly perceive? Are the two answers the same? If they are different, what effect, if any, do any higher dimensions have on us?

# What Is a Dimension?

We exist in physical space. In this space, every point or location can be labeled by a set of *coordinates* in a *coordinate* *system*. There can be more than one coordinate system for the same space. But once we choose a system, two points have the same coordinates if and only if they are the same point in space. *The number of independent coordinates needed to label a point is the number of spatial dimensions*. This dimensionality is a geometric property of space and is the same in every coordinate system for that space. An N-dimensional space has N degrees of freedom, requiring N coordinates to locate a point in it.

# Point Land: 0D

Let’s build spatial dimensionality from the ground up, starting with 0 dimensions. Is there nothing in 0-dimensional space? Actually, there is something, but it is sizeless and shapeless. There is just one single point. This *Point Land* requires no coordinates whatsoever to identify a location in it because there is only one location.

Not much can happen in Point Land. Nothing can move, there is nowhere to go. Nothing can take shape or form, there is no extent. Clearly, we don’t live in Point Land. Let’s move on up a dimension.

# Line Land: 1D

A point in 1-dimensional space requires a single coordinate to uniquely identify it. We can visualize such spaces as *lines*. Depending on the space, the line can be straight or curved, bounded or unbounded.

It’s amusing to imagine hypothetical creatures living in a *Line Land*. Quirks abound. For example, a creature would be a line segment and then would be able to interact only with its immediate neighbors (since there’s no way to go around someone). Thankfully we don’t live in a Line Land, so no need to fret about the loneliness and other maladies rife in such spaces.

# Surface Land: 2D

A 2-dimensional space has 2 degrees of freedom, requiring 2 coordinates to label points in it. Such spaces would appear as *surfaces* to us. *Surface Lands* could range from a flat plane to the surface of a sphere to an arbitrarily curvy surface.

Life in a 2-dimensional world would be very different from what we’re used to. For an amusing take on it, read the satire “*Flatland: A Romance of Many Dimensions”*. Oddities would include the inability to tie shoelaces (try tying a knot without lifting any part of a string off a surface), the problem of keeping our bodies intact despite a digestive tract cleaving us in two, and so on. Thankfully we don’t encounter such issues because we’re not constrained to live in a Surface Land.

# Human Land: 3D

Going up to 3 dimensions, finally we arrive at *Human Land*, the space in which we humans perceive ourselves to exist. Our every day experience is of a flat 3-dimensional space (known as Euclidean space). In this space a point is labeled by 3 orthogonal coordinates: one for left/right, one for forward/back and one for up/down. Unlike Point Land, we have shapes and forms. Unlike Line Land, we can befriend people we like and avoid people we don’t. Unlike Surface Land, we can tie our shoelaces and digest food without the risk of splitting apart.

Flat 3-dimensional space is not the only possible Human Land. There are other 3-dimensional spaces possible. Not only possible, but experimentally proven to exist. Einstein’s General Theory of Relativity formulates gravity as a curvature in 3-dimensional space caused by mass (well, actually 4-dimensional spacetime, but let’s not go down that rabbit hole). We find it hard to visualize curved 3-dimensional spaces. But mathematically they are quite easily defined. For example the surface of a sphere in 4 dimensions is a curved 3-dimensional space. Its points can be labeled by 4 Cartesian coordinates x, y, z, w, with only 3 of them being independent because of the constraint that x² + y² + z² + w² = (radius)².

# Superhuman Lands: Beyond 3D

In principle, there’s nothing stopping us from going beyond 3 spatial dimensions. Mathematically we can easily define higher dimensional spaces as a simple extension of 1, 2, and 3 dimensional spaces. For example, a 4 dimensional Euclidean space (flat space) would require 4 coordinates (in the x, y, z, and w “directions”). Such higher dimensional spaces are frequently used in abstract ways. For example, in machine learning categorization (such as speech recognition), an input (such as a speech sample) is mapped to a set of salient features (such as sound frequencies). These features of an input define a vector in a high-dimensional space and vector analysis can be used to determine which category a vector is closest to or how similar two feature vectors are.

Conceiving of higher spatial dimensions brings us into the realm of *Superhuman Lands*. We, humans, lack the biological ability to perceive higher than 3 spatial dimensions. Where else can you go apart from left/right, forward/back, up/down? Does the fact that we can’t directly perceive anything higher than 3 dimensions preclude the existence of any higher dimensions? Or is it just a biological restriction and there are other ways we can indirectly observe their effects (if they exist)? If we had always relied merely on our senses, we would never have worked our way down to “see” atoms, nuclei, protons, quarks, and leptons. Nor worked our way up to “see” the large scale structure of the universe with black holes and dark matter and dark energy.

If higher dimensions exist, and our biology prevents us from directly observing them, how then could they manifest themselves to us? To answer the above question we can turn to our knowledge of Physics and what measurable predictions we can make about the existence and properties of any extra dimensions.

# A Particle In 4D

Consider a particle with mass ‘*m₀’* moving freely in a 4-dimensional space made up of our usual 3 dimensions and one extra dimension of length ‘*d’*. If the particle is not moving in any of the 4 dimensions, it will of course appear completely stationary to us. The energy *‘E’ *of such a particle at rest is the mass of the particle times the square of the speed of light (from the well known and experimentally verified mass-energy equivalence in the Special Relativity Physics of fast-moving objects):

E = m₀ c²

Now, suppose that the particle is moving (with momentum *‘p’*), but only in the 4th dimension, not in any of our 3 usual dimensions. That particle will still appear to be stationary to us because we can’t directly see the 4th dimension. But now the mass-energy equivalence will look slightly different (from the usual generalization of the mass-energy equivalence for moving particles):

E² = (m₀² + p²/c²)c⁴

So we would observe a stationary particle and the motion in the extra dimension would appear to us as if the particle has an increased mass ‘mₚ’, given by the following:

mₚ² = m₀² + p²/c²

We know from Quantum Mechanics, which is the Physics of small scales, that if a particle is constrained in a dimension then its allowed momenta will be quantized i.e. the momentum will take on discrete values instead of a continuum of values. We need to use Quantum Field Theory, which is the Physics of the small and fast, to rigorously derive the exact quantization of the momentum. But that is beyond the scope of this article. So instead let’s use arguments from the early interpretation of Quantum Mechanics. Specifically, the notion of wave-particle duality i.e. a particle can also act as a wave. The momentum *‘p’* of a particle is related to its wavelength *‘λ’* via Planck’s constant *‘h’*:

p = h / λ => mₚ² = m₀² + (h/λ)²/ c²

Since the extra dimension has length *‘d’*, the Quantum wavelength of the particle must be such that half the wavelength fits an exact number of times in *‘d’*. This is precisely analogous to standing waves in a guitar string that is tied at both ends. This constraint quantizes the allowed wavelengths, and in turn the observed particle masses, in the following way (for n = 0, 1, 2, …):

*n λ / 2 = d => mₙ² = m₀² + n²h²/(2cd)²*

We started out with a particle of mass *m₀* and calculated using Special Relativity and Quantum Mechanics that if there is an extra spatial dimension of length ‘d’, we should expect to find heavier copies of the particle due to its motion in the extra dimension. When properly computed using Quantum Field Theory, these heavier copies of known elementary particles are called Kaluza-Klein excitations.

If *m₀* is much smaller than *h/(2cd)*, then we expect a copy at the approximate mass intervals of *h/(2cd).*

# So, How Many Dimensions Are There?

Short answer: at least 3, possibly more.

Long answer: we have not yet seen any heavier copies of known elementary particles in our particle accelerators. Since we have thus far probed energy scales of ~1 TeV (that’s a terra-electronvolt) and not found any Kaluza-Klein excitations, we can put an upper limit on the possible sizes of any extra dimensions:

hc²/(2cd) > 1 TeV => d < hc/(2 TeV) => **d < 10⁻¹⁸ meters**

So, if any higher dimensions exist they must be smaller than the size of a proton. No wonder they don’t affect our everyday lives.

As we build more powerful particle accelerators and probe for new physics beyond the Standard Model, we hope to see signatures of extra dimensions, supersymmetry, and other exciting ideas in modern theoretical physics.