Can We See in 4-D?
Although most of present-day mathematics is limited to analysis in one, two or three dimensions, the fact that multidimensional analysis will have equal (or perhaps greater) importance in the coming years is inevitable. As a matter of fact, four-dimensional modular forms were an integral part of Andrew Wiles’ proof of the famous Fermat’s Last Theorem. The mathematicians today make use of complicated equations to describe these multiple dimensions since visualization beyond three dimensions continues to seem impossible. This article aims at explaining the fourth dimension and a few of its properties by drawing an analogy with 1-D, 2-D and 3-D analysis, taking aid of a method described in the book “Weird Maths” written by Indian mathematics prodigy Agnijo Bannerjee under his mentor David Darling.
Imagine yourself as being constrained to one dimensional motion, i.e., you are standing on a line segment and all you can do is move forward or backward. Now let us curve this line segment into a circle. Your motion will still be one dimensional along the circumference of this circle (considering you are not allowed to move in the interior of this ring). However, this circle, as we all know, exists in a 2-D plane. Curving the 1-D line segment gave us an object (circle) that exists in 2-D plane. Similarly, curving a 2-D plane gives us a sphere, an object that exists in 3-D space. Continuing this exercise, curving a 3-D volume should give us an object that would exist in 4-D. Although visualization is still beyond our thinking capacity, we can now grasp the fact that 4-D is basically curved volume, whatever that may look like. Let us similarly try to explore a few more properties of the 4-D world.
Imagine you wake up one day to realize that you have become your mirror image. That would mean your heart is slightly shifted to your right, as opposed to humans in the normal non-mirror world. What if I tell you that any hypothetical being in 4-D has the power to flip any 3-D human into his/her mirror image?! If you can’t grasp this, you’ll be able to do so in a short while, following our famous exercise of drawing an analogy with 2-D and 3-D. Imagine a right shoe sole on a 2-D plane with no thickness (0 height, since it’s 2-D!). Also imagine Mr.X, a hypothetical 2-D being constrained to live in the 2-D world. Now, I (a 3-D being) flip the shoe sole (for which I’ll have to temporarily lift it out of the 2-D plane) such that it becomes a left shoe sole. For a 2-D being who can’t visualize the third dimension of height and can’t see what happens outside the plane, the process of flipping is impossible to visualize (sort of invisible) and hence how the sole got flipped remains a mystery to Mr.X. Similarly, the flipping of 3-D objects into their mirror images could be possible in the fourth dimension which we can’t visualize. Fortunately, no 4-D being has been found to exist and we seem to be safe!
Another rather interesting theory is that we humans could be 4 dimensional in reality. Where did this theory arise from? Let us imagine a solid sphere resting on a 2-D plane. The sphere now starts moving vertically downward, through the plane. What the above-mentioned Mr.X sees is a 2-D disc (initially a point disc) increasing in size till it becomes a disc with diameter equal to the diameter of the sphere and then decreasing in size till it becomes a point disc again.
Now, if this sphere stops in one of the stages of the vertical downward movement and starts moving horizontally along the plane, what Mr.X sees is a disc of certain diameter moving along the 2-D plane. So, this moving solid 3-D sphere was seen by Mr.X as a 2-D object (disc) moving in the 2-D plane. Extending this analogy and considering our consciousness and vision to be constrained to 3 dimensions, we might be 4-D beings moving through the 3-D world and while doing so, making an impression that we are moving 3-D beings.
One inexplicit consequence of the above properties is the linking of the idea of higher dimensions with philosophy and spiritualism. Most of us have seen or at least heard about humans performing certain magical tricks with no logical explanation. Although most of these are simple illusions, it is quite possible that a few humans have developed the capacity to see in 4-D and make use of this fourth dimension to perform the above tricks (the way flipping of the shoe sole is accomplished by 3-D beings). The ideas suggested by spiritualists and condemned by scientists could very well be true in these higher dimensions which humans can’t see.
The study of higher dimensions is a vast topic spanning across a multitude of areas in mathematics and physics. It has huge scope and believe it or not, the knowledge of higher dimensions serves as a powerful tool in various fields, including medicine, education and most importantly, in the 21st century physics: string theories. (A version called superstring theory requires a total of 10 dimensions, M-theory involves 11 and the bosonic string theory demands 26 dimensions.)