Less obviously, we can consider time as an additional, fourth dimension, as Einstein famously revealed. But just as we are becoming more used to the idea of four dimensions, some theorists have made predictions wilder than even Einstein had imagined. String theory intriguingly suggests that six more dimensions exist, but are somehow hidden from our senses.
They could be all around us, but curled up to be so tiny that we have never realized their existence. Dimensions are really just the number of co-ordinates we need to describe things. We can compare this to a tightrope walker travelling along a rope.
For the acrobat there is only one dimension — forwards or backwards, and we can state her or his position with just one number. If we zoom in even further, for atoms inside the rope, the world would be in three dimensions, the x, y and z of everyday coordinates. Who is to say that as we go smaller and smaller the number of directions to travel in, the number of dimensions, does not increase even further? Some string theorists have taken this idea further to explain a mystery of gravity that has perplexed physicists for some time — why is gravity so much weaker than the other fundamental forces?
Why do we need objects the size of planets in order to feel its force when we can experience the electromagnetic force with just a small magnet? From the perspective of the 21st century, this seems almost self-evident. Yet the notion that we inhabit a space with any mathematical structure is a radical innovation of Western culture, necessitating an overthrow of long-held beliefs about the nature of reality. Although the birth of modern science is often discussed as a transition to a mechanistic account of nature, arguably more important — and certainly more enduring — is the transformation it entrained in our conception of space as a geometrical construct.
Over the past century, the quest to describe the geometry of space has become a major project in theoretical physics, with experts from Albert Einstein onwards attempting to explain all the fundamental forces of nature as byproducts of the shape of space itself.
While on the local level we are trained to think of space as having three dimensions, general relativity paints a picture of a four-dimensional universe, and string theory says it has 10 dimensions — or 11 if you take an extended version known as M-Theory. There are variations of the theory in 26 dimensions, and recently pure mathematicians have been electrified by a version describing spaces of 24 dimensions.
And what does it mean to talk about a dimensional space of being? In order to come to the modern mathematical mode of thinking about space, one first has to conceive of it as some kind of arena that matter might occupy.
Obvious though this might seem to us, such an idea was anathema to Aristotle, whose concepts about the physical world dominated Western thinking in late antiquity and the Middle Ages. Think of a cup sitting on a table. For Aristotle, the cup is surrounded by air, itself a substance. In his world picture, there is no such thing as empty space, there are only boundaries between one kind of substance, the cup, and another, the air.
Or the table. But Aristotle rejected atomism, claiming that the very concept of a void was logically incoherent. Not until Galileo and Descartes made extended space one of the cornerstones of modern physics in the early 17th century does this innovative vision come into its own. L ong before physicists embraced the Euclidean vision, painters had been pioneering a geometrical conception of space, and it is to them that we owe this remarkable leap in our conceptual framework.
Hence, if artists wished to portray it truly, they should emulate the Creator in their representational strategies. In the process, they reprogrammed European minds to see space in a Euclidean fashion. What is so extraordinary here is that, while philosophers and proto-scientists were cautiously challenging Aristotelian precepts about space, artists cut a radical swathe through this intellectual territory by appealing to the senses.
The illusionary Euclidean space of perspectival representation that gradually imprinted itself on European consciousness was embraced by Descartes and Galileo as the space of the real world.
Worth adding here is that Galileo himself was trained in perspective. His ability to represent depth was a critical feature in his groundbreaking drawings of the Moon, which depicted mountains and valleys and implied that the Moon was as solidly material as the Earth. By adopting the space of perspectival imagery, Galileo could show how objects such as cannonballs moved according to mathematical laws.
The space itself was an abstraction — a featureless, inert, untouchable, un-sensable void, whose only knowable property was its Euclidean form. By the end of the 17th century, Isaac Newton had expanded this Galilean vision to encompass the universe at large, which now became a potentially infinite three-dimensional vacuum — a vast, quality-less, emptiness extending forever in all directions.
In the process, he formalised the concept of a dimension, and injected into our consciousness not only a new way of seeing the world but a new tool for doing science. By definition, the Cartesian plane is a two-dimensional space because we need two coordinates to identify any point within it.
Descartes discovered that with this framework he could link geometric shapes and equations. One way to understand calculus is as the study of curves; so, for instance, it enables us to formally define where a curve is steepest, or where it reaches a local maximum or minimum.
When applied to the study of motion, calculus gives us a way to analyse and predict where, for instance, an object thrown into the air will reach a maximum height, or when a ball rolling down a curved slope will reach a specific speed. Since its invention, calculus has become a vital tool for almost every branch of science.
Thus with an x, y and z axis, we can describe the surface of a sphere — as in the skin of a beach ball. With three axes, we can describe forms in three-dimensional space. But why stop there? But in fact, theoretical physicists have been discussing such extra dimensions for a good number of years now. Viewed from a distance, the rope looks like a one-dimensional object, akin to a line, with just one possible set of directions of motion: forward or backward along the line represented by the cyan double arrow :.
Only if you look closer you will notice that the rope has a certain circumference — a two-dimensional surface. An intelligent ant might also notice that this second dimension is rolled up, so to speak: If they wander straight along the circumference without changing their direction, they will end up where they started — in an ordinary plane, that could never happen. The picture of the ant on the surface of the rope can be generalized to three-dimensional space.
Experience tells us that there are no more than three extended dimensions. There are, in fact, physical theories that predict the existence of such extra dimensions. In string theory, for instance, the natural number of space dimensions for our universe is nine or even ten. In , Einstein published the general theory of relativity, which applies to frames that are accelerating with regard to each other.
Time does not pass at the same rate for everyone. A fast-moving observer measures time passing more slowly than a relatively stationary observer would. This phenomenon is called time dilation. A fast-moving object appears shorter along the direction of motion, relative to a slow-moving one. This effect is very subtle until the object travels close to the speed of light. Mass and energy are different manifestations of the same thing. The increase in mass is the reason that Einstein says that matter cannot travel faster than light.
The mass increases with velocity until the mass becomes infinite when it reaches light speed. An infinite mass would require infinite energy to move, so this is impossible.
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