*Sujan Dhakal*

*M.Sc. Physics 3 ^{rd} semester*

Mathematically, “dimension” refers to the number of coordinates needed to describe a point, or equivalently the degrees of freedom of motion in a space. A line is one-dimensional because a point in the line needs only one coordinates for its description. You could say “dang, there are a hundred people in front of me,” which describes very well your sorry position in line. As another example, the volume of sound is a one-dimensional concept. A particular volume needs only one number to describe it, possibly from the scientific decibel scale or maybe on the stereo knob “turn it up to 10” scale.

A two-dimensional space needs two numbers for each point. The flat, infinite plane from high-school geometry is the prime example, with each point given an x and a y coordinate. The surface of a sphere is also two-dimensional; for example, points on the Earth are described by longitude and latitude. Though we spend most of our days wandering the two-dimensional surface of the Earth, our space is in fact three-dimensional, which means we can move on three axes, North-South, East-West, and Up-Down. Describing points in space requires three coordinates: to spot an airplane, you need longitude and latitude, plus elevation.

The next step is the fourth dimension. Mathematically, it’s no problem to define four-dimensional space, or “hyperspace.” It’s just an abstract space that needs four coordinates to describe each of its points, which works very well for computations, but is not much help in visualization. Trying to think in four dimensions is a serious challenge, and requires a complicated collection of mental crutches to make any progress.

The tesseract, or “hypercube,” is the most accessible four-dimensional object, so it’s worth trying to understand. We work by inductive reasoning, starting with a point, and dragging it to trace out a segment. Then, drag the segment to trace a square, and drag the square to trace a cube. The next step is to drag the cube in a fourth direction, perpendicular to all edges of the cube, resulting in a tesseract or “hypercube.” The last step, as usual, is difficult to imagine because it requires the fourth dimension. We get the flavor with some drawings:

In mathematical physics, Minkowski space or Minkowski space time is a combination of Euclidean space and time into a four-dimensional manifold where the space time interval between any two events is independent of the inertial frame of reference in which they are recorded. Although initially developed by mathematician Hermann Minkowski for Maxwell’s equations of electromagnetism, the mathematical structure of Minkowski space time was shown to be an immediate consequence of the postulates of special relativity.

Four-dimensional Minkowskispacetime is often pictured in the form of a two-dimensional lightcone diagram, with the horizontal axes representing “space” (x) and the vertical axis “time” (ct). The walls of the cone are defined by the evolution of a flash of light passing from the past (lower cone) to the future (upper cone) through the present (origin). All of physical reality is contained within this cone; the region outside (“elsewhere”) is inaccessible because one would have to travel faster than light to reach it. The trajectories of all real objects lie along “worldlines” inside the cone (like the one shown here in red). The apparently static nature of this picture, in which history does not seem to “happen” but is rather “already there”, has given writers and philosophers a new way to think about old issues involving determinism and free will.

Modern string theories suggest a whole bunch of dimensions on the sub-atomic scale. But none of this precludes another direction, perpendicular to space, in which we could move if we only knew how.

Rotation in four-dimensional space

https://youtu.be/R1rzI0Y_d3I

The 5-cell is an analog of the tetrahedron.

https://youtu.be/BjvdrhK8yws

Tesseract is a four-dimensional hypercube – an analog of a cube.

https://youtu.be/Pa0c7M4lZv0

The 16-cell is an analog of the octahedron.

https://youtu.be/np0ZxC1wXqc

The 24-cell is one of the regular polytope.

https://youtu.be/T01qw0_qitI

A hypersphere is a hypersurface in an n-dimensional Euclidean space formed by points equidistant from a given point, called the center of the sphere.