Tesseract or time which is the fourth dimension

[mitch bass]

I am hoping someone can help me to develop a working definition for the term "dimension". In the past I was taught that there is the first dimension which is a single point, the second dimension is defined as this single point being extended in such a way that a second dimensional object would consist of a point that extends into a line which ends in a right angle and that right angle ends in a right angle and that right angle ends in a right angle until the fully defined second dimensional shape is a plane, the third dimension continues working with this concept so that every end point of the plane extends into a right angle so that the plane becomes a cube. Going with this concept a fourth dimensional object I have heard called the name tesseract. The tesseract being a third dimensional object which is extended into right angles at every end point like the two dimensions before it.

My confusion is as follows:

I know "time" is often called THE FOURTH DIMENSION. Is this because one working definition for a dimension is a way to define a position? For example: "A" exists at x, y, z coordinates WHEN -- and then a time is given without which the location could fall short of indicating position (unless object "A" somehow exists from the beginning of time for all eternity, that is the only exception I can think of where time is not required to pinpoint location...but even then perhaps. .. .


Part of my confusion in understanding how the term dimension is defined is a result of what I have heard about string theory in which it is suggested that if string theory has any validity it requires 32 dimensions. Are these dimensions an extension of the tesseract, going with the idea that each higher dimension is generated by extending the previous dimensions corners into right angles to itself? But then if this is so, how does time come under the definition of a fourth dimension?

If string theory requires 32 dimensions, and having 32 dimensions is something considered, than even if the fourth dimension is time, how does one define the fifth dimension?

 

[arcnets]

Hi mitch bass,
I think '4 dimensions' or 'the 4th dimension' is loose speaking. A correct statement is "The dimension of this vector space is 4", or "This is a 4-dimensional vector space".

 

[arcnets]

I'll try to answer the original question:

You can identify space with a 3-dim vector space. Because 3 real numbers x,y,z (called coordinates) are enough to locate any point in space. Let's write down the corners of a cube:
(0,0,0)
(0,0,1)
(0,1,0)
(0,1,1)
(1,1,0)
(1,0,1)
(1,1,0)
(1,1,1)

We can, of course, think of a vector space that requires 4 coordinates. Without using imagination, we know that a 'cube' in this space would look like this:
(0,0,0,0)
(0,0,0,1)
(0,0,1,0)
...
This object is called a 'hypercube' or tesseract.

Now to physics. Of course, not any object that has 4 components is a vector. It must have certain properties, which I will not go into now. However, consider an event that takes place at the position (x,y,z) in space, and at a time t. In Special Relativity, it has been found out that the object
(x,y,z,ict)
can be treated as a vector. Where i is the imaginary unit (i^2=-1), and c is the vacuum lightspeed.
From this stems the statement that "spacetime can be identified with a 4-dimensional vector space".

 

[HallsofIvy]

"Dimension" is the number of coordinates necessary to specify an object of interest. If we are working in plane geometry, we can set up an x-y coordinate system and identify every point with two coordinates: two dimensions. In solid geometry we need three. Physicists are interested in "events"- something that happens at a specific place in a specific time. Since it take three coordinates to identify the place and one to identify the time, physicists normally work in 4 dimensions.

On the other hand, in thermodynamics, it is not unusual to work in "3n" dimensions where n is the number of individual particles involved: that is one "point" specifies the position of every particle.