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Enola Straight
09-15-2017, 08:54 AM
The complex plane...a real number line and an imaginary number line perpendicular...was found to calculate rotation of a two dimensional object.

Rotating a three dimensional object with a three dimensional complex framework wasn't working...until 1843 Hamilton introduced a 4D quaternion.


When Einstein could not integrate gravity AND electromagnetism in his 4D Relativity framework, Kaluza and Klein took the rules of relativity and extended it to 5 dimensions.



When a number of 10D grand unified theories were in competition with each other...and were mathematically incompatible...an 11th dimension was added to the framework to create M Theory, which described the 10D theories as facets of a gem.



What other problems have been/could be solved by adding an extra dimension to the mathematical framework describibg the problem?

Chronos
09-15-2017, 10:44 AM
Almost all of them, for suitable definition of "dimension". But the original dimensions to the problem might not have been the familiar length, width, and height that you're thinking of, nor anything particularly reminiscent of them.

In any scientific or mathematical problem, you've got some set of objects that you need to describe. And you can describe each object using some set of numbers. The number of dimensions is just a count of how many numbers you need to describe each object. If, for instance, the object is the image on your computer screen, then you need millions of numbers to describe it, one for the value of each pixel. So you can say that your computer screen is a multi-million-dimensional object.

Hari Seldon
09-15-2017, 05:05 PM
It is really misleading to say that Hamilton "added a dimension". He was trying to describe the rotation group of a 2-sphere. He knew the group was three-dimensional and he tried for years to multiply triplets of numbers, without success. Finally he discovered how to multiply quadruplets of numbers and thus were born quaternions with one real and three imaginary dimensions. The rotation group (which had been fully described around 1840 by a French banker and amateur mathematician named Olinde Rodrigues five years before Hamilton) is still three dimensional, essentially the quaternions of absolute value 1.

Buck Godot
09-16-2017, 04:00 PM
I gotta agree that living in a 4D universe would have been a big help in visualizing functions when I was studying complex analysis. Although it also may have made the evolution of life impossible, so I'm probably actually glad that I don't.

Trinopus
09-16-2017, 05:59 PM
An exercise I did in college was fun: you can rotate a semi-circle around the flat part to get a sphere. Well, you can rotate a hemisphere around the base to get a hypersphere. And from there, you can rotate a hemi-hypersphere around it's base to get a 5-dimensional sphere. It's very straightforward numerical math.

It doesn't have any physical meaning in our universe, but it's fun, and something a college frosh can pull off!