Two Dimensions

A diagram showing the first four spatial dimensions.

 The concept of dimension is not restricted to physical objects. High-dimensional spaces occur in mathematics and the sciences for many reasons, frequently as configuration spaces such as in Lagrangian or Hamiltonian mechanics; these are abstract spaces, independent of the physical space we live in.

 ***

Big Bang, Classic Confusions-

One of the most confusing things about the Big Bang is that it involves an expanding universe. Any reasonable person, hearing about the Big Bang, will imagine something that he or she has seen expanding: a cloud of smoke exploding outward, or a balloon expanding as it is filled with air. This is very natural. And having imagined this, the reasonable person will ask, “But what is the universe expanding into?”

 ***

 

Let's now start analysing a 2D case, that of the classic Flatland example, in which a person lives in a 2D universe and is only aware of two dimensions (shown as the blue grid), or plane, say in the x and y direction. Such a person can never conceive the meaning of height in the z direction, he cannot look up or down, and can see other 2D persons as shapes on the flat surface he lives in.

We cannot directly visualize a hypersphere for the very reason that it is a 4-dimensional object and goes beyond our senses. What we can visualize, however, is a hypersphere in the form of 3-dimensional slices (as is displayed to the left). A hypersphere is in essence an array of 3 dimensional solid spheres that increase and then decrease in size. This would represent our basic conception of the hypersphere, and is shown in the animated picture here.-

Understanding 4 dimensional space

Dimension (n)	Shape	Volume	Surface Area
2	circle	π r2	2πr
3	sphere	(4/3)π r3	4πr2
4	4-sphere	(1/2)π2 r4	2π2 r3
5	5-sphere	(8/15)π2 r5	(8/3)π2 r4
6	6-sphere	(1/6)π3 r6	π3 r5
7	7-sphere	(16/105)π3 r7	(16/15)π3 r6

See: Spacetime 101