Where is the centre of the universe?
There is no centre of the universe! According to the standard theories of cosmology, the universe started with a "Big Bang" about 14 thousand million years ago and has been expanding ever since. Yet there is no centre to the expansion; it is the same everywhere. The Big Bang should not be visualised as an ordinary explosion. The universe is not expanding out from a centre into space; rather, the whole universe is expanding and it is doing so equally at all places, as far as we can tell.
The Famous Balloon Analogy
A good way to help visualise the expanding universe is to compare space with the surface of an expanding balloon. This analogy was used by Arthur Eddington as early as 1933 in his book The Expanding Universe. It was also used by Fred Hoyle in the 1960 edition of his popular book The Nature of the Universe. Hoyle wrote "My non-mathematical friends often tell me that they find it difficult to picture this expansion. Short of using a lot of mathematics I cannot do better than use the analogy of a balloon with a large number of dots marked on its surface. If the balloon is blown up the distances between the dots increase in the same way as the distances between the galaxies."
The balloon analogy is very good but needs to be understood properly—otherwise it can cause more confusion. As Hoyle said, "There are several important respects in which it is definitely misleading." It is important to appreciate that three-dimensional space is to be compared with the two-dimensional surface of the balloon. The surface is homogeneous with no point that should be picked out as the centre. The centre of the balloon itself is not on the surface, and should not be thought of as the centre of the universe. If it helps, you can think of the radial direction in the balloon as time. This was what Hoyle suggested, but it can also be confusing. It is better to regard points off the surface as not being part of the universe at all. As Gauss discovered at the beginning of the 19th century, properties of space such as curvature can be described in terms of intrinsic quantities that can be measured without needing to think about what it is curving in. So space can be curved without there being any other dimensions "outside". Gauss even tried to determine the curvature of space by measuring the angles of a large triangle between three hill tops.