Physicists compare gravitational theories in curved anti-de-sitter spaces on the one hand, and quantum field theories in spaces with one fewer dimension on the other. The former have peculiar properties: they are negatively curved and any object thrown away on a straight line will eventually return. But our world and our bodies do not exist in such an ‘anti-de-sitter-space’ – that is the domain of another dimension. The visible universe, in contrast, is quite flat – and has positive curvature. What we perceive as three dimensional may just be the image of two dimensional processes on a huge cosmic horizon - think of the holograms on credit cards or banknotes. They are two dimensional, but to us they appear three dimensional. So, think of our dimension as a flat one incorporating a hologram, and the next one – call it the fourth if you like – as the reality sitting above us that truly does extend in all directions.
(For more on the holograph, see: http://thewatchers.adorraeli.com/2015/05/03/is-the-universe-a-hologram/)