A balloon is used as analog to explain how the universe is expanding
because it appears to be a 2 spatial dimensional object expanding into
3 spatial dimensions.  But then, when they talk about the actual
universe, they say it is not really 3 spatial dimensions expanding
into 4 spatial dimensions.  Can you tell me (or tell me an accessable
source) where the balloon analog fails?

I've never heard the balloon analogy used that way, it's supposed to
show that an observer at any point on the balloon (universe) sees all
the other points moving away.  A similar analogy is cherries in a
fruit cake - as the cake rises each cherry sees all the other cherries
moving away.

Ian G.

The balloon analogy isn't thoroughly analogous to the way spatial
expansion is thought to be by relativists. It does fail if mistakenly
used as an actual model of the Universe. The balloon analogy is really
only a handy illustration of certain aspects of big-bang cosmology.

   It's similar to the rubber sheet illustration of curved space as
gravity, which itself uses pre-existing gravity generated weight to
induce the curvature. This also is *only* an illustration.

The balloon analogy fails because you are trying to represent a 4-D
process with a 3-D illustration and to our 3-D brain. We simply cannot
imagine a creation and "expansion" of a 4-D entity. You see, the dots
analogy resides on just the surface of the balloon, in effect limiting
it to 2-D, while the space expands in at least 3-D. (4-D if you count
time)