Question: As we know, (1) the macroscopic spatial dimension of our universe is 3 dimension, and (2) gravity attracts massive objects together and the gravitational force is isotropic without directional preferences. Why do we have the spiral 2D plane-like Galaxy(galaxies), instead of spherical or elliptic-like galaxies?

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Gravity is (at least, seems to be) isotropic from its force law (Newtonian gravity). It should show no directional preferences from the form of force vector $\vec{F}=\frac{GM(r_1)m(r_2)}{(\vec{r_1}-\vec{r_2})^2} \hat{r_{12}}$. The Einstein gravity also does not show directional dependence at least microscopically.

If the gravity attracts massive objects together isotropically, and the macroscopic space dimension is 3-dimensional, it seems to be natural to have a spherical shape of massive objects gather together. Such as the globular clusters, or GC, are roughly spherical groupings Star cluster, as shown in the Wiki picture:

However, my impression is that, even if we have observed some more-spherical or more-ball-like Elliptical galaxy, it is more common to find more-planar Spiral galaxy such as our Milky Way? (Is this statement correct? Let me know if I am wrong.)

Also, such as this NGC 4414 galaxy:

Is there some physics or math theory explains why the Galaxy turns out to be planar-like (or spiral-like) instead of spherical-like?

I don't think I'd say that spirals are more common than ellipticals. It depends on where in space and in cosmic history you look, but ellipticals certainly aren't rare (though they don't get as much publicity with spectacular photographs as spirals because they're well... kind of boring).
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KyleJan 15 '14 at 19:57

2 Answers
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To understand how, let us as a starting point look at Wikipedia's sketch of the structure of a spiral galaxy:

A spiral galaxy consists of a disk embedded in a spheroidal halo. The galaxy rotates around an axis through the centre, parallel to the GNP$\leftrightarrow$GSP axis in the image. The spheroidal halo consists mostly of Dark Matter (DM), and the DM makes up $\sim90\%$ of the mass of the Milky Way. Dynamically, it is the DM, that, ehrm, matters. And DM will always arrange itself in a ellipsoid configuration.

So the question should rather be: Why is there even a disk, why isn't the galaxy just an elliptical? The key to answering this lies in the gas content of a galaxy.

Both stars and Dark Matter particles - whatever they are - are collisionless; they only interact with each other through gravity. Collisionless systems tend to form spheroid or ellipsoid systems, like we are used to from elliptical galaxies, globular clusters etc.; all of which share the characteristic that they are very gas-poor.

With gas it is different: gas molecules can collide, and do it all the time. These collisions can transfer energy and angular momentum. The energy can be turned into other kinds of energy, which can escape, through radiation, galactic winds etc., and as energy escapes, the gas cools and settles down into a lower energy configuration. The gas' angular momentum, however, is harder to transfer out of the galaxy, so this is more or less conserved. The result - a collisional system with low energy but a relatively high angular momentum - is the typical thin disk of a spiral galaxy. (Something similar, but not perfectly analogous, happens in the formation of protoplanetary disks).

Stars also do not collide, so they should in theory also make up an ellipsoid shape. And some do in fact: the halo stars, including but not limited to the globular clusters. These are all very old stars, formed when the gas of the galaxy hadn't settled into the disk yet (or, for a few, formed in the disk but later ejected due to gravitational disturbances). But the large majority of stars are formed in the gas after it has settled into the disk, and so the large majority of stars will be found in the same disk.

Elliptical galaxies

So why is there even elliptical galaxies? Elliptical galaxies are typically very gas-poor, so gas dynamics is not important in these, they are rather a classical gravitational many-body system like a DM halo. The gas is depleted from these galaxies due to many different processes such as star formation, collisions with other galaxies (which are quite common), gas ejection due to radiational pressure from strongly star forming regions, supernovae or quasars, etc. etc. - many are the ways for a galaxy to lose its gas. If colliding galaxies are sufficiently gas-depleted (and the collision results in a merger), then the resulting galaxy will not have any gas which can settle into a disk, and kinetic energy of the stars in the new galaxy will tend to be distributed randomly due to the chaotic nature of the interaction.

(This picture is simplified, as the whole business of galactic dynamics is quite hairy, but I hope it gets the fundamentals right and more or less understandable).

I do not think the assumption of a random 3D environment is justified, galaxies are not separated from their environment.
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ThrivethOct 6 '14 at 12:17

Besides @HelderVelez , I don't see what the answers in the question you link to has to do with my answer or, for that matter, your own comment...?
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ThrivethOct 6 '14 at 12:25

"Stars also do not collide"? How so? Even if they don't collide head-on, but pass each other somewhat closely, they will both be deflected by each other's gravitational field.
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endolithNov 25 '14 at 23:28

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@endolith Yes, but gravitational tugging is a very different process from a collision and has different consequences - for example, equipartition of energy does not happen, thermodynamic equilibrium does not happen.
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ThrivethNov 26 '14 at 2:19

A less popular version of physics.stackexchange.com/a/148423/56960 , but needs downvotes as well. Not all galaxies have (prominent) nuclei, not all galaxies have a noticeable total angular momentum (although most of them indeed have), and survival of a huge stellar system for billions years isn’t necessarily determined by its rotation.
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Incnis MrsiNov 24 '14 at 17:32