The break cue is used for the break in billiards, but under most rules, is not any different than any other cue. A player wants to use a very hard tip and a light cue for a break stick, and may also want a different balance point than for the play cue. The break cue is often also used for jump shots.
'Spider' is the most common, or the 'extended spider', or a 'swan neck' rest.
billiards
There are generally two terms used for billiards - English billiards or pocket billiards. Pocket billiards is also called pool. Billiards is a popular bar game that has a number of different names that it has been called over the years but the most notable and popular is simply that of pool.
In American pool, break is used to describe the first shot of the game where all of the object balls are racked together and the cue ball 'breaks' them up. In snooker, a chiefly British game, break is used to describe a series of consecutive pots made by the player. This is similar to what would be called a 'run' in American pool.
Billiards is not directly related to math, but is related to advanced math - Geometry, trigonometry, calculus and physics can all be used and demontrated with the complex actions that occur on a billiards table. At least one US professor uses billiards as a part of complex physics demonstrations.
There is no record of the first billiards games. However it is documented it was being played in the 1420's on tables similar to those used today.
Two. There are two white and one red ball used for billiards.
The movement of the balls, after collision, obeys the laws of mechanics (a branch of mathematics).
Billiards can be a generic term used to catagorize all cue sports. It is also the name of a specific game which is played with only 3 balls and on a table with no pockets. It is also the term used to describe the action of glancing, or carom, off a ball to make contact with another or to pocket a ball.
The plow. At first, it was only made of wood, but as technology advanced and forging iron became possible, plows became more efficient.
There are no points for pocketing a particular color ball in billiards. However, snooker awards 1 point for potting (the term used for pocketing by snooker players) a red ball.
"Cue" can refer to a signal or prompt to start or do something. It is commonly used in the context of giving an indication or prompting a specific action. In media, a cue can also refer to a signal for an actor or crew member to perform a specific task.