Point (geometry)
In modern mathematics, a point refers usually to an element of some set called a space.
More specifically, in Euclidean geometry, a point is a primitive notion upon which the geometry is built. Being a primitive notion means that a point cannot be defined in terms of previously defined objects. That is, a point is defined only by some properties, called axioms, that it must satisfy. In particular, the geometric points do not have any length, area, volume, or any other dimensional attribute. A common interpretation is that the concept of a point is meant to capture the notion of a unique location in Euclidean space.
Points in Euclidean geometry
Points, considered within the framework of Euclidean geometry, are one of the most fundamental objects. Euclid originally defined the point as "that which has no part". In two-dimensional Euclidean space, a point is represented by an ordered pair (x, y) of numbers, where the first number conventionally represents the horizontal and is often denoted by x, and the second number conventionally represents the vertical and is often denoted by y. This idea is easily generalized to three-dimensional Euclidean space, where a point is represented by an ordered triplet (x, y, z) with the additional third number representing depth and often denoted by z. Further generalizations are represented by an ordered tuplet of n terms, (a1, a2, … , an) where n is the dimension of the space in which the point is located.
Geometric group action
In mathematics, specifically geometric group theory, a geometric group action is a certain type of action of a discrete group on a metric space.
Definition
In geometric group theory, a geometry is any proper, geodesic metric space. An action of a finitely-generated group G on a geometry X is geometric if it satisfies the following conditions:
Uniqueness
If a group G acts geometrically upon two geometries X and Y, then X and Y are quasi-isometric. Since any group acts geometrically on its own Cayley graph, any space on which G acts geometrically is quasi-isometric to the Cayley graph of G.
Examples
Cannon's conjecture states that any hyperbolic group with a 2-sphere at infinity acts geometrically on hyperbolic 3-space.
References
American football rules
Game play in American football consists of a series of downs, individual plays of short duration, outside of which the ball is dead or not in play. These can be plays from scrimmage – passes, runs, punts, or field goal attempts (from either a place kick or a drop kick) – or free kicks such as kickoffs and fair catch kicks. Substitutions can be made between downs, which allows for a great deal of specialization as coaches choose the players best suited for each particular situation. During a play, each team should have no more than 11 players on the field, and each of them has specific tasks assigned for that specific play.
Objective of the game
The objective of this game is to score more points than the other team during the alotted time. The team with the ball (the offense) has 4 plays (downs) to advance at least 10 yards, and can score points once they reach the opposite end of the field, which is home to a scoring zone called the end zone, as well as the goal posts. If the offense succeeds in advancing at least 10 yards, they earn a "first down" and the number of tries allotted is reset and they are again given 4 tries to advance an additional 10 yards, starting from the spot to which they last advanced. If the offense does not advance at least 10 yards during their 4 downs, the team without the ball (the defense) regains control of the ball (called turnover on downs).
Point (mortgage)
Points, sometimes also called "discount points", are a form of pre-paid interest. One point equals one percent of the loan amount. By charging a borrower points, a lender effectively increases the yield on the loan above the amount of the stated interest rate. Borrowers can offer to pay a lender points as a method to reduce the interest rate on the loan, thus obtaining a lower monthly payment in exchange for this up-front payment. In United States, for each point purchased, the loan rate is typically reduced by anywhere from 1/8% (0.125%) to 1/4% (0.25%).
Selling the property or refinancing prior to this break-even point will result in a net financial loss for the buyer while keeping the loan for longer than this break-even point will result in a net financial savings for the buyer. The longer you keep the property financed under the loan with purchased points, the more the money spent on the points will pay off. Accordingly, if the intention is to buy and sell the property or refinance in a rapid fashion, paying points is actually going to end up costing more than just paying the loan at the higher interest rate.
Point (ice hockey)
In ice hockey, point has three contemporary meanings:
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