# X,y,z,t: dimensions of science fiction - X, Y, Z, T: Dimensions of Science Fiction by Damien Broderick

Three-dimensional space (also: 3-space or, rarely, tri-dimensional space ) is a geometric setting in which three values (called parameters ) are required to determine the position of an element (i.e., point ). This is the informal meaning of the term dimension .

In mathematics, analytic geometry (also called Cartesian geometry) describes every point in three-dimensional space by means of three coordinates. Three coordinate axes are given, each perpendicular to the other two at the origin , the point at which they cross. They are usually labeled * x* , * y* , and * z* . Relative to these axes, the position of any point in three-dimensional space is given by an ordered triple of real numbers , each number giving the distance of that point from the origin measured along the given axis, which is equal to the distance of that point from the plane determined by the other two axes. [1]

Other popular methods of describing the location of a point in three-dimensional space include cylindrical coordinates and spherical coordinates , though there are an infinite number of possible methods. See Euclidean space .

Two distinct points always determine a (straight) line . Three distinct points are either collinear or determine a unique plane. Four distinct points can either be collinear, coplanar or determine the entire space.

Two distinct lines can either intersect, be parallel or be skew . Two parallel lines, or two intersecting lines , lie in a unique plane, so skew lines are lines that do not meet and do not lie in a common plane.

Two distinct planes can either meet in a common line or are parallel (do not meet). Three distinct planes, no pair of which are parallel, can either meet in a common line, meet in a unique common point or have no point in common. In the last case, the three lines of intersection of each pair of planes are mutually parallel.

Three-dimensional space (also: 3-space or, rarely, tri-dimensional space ) is a geometric setting in which three values (called parameters ) are required to determine the position of an element (i.e., point ). This is the informal meaning of the term dimension .

In mathematics, analytic geometry (also called Cartesian geometry) describes every point in three-dimensional space by means of three coordinates. Three coordinate axes are given, each perpendicular to the other two at the origin , the point at which they cross. They are usually labeled * x* , * y* , and * z* . Relative to these axes, the position of any point in three-dimensional space is given by an ordered triple of real numbers , each number giving the distance of that point from the origin measured along the given axis, which is equal to the distance of that point from the plane determined by the other two axes. [1]

Other popular methods of describing the location of a point in three-dimensional space include cylindrical coordinates and spherical coordinates , though there are an infinite number of possible methods. See Euclidean space .

Two distinct points always determine a (straight) line . Three distinct points are either collinear or determine a unique plane. Four distinct points can either be collinear, coplanar or determine the entire space.

Two distinct lines can either intersect, be parallel or be skew . Two parallel lines, or two intersecting lines , lie in a unique plane, so skew lines are lines that do not meet and do not lie in a common plane.

Two distinct planes can either meet in a common line or are parallel (do not meet). Three distinct planes, no pair of which are parallel, can either meet in a common line, meet in a unique common point or have no point in common. In the last case, the three lines of intersection of each pair of planes are mutually parallel.

The particle motion of a Love wave forms a horizontal line perpendicular to the direction of propagation (i.e. are transverse waves ). Moving deeper into the material, motion can decrease to a "node" and then alternately increase and decrease as one examines deeper layers of particles. The amplitude , or maximum particle motion, often decreases rapidly with depth.

Since they decay so slowly, Love waves are the most destructive outside the immediate area of the focus or epicentre of an earthquake. They are what most people feel directly during an earthquake.

In the past, it was often thought that animals like cats and dogs could predict an earthquake before it happened. However, they are simply more sensitive to ground vibrations than humans and able to detect the subtler body waves that precede Love waves, like the P-waves and the S-waves. [1]

where i = − 1 {\displaystyle i={\sqrt {-1}}} . The stresses caused by these displacements are

If we substitute the assumed displacements into the equations for the conservation of momentum, we get a simplified equation

The above equations describe an eigenvalue problem whose solution eigenfunctions can be found by a number of numerical methods . Another common, and powerful, approach is the propagator matrix method (also called the matricant approach)