Space We Live In

Humans tried to understand the space in which they and the objects in their world were situated. They realized that the space itself, though it was largely empty, can also be thought of as having 3 dimensions

Understanding Dimensions

The idea of dimensions is a very abstract & difficult concept. Let us try and give a simple explanation.

Imagine that you have to fix a fan on the ceiling of a room. How do you tell the electrician where exactly to hang the fan? In this particular case, it may be easy to take the electrician to that particular room and point out the hook on the roof with your finger or a stick.

But this way of pointing out may not be feasible in all cases, like if that point is so far away that it is not visible. In most cases, the exact location may have to be specified in writing or in a drawing to avoid any confusion.

The fan can be located with certain movements with associated distances. For standardizing these movements, we have fix the directions in which the movements take place. Geometers realized that these moves have to be in 3 directions which are perpendicular to one another.

First move would be in any one of the directions along the floor for a certain distance. Second move would again be on the floor for a certain distance but in a direction perpendicular to the first move. These 2 directions and the distances have to be coordinated in such a way that we reach a point on the floor directly below the fan hook. The third move would be vertically up for a certain distance until we reach the hook in the ceiling where the fan needs to be fixed. Hence we see that this requires a minimum of 3 measurements in 3 mutually perpendicular directions.

This idea was generalized that any location in space, can be reached by a series of 3 moves for certain distances in 3 mutually perpendicular directions. Hence the space in which we exist was called 3 dimensional.

The same idea was used by mathematician Rene Descartes in devising a system of 3 mutually perpendicular axes as the framework for Coordinate Geometry which paved the way for combining Geometry with Algebra. The coordinate system is also called Cartesian Coordinates.

The 3 "distances" at which any point in space is located from the 3 mutually perpendicular axes are called the "coordinates" of the point. Together they fix the location of that point in space.

We will see this in details in chapter 28.11.

Polar Coordinates

In the 18th century, Gregorio Fontana gave the term "polar coordinates" to another system which also helped locate a point in space with 3 numbers.

This system was based on 2 angles which a line joining a point and the origin makes with two orthogonal axes along with the distance of the point from the origin.

3D Objects

Similarly it was also realized that all the objects in this world, including living beings can be grasped by our hands. Since our world was filled with solids, we started studying their shapes & properties. We call them as solids.

All solids cast a shadow on surfaces. These shadows are spread in 2 directions that we can think of as length & breadth. But shadows do not have any thickness which rises above the surface on which they rest. Hence a shadow could be thought of as having 2 dimensions. It cannot be grasped physically by our hands.

In contrast, the solid object which casts a 2D shadow, also has a vertical extension in which it exists above this base. We call this as thickness or height. This extension could be thought of as an additional dimension which is independent of the 2 dimensions of the 2D space.

Hence all objects in this world can be thought of as having 3 dimensions. They were called 3D objects.