*- Ramanraj K*

14^{th}July, 2017

#### Introduction

2-D and 3-D co-ordinates can be plotted with absolute certainty. However, 4-D and higher dimensions in n-D spaces are not clearly defined and their co-ordinates cannot be plotted with mathematical certainty. A definition of dimensions higher than three is necessary for clarity and use in mathematics, physics and computing. This would help both man and machine to describe and visualise virtual models of the world.#### 1-D Space

The one dimensional space is defined as a stright line along x-axis with a single vertex O in the centre.

Example: A railway track with no branches.

#### 2-D Space

Two dimensional space is defined by two intersecting perpendicular straight lines x and y with a single vertex O.

Example: The screen frame, with top and left as co-ordinates.

#### 3-D Space

Three dimensional space is defined by two intersecting perpendicular straight lines x and y with a single vertex O, and a third straight line z perpendicular to both x and y passing through O.

Example: A train engine, a ball, earth, sun, and moon in space.

#### 4-D Space

Four dimensional space is the 3-D space defined by lines x, y and z as above whose vertex O_{1} lies on a straight line w with vertex O. Therefore, in 4-D space at least two vertices co-exist and the distance between the vertices O and O_{1} is defined by the 4th dimension w. The lines x and w coincide. Let a 3-D space with vertex O_{1} be defined as follows:

Then, the 4-D space is as follows:

In the above figure, w = 4, and the distance between the vertices O and O_{1} is 4 units. This is useful to define the relationship between a 3-D object with a given vertex O and another vertex O_{1}. Any number of 3-D objects may lie on line w and the distance between O and the vertices O_{1} to O_{n} may be defined by w.

Example: A train engine defined by 3-D co-ordinates on a track, with no branches. The track is the 4th dimension, and if a station is located on the vertex of the straight line, then the distance between the train and the station is given by the 4th dimension.

#### 5-D Space

Five dimensional space is the 3-D space defined above whose vertex lies in a 2-D plane. Therefore, in 5-D space at least two vertices exist and the distance between them is defined by the 4th and 5th dimensions.

#### 6-D Space

Six dimensional space is the 3-D space whose vertex lies in another 3-D space. In 6-D space at least two vertices exist and the distance between the two is defined by the 4th, 5th and 6th dimensions.

#### 7-D Space

In seven dimensional space, at least three vertices exist, with a six dimensional space having at least two vertices O_{1} and O_{2} where O_{1} lies along on a straight line s with vertex O. Let the following be a six dimensional space:

Then, the 7-D space is as follows:

#### 8-D Space

In eight dimensional space, at least three vertices exist, with a six dimensional space having at least two vertices O_{1} and O_{2} where O_{1} lies along on a plane with vertex O, and axis s and r.

#### 9-D Space

In nine dimensional space, at least three vertices exist, with a six dimensional space having at least two vertices O_{1} and O_{2} where O_{1} lies on the 3-D space with vertex O and axis s, r and q.

#### n-D Space

The above pattern can be nested to infinite levels, in sets of three. The sun, earth and moon can be plotted without any ambiguity in nine dimensions. Other dimensions may be added in sets of three, down to fundamental particles, or higher up to other galaxies, and their relationships may be mapped and studied with more clarity.

For example a 12-D space would be as follows:

A higher number of dimensions can always be expressed in lesser number of dimensions. For example, no matter how many levels deep, they can be plotted in a 2-D plane of the screen, or a 1-D line of bits and components.

Arrays can be used to efficiently store information about the dimensional space. The dimensions higher than 3 spatially relate to mod 3 and can be represented as a graph with parent and child nodes. Any two nodes may be singled out as frames of reference.

Last modified: Fri Jul 14 05:21:45 IST 2017