Let us next direct our attention to ${\displaystyle 3}$-space, an inhabitant of which we might call an animal, but which, to continue the nomenclature adopted, we shall sometimes in a general way speak of as a tridim. Here freedom of life is much more augmented, even more so than in Fig. 7. passing from ${\displaystyle 1}$-space to ${\displaystyle 2}$-space. For here we have added the up-and-down motion to the right-and-left and the forward-and-backward motions. Here any point is located by means of its distances from three mutually perpendicular planes, each plane being formed by two of the three lines that can be drawn mutually perpendicular to one another. In Fig. 7, ${\displaystyle Ox,Oy,Oz}$—representing directions to the right, hitherward and upward, respectively—are the axes of reference, each being perpendicular to the other two, forming the mutually perpendicular planes, ${\displaystyle xOy,yOz,zOx}$.

We saw that in ${\displaystyle 2}$-space the axes ${\displaystyle xx',yy'}$ divided the space into four equal parts of indefinite extent. A straight line in ${\displaystyle 2}$-space divides that space into two parts. In ${\displaystyle 3}$-space, it is evident that the coordinate planes divide space into eight equal parts of indefinite extent. Any point in ${\displaystyle 3}$-space is definitely determined when its distances from the three planes of reference is known. Distances perpendicular to the ${\displaystyle yz}$ plane, denoted by ${\displaystyle x}$, are positive if measured to the right, negative if measured to the left; distances perpendicular to the ${\displaystyle xz}$ plane, denoted by ${\displaystyle y}$, are positive if measured towards us, negative if measured away from us;