Mapping the third dimension involves extending our mathematical landscape from the flat $\mathbb{R}^2$ plane into $\mathbb{R}^3$ by establishing three mutually perpendicular directed lines (the x, y, and z-axes) that meet at the origin $O$.
Just as we use the Maclaurin series for the exponential function, $e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}$, to build complex functions from simple polynomial terms, we build 3D space by partitioning it into eight octants using three intersecting coordinate planes (xy, yz, and xz). This transition allows us to locate any point P as an ordered triple (a, b, c), representing its directed distances from these planes—moving from the "infinite complexity" of a 2D snowflake curve to the structured volume of the physical world.
The Geometry of $\mathbb{R}^3$
To identify points in space, we fix three directed lines through $O$ that are perpendicular to each other, called the x-axis, y-axis, and z-axis. Their orientation follows the Right-Hand Rule: if you curl the fingers of your right hand from the positive x-axis toward the positive y-axis, your thumb points toward the positive z-axis (Figure 2).
The three coordinate axes determine the three coordinate planes: the xy-plane ($z=0$), the yz-plane ($x=0$), and the xz-plane ($y=0$). These planes divide space into eight parts called octants. The first octant is where all coordinates are positive.
For any point $P$, the triple $(a, b, c)$ contains the x-coordinate ($a$), y-coordinate ($b$), and z-coordinate ($c$). These are the directed distances from the yz, xz, and xy planes, respectively.
Mathematical Mapping Analogy
Locating a point $P(a, b, c)$ by summing components is conceptually similar to summing the terms of a series. Consider finding the sum of the series $\sum_{n=0}^{\infty} \frac{(x+2)^n}{(n+3)!}$. This requires recognizing the familiar pattern of the $e^x$ Maclaurin series.
The series $\sum_{n=0}^{\infty} \frac{(x+2)^n}{(n+3)!}$ is related to $e^{x+2} = \sum_{n=0}^{\infty} \frac{(x+2)^n}{n!}$. To solve this, we manipulate the index to match the familiar form:
$$\sum_{n=0}^{\infty} \frac{(x+2)^n}{(n+3)!} = (x+2)^{-3} \left[ e^{x+2} - 1 - (x+2) - \frac{(x+2)^2}{2!} \right]$$
Just as we identify ingredients in a power series, we identify axes and planes to determine spatial position.
The Pitfall of Dimension
Note: When an equation is given, we must understand from the context whether it represents a curve in $\mathbb{R}^2$ or a surface in $\mathbb{R}^3$.
- Equation $y=5$: In $\mathbb{R}^1$, it's a point. In $\mathbb{R}^2$, it's a horizontal line. In $\mathbb{R}^3$, it is an entire plane parallel to the xz-coordinate plane (Figure 7).
- Equation $y=x$: In $\mathbb{R}^3$, since $z$ is "free," this equation represents a vertical plane passing through the z-axis, cutting the xy-plane along the line $y=x$.