## Point on a Line, a Line on a Plane, and a Plane in Space

Each point has a specific location. Two points determine a line. Three points determine a plane. Let us consider some simple math derivations to arrive at a format for each. For simplicity’s sake, we will use the familiar x, y, z Cartesian coordinate system. We begin with a point on a line. First, Point on a Line In space, a single point has an x value, a y value, and a z value. If the coordinate system chosen for the point is a simple 1-D line, then only one variable – say x – is needed to describe it. Then, since there is no y or z to consider, the mathematical description of the point is x = c But let us, for reasons that will be understood later, write…

## Collapsing then Expanding the Equation for a Sphere

[caption id="attachment_8626" align="alignright" width="480"] How simple is a sphere?[/caption] Equation for the Simplest Sphere The equation for a sphere with its center at the origin is: x2 + y2 + z2 = c2 Where c is a positive constant. For simplicity, let's choose a positive constant, k, such that k = c2. Equation for a Circle by Collapsing a Sphere Collapsing it in one dimension generates the equation of one of three circles: x2 + y2 = k x2 + z2 = k y2 + z2 = k Equation for a Point by Collapsing a Circle Collapsing the three circles in one dimension generates two equations representing precisely two points for each of them: For x2 + y2 = k, x²2 = k y2 = k For x2 + z2…

## What is a Negative Number?

Maybe a negative number is simply a number headed in a different direction. After all, an imaginary number is real. Opposite charges don’t cancel. Antimatter isn’t the opposite of matter. Male and female compliment each other. Instead of three dimensions, maybe there are six dimensions. Or if you include time, maybe there are seven dimensions. Coordinate Systems If there truly is no such thing as a negative number, then both halves of the x-axis, at 180º with respect to each other are positives headed in different directions. What would stand in the way of there being an infinite number of positive directions in between what was formerly negative x and positive x, each rotated a miniscule amount? Waves Even waves that cancel one another at various points may sing a…