Collapsing then Expanding the Equation for a Sphere

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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,

2 = k

y2 = k

For x2 + z2 = k,

x2 = k

z2 = k

For y2 + z2 = k,

y2 = k

z2 = k

Thus a 3-D figure transforms into three 2-D equations. A 2-D equation transforms into two 1-D equations. A 1-D equation may form only 1 0-D equation.

Expanding the Equation of a Sphere

Reversing the list just above, we see that:

Two 1-D equations become one 2-D equation.

Three 2-D equations become one 3-D equation.

Four 3-D equations lead to one 4-D equation.

Allow Me to Introduce Time

Following the pattern established throughout this article, let’s choose T as the fourth variable. We’ll call it “time,” since in our modern era, time has been called the fourth dimension.

If we replace each of the variables with T, we get:

t2 + y2 + z2 = k

x2 + t2 + z2 = k

x2 + y2 + t2 = k

Including the above three equations with the original sphere equation:

x2 + y2 + z2 = k

We can reverse the dimension collapsing process to get:

x2 + y2 + z2 + t2 = k

If this last equation is a valid one, then so are equations 1 – 4 resulting from its collapse.

Introspection AKA Brain Burnout

The four equations can be understood, since this appears to humans to be a 3-D world we live in. Equation 4 we already comprehend. Equations 1 – 3 include two spatial dimensions plus time. As time deviates from zero in either direction (whether or not there is such a thing as negative time), the circle (if it can be visualized as a circle, associated with this kind of 3-D object) gets smaller, until it reaches the square root of k, whose absolute value is positive (equal to c²).

The spatial part of the sphere comes and goes. And so it would be for the other two space-time spheres. Only the strictly spatial sphere remains unchanging. If any of what I have said above makes sense (and I’m not certain that it does), I have reached my endpoint, at least for now. Any comments from those who specialize in mathematics?

Note: You might also enjoy Too Quirky Even For Me: The Mathematics of a Tornado