Sphere Reciprocal? Not Inside Out, But Equation Inverse?

One way of mathematically representing a very simple sphere in 3D space is,

r² = x² + y² + z²

where r equals a radius of the sphere. Solving in terms of x, y, and z, we get,

x = √(r² — y² — z²)
y = √(r² — x² — z²)
z = √(r² — x² — y²)

A Sphere Reciprocal

Now a sphere may be the most aesthetically pleasing of the simple geometric curves. So it is natural to wonder, concerning a sphere, what if…? So what if we convert the equation into an equation for a sphere reciprocal? No, not turn the sphere inside out. Rather an inverse of the equation of a sphere? What is the graph of,

r² = 1/( x² + y² + z²) ?

Is it ordinary, or could it be something “romantic,” perhaps something like the popular image of a black hole? Do you have the mathematical skills and (hopefully) the graphing software so you can produce a visual of the curve? If so, and you are willing to share the image, please do. Click Contact Us and attach it along with any narration. I would be pleased to credit you for the image and include it with this article.

Note: You might also enjoy Collapsing Then Expanding the Equation for a Sphere

References:

• University of Illinois at Urbana-Champaign: History of Sphere Eversions

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