# What is an inverted sphere?

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A sphere. A sphere is round no matter what. Therefore a sphere, when it is inverted, is still a sphere.

A sphere turned inside out. It’s like a circle: it only has 2 sides.

That is just freaking amazing!

I tried picturing it somehow collapsing into its center and becoming negative space. Couldn’t make that work for me.

Don’t ask me to do the math or explain the math behind it. It makes my brain hurt and then it turns into an Escher drawing. The days of doing that sort of brain exercise is over for me, I’m afraid. But I can call on the concepts, still, and the terminology.

Technically speaking, we are talking about the eversion of a sphere. I found this link. Interestingly, the article says that there is no way of turning a circle inside out.

A spherical inversion is different from eversion. In 3 dimensions, an inversion can be thought of as the reflection in a spherical mirror. It maps points on the sphere to themselves. Points inside the sphere are mapped to the outside of the sphere and vice versa.

I looked on the Web for a simple explanation but could not find one, so here is how it works. For any point p, let d be the distance from p to c, the center of the sphere. Consider the value d/r, where r is the radius of the sphere. For p outside the sphere, d/r > 1 and for p inside the sphere, d/r < 1. The effect of the spherical inversion is to move p along the ray joining p to c in such a way that the new distance d’ satisfies d’/r is one over d/r. If d/r was 3 then d’/r is ⅓.

An inverted sphere is the inversion of one sphere in another. In two dimensions, the circular inversion of a circle is a circle, provided that the circle does not pass through the circle of inversion. I would guess that in a similar fashion, the inversion of a sphere would be another sphere.

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