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Take a sphere or ellipsoid or similar (hereafter just called sphere) ... and imagine pinching it in the middle, deforming it by moving two points that were on opposite sides of the sphere inward until they overlap, becoming one point.

The resulting shape is, as I said in the title, among those things you could imagine meant by "three dimensional figure 8" ... It could also be described as two raindrop lobes connected at a point to form one shape. Or, what you (kind of) make if you take a flexible round balloon and make a twist in one place.

I have been assuming that this meets the criteria to be a shape, correct me if I am wrong there.

If it is a shape, I want to know what its name(s) are, and anything else you might know about it, mathematically.

The name is the important part, though, because I can use a name to look up more about it.

Double lobe, or bilobe, maybe... I could make up a name, but would rather know what mathematicians call it.

In my search, I have found lemniscates, but those do not seem right. They are more... membrane-like, not filled out. In pictures they appear dissimilar to the shape I'm asking about here.

I am most knowledgable in biology, not math, or even physics, and don't really know where or how to look ...

So, thank you very much to anyone who addresses or answers my question.

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I don't believe such a shape has a single-word name like "sphere" or "cube" associated to it. However, in mathematics we can characterize such a shape as "the wedge of two spheres" and write it symbolically as $$S^2\vee S^2$$ $S^2$ denotes the "$2$-sphere" (Wikipedia link). Note that in mathematics this refers specifically to the "hollow" sphere; if you meant in your question to refer to a "filled-in" sphere, then the correct mathematical word is "$3$-ball" (Wikipedia link) and you would write $B^3\vee B^3$ instead.

The $\vee$ in the middle is the "wedge sum" operation (Wikipedia link). It takes two "shapes" (i.e., topological spaces) and glues them together at a single point. But of course, taking two spheres and attaching them at a single point produces the same shape as starting with one sphere and pinching it in the middle.

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It could be a lemniscate (https://en.wikipedia.org/wiki/Lemniscate). Lemniscate is technically 2D and you may want a 3D shape, but you can obtain that by naming "3D lemniscate" the surface obtained by a revolution of the lemniscate around its (pinched) center.

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