## the fourth tapered torus

Discussion of shapes with curves and holes in various dimensions.

### the fourth tapered torus

At the moment, we can produce three different objects by tapering a torus. We make torus -> circle by reducing the minor radius to zero. We make torus -> sphere by reducing the major radius. And we make torus -> point by reducing (or increasing) both radii simultaneously.

Now, there is a fourth object we can make, by reducing one radius while simultaneously increasing the other.

We would start with a circle, i.e. a torus with a minor radius of 0. Then we increase the minor radius and decrease the major radius while going up the w-axis. Eventually we end up with a sphere, i.e. a torus with a major radius of 0.

I don't think it's a good idea to add this to the official list of tapertopes, but it's an interesting object. It's RNS representation might be (2<sup>1</sup>1)<sup>-1</sup> or (2<sup>-1</sup>1)<sup>1</sup>. It could also be called "circle -> sphere via torus"

PWrong
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### Re: the fourth tapered torus

Why didn't you post this in cone-like objects?

PWrong wrote:(2<sup>1</sup>1)<sup>-1</sup> or (2<sup>-1</sup>1)<sup>1</sup>

I wouldn't agree with either of those. How can you taper negative times?

Keiji
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Why didn't you post this in cone-like objects?

I just didn't want to change the subject.

I wouldn't agree with either of those. How can you taper negative times?

It just means you're tapering once in the opposite direction.

PWrong
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Still makes no sense.

Keiji
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Well, we don't have to use that notation. The object itself is interesting.

Anyway, I noticed something interesting about the torus->sphere. Suppose the angle at the base is pi/4. Then take a cross section, so the torus becomes a pair of circles. The object will be a pair of cylinders intersecting at right angles. So the torus->sphere is somehow related to the crind.

PWrong
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