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Forgotten 4D object: The duo-cone

May 1, 2012

This page: http://eusebeia.dyndns.org/4d/geom.html discusses a lot of different 4D objects. One of them is called a “cylindrone“, which starts with a cylinder in 3D, which is extruded into the 4th dimension and tapered to a point. A lower dimensional analogy would be taking a circle on a paper, and raising it up into the third dimension (extruding), but at the same time shrinking it to a point as it rises (tapering). This process generates a cone. So if you do this with a 3D cylinder into the fourth dimension, you are creating a kind of hypercone, but with a cylindrical base. So this page had an illustration of this, but the description was originally that of tapering a cone into 4D. That’s a different object. That would be a pure hypercone; a 4D cone with a conic base (as opposed to another analogue, a spherone, which tapers a sphere into 4D, which would then be a hypercone with a spherical base).
So I drew this image of the object, which I call a “duo-cone“, since it is conic in both the 3D hyperplane, and in the 4D plane. I mailed him about this with the image, and he said he would look into it. He since corrected the description and redid the page with shiny new transparent colored images, but still left off this one. So I might as well host it for now.
When the initial cone begins to be moved into 4D, the circlular base generates a new cone sharing the same base, the apex generates a ridge connecting the apices of both cones, and the sloped, curved sides sweep out a kind of hypertorus that “fills” the figure in the image.

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