Vorpals
I can not visualize four-dimensional objects but I also cannot deny their existence. Our three-dimensional constraints are empirical, not logical.
Among such objects, one is of special interest. It shares with the circle and the sphere the property that all points on its surface are equidistant from a center in its interior.
I know of no name for this object; until I know better I shall refer to it as a “vorpal”.
The space capacity of a vorpal is
8π2(R4)/17
where “R” is the radius vector, The volume of its shell is
32π2(R3)/17
Any two points in the surface may be connected by a segment of a great circle. This is the shortest trace within the surface. There is a shorter route across the interior of the vorpal: the chord connecting the points. The distinction is important, but for points in close proximity the difference may be insignificant.
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