If a tesseract is to a cube what a cube is to a square, what is to a sphere as a sphere is to a circle? What about rectangle, ring, triangle? I'm trying to come up with sensible names for a programming library I'm putting together. One minor part of this library is the generation of shapes of varying dimensions.
Basically I'm just trying to fill in the blanks below but I'm having awful trouble finding answers as it seems I need to know the name of the shape to find anything about it. The last column is the name I'm giving each group at the moment.

____________________________________________________
|    2D    |      3D     |      4D     |     Name    |
|----------------------------------------------------|
|  Square  |     Cube    |  Tesseract  |   Hypercube |
|  circle  |    Sphere   |    Glome    |   n-Sphere  |
|  Ring    |     Torus   |             | Hypertorus? |
| Triangle | Tetrahedron |             |   Simplex   |
|   Bar    |   Cylinder  | Duocylinder |             |
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On the chance that these dimensional shapes don't exist, or maybe they don't have names, any suggestions on what to actually name each group would be appreciated.
 A: I can suggest some names, which I have learned from others who research these things.

*

*A 4-simplex, or 4D analogue of a triangle, is called a Pentachoron , describing a regular, 5-sided 4D polytope. Also called a 5-cell. These are the n-simplex.

There are 4 types of ring torus objects in 4D, which can be seen visualized here. The general name of hypertorus works well:

*

*Spheritorus : sphere-bundle over the circle :$S^2$ x $S^1$ $$\left(\sqrt{x^2+y^2} -a\right)^2 +z^2+w^2 = b^2$$


*Torisphere : circle-bundle over the sphere : $S^1$ x $S^2$
$$\left(\sqrt{x^2+y^2+z^2} -a\right)^2 +w^2 = b^2$$


*3-torus : circle over circle over circle : $T^3$
$$\left(\sqrt{\left(\sqrt{x^2+y^2}-a\right)^2+z^2}-b\right)^2+w^2 = c^2$$


*Tiger : circle-bundle over the flat 2-torus (Clifford torus)
$$\left(\sqrt{x^2+y^2} -a\right)^2 +\left(\sqrt{z^2+w^2} -b\right)^2 = c^2$$
As for the bar -> cylinder -> duocylinder, I'm not sure exactly what sequence you are using here. The best fit I can see is describing a specific bisecting rotation around an n-1 plane into n+1 dimensions. In this case, the next 5D shape is called a Cylspherinder , a cartesian product of a $D^2$ and $D^3$ (solid disk times solid sphere). But, you can also make a Spherinder (sphere prism, another type of 4D cylinder) from a rotation of a cylinder into 4D.
Cylinder: $\left|\sqrt{x^2+y^2} -z\right|+\left|\sqrt{x^2+y^2} +z\right| = a$
Duocylinder: $\left|\sqrt{x^2+y^2} -\sqrt{z^2+w^2}\right|+\left|\sqrt{x^2+y^2} +\sqrt{z^2+w^2}\right| = a$
Spherinder : $\left|\sqrt{x^2+y^2+z^2} -w\right|+\left|\sqrt{x^2+y^2+z^2} +w\right| = a$
Cylspherinder : $\left|\sqrt{x^2+y^2+z^2} -\sqrt{w^2+v^2}\right|+\left|\sqrt{x^2+y^2+z^2} +\sqrt{w^2+v^2}\right| = a$
I guess you can call these n-cylinders, but there are even more types of these than just product of n-balls and n-cubes. You can also include the product of n-ball and n-simplex as well. In fact, any shape with both flat and curved cells can fit into this group (product of 2-ball (and higher) with any genus-0 object)
Cyltrianglinder : $\left|\big||x|+2y\big|+|x| -2\sqrt{z^2+w^2}\right|+\left|\big||x|+2y\big|+|x| +2\sqrt{z^2+w^2}\right| = a$
