While writing some code to explore abstract polytopes, I've noticed that many intuitive polytope operations (like the Conway polyhedron operations) can be obtained by defining the output polytope $Q$ as a set of sections of the input polytope $P$ ordered by inclusion (i.e. pairs $x,y\in P$ with $x\le y$, letting $(x_1,y_1)\le(x_2,y_2)$ iff $x_2\le x_1\le y_1\le y_2$), subject to varying restrictions on which sections are included. I'm wondering if there's a way to unify and generalize these beyond the six types I've found.
With no additional constraints, we get an operation (which I call hasse) that increases dimension by 1 and produces a polytope that looks like the Hasse diagram of the input. For example, it turns a triangle into a cube.
An optional additional constraint is to require our sections to use only proper elements of $P$. (If we do this, we add a new "synthetic" top element to $Q$ in place of the section $F_n/F_{-1}$. A synthetic bottom element is needed either way.) On polyhedra ($n=3$), this gives us the Conway ortho operation.
With or without the above restriction, we can also choose one of these:
- Include only sections $y/x$ with $\operatorname{rank}(x)\lt r\le\operatorname{rank}(y)$, for some fixed $r$. When $r=1,n=3$ (and allowing improper elements of $P$ again), this gives truncate, so I call it $\operatorname{trunc}_r$.
- Include only sections $y/x$ with either $\operatorname{rank}(x)\lt r\lt\operatorname{rank}(y)$ or $\operatorname{rank}(x)=\operatorname{rank}(y)=r$, for some fixed $r$. When $r=1,n=3$ (and allowing improper elements of $P$), this gives ambo, so I call it $\operatorname{ambo}_r$.
The versions of these operations with the "proper elements only" restriction are dimension-reducing:
- The truncate-related one I call $\operatorname{hollow}_r$, and the output poset is only a polytope if $2\le r\le n-2$ (implying $n\ge 4$). $\operatorname{hollow}_2$ operates on a 4-polytope $P$ by carving out a vertex-figure-shaped "room" at each vertex, letting adjacent rooms intersect slightly to make a polygonal doorway corresponding to each original edge; the output 3-polytope is the interior of this room complex. (Slightly easier to visualize is $\operatorname{dual}\circ\operatorname{hollow}_2\circ\operatorname{dual}$, which shrinks the cells of $P$ into cell-shaped rooms and extrudes their faces to form "hallways" connecting them. I call this "splay".)
- The ambo-related one I call $\operatorname{vex}_r$, and it requires $2\le r\le n-3$, so $n\ge5$. This makes it harder to visualize, but roughly, $\operatorname{vex}_2$ on a 5-tope shrinks 4-cells away from each other, modifies them by taking vertex figures, ambo-ing each, and joining them along the new faces (similar to a $\operatorname{hollow}_2$), leaving unmatched 2-faces along which we re-glue adjacent 4-cells.
(I find these two interesting because they erase the higher-dimensional figure while retaining many of its symmetries and therefore in some sense preserving its topology. For example, a hemi-24-cell is a regular tiling of $\Bbb{RP}^3$; its splay is a regular complex of square sectors on which we can "walk around in projective 3-space". This idea is what motivated this whole exploration.)
Proving that each of these constructions satisfies the required poset diamond criterion involves special cases at the high and/or low ranks of the output polytope, and the logic is slightly different for each constraint choice. Relatedly, my computer implementations of the constructions are very similar (in how they represent flags and implement exchange maps for middle ranks) but differ in a couple of places. I don't really see a pattern among the few constraint types I've discovered so far, so I have no idea which others exist. Is there a way of looking at these operations that clarifies their relationships and reveals other ones?
By the way, I've noticed that the Conway meta operation generalizes to taking chains of $P$ rather than sections. I wonder if this hints at a scheme that encompasses the section-based constructions.