There is the matter of 'progressions'. This is a kind of additive process that changes one polytope into another. It is used in figuring out what sort of things make intermediate sections.
For example, the progression between the icosahedron and dodecahedron, the first and second vertex-first sections of {3,3,5}, runs like this.
o5o3x becomes o3o5x by way of xa3o5xb (a+b=1). This polytope is a rhombo-icosadodecahedron. The pentagons expand, while the triangles shrink. This is what one expects as one descends a pentagonal pyramid, and ascends a triangle pyramid. What also happens, is there is a series of rectangles, of constant perimeter 2(a+b), which correspond to the sections of the line-line pyramid. These are the 30 tetrahedra whose bases are in fact edges of x5o3o and of x3o5o.
There is also an apiculation, or pyramid-raising, on the pentagons. This is caused by edges that run from o5o3x to o5o3f (girthing decagons). This edge is one of the bases of a tetrahedron, the other base is the edge of the dodecahedron x5o3o. This gives sections that raise pyramids on the twelve faces of the dodecahedron (and the intermediate xa3o5xb, as well).
Progressions can be used in some form of generalised lace-prism, which i origionally implemented as an extended pyramid product.
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A2 B1 B2 B3 (A) B1 B2 B3 (A) X Y Z
a x3o o3o o3o a x3o o3x o3o a x f o
b o3o x3o o3o b o3o x3o o3x b o x f
c o3o o3o x3o c o3x o3o x3o c f o x
simplex in 8 dim The 2_21, or /4B The icosahedron
We see here that the altitude A can be multi-dimensional. In the first, A is a smallish equalateral triangle, perpendicular to the spaces B1, B2, and B3. The combination of A2 + B1 + B2 + B3 gives eight dimensions.
In the second and third examples, the altitude is 0, so one gets a figure inscribed in another (like the hexagonal cupola ox6xx, which is flat in the plane). The second are a set of coordinates for a complex polytope 3{3}3{3}3 in eisenstein complex integers. It maps neatly without distortion to a tripple {3,6} coordinate system for the 2_21, which is what is given here. One can see that in B1, the figure given is a oxo3oox&x, a segmentotope corresponding to an octahedral pyramid. In practice, the two tetrahedra are the opposite faces of an octahedron, which appears in flattened pyramid (ie tegum) product, with the nine squares that form in the bi-triangular prism in x3o o3x. The squares are actually 2d tegums, which join up with the 3d tegum to make 9 of the 27 2_11 faces of this polytope.
The third is a crossing of three golden triangles to make an icosahedron.
The finding of the relevant faces happens in the same way in all cases. One draws faces from one section onto faces of an other, giving rise to prism and pyramid products, as the elements align or fall against each other.
Most of this i figured out long before i understood the dynkin symbol.