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Vertex Figures, etc.


Vertex Figures

Take a small (hyper)sphere situated at a vertex, then the intersection of its surface with the polytope defines the vertex figure. Often this (hyper)spherical tesselation is represented by a corresponding polytope. Topological this figure is equivalent to the (sub-dimensional) dual of that facet of the dual polytope which corresponds to that chosen vertex (of the original polytope).

As the edges incident to the chosen vertex show up in the vertex figure as its vertices, the faces incident to the chosen vertex show up as its edges, etc. the numbers of that figure show up in the incidence matrix of the original polytope as the superdiagonal part of the matrix row which corresponds to the chosen vertex.

In a stricter sense for spherical space uniform polytopes, when embedded into euclidean space of one dimension plus, one could consider an orientation of the polytope with one vertex being top-most. Then the vertex figure can be derived as the section of that polytope with an affine hyperplane, situated at the one but top-most vertex layer. - This in fact is nothing but the sefa (sectioning facet) underneath a vertex. And this too is what will be reffered to in the next section as the metrically correct vertex figure, when being derived by means of the Dynkin symbol.

Using this metrically correct vertex figure, then there can be derived a nice formula, relating the circumradii of the original polytope (R) and of the vertex figure polytope (r): From the graphic at the right one easily derives (using the edge length x): r2 = x2 - p2. Similarily one has r2 = R2 - (R - p)2 = 2Rp - p2. Therefrom one freely gets x2 = 2Rp. Now solving for p and then inserting that into the first equation results in r2 = (4R2 - x2) x2 / 4R2. If we use unit edge lengths, then this relation even can be simplified to r2 = 1 - 1 / 4R2.

Note that for spherical cases R generally is real. – Accordingly (as 2R always is ≥ x = 1) r will be real, and 0 ≤ r < 1.
For euclidean cases R gets infinite. – Accordingly r will be = 1.
For hyperbolic cases R is purely imaginary. – Then r would get > 1.
In rare hyperbolic cases R is itself 0 i. – Then we get r being positive infinite.



Deriving them from Dynkin diagrams   (up)

For quasiregular polytopes the vertex figure is easily derived. Just omit the ringed node and the incident links. Ring instead the previously neighboring nodes. More precisely the edges, corresponding to those new ringed nodes, will have to be sized differently, each according to the vertex figure of the sub-diagram consisting out of the original node, the neighboring node under consideration, and the (now deleted) link in between.

How this works is shown for x3o4o (oct), in order to demonstrate that this concept also works for quasiperiodic polytopes independent of whether the links marked 2 are used or omitted in Dynkin diagrams. The direct application of the above said demand the vertex figure to be x(3)-4-o. Here the number in parentheses represents the mark of the omitted link. Quite generally the function x(.) results in (an edge of) the length of the first chord of the corresponding polygon, i. e. represents the vertex figure of that specific polygon. For instance the vertex figure of a triangle is just the opposite edge, thus x(3) = x. Therefore the above derived vertex figure just is x4o. More generally, these length values are given by

  x(2) : x  = 0
  x(3) : x  = 1
  x(4) : x  = 1.414214
  x(5) : x  = 1.618034
  x(6) : x  = 1.732051
  x(∞) : x  = 2
  x(p) : x  = sin(2π/p)/sin(π/p) for p>1

Some special cases Miss Krieger has given single characters just for abreviation (cf. also notation elements of Dynkin symbols). These are:

  x(3)   = x
  x(4)   = q
  x(5)   = f
  x(5/2) = v
  x(6)   = h
  x(∞)   = u

Note, the "full" diagram of the above used starting polytope (oct) would be x3o4o2*a, if no links would be sub-pressed. Its vertex figure thence would be accordingly x(3)-4-x(2). But the vertex figure of a dyad is just its opposite vertex, therefore the side corresponding to x(2) is degenerate, having zero length, cf. the above listing. Thus this edge could be neglected as well, and the incident vertices could be identified. Then we get again, just as already followed above from the reduced diagram, as vertex figure the square x4o.

As alternate example consider o3o3x4o (rit). Its vertex figure, in this metrically correct notation, will be given accordingly by the diagram o-3-x(3) . x(4). Thus it is essentially a triangular prism, but the base faces are scaled as having unit edges (vertex figure of triangle), contrasting to the lacing edges, which are of size sqrt(2) (vertex figure of square).

For polytopes with Dynkin diagrams, where more than just one node is ringed, things get a bit more complicate. This is where the concept of lace simplices needs to come in. This concept (up to my knowledge) is due to Miss Krieger.

First one splits the diagram into several sub-diagrams by deleting all but one ringed node each and omits further all links incident to the deleted nodes. The vertex figures of all those subgraphes (derived in the above sense, as those now represent quasiregulars only) are then to be used as the parallel layers of the lace simplex. Make sure to derive those metrically exact. These derived layers will be connected pairwise by lacings which in turn are exactly the vertex figures of the subgraphs consisting out of any 2 of the original ringed nodes (plus the connecting link, if existent).

As a first easy example, the vertex figure of x3x4o (toe) is derived. The layers would be verf(x . o) and verf(. x4o). The vertex figure of a dyad is just a point, that of the square a line of length sqrt(2). In this example we further have just one kind of lacings: verf(x3x .), i.e. a line of length sqrt(3). Thus we got a triangular vertex figure which is a point above a sqrt(2) base, laced by sqrt(3) sides (which in this trivial example could have been derived directly as the 3 vertex figures of the faces incident to each vertex).

Again giving a more complex alternate example. The vertex figure of o3x3x4x (grit) is derived by the layers verf(o3x . .), verf(o . x .), and verf(o . . x). Thus those layers are a unit length edge (vertex figure of the triangle) and 2 points (vertex figures of the dyads). These layers now are laced by verf(. x3x .), verf(. x . x), respectively verf(. . x4x) edges, that is the former unit edge is joined to the first point by sqrt(3) edges (vertex figure of hexagon), to the second point by sqrt(2) edges (vertex figure of square), and the 2 points are joined by an edge of length 2*sin(135°/2) ≈ 1.847759 (vertex figure of octagon).

Not totally different, but kind of an extreme case, is the consideration of the vertex figure of a cross product polytope. Consider for example the duoprism x3o x5o (trapedip). The layers are clearly verf(x3o . o) = . x(3) . . respectively verf(. o x5o) = . . . x(5), ie. edges of length 1 respectively tau. And the lacing edges here are verf(x . x .) only, ie. edges of length sqrt(2). What is more interesting, those top and bottom edges do not belong to the same 2-dimensional subspace, as can be seen from the given Dynkin diagrams, ie. the vertex figure becomes not a flat trapezium, they rather are to be placed orthogonally, so that the vertex figure becomes a kind of digonal antiprism. More generally any duoprism P x Q has as vertex figure a wedge with layers verf(P) respectively verf(Q), and those layers are placed within the mutually perpendicular subspaces of P respectively Q. Especially, the latteral facets are pyramides only, connecting facets of P to vertices of Q and vice versa.

Theory could be expanded even further. The vertex figures in turn of those latter rather special lace prisms with orthogonal arranged layers can be easily derived as well, as the used cells are either the top or bottom figure or pyramids. So they clearly are either verf(verf(P)) atop orthogonal verf(Q) or the other way round. And as edge figures are nothing but the vertex figures of vertex figures, and so on for even higher figures, the set of all those figures for quasiregular polytopes is easily derivable. Consider for example the diagram o3o3x3o3o3o (bril), if "&#x" is used to denote the lacing edges, respectively "||" is used to denote 'atop' (see here):

. . . . . . -figure of  o3o3x3o3o3o  is  o3x . x3o3o              =  o3x . . . . x . . . x3o3o

. . x . . . -figure of  o3o3x3o3o3o  is  xo .. .. .. ox3oo&#x(4)  =  x . . . . . || . . . . x3o

. o3x . . . -figure of  o3o3x3o3o3o  is  .. .. .. .. ox3oo&#x(4)  =  . . . . . . || . . . . x3o
. . x3o . . -figure of  o3o3x3o3o3o  is  xo .. .. .. .. ox&#x(4)  =  x . . . . . || . . . . . x

o3o3x . . . -figure of  o3o3x3o3o3o  is  .. .. .. .. .x3.o        =                . . . . x3o
. o3x3o . . -figure of  o3o3x3o3o3o  is  .. .. .. .. .. ox&#x(4)  =  . . . . . . || . . . . . x
. . x3o3o . -figure of  o3o3x3o3o3o  is  xo .. .. .. .. oo&#x(4)  =  x . . . . . || . . . . . .

o3o3x3o . . -figure of  o3o3x3o3o3o  is  .. .. .. .. .. .x        =                . . . . . x
. o3x3o3o . -figure of  o3o3x3o3o3o  is  .. .. .. .. .. ..&#x(4)  =  . . . . . . || . . . . . .
. . x3o3o3o -figure of  o3o3x3o3o3o  is  x. .. .. .. .. ..        =  x . . . . .


Notation used for vertex figures of polyhedra   (up)

For polyhedra the vertex figures become especially simple denotable, as the faces become represented by lines of different length, and exactly 2 such lines connect at those points, which are the cross-sections of the edges incident to the chosen vertex. That is those face-representing lines make up circuits (which well can have multiple windings, crossings, etc.). Thus vertex figures of polyhedra can be denoted by sequences of incident face types. Most often this looks like

[3,4,5,4]

for the small rhombicosidodecahedron (sirco). If sub-sequences are repeated, this can be denoted using a power notation, such as

[(3,4)2]

for the cuboctahedron (co). Further the winding number has to be considered. There are sequences that spool several times around the vertex. As winding numbers of faces are denoted by quotients, this notation is taken over to vertex figures as well. Thence

[35]/2

denotes the vertex figures of the great icosahedron (gike). On the other hand it might even wind to and fro with a total being null. Here we have to be a bit careful. Consider a crossed trapezoid where the parallel lines remain. Then the center with respect to the vertices is completely within one loop and the winding number nevertheless is 1. But if we consider a crossed trapezoid where the lacings remain that vertex center is completely outside the figure, and we have a total winding number of 0. For the intermediate case of a crossed rectangle, where the diagonal lines are incident to the center, both views could be applied as limiting cases, but as a divisor being 0 looks a bit strange, in those cases we favor the number 1. Thus we denote

[3/2,43] resp. [3/2,4,3,4] but [8/7,4/3,8,4]/0

for the vertex figures of the quasirhombicuboctahedron (querco), the octahemioctahedron (oho), respectively the small rhombihexahedron (sroh). And finally vertices might coincide (without necessarily making the solid figure itself to a compound). Here an additive notation is chosen. For instance the Grünbaum polyhedron which looks like an cubohemioctahedron plus 4 tetrahedrally arranged {6/2} (cho+4{6/2} (?)) would have as vertex figure

2[6/2,4,6]


Truncated Polytopes   (up)

Truncation is a physical process applicable to any polytope, which is ment as cutting off some or all vertex pyramids more or less deepely. (Having spoken of vertex pyramids, it becomes clear that – for this very reason – truncation depth will be restricted such that this intersection (hyper)plane still intersects the vertex emanating edges. – In more specialised cases some deeper truncations will be considered below.) In the context of a vertex transitive symmetry, an according application to all vertices is understood. (Therefore alternatively it could be understood to be the intersection with an apropriately scaled dual polytope.) Moreover it is obvious that those cut-off pyramids then will have to be upright, i. e. the base would be orthogonal to the axis of symmetry. This base polytope of the pyramid further more will be the vertex figure of the polytope to be truncated. Note that independently of the depth of truncation the geometry of the additional facets introduced by truncation is fixed (up to size), at least as long as those from different positions do not intersect. In opposition to that, the length of the former edges clearly gets smaller and smaller, depending on the depth of truncation. The limiting case, where the intersections of the truncating hyperplanes of neighbouring vertices will just meet at the middle of the former edge, further is called rectification.

Even so one is used to apply this operation most generally to highly symmetrical polytopes like regulars or quasiregulars (see below), it well can be applied too at other polytopes as well. In this generality however, it leaves the realm of uniform polytopes. Examples would be the truncation resp. the rectification of sadi, i.e. tisadi resp. risadi.

With respect to quasiregular polytopes, truncation in any depth (down till their next vertex intersection) can be given explicitely in the Dynkin diagramatical description with full metrical correctness. Consider here as an example the rectified hecatonicosachoron o3o3x5o (rahi). The vertex figure of which according to the above is o-3-x(3) . x(5). Here it follows what is derived:

  o-3--o--3--x--5--o  : starting polytope
  o-3-x(3)   .   x(5) : vertex figure
  o-3-x(3)-3-t-5-x(5) : truncated polytope

Here the relative depth of truncation is modelled by the size of the edge marked t. For t = 0 the former edges would be reduced to nothing, thereby representing the limiting case of the rectified polytope. While on the other hand for t = ∞ the length of the former edges would overcome any finite length by far, therefore representing the untruncated polytope again.

Note that allthough this t can be chosen independingly, and thus could be used to get the uniform representant (which has all edges of the same length), this will not be possible in this case here, because of the independant non-uniform geometry of the vertex figure itself. None the less there is a representant in the same topological equivalence class, even having all facet planes parallel to the true truncate, which is uniform. That one can be obtained by replacing both, all x(p) and t, by the unit edges x. Thus in the case of consideration that uniform representant of the truncate would be o3x3x5x (grahi). On the other hand, the uniform representant of the rectified version would be o3x3o5x (srahi). This btw. is the deeper sense because e.g. the great rhombicuboctahedron (girco) also is known as the "truncated cuboctahedron".

For regular polytopes even deeper "truncations" can be considered as well. Then the dual polytope, the one which is used for intersection, is regular as well, and can be given within the same symmetry group (i.e. its Dynkin diagram will have the same structure, only that the opposite end node is the one being ringed instead). This gives rise to a complete truncational series, best being described by an explicite example:

x3o3o5o  - regular base polytope (ex)
x3x3o5o  - truncation (tex)
o3x3o5o  - rectification (rox)
o3x3x5o  - bi-truncation (xhi)
o3o3x5o  - bi-rectification (rahi)
o3o3x5x  - tri-truncation (thi)
o3o3o5x  - (... and finally:) dual (hi)

Obviously one-ringed and two-adjoined-ringed states do alternate. In fact, the n-truncational states can be made also continuously filling the gaps between the n-rectates, just by applying different edge scales to those two-ringed states (i.e. applying 2 variables x and y instead), running from one-to-zero up to zero-to-one.

This sequence of different polytopes of that truncational series finally could further be understood as an sectioning series perpendicular to an additional dimension, i.e. of a polytope within one dimension higher. That very higher-dimensional polytope then technically would be an antitegum, i.e. the dual of that antiprism, which in turn is derived as the segmentotope "regular polytope || dual polytope".

But there is a different extension of the truncational process as well. That different one moves the intersection points of the truncating (hyper-)plane along the arbitrarily extended edge-lines of the starting figure. The difference here is that both, beyond the starting figure (negative positions, outside of the starting polytope), as well as positive positions beyond the rectified polytope, would lead to non-convex shapes. Here the positions can be moved from -∞ to +∞. And, if additionally the overall size all the way through is scaled down to a given circumradius, then the finite size of the former figure would get down to zero in those limiting cases. Doing so would lead to the so called truncation rotation, i.e. a process which would close projectively those limiting points into one.

That truncation parameter would be z=±∞ at the left, would be z=0 (i.e. no truncation at all) at the top, would be z=1/2 at the right, and z=1 (i.e. the intersection points reach the opposite end of the former edges) at the bottom in the diplay of the nearby shown picture. – In fact, the top-right realm from z=0 to z=1/2 clearly is the usual truncation, as already described above. The bottom-right realm from z=1/2 to z=1 is known as hypertruncation. The bottom-left one from z=1 to z=+∞ is known as quasitruncation. Finally the remaining top-left realm of negative z-parameters is known as inflected truncation.



Subsymmetrical Diminishings

Truncation is meant to replace any vertex by a more or less small copy of its vertex figure. (For the cases of larger copies those even might touch, or, instead of intersect, themselves will get truncated by one another.) But surely one could ask for subsets of vertices likewise, which are to be replaced only.

sadi
The most famuous application here are the snubs, which in fact can be considered as a vertex alternated faceting. – Just one examplifying case here is the snub 24-cell (sadi), the alternated faceting of tico. That one also can be seen as an even lesser partial diminishing at non-neighbouring vertices of ex, introducing there additional ike (replacing some of the tet), while other tet remain in place: Here those former ike vertex figures become subsymmetrically diminished to teddi vertex figures.
gap
Accordingly one might ask for non-snub examples (using non-alternating vertices only). Those would be especially interesting for sure, if they still result in uniform figures. – Such an other example, starting with the same polychoron, ex, diminishes 2 bands of 10 neighbouring vertices each, giving rise to additional pap, while again other tet remain in place: Here those former ike vertex figures become subsymmetrically faceted by 2 adjacent x-x-x-f trapeziums. The resulting polychoron is the single extraordinary convex one, gap.
bidex
A simmilar case to sadi was found by A. Weimhold in 2004. Again the icoic subsymmetry of the hyic symmetry is used. In fact 2 mutual coset classes thereof: instead of diminishing ex at the vertex directions of an inscribed ico (which results in sadi), a diminishing by 2 such inscribed icos is applied. (In fact, there is a chiral compound of 5 icoes (chi) which can be vertex inscribed into a (re-)scaled ex. Two of those icoes will be used here.) Here the sections already do intersect themselves, resulting in diminished icosahedra, in fact in teddi. The resulting (overall) figure here is bidex, which only is scaliform, but on the other hand comes out to be a noble, swirl-symmetric and convex polychoron. The new vertex figure is a (subsymmetric) faceting of ike having two x-x-x-f trapeziums and four x-f-f triangles. Moreover that very vertex figure comes out to be self-dual.
tes & hex
Clearly the 2 cases of a subsymmetrical diminishing of ico itself could be stated here: diminishing at the vertex directions of an inscribed hex result in a tes. A similar diminishing at the vertex directions of 2 inscribed hex results in a third hex. – Conversely we could state: there is a compound of 2 hex (haddet) vertex inscribed into tes, resp. a compound of 3 hex (stico) vertex inscribed into ico.
spd{3,5,3}
In 2011 W. Krieger found 2 other examples, both being such subsymmetrical diminishings of the hyperbolic honeycomb x3o5o3o. Sure, the vertex figure of that honeycomb itself is the doe. In the first case the vertices to be replaced by their vertex figures are selected such, that the former vertex figure doe will get faceted at a pair of antipodal vertices only. In fact that doe-faceting has the depth of just one reduced edge set, down to the neighbouring vertices (resulting in fxfo3ofxf&#xt). The obtained honeycomb is spd{3,5,3}, the semi partially diminished {3,5,3}, as she calls it, has for cells doe, pap, and (remaining) ike.
pd{3,5,3}
The other one uses a slightly more often truncation, so the qualifier "semi" was omitted here: pd{3,5,3}, the partially diminished {3,5,3}. In fact, the faceting planes to the vertex figure have the same depth, but will not take place in an antipodal way, but rather in a tetrahedral vertex subset. Accordingly, the resulting vertex figure gets chiral. In that honeycomb cells are doe and pap, i.e. the original ike get completely replaced.
spd{3,3,3,5}
More recently, in 2013, W. Krieger came up with a similar subsymmetrical diminishing of the hyperbolic tiling {3,3,3,5}. There she uses instead of the (former) vertex figure ex already the above described bidex. It turns out, she states, that the resulting tiling spd{3,3,3,5} will be noble, using only sadi for tiles.
pd{3,3,3,5}
W. Krieger moreover continues: Like ex has ike for vertex figure, like sadi has teddies for vertex figure and ike for (some) facets, and like bidex has that highly subsymmetrical ike-faceting for vertex figure and teddi for facets (so far: see above), there ought to be some tridex (tri-icositetra-diminished ex, i.e. at the vertex directions of 3 inscribed icoes) too, then having those (not even scaliform) bidex vertex figures for cells now, only. That one then she uses to build up the figure pd{3,3,3,5}: It should have that tridex for vertex figure and should use bidex for tiles only.

If being applied to convex polytopes only, that research for subsymmetrical diminishings clearly is contained within the broader research for CRF.



Centers Hulls   (up)

In the sub-case of convex polytopes one might ask for any kind of sub-elemental class: what would be the shape of the convex hull of the centers of these elements? – Sure, using the vertices here, the outcome will be the starting polytope again, so that special case would not be too interesting. But beyond? I.e. for any symmetry equivalent class of edges, any such class of faces, etc.? – Interestingly, this question can be answered uniformely for any Dynkin diagram derived convex quasiregular polytope (i.e. having exactly one node ringed and no rational link marks).

Here is how it works technically. – In order to explain it in parallel at some example, we will use rico, i.e. o3x4o3o, and as relevant sub-element we will consider its edge centers.
Step 1: Consider the Dynkin diagram of the very sub-element, taken within the symmetry of the starting figure itself. – In our example: . x . . would be required.
Step 2: Construct the Dynkin diagram of the same symmetry with all nodes remaining un-ringed, except for the ones next to the node positions being used in step 1. – In our example: x3o4y3o where 2 different letters x and y are used here, as the relative length scale of these edges so far is not being determined. (Thus we know already that the desired hull will be a variation of srico.) That relative scaling will be the topic of
Step 3: If required, the relative length scale of the to be applied edge types can be deduced again from the being crossed link marks between the sub-diagram of the sub-element and the neighbouring node, which according to step 2 has to be ringed: in fact, if that crossed link would have the mark n, the required size would be x(n). (Sure, in case there would be just a single node position to be ringed in step 2, we well could apply this rule of step 3; but there will be nothing to relate that very scaling to, therefore in that special case we could stay with a simple x, no matter what link mark has been crossed.) – Therefore, in our example, the final diagram for the to be derived convex hull of the edge centers of our starting figure (rico) would just read x3o4q3o.



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