## A bit deep think on terms of truncation operators

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### A bit deep think on terms of truncation operators

Recently, I thought of a few of terms of truncation operations.

Truncation : Derived from Kepler's name for Archimedean Solids.

Bitruncation : If we treat "complete-truncation" as "truly-once-truncation",
we can think this operation as "continued" truncation after "truly-once-truncation".
So (truly once) truncation + (continued) truncation → twice truncation = bitruncation

Birectification : this may equal twice complete-truncation (another continued truncation after bitruncation)

and these can be extended to n-truncation and n-rectification...

Is all of my think correct?

If my think is exactly wrong, could you correct?

Ah, Cantitruncation equals "Cantellation after truncation" and "truncation after cantellation"?
And this can be extended to higher operations (like Runcicantitruncation)?
hy.dodec
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### Re: A bit deep think on terms of truncation operators

Truncation is simply the operation performed by truncating the vertices of the polytope, which is then replaced by its vertex figures as new facets, and where they do not yet meet each other by its vertices. The original facets are converted into their truncated variants.

Take a 24-cell (ico) with 24 octahedra and a cubic vertex figure. Truncating it replaces its vertices with new cubic cells, and the octahedra are truncated, creating truncated octahedra. You now have 24 cubes (from the 24-cell's 24 vertices) and 24 truncated octahedra (from the 24-cell's 24 octahedra).

Taking that further, we have rectification. Basically, the new facets touch each other by a single vertex. To visualize, imagine the 24-cell being rectified. This time, the new cubic cells touch each other exactly by its vertices, Instead of having truncated octahedra from octahedra, you have cuboctahedra from octahedra. This is because rectification involves putting a vertex at every edge, which is a topologically identical process.

Going further, we have bitruncation. It is deeper than rectification. In bitruncation, the vertex figures overlap, The new facets derived from the vertex figures are truncated variants of those produced by truncation. The original facets are bitruncated (if they are 3D, they are the same as the truncate of their duals). If a 24-cell undergoes bitruncation, the cubic vertex figure gets truncated, forming a truncated cube in its place. For the octahedra, they are bitruncated, turning them into truncated cubes also. In fact, the figure being constructed (cont) can have one cell type under contic symmetry (order 2304 = 2*ico).

Birectification is deeper than bitruncation, and instead of having truncated vertex figures, we have rectified vertex figures. Also, if the facets are 3D, they are dualized. The birectified 24-cell (same as the rectified 24-cell because in 4D, it is the rectification of its dual) has cuboctahedra in place of vertices and cubes in place of octahedra.

In fact the sequence of such operators is infinite.

And yes, combination of operators usually involve facet components. So for cantitruncation, you shrink its faces and truncate them. For runcicantitruncation, you shrink the cells and cantitruncate them. They can also be extended infinitely.
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### Re: A bit deep think on terms of truncation operators

You have to think of Coxeter symmetries as symmetries, which are generated by reflections. And all these reflections do keep a single point of space fixed. So, in 3D you might think about their actions as being visualised by the intersections of these mirror planes with the sphere around this fix point, i.e. the according great circles. All these mirror symmetries thus happen to produce kind a net on the sphere. And all their loops happen to be spherical triangles. In higher dimensions the same idea could be applied, again the mirror hyperplanes do intersect the hypersphere into spherical simplices, which once where called orthoschemes. That name derives its ortho part from the fact that lots of dihedral angles happen to be orthogonal.

The next idea then is to dualise these orthoschemes into Dynkin diagrams. I.e. every bounding facet of the orthoscheme, which in fact is nothing but a part of the original mirroring hyperplane, is represented by a node, and every intersection of 2 such facets is represented by a link between the 2 involved nodes. As you might know, the symmetry asks for dihedral angles, which are submultiples of pi (alpha = pi/k), thus this divisor plays a prominent role. This is why the links of the Dynkin diagrams get labeled by that divisor. Because most dihedral angles are 90 degrees, the according link usually is just omitted at all in the Dynkin diagram. And as the second most frequent dihedral angle of orthoschemes is 60 degrees, the according link then is provided without an additional labeling.

Wythoffian polytopes then can be easily constructed by providing a seed point somewhere in the interior or at the boundary of an orthoscheme. That seed point in fact represents one of the (all being equivalent) vertices of the corresponding polytope. In fact, all the other vertices can be constructed by mirror reflection of the seed point at any of the boundary facets of the orthoscheme (resp. the mirror hyperplanes). Moreover any 2 points (vertices), which belong to 2 neighbouring (mirrored) orthoschemes get connected by an edge. This same kaleidoscopical construction then can be applied all the way up through the dimensions, so that finally all polytopal elements get provided.

You furthermore have the possibility, that the seed node itself is situated on or off any of the bounding facets (mirror hyperplanes). Accordingly the corresponding node of the diagram remains unringed resp. gets ringed.

For regular polytopes it comes down that the according Dynkin diagram is a linear graph, i.e. all dihedral angles of the orthoscheme are orthogonal except of a connected path of sharper angles each. And that just one of the end nodes is ringed, while all others are unringed. The single type each of any subpolytopal element of a regular polytope is a regular polytope again. In fact it can be constructed from a similar dynkin diagram. The relation of these diagrams is very easy, as those just happen to be subdiagrams of the former (while keeping the single ringed end node), i.e. just chopping off at the opposite end. The vertex figure of a regular polytope too is a regular polytope. Thus that one too has an according Dynkin diagram. That one could be obtained from the former by chopping off the single ringed node, and ringing the neighbouring one instead (which now is an ending node of that diagram).

Next, when applying truncation onto some regular polytope, this amounts in chopping off the vertices and inserting there instead a tiny copy of the vertex figure instead. For all subpolytopal elements the effect of truncation also is a truncation of the facet polytope. Thus all former facets now are truncations of regular polytopes, while the now inserted vertex figures are new regular subpolytopes. In terms of dynkin diagrams a truncated regular polytope is a linear graph having one end node ringed and the neighbouring node being ringed as well. The above described facets of that truncate can be described in terms of subdiagrams, deleting either of the end nodes. In fact these thus have either the form of a truncated regular polytope (still 2 nodes being ringed) or of a regular one (now just a single ringed node is being left).

When considering the truncational process as a process of increasing depth, then you could distinguish 2 edge types. On the one hand there are those edges, which are more and more forshortened versions of the old regular polytope's edges, and then there are the additional increasing edges of the added vertex figure. The former ones are represented by the ringed end node, while the latter ones are represented by the neighbouring ringed node of the diagram. So you could consider this process of gaining depth as a one parameter linear decrease of the edge size of the end node type and a meanwhile linear increase of the other edge size of the second node type. That is, starting at the untruncated regular polytope, the first type has full size and the other one is zero. Then, while going deeper, it comes to a point, where both edge types have the same size. And finally you'll come to a point, where the former edges of the regular polytope getting truncated will be eaten completely, and only the new edges of the vertex figures remain. This then is the point, where the truncate becomes the rectificated polytope.

In terms of Dynkin diagrams this dynamical process could be described as well. But then you'd have to get into play that additional parameter. Thus instead of simply drawing ringed nodes in these diagrams you'd to provide there edge size acronyms instead. Thus the formerly ringed end node has a decreasing size acronym, while the neighbouring one has an increasing edge size acronym. And this gaining truncational depth process here thus ends in that diagram, where that formerly being ringed end node now has got completely unringed, while the neighbouring one is ringed and the corresponding edge size has got its maximum.

This process at the Dynkin diagram, that one node is fading out while the neighbouring one gets more and more illuminated, does continue from now on, when applying deeper and deeper truncations beyond the state of the rectified polytope. There the next state with equal edge sizes is the bitruncate polytope, the next state with once again just a single remaining edge type would be the birectified polytope. Then you'll pass the tritruncate and then the trirectified, etc. And finally you would drilled the truncational depth so far, that just the dual figure would remain, i.e. wherever once has been a vertex, there now would be a facet (and vice versa). And indeed, in the diagram too, the whole fading and illuminating shift process would end on that linear Dynkin diagram at the opposite end, when that other end node is being ringed solely.

--- rk
Klitzing
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### Re: A bit deep think on terms of truncation operators

In the above so far described Dynkin diagrams just a single node was being ringed (n-rectifications) or 2 neighbouring nodes were being ringed (n-truncation).

But is is obvious that you might consider positions of the seed point, which is off from more than just one or two boundary facets of the orthoscheme. Then the corresponding kaleido-constructed Wythoff polytope is represented by a Dynkin diagram with more than just one ringed node, which moreover may be spread out at any node position whatsoever. In fact, when the seed point would happen to lie somewhere in the interior of the orthoscheme, then it is off from all boundaries. Accordingly the corresponding Dynkin diagram would be represented by having all its nodes being ringed.

So consider for instance a 5D setup with symmetry
Code: Select all
o---o---o---o---o
P   Q   R   S

where the labels P,Q,R,S are integrs larger or equal to 2.

Then we have the following possibilities, which correspond to the according polytopes:
Code: Select all
(o)--o---o---o---o   = regular {P,Q,R,S}
P   Q   R   S

o--(o)--o---o---o   = rectified {P,Q,R,S}
P   Q   R   S

o---o--(o)--o---o   = birectified {P,Q,R,S}
P   Q   R   S

(o)-(o)--o---o---o   = truncated {P,Q,R,S}
P   Q   R   S

(o)--o--(o)--o---o   = small rhombated {P,Q,R,S} (AKA cantellated {P,Q,R,S})
P   Q   R   S

(o)--o---o--(o)--o   = small prismated {P,Q,R,S} (AKA runcinated {P,Q,R,S})
P   Q   R   S

(o)--o---o---o--(o)  = small cellated {P,Q,R,S} (AKA stericated {P,Q,R,S})
P   Q   R   S

o--(o)-(o)--o---o   = bitruncated {P,Q,R,S}
P   Q   R   S

o--(o)--o--(o)--o   = small birhombated {P,Q,R,S} (AKA bicantellated {P,Q,R,S})
P   Q   R   S

(o)-(o)-(o)--o---o   = great rhombated {P,Q,R,S} (AKA canti-truncated {P,Q,R,S})
P   Q   R   S

(o)-(o)-(o)-(o)--o   = great prismated {P,Q,R,S} (AKA runci-canti-truncated {P,Q,R,S})
P   Q   R   S

etc., e.g.

(o)--o--(o)-(o)--o   = prismatorhombated {P,Q,R,S} (AKA runci-cantellated {P,Q,R,S})
P   Q   R   S

--- rk
Klitzing
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### Re: A bit deep think on terms of truncation operators

Thanks for all replies!!!

Ah... How I interpret "Omnitruncation?"

Truncation means "cutting vertices", but edges and faces are cut if we cut vertices

Similarly, If we bevel edges and vertices (cantellate), faces are cut, also..

So how I should explain omnitruncation more deeply...

+ I wanna translate these operations to our language, that's why I do this...
hy.dodec
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### Re: A bit deep think on terms of truncation operators

What Klitzing describes of are just illustrations of operations based on Dynkin diagrams. In fact, it's a very useful tool to categorize isogonal polytopes through their Wythoff constructions. Say, the base polytope is x-o-o-o-o, which represents some 5D polytope or 4D honeycomb. For regular operations (without bi-, tri-), you have the ringed node (x) on the first circle. From that, we have truncation x-x-o-o-o, cantellation x-o-x-o-o, runcination x-o-o-x-o, and sterication x-o-o-o-x. In fact, counting from the additional node being marked after the first node, you have the sequence I described earlier. But for bitruncation, you have to place the first ringed node on the second node, such as o-x-x-o-o. Since the distance between the two is 1, it is considered a truncation and not a cantellation. For rectates, you only need to ring one node such as o-x-o-o-o. Note that you have to start one node further for rectates, so birectification is o-o-x-o-o (tritruncation is o-o-x-x-o).

For bicantellation you have o-x-o-x-o and for biruncination you have o-x-o-o-x (same as runcinated dual in 5D). In fact there is a symmetry associated with it, so the diagram can be reversed and re-interpreted. Remember, for bi- prefixes you ring the second node first and place the others after it, considering the distance from the second node from the left, and for tri-, do it on the third node first.

For example, you have for a 6D polytope or 5D honeycomb a tritruncation o-o-x-x-o-o, a biruncicantitruncation o-x-x-x-x-o, and trirectification o-o-o-x-o-o.

Omnitruncation is literally "all-truncation". In the sense of geometry it is an operation that combines all operations (truncation, cantellation, runcination, sterication, pentellation, and so on) to produce a figure that under symmetry, is the maximized form of the polytope. It is the equivalent of marking all nodes (like x-x-x-x-x-x). In constructing an omnitruncation, you have to consider the facets. You then begin to truncate (expand the edges), after that, cantellate (expand the faces), runcinate (expand the 3D cells), stericate (expand the 4D terons) until you get to expand the facets. The faces will always be even (in fact, inversion-symmetric) and can be snubbed (generally not uniform from 4D onwards, but exceptions exist).

Let's say you want to omnitruncate a penteract (5D cube). You start with the edges, expand them (truncate) to get a truncated penteract, next, expand its faces to get the cantitruncated penteract, expand its cells to get the runcicantitruncated penteract (not an omnitruncate), and finally expand its facets (terons) to get the omnitruncated or steriruncicantitruncated penteract, which is x4x3x3x3x. From there, you have these topological facets:
x4x3x3x - omnitruncated tesseract
x4x3x x - great rhombicuboctahedral prism
x4x x3x - 6-8 duoprism (with 3-4 duoprism symmetry)
x x3x3x - truncated octahedral prism (has tetrahedral prism symmetry)
x3x3x3x - omnitruncated 5-cell (does not carry over the extended A5 symmetry).

It's actually easy to label the facets of any Wythoff polytope, first, you have to delete an o/x and the numbers neighboring it to get one facet; if it leaves a hole, then it is considered as a duoprism of the first and second components (like x4x (3x3) x3x = x4x x3x = duoprism of octagon and hexagon). This is what makes the Dynkin diagram a useful tool to label a polytope without having to look at its form.
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### Re: A bit deep think on terms of truncation operators

The various operators do not need to be connected with the CD diagram, but rather one can think in terms of polytopes interacting etc.

Truncation + Rectification.

The process here is to imagine a polytope getting bigger, and the dual getting smaller. This is the 'decent of the dual' The particular truncates + rectates are then points in this process.

0 truncate, 0 rectate. The figure is entirely inside the dual, and at the rectate, the vertices of the figure touch the faces of the dual.

1 truncate, 1 rectate. As the figure rises above the dual, the overtices of the truncate now travel down the edges of the original figure. The rectate happens when the edges are just consumed.

2 truncate, 2 rectate. The vertices of the 2truncate head towards the centre of the hedra (2faces), eventually meeting in the centre of the 2faces (the 2rectate).

&c.

Apiculate + Surtegmate.

This is the dual of the previous process. It produces a convex hull of duals expanding around a figure.

0-apiculate, 0-surtegmate. The dual is expanding inside the figure, (apiculate), reaching the the surtegmate when the vertices of the dual touch the face-centres of the polytope.

1-apiculate, 1-surtegmetate. As the dual expands, it raises peaks on the faces of the figure, these peaks end in this process, when the full edge of the dual is on the surface, the peaks join by pairs in the 1-surtegmate.

2-apiculate, 2-surtegmate. As the dual continues to expand, the edges of the dual are now exposed as the tops of the peaks. This ends when the polygons of the dual are now fully on the surface.

&c.

apiculate = to raise peaks on the
surtegmate = surface tegums (a convex cover of a two orthogonal polytopes, like a rhombus is a cover of its two diagonals).

runcinate, strombiate

The runcinate process, is to push each face outwards, without making them any larger. For example, if you imagine a cube inside a skin, and then pump the cube up, the faces would remain rigid, but move outwards. The skin would expand over the gaps, so you get a rhombo-cuboctahedron. The edges have become squares, and the vertices have stretched into triangles.

In a construction, you take a figure and its dual, and construct it so that each vertices of the figure are replaced with the faces of the dual. The gaps in between are filled with prisms of the surtope of the figure, and the matching surtope of the dual. eg in the runcinated {5,3,3}, the dodecahedra and tetrahedra meet at corners, the pentagons of the dodecahedra form prisms, with the height as the edges of the tetrahedra, the faces of the tetrahedra form prisms with the edges of the dodecahedron.

The strombiate is formed by putting the figure and its dual on the same sphere. The faces of the strombiate are then the the intersections of the figure's faces and its dual. These are generalised rhombuses, or by John Conway, 'strombuses'. The faces are actually the dual of the antiprism of a figure and its dual, or the anti-tegums.

omnitruncate, vaniated.

A pair of duals here. A 'flag' is a triangle or simplex, whose vertices are the centres of the vertex, edge, 2face, 3face, ... Vane = flag (eg a weather-vane is a metal flag that rotates to point to the weather).

vaniated X is to replace the surface of X, with the flags of the surface.

omnitruncate, in the sense of Stott, is to mark every node of the dynkin graph, but here, is to 'expand' a polytope, by pushing the vertex against every surtope.
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wendy
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### Re: A bit deep think on terms of truncation operators

Umm... Due to not studied topology yet ( I'm high school student yet), almost of replies are hard but thanks for all replies!!

I'll see all of replies after studying advanced maths...

( I'm not sure if you understand what I ask, due to my bad english... If you felt this sentence rude, I'm so sorry)
( By the way, is etymology ( or originate) of operations in replies? I asked how that terms derived...)
( Sorry for my bad comprehension )
hy.dodec
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### Re: A bit deep think on terms of truncation operators

hy.dodec wrote:Umm... Due to not studied topology yet ( I'm high school student yet), almost of replies are hard but thanks for all replies!!

I'll see all of replies after studying advanced maths...

( I'm not sure if you understand what I ask, due to my bad english... If you felt this sentence rude, I'm so sorry)
( By the way, is etymology ( or originate) of operations in replies? I asked how that terms derived...)
( Sorry for my bad comprehension )

Don't worry, I don't understand a lot of terms being thrown around. Another thing interesting about these processes is that you can also have antitruncation, hypertruncation, and quasitruncation which are used less than the other terms. Considering it's easier to visually learn what it is, I have a picture.

ubersketch
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