## Constructing uniform polytopes (split from new nomenclature)

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

### Constructing uniform polytopes (split from new nomenclature)

quickfur wrote:
Keiji wrote:I know. I refer to ox+o (where x+ = one or more 'x', as in regexp) as rectate for any dimension because of that property, and you can see my names for them in the "conventional" column here. That said, I'm not particularly convinced that my scheme gives good names for the others, but this is because I can't properly visualize most of the non-parent polytopes (yet) so I have no way of knowing what the most natural scheme would be.

Well, all the uniform polytopes arise from the possible combinations of circled and non-circled nodes in the Coxeter-Dynkin diagram for the parent polytope. So that gives us a way of checking whether our naming schemes cover all the possibilities. As for visualizing them... it's really not that hard in the case of 4D; take a look at http://en.wikipedia.org/wiki/Uniform_polychoron - many of the uniform polytopes, especially the pentatopic/tesseractic ones, have nice projection images on their respective pages.

Yes, I've looked all over Wikipedia's uniform polychora pages and images and I still haven't been able to find a scheme which includes all possibilities, has rectates be self-dual and makes sense to the look of the objects. I don't understand, or agree with (due to rectates, if nothing else) the one that's officially accepted and I'm certain there are problems with the one I made up about, what, two and a half years ago. So I've been avoiding really doing much with them since.

I've made a table of the regular polytopes on the wiki now - next is to either rename the appropriate shape pages, or redirect the redlinks to them.

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### Re: New nomenclature for regular polytopes

Keiji wrote:
quickfur wrote:[...]
Well, all the uniform polytopes arise from the possible combinations of circled and non-circled nodes in the Coxeter-Dynkin diagram for the parent polytope. So that gives us a way of checking whether our naming schemes cover all the possibilities. As for visualizing them... it's really not that hard in the case of 4D; take a look at http://en.wikipedia.org/wiki/Uniform_polychoron - many of the uniform polytopes, especially the pentatopic/tesseractic ones, have nice projection images on their respective pages.

Yes, I've looked all over Wikipedia's uniform polychora pages and images and I still haven't been able to find a scheme which includes all possibilities, has rectates be self-dual and makes sense to the look of the objects. I don't understand, or agree with (due to rectates, if nothing else) the one that's officially accepted and I'm certain there are problems with the one I made up about, what, two and a half years ago. So I've been avoiding really doing much with them since.

I've made a table of the regular polytopes on the wiki now - next is to either rename the appropriate shape pages, or redirect the redlinks to them.

There's no shame in having slightly redundant terminology for stuff like your "rectates" (which technically speaking are more like mesotruncates - i.e., truncate halfway between a polytope and its dual), overlapping with a Coxeter-Dynkin-based scheme for naming the other stuff.

The thing to note about the circled Coxeter-Dynkin diagrams is that the configuration of circles correspond with various elements of the underlying symmetry being "expanded". For example, in 3D, cubic symmetry has 6 faces (call it A), 8 vertices (call it B), and 12 edges (call it C). If B is expanded into faces and A reduced to vertices, then you get an octahedron. If both A and B are expanded into faces, you get the truncated cube. If you reduce C to vertices, it becomes a cuboctahedron. And so forth.

So one way to approach this is to forget about truncation, and instead focus on how the surtopes correspond with elements in the underlying symmetry.

Of course, this scheme doesn't cover all possibilities: in each dimension, there may be sporadic uniform polytopes that can't be easily classified. For example, the snub 24-cell is a sporadic that just happens to be uniform in 4D; the 5D equivalent is not uniform, and there may not even be a well-defined 6D equivalent; the 3D equivalent is the regular icosahedron. The grand antiprism can't even be expressed with Coxeter-Dynkin diagrams, and is another sporadic. Gosset's E polytopes may also be considered as sporadics, although in lower dimensions they coincide with non-sporadic polytopes. There are also other sporadics in very high dimensions, corresponding with mathematical objects of high symmetry (Jonathan Bowers mentioned them to me once, but I forgot what they are) with no lower-dimensional equivalents.
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### Re: New nomenclature for regular polytopes

quickfur wrote:The thing to note about the circled Coxeter-Dynkin diagrams is that the configuration of circles correspond with various elements of the underlying symmetry being "expanded". For example, in 3D, cubic symmetry has 6 faces (call it A), 8 vertices (call it B), and 12 edges (call it C). If B is expanded into faces and A reduced to vertices, then you get an octahedron. If both A and B are expanded into faces, you get the truncated cube. If you reduce C to vertices, it becomes a cuboctahedron. And so forth.

Yes, I've known about this for quite a while. It still doesn't help me understand the higher dimensions truncations, though. At least, not any more to be able to make justified analogs between the dimensions.

Ignoring snubs, the grand antiprism and whatever other exotic solutions there are, the patterns I completely understand so far are (and forgive me if you aren't familiar with regexp):
• xo* (parent),
• reversing the string (dual),
• xxo* (lowest-order truncate),
• xo*x (highest-order truncate),
• xxx* (omnitruncate).

For rectification, I can't tell whether the pattern should be simply ox+o (all mirrors apart from end ones set), or onxx?on (i.e. the middle mirror set in odd dimensions, and the middle two mirrors set in even dimensions). These two possibilities are identical in both 3D and 4D, so I'd need to look to 5D for a decision-maker. But my visualization skill in 5D is limited to that dimension's hypercube and simplex only (which are easy to visualize in any dimension, really).

However, I have no idea where to begin for the polychora with Dx 5, 7 or 11. I understand how they are constructed, but I don't know what "class" to put them in. It's obvious they don't have 3D analogs, as I've already exhausted the possibilities of 3D. I know some would be quick to consider 5 and 7 analogs of 3D's 5 and 7 (probably the same people who labeled Dx 6 as bitruncate instead of rectate in 4D), but I don't think you can just arbitrarily append 'o's to the CD string to make analogs.

Another question is the double rectifying of a parent to form a highest-order truncate. This applies in 3D and 4D, but I have no idea whether it would apply in higher dimensions.

To me, I can't really come up with, or accept, any completely general naming scheme for arbitrary Dx-series uniform polytopes, until I have and understand some kind of algorithm to group every one of them into "classes".

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### Re: New nomenclature for regular polytopes

Forget about truncations for the moment. One very intuitive way to think about Coxeter-Dynkin diagrams is to read them in the following way:

- Start with the terminal node that is connected to the numbered edge (there is no numbered edge for the simplices, but it doesn't matter 'cos they're self-dual and the order you read the diagram doesn't matter), and step through each node, noting at each node whether it's circled.

- The first node corresponds with dimension 1, and each subsequent node corresponds with the next higher dimension. At the first node, we begin with a single point. If the node is circled, that means we "expand" the point along 1D space, giving us an edge. Otherwise, it remains a point, and we move on to the next node.

- At each subsequent node, if it's not circled, that means we attach instances of the shape we have so far to each other according to the symmetry group we're working with, without adding extra space between them. If the node is circled, then we "expand" the shape outwards as we fit them together, filling in the gaps in between with the appropriate prisms. Either way, this produces a polytope of the next higher dimension.

Maybe an example will make this clearer. Let's take cubic symmetry for simplicity. So the Coxeter-Dynkin diagram has an edge marked 4 at one end. We start from that end, and read each node until we reach the other end.

Take for example, the cube. The diagram is oxx. We start with a single point. The first node is circled, so we "expand" the point into an edge. The second node is not circled, so we attach 4 edges to each other with no intervening space, producing a square. The third node likewise is not circled, so we attach 6 squares to each other with no intervening space, producing a cube.

Now take a diagram like xoo. We start with a single point. The first node isn't circled, so the point remains a point. The second node is circled, so we "expand" the point outwards according to square symmetry (remember the 2nd node == 2D) and connect them with edges. This makes a diamond shape. (Remember that each diamond is oriented and placed according to square symmetry, so it is distinct from a square.) The third node is likewise circled, so we join the diamonds together in cubic symmetry like a cuboctahedron, but expand it outwards, thus forming a truncated octahedron.

In other words, the Coxeter-Dynkin diagram describes exactly how the polytope is built. Take a 5D hypercubic polytope like oxoxo, for example. The first node gives an edge, which becomes a square in the second node, which then gets expanded in the third node, forming a small rhombicuboctahedron, which becomes a 4D polychoron having small rhombicuboctahedra as cells (touching each other by their axial square faces---because the 4th node is not circled so no space is introduced between them) -- i.e., the cantellated tesseract. The last node turns this into a polyteron with cantellated tesseract facets that are "expanded": there are rhombicuboctahedral prisms connecting them.

Don't ask me what the name of this thing is, I just know how to read the diagrams, I don't know the standard naming convention. The point is that the Coxeter-Dynkin diagram gives you a precise way to construct these things, with each node specifying whether to simply attach the previous surtopes together with no intervening space, or to "expand" them outwards and attach them via the appropriate prisms.

So far I've only illustrated using hypercubic symmetry, but the same principle goes for the other symmetries. E.g., for icosahedral symmetry, the 2nd node will place the points/edges in pentagonal symmetry, the 3rd node places the resulting faces in icosahedral symmetry, and the 4th node places the cells in 120-cell symmetry.

IOW, instead of thinking about what results from truncating hypercubes and cross polytopes, which frankly isn't all that obvious unless you're Wendy, you build up the uniform polytope directly from its Coxeter-Dynkin diagram, which immediately tells you what kind of facets it has. E.g. the 5D oxoxo polytope I gave above has cantellated tesseracts as facets (we know this just by what kind of shape we had by the 4th node), with rhombicuboctahedral prisms joining them (we know this 'cos the last node is circled). The remaining facets are simply simplicial 4D polytopes whose Coxeter-Dynkin diagram is circled the same way as the last 4 nodes of this polytope. In general, the first N-1 nodes and the last N-1 nodes tell you the shapes of two classes of its facets, and the last node tells you whether prisms are inserted between those facets. (Except for certain special cases, all facets corresponding to surtopes that aren't vertices or facets in the base regular polytope must be prisms, since otherwise the polytope won't be uniform.)
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### Re: New nomenclature for regular polytopes

P.S. The fact that there are potentially 2^N uniform polytopes per symmetry group in N dimensions (a little less for self-dual symmetries) means that if you want a generic naming convention, it'd better be able to handle exponential growth in the number of possibilities as you go up the dimensions.
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### Re: New nomenclature for regular polytopes

In N dimensions, there are these forms, so to exclude duplicates (eg x3o3o and o3o3x are duplicates)

{3..3,4} 2^N - 1 , also {3..3,5}
{3..3,3} (2^N + 2^((N+1)DIV 2) -2 )/2, also {3,4,3}
{3..3,A} 2^(N-2), [N=5 and larger]

For the gosset-groups: 2_21, 3_21 and 4_21, one has (64+16-2)/2 = 39, 127 and 255 respectively

These evaluate to: 3D: 7,5 ; 4D, 15, 9 ; 5D: 31, 19, 8, ; 6D: 63, 35, 16 , 39; 7D 127, 71, 32, 127; and 8D 255, 143, 64, 255, and so on.

One reduces according to the equities not counted here, and add further constructions (viz snubs)

all [-] The measure polytope is a prism-product of lesser figures.
3D [-] a3c3a = c3a4o, three examples, reduces 5 to 2
[+] sPsQs, 2 examples, the equity s3s3s = s3s4o = x3o5o
4D [-] a3c3a4o = c3a4o3o : three examples
[+] s3s4o3o, grand antiprism [non-wythoff construction]: 2

When one is enumerating all of the uniform polytopes, one also eliminates those that are derived by uniform product (prisms), to prevent things like the cube being counted also as a square-prism. To this end, the cube is a uniform product of the line-segment, and is counted at this point. One then gets:

In this list, the simple line prism, like "pentagonal prism" is counted as a pentagon. In this way, the list to N dimensions includes everything before it. One notes the equity of cube = square.line prism, square = line.line prism, means that the square, cube, tesseract, &c are counted at the very top of the list, under line. The other prism-powers (simplex and cross), are generated by products that do not generate uniform figures automatically, are counted separately for each power.

1D 1 point, line, square, cube, tesseract

2D oo polygons, except the square; triangle, pentagon, hexagon, etc , and their prisms, square-prisms, etc.

3D oo The infinite family of antiprisms, excluding the cases for 2 (tetrahedron) and 3 (octahedron)
17 being, 5+7+7 - 4 + 2 those from {3,3}, {3,4}, {3,5}, less equities, plus non-wythofian

4D oo the product of polygons, except those involving the square
46 9 + 15 + 9 + 15 - 4 + 2 , being {3,3,3}, {3,3,4}, {3,4,3}, {3,3,5}, less equities, plus non-wythoffian

5D oo, being polygon-polyhedra
57 = 19+8+31-1 ; being {3,3,3,3}, {E,3,3,3}, {4,3,3,3}, less equities.

6D oo polygon-polygon-polygon prisms
oo polygon-polychoron psisms
153 polyhedron-polyhedron prisms
152 = 35+16+63+39-1 being new non-prism wythoffians, less equities.

7D oo polygon.polygon.polyhedron
oo polygon.polyteron
782 polyhedron-polychora
356 = 71+ 32+127+127-1 new figures, less euities.

8D oo polygon.polygon.polygon.polygon
oo polygon.polygon.polychora
oo polygon.polyhedra.polyhedra
oo polygon.polypeta.
969 polyhedra.polypeta
1081 pelychora.polychora
717 = 143+64+255+255 new wythoffs, less equities.

and so on.
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### Re: New nomenclature for regular polytopes

quickfur wrote:Forget about truncations for the moment. One very intuitive way to think about Coxeter-Dynkin diagrams is to read them in the following way: [...]

Alright, that does help. It helps me understand how they are constructed and hopefully it will help me visualize them. However, it doesn't relate to the patterns I found, for example, it doesn't tell you whether it's self-dual or not, since you're only taking one node at a time.

quickfur wrote:[cube = oxx, cuboctahedron = xoo]

Hmm, I'm used to reading those the other way round - x for a circled node and o for a non-circled node. If you look on the wiki, all the CD strings will be the inverse of the ones you're posting...

quickfur wrote:In general, the first N-1 nodes and the last N-1 nodes tell you the shapes of two classes of its facets, and the last node tells you whether prisms are inserted between those facets. (Except for certain special cases, all facets corresponding to surtopes that aren't vertices or facets in the base regular polytope must be prisms, since otherwise the polytope won't be uniform.)

This sounds like a useful interpretation. I shall have a look at them bearing this in mind and see if I can come up with anything.

quickfur wrote:P.S. The fact that there are potentially 2^N uniform polytopes per symmetry group in N dimensions (a little less for self-dual symmetries) means that if you want a generic naming convention, it'd better be able to handle exponential growth in the number of possibilities as you go up the dimensions.

I already knew I would have to arrive at an infinite number of classes, with a certain method to enumerate them. However, what I had in mind was dividing them up so that the most important classes come first, and the more arbitrary ones come much later.

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### Re: Constructing uniform polytopes (split from new nomenclature)

BTW, I've just found the pattern for generating the entire series of n-cube uniform truncations. It is as follows:

2D:
xo - square (sorry for the reversed notation last time, I'll try to stick to the convention of x=circled node and o=uncircled node from now on)
xx - octagon (=mesotruncated square) (*)
ox - diamond (dual square)

3D:
xoo - cube
xxo - truncated cube
oxo - cuboctahedron (rectified cube - or "hemified" in your terminology) (*)
oxx - truncated octahedron
oox - octahedron

4D:
xooo - tetracube
xxoo - truncated tetracube
oxoo - rectified tetracube (note the cuboctahedral cells)
oxxo - bitruncated tetracube (= mesotruncated 16-cell) (*)
ooxo - rectified 16-cell (=24-cell)
ooxx - truncated 16-cell (note the octahedral cells, which are vertex figures of the 16-cell)
ooox - 16-cell

5D:
xoooo - 5-cube
xxooo - truncated 5-cube
oxooo - rectified 5-cube
oxxoo - bitruncated 5-cube
ooxoo - mesotruncated 5-cube (*)
ooxxo - bitruncated 5-cross
oooxo - rectified 5-cross
oooxx - truncated 5-cross
oooox - 5-cross

I think you're beginning to see a pattern emerge. In each dimension N, the truncation series is basically the truncation series of dimension (N-1) suffixed with o's, plus two new entries consisting of o+xx followed by o+x (using your regex notation). Here's the "intuitive" justification of it:

As you truncate a polytope P, its facets undergo truncation just like in the previous dimension (a truncated tesseract has truncated cubes as cells, for example, and the rectified tesseract has rectified cubes (=cuboctahedra) as cells, and so forth). When the facets eventually get truncated to the dual shape, that's precisely the point where you have reached the rectified dual of P, because the vertex figure of the dual of P is simply the dual of P's facets. Truncating it further results in the reduction of the facets' dual shape, but doesn't change the shape 'cos you've already reached the dual; they will merely shrink. Since shrinking them introduces gaps which must be filled by other facets, the last node in the Coxeter-Dynkin diagram must be circled (remember "expansion" - shrunk facets bridged by other facets = circled node). So the second last entry must be of the form o+xx. Once the original facet is finally reduced to a point, you have reached the dual of P, by the definition of dual. So the last entry must be o+x.

So which one is the mesotruncate (your "rectified" polytopes)? Precisely the middle entry in each list. So, by dimension, their respective Coxeter-Dynkin diagrams are:

xx (e.g. octagon)
oxo (e.g. cuboctahedron)
oxxo (e.g. bitruncated tetracube)
ooxoo
ooxxoo
oooxooo
oooxxooo
...

You get the idea.
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### Re: Constructing uniform polytopes (split from new nomenclature)

quickfur wrote:oxo - cuboctahedron (rectified cube - or "hemified" in your terminology) (*)

Hemication is a term I made up to refer to oxoo so that oxxo could be called the rectate. There is no 3D hemication, the cuboctahedron is a rectate.

I had noticed this pattern before, but I figured maybe it didn't generalize to higher dimensions so I didn't do anything with it. I'm surprised that the 5D rectate is ooxoo instead of oxxxo though - while I said I didn't know which it would be, I was expecting it to be ox+o because that pattern is simpler.

So, where do the other strings come into play then?

i.e. xox, xxx in 3D, xoox, xoxo, xxox, xxxo, xxxx in 4D, etc.

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### Re: Constructing uniform polytopes (split from new nomenclature)

Keiji wrote:
quickfur wrote:oxo - cuboctahedron (rectified cube - or "hemified" in your terminology) (*)

Hemication is a term I made up to refer to oxoo so that oxxo could be called the rectate. There is no 3D hemication, the cuboctahedron is a rectate.

Well, the term "rectate" comes from 3D, and it so happens that in 3D the mesotruncate is equivalent to truncation at the midpoint of edges. But in higher dimensions these two become distinct, and you have to choose which of the two will inherit the classification of "rectate".

I had noticed this pattern before, but I figured maybe it didn't generalize to higher dimensions so I didn't do anything with it. I'm surprised that the 5D rectate is ooxoo instead of oxxxo though - while I said I didn't know which it would be, I was expecting it to be ox+o because that pattern is simpler.

Well, like many polytopic patterns, many things coincide in lower dimensions but diverge in higher dimensions.

Taking n-cube symmetry as an example, we have the following:
In 3D, oxo is the cuboctahedron, equivalent to the mesotruncate and the rectate.
In 4D, oxxo is still equivalent to the mesotruncate, although the rectate diverges at this point.
In 5D, oxxxo is something with oxxx as facets: oxxx is a 4D uniform with truncated octahedra as cells, joined to each other via square prisms. Its other cells are truncated tetrahedra (o3x3x). Looking it up in Wikipedia, it's the cantitruncated 16-cell, which happens to be the same as the truncated 24-cell. At this point, the ox+o polytopes diverge from the mesotruncates: the large number of circled nodes make these polytopes contain more "expanded" elements than can be obtained just by truncating the regular polytopes (see below).
In 6D, oxxxxo is something with oxxxx as facets. oxxxx is a 5D polytope with cantitruncated 16-cells (truncated 24-cells) as facets, separated by truncated octahedral prisms (don't know what it's called in the "standard" terminology).

and so on.

So, where do the other strings come into play then?

i.e. xox, xxx in 3D, xoox, xoxo, xxox, xxxo, xxxx in 4D, etc.

Well, xxx, xxxx, xxxxx, ... etc., are the omnitruncates. The others are analogs of the rhombicuboctahedron.

Again, remember that the Coxeter-Dynkin diagram tells us how to construct these things; each subsequent node tells you whether the element corresponding to that dimension is "expanded" or not. Expanding all elements in every dimension (i.e., xxxxx) gives you a "fully-expanded" polytope, i.e., an omnitruncate. Expanding only a subset of them gives you a "partially-expanded" polytope, with various expanded and non-expanded elements. When there is only one x, you get the "least expanded" polytopes: the regular polytopes themselves, and half of the elements in the "main truncation sequence". The "expanded" elements are things like octagons, truncated cubes, truncated octahedra, etc., where there are prisms connecting the faces; the non-expanded elements are like cuboctahedra (no prismic elements). Surtopes of higher dimensions may have their own combination of which elements are expanded. Which combination of elements are expanded, is specified by which node is circled in the Coxeter-Dynkin diagram. As you go up in dimension, you get more and more possible combinations of expanded or non-expanded elements, each combination leading to a distinct uniform polytope.

In other words, circled nodes tell you whether elements of the corresponding dimensions are expanded or non-expanded: it's a description of how each aspect of the underlying symmetry group is expressed.

To elaborate: take for example cubic symmetry. It has 3 classes of elements: a 6-fold symmetry corresponding to a cube's faces, an 8-fold symmetry corresponding to the cube's vertices, and a 12-fold symmetry corresponding to the cube's edges. If only the 6-fold symmetry is expressed, you get a cube. If only the 8-fold symmetry is expressed, you get an octahedron. If both the 8-fold and 6-fold symmetries are expressed, you get truncated cubes (if the 6-fold symmetry is "more dominant" in expression), truncated octahedra (if the 8-fold symmetry is "more dominant"), and cuboctahedra (both 8-fold and 6-fold symmetries are "equally expressed"). If 12-fold symmetry is also expressed, you get things like the small rhombicuboctahedron (all 3 symmetries "equally" expressed) or the great rhombicuboctahedron (8-fold and 6-fold symmetries are "more dominant" than the 12-fold symmetry). For a non-uniform example, if only 12-fold symmetry is expressed, you get a rhombic dodecahedron. Of course, Coxeter-Dynkin diagrams only describe the uniform subset of the full set of possibilities, but still, it does capture some of that structure. So you may think of circled nodes as increasing the "dominance" of a particular aspect of the underlying symmetry, and uncircled nodes as "less dominant" expressions of that aspect of the symmetry.
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### Re: Constructing uniform polytopes (split from new nomenclature)

P.S. Here's an interesting tidbit about omnitruncated n-cubes (the x+ polytopes with n-cube symmetry):

You can construct their Cartesian coordinates as (1, 1+sqrt(2), 1+2*sqrt(2), ... 1+n*sqrt(2)). (To see this, consider the meaning of the "expand" operator: to expand a square into an octagon, we add/subtract sqrt(2) from the coordinates of the vertices. To expand a truncated cube into a great rhombicuboctahedron, we add/subtract sqrt(2) to the coordinates of the truncated cube, etc.. Each time, sqrt(2) is added. Since n-cube symmetry entails taking all permutations of coordinates and sign, we can arrange the coordinates as shown above.)

Since the last n+1 nodes in the Coxeter-Dynkin diagram are all ringed, that means its facets lying in the hyperplanes parallel to the n-cross's facets are omnitruncated (n-1)-simplices. We can select the omnitruncated simplex orthogonal to (1,1,1,1...,1), whose coordinates are all positive: its vertices therefore are simply all permutations of coordinates (but not of sign) of (1, 1+sqrt(2), 1+2*sqrt(2), ..., 1+n*sqrt(2)). Since translation and scaling preserves the shape of the polytope, we can subtract (1,1,1,1...) from the coordinates and divide them by sqrt(2), giving us (1,2,3,4,...n).

The polytope whose coordinates are all permutations of coordinates of (1,2,3,4,...,n) is known the permutahedron of order n. By the above derivation, we conclude that the permutahedron of order n is simply the omnitruncated (n-1)-simplex.
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### Re: Constructing uniform polytopes (split from new nomenclature)

The family of rectates and truncates, can be done by descent of faces of the dual. This means by intersecting of a figure and its dual as the dual goes to zero size. In the wythoff sequence, one has eg xoooo -> xxoooo -> oxoooo -> oxxoooo -> &c.

The names like truncate and rectate, refer to the last 'x', counting from the head as 0,1,2,3. So the bitruncate is oxxooo...

Note that the vertices of the n-rectate fall on the surtopes of dimension n, and the n=truncate falls on the line connecting the centres of n-1 and n surtopes.

There are two other simple families.

1. The runcinate has a vertex falling on the line between the centre and vertex of a face.

2. The n-cantellates and n-cantetruncates, falls by respectively rectifying and truncating the n-rectates. For sequence with one marked node, the cantellates are the figures which mark those nodes that are connected to the marked node (but not the marked node itself), and the cantetruncate is the same, including the original node.
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