## For Beginners: Plane vs. Hyperplane

Ideas about how a world with more than three spatial dimensions would work - what laws of physics would be needed, how things would be built, how people would do things and so on.

### For Beginners: Plane vs. Hyperplane

One of the things that confused me most when getting started with this was the plane. I'm used to planes being flat, being surfaces, and partitioning spaces. In 4D, things are different. Eventually I realized what was confusing me was the distinction between planes and hyperplanes. A plane has dimension 2 while a hyperplane has dimension N-1, with N the number of spatial dimensions. Only in 3D are the plane and the hyperplane one and the same.

A hyperplane is a surface and may be a partition.

A plane is only a partition or surface in 3D. In other-dimensioned spaces it remains useful to specify subspaces such as a plane of rotation.

A hyperplane may have fewer dimensions than a plane. In 2D a plane is 2D, but a hyperplane is 1D. In 1D there are no planes and a hyperplane is zero dimensional. Surfaces and partitions are zero dimensional.

Then came the question of flatness. Is a plane flat? Is a hyperplane flat? There is no real answer. The best I was able to do was to think of a hyperplane as flat from one point of view but having a volume from another point of view, and switch back and forth rapidly. In an N dimensional space this is true of all objects of fewer than N dimensions.

Planes are always flat I think, but in higher spatial dimension they have a sort of skinny flatness, like a line has in 3D. The easiest thing was to think of anything that can be embedded in an N-1 dimensional space as flat.
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### Re: For Beginners: Plane vs. Hyperplane

I think the term hyperplane just means a general n-plane of arbitrary dimension. Like hypercube is the same as saying n-cube. You don't get the distinctive flatness until comparing n-1 planes with n-planes. One dimension less is a flat space that divides. An n-2 plane is comparable to the thinness of a needle. Two dimensions less, and you get a 'line-poking' tendency, even when graphing 5D things in 3D slices.

And, for n-3 planes, well, our closest analogy is a singular point-cut, which does actually play the role of a point, but contains more than zero dimensions. For instance, a 6D cube array of 64 objects can be decomposed into a 2x2x2 cube array of identical 2x2x2 cube arrays, where the 8 of them are stacked in 3 new, orthogonal directions.

If you tried to approximate what this would look like, you'd probably imagine a cube array of smaller cube arrays of 64 objects. But, this is not actually the case, since this is still thinking in 3D terms. The eight 2x2x2 cube arrays are in fact layered against each other in a higher dimension, as if they were flat (which they are!). The layout of there being a 2x2x2 cube array of 2x2x2 cube arrays is correct, where you can place the point-like 3-plane at any of these 8 locations, and see a 2x2x2 cube array.
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### Re: For Beginners: Plane vs. Hyperplane

Words have etymologies, and a good number of us were taught these at school.

You will find in Mathematics and Sciences, that words are attached to different meanings, to the extent that it causes a lot of confusion as students go from class to class. This in general will give absurd meanings like (upstairs = floor 2), being used when spoken on floor 6. There are numerous examples where words like 'cell' take on quite bizaar meanings, and a new word has to be invented for the original.

The remedy is to get someone who understands language, and prepared to work across the whole glossary, to fix things. You should note that because 'plane' has a meaning in 3D, that it equates to '2D', it does not mean it is appropriate to use that word for a 2d space in 4D. Plane has a number of loaded meanings which are transferred to different things in 4D.

Even 'hyperspace' etc, look rather stupid when you think that hyper-space (ie the space above all space), is the sort of thing you make monkey-bars out of in six dimensions. Hyper- means 'over', so if you are talking in six dimensions, hyperspace is seven dimensions. If you are talking in two dimensions, hyperspace is three.

An example of the remedy is my 'polygloss' http://www.os2fan2.com/glossn/index.html .

You have in the higher dimensions, that subspaces are constituted of + signs (number of orthogonal lines), and = signs (number of equities to define). One fills out + signs to the dimension of the subspace, and equal signs to the containing space.

A plane is simply something one stands on, and is thus represented by one equal sign. So a plane is h=0, is 'one equal sign'. In 3d it comes ++= (2d), in 4d it is +++= (3d), and so forth. The thing that makes the monkeybars is a marginix, or two equal signs, ie in 3d it is ==+ (a line), in 4d, it is ==++ (a hedrix).

A set of words has been created to name dimensions by number, ie

+ line, ++ hedrix, +++ chorix, ++++ terix, +++++ petix, 6+ ectix, 7+ zettix, 8+ yottix.

If you want to make patches from these fabrics, replace the -ix with -on, eg choron. So a twelftychoron, is a figure made out of 120 3d patches, or a 4D version of the dodecahedron.

All space itself is a hyperplane, when one finds the need to exceed the space by an additional number of dimensions.
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### Re: For Beginners: Plane vs. Hyperplane

The problem with standard terminology such as "plane" is that it's ultimately 3D-centric, and carries with it 3D-specific connotations that may not generalize to higher dimensions.

In fact, there are two different functions of planes in 3D, which are conflated in 3D but become distinct in higher dimensions. In 3D, planes serve both as a division of space (a 2D plane cutting through 3D space divides the space into two half-spaces, one on either side), and also as the basis of phenomenon like plane rotations (points on a rotating object trace out circles, which therefore define a plane of rotation).

In 4D, however, these two functions split: a 2D plane in 4D no longer divides space. It amounts to what in 3D would be a 1D line, stretching indefinitely through space but without dividing the space into two halves. However, a 2D plane in 4D still serves as the basis for plane rotations. In order to divide 4D space, you need a 3D hyperplane. This is why 4D objects (that occupy non-zero 4D hypervolume) have surfaces that are 3D manifolds. A 2D surface, like that of 3D objects, is insufficient to partition the space into "inside" and "outside".

So basically we are looking at two separate things: the (n-1)-dimensional plane, which serves as a divider of space, and the 2-dimensional plane, which serves as the basis for plane rotations. In 3D, these two things coincide; however, in higher dimensions, they are different things, and therefore must be understood separately.
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### Re: For Beginners: Plane vs. Hyperplane

This distinction then happens to be essential whenever transporting concepts of 3D into higher dimensions.

One best-known example are the well-known Johnson solids: the set convex polyhedra with regular faces, once enumerated by Norman Johnson.
So what shall mean that "faces" within higher dimensional setups?
• The first consideration, then enumerated by the Blind couple, was: "face" ought mean then (n-1)-face.
• But the newer research assumes 2-face instead. This is what CRF stands for: convex regular-(2-)faced polytopes.
--- rk
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### Re: For Beginners: Plane vs. Hyperplane

http://www.os2fan2.com/glossn/updown.html

A word acquires many meanings, and some of these meanings enter popular use. My doctor related to me, that 'genetic disorders', has nothing to do with 'genes', but derive from some earlier meaning of the word (The root means 'born'). The same is true in many of the sciences one encounters, the science uses these as a variable, and the student is supposed to go from room to room, supposing that all arts of science are agreeing and describing the truth. Neither are true.

Polytopic in biology, has nothing to do with polytopes, but is truer to the roots than the geometric one. Poly = many, and tope = place. In biology, polytopic means a feature or gene that has come in the same form in many different places. For example, the same mutation might be observed into isolated populations. Topic is derived from tope, refers to along the lines between local gossip to information on the local transport.

One might note that we live in a reality loosely based on E2H. That is, we live in a euclidean 2d space, with height. This is the flat-earth theory. We can refine this theory, to S2H, &c, but it serves no end to this discussion. So we walk at the bottom of an ocean of air, on a fairly extensive plane. When we use words, they are fixed to this model, but E2H is essentially E3, the words still have some meanings. Euclid's plane geometry deals with the ground, and solid geometry deals with buildings.

When we go randomly into 4D, then that is space over 3D, and 'hyperspace' is largely correct. It's like 'outside' if you never been there. Or 'upstairs'. But a good portion of the art is given to spending a good deal outside, and 'hyperspace' and 'planes' no longer are befitting of their 3D meanings.

To make the student's life less miserable, one ought tease apart the most significant parts of the word from their perception, and try to keep that meaning in higher dimensions.

A thing has both consist and context. An edge of a blade cuts in three dimensions, and we might suppose it consists of 1D element is the cause. But the same blade in four dimensions, would require an edge of two dimensions to cut. In essence, a blade divides space on motion, ie is solid space ÷ cutting ÷ sweeping, or 2d less than solid.

A plane, in the sense of the ground, is represented by h=0, as such, is 1D in 2d space, 2D in 3d space, and 3D in 4d space.

We can now create symbols representing consist (ie number of included dimensions), and context (relation to solid space), along with a special mark for something at the bottom of the ocean of air. The dimensions we mark +, the context (by equal signs like h=0), by =, and the ground itself is ~.

A line is 1D, and this is preserved in shipping line, bus line, bee line. But a line in the sand, to 'toe the line', a ship of the line, dead-line, are actually an equals on the ground, ie =~. Here the line is not to get to get you from A to B (as + infers), but to prevent you from crossing it. (ie =~ a division on the ground). The edge of a blade is not + (1D), but == (a cut by sweeping).

When these are fleshed to four dimensions or higher, one adds as many of the + or = as not mentioned in the definition. So a plane as = becomes +++= (a three dimensional thing), and a blade ( ==) comes to be ++== . The bee-line + is +==~ something that requires two equations as well as h=0, to define.

Hyper simply means 'over', and we add an extra & to the front of the equation, to say that we normally restrict ourselves to 4D, but we need to go over our needs with 5D.

Space is (=) comes +++. Hyperspace is then &+++, but in 4D, it's &++++. A hyperplane is simply &=, gives &++= in 3D, and &+++= in 4D. In other words all space is a hyperplane! 3d space is hyper-space to 2d space.

Space is made of fabric, and we have words to denote fabric (infinite or indefinite boundary), against patches (a definite boundary). Patches are sewn together to make things.

A poly+hedr+on is many 2d patches sewn together to make a closed bag. So poly means many, hedr is taken as 2d, and on as patch. We can simply create huge numbers of words by fixing hedr and on to other things. Putting -ix, -ices, will make a fabric out of the patches. Hedra are cut from a hedrix or hedrid sheet. -id comes from solid, here it means solid in 2d space. -ous means roughly, like like calling a snake a latrous thing (1d-like thing).

Latr-, hedr-, chor- and ter- are the 1d, 2d, 3d, 4d roots. You make fabric by -ix, and patches by -on. From there you add prefixes to say what you mean. Poly means a closed many. Apeiro- means without any boundary in the space they're in (ie a tiling). An apeirochoron is a tiling of 3d patches, literally 'without a fence, 3d patches'.
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### Re: For Beginners: Plane vs. Hyperplane

Thank you very much for taking the time to write this. I'll absorb it later.

wendy wrote:http://www.os2fan2.com/glossn/updown.html

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### Re: For Beginners: Plane vs. Hyperplane

I'm pretty sure wendy has an alternate version of the glossary where th is used in place of thorn. Just ask. :-)
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### Re: For Beginners: Plane vs. Hyperplane

@klitzing: "face" is one of the most ambiguous and troublesome terms when it comes to higher-dimensional geometry, because of the conflation of (n-1)-dimensional element with 2D polygon, and also (n>1)-dimensional element in general. As far as I know, there are at least the following uses of the term (there are probably more):

1) An (n-1)-dimensional element of an n-dimensional polytope. Due to ambiguity, this usage is fortunately relatively rare except in the case of 3D polytopes, though it still does occur. Personally, I use the term "facet" to refer to (n-1)-dimensional elements in order to distinguish it from the below usages.

2) A 2D element of an n-dimensional polytope. This occurs in some of the literature speaking about 4D polytopes, and causes confusion because "face" carries the intuitive connotation that the element has a facing direction, like in 3D, and constitutes one of the surface patches that divide the inside of the polytope from the outside surrounding space. It also adds confusion with the actual "faces" (in the sense of a space-dividing surface patch) of the polytope. Thus one hears of the tesseract having 24 faces vs. 8 cells (which ought to be called "faces" in this sense!). It wouldn't be so bad if "face" was reserved exclusively for 2D elements, but unfortunately the next usage compounds this problem.

3) A general j-dimensional element of an n-dimensional polytope, where j ≤ n. Usually written as j-face (as in, 1-face, 2-face, 3-face, etc.), but in some contexts may appear without the disambiguating prefix. Thus one sometimes reads of a polytope having 80 faces (meaning 80 elements of various dimensions), giving the misleading idea that there are 80 facets, or perhaps 80 2D elements, but it's actually referring to the sum of all elements from 0D to (n-1)-D. As to elements of which dimension "face" might refer to, one must read the context, though even then it is not always clear!

In the acronym CRF, we have chosen usage (2), though arguably it is quite ambiguous, as it can refer, as you said, to the Blind polytopes or to what we currently call CRFs. Perhaps it would have been wiser to use CR2F as the acronym instead, but I suppose it's far too late to change that now (and it would only cause even more confusion when it gives the wrong impression that CR2F is somehow different from CRF).

In light of these conflicting uses of terminology and the ensuing confusion, one can understand why wendy invented her polygloss!
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### Re: For Beginners: Plane vs. Hyperplane

quickfur wrote:@klitzing: "face" is one of the most ambiguous and troublesome terms when it comes to higher-dimensional geometry, ...

In the acronym CRF, we have chosen usage (2), though arguably it is quite ambiguous, as it can refer, as you said, to the Blind polytopes or to what we currently call CRFs. Perhaps it would have been wiser to use CR2F as the acronym instead, but I suppose it's far too late to change that now (and it would only cause even more confusion when it gives the wrong impression that CR2F is somehow different from CRF).

In light of these conflicting uses of terminology and the ensuing confusion, one can understand why wendy invented her polygloss!

In fact, I remember that in the early days, once Wendy understood what CRF ought mean, she aimed to propagate CRH (convex regularly hedrated polytope) instead ... - But since it never made its way.

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### Re: For Beginners: Plane vs. Hyperplane

The polygloss exists in two forms, because there are people who read the þ version and people who read the th version. I found this out when I got complaints from people when i uploaded the th version over the þ version.

Both are compiled from the same source, it amounts to fiddling with the header in pgloss.kml or something. But the þ version is updated more often than the th version is.

The th version has /glosss/ and the þ version has /glossn/ in the URL. I checked the th version, and this page is not yet in it. It is something that came out of a discussion between a number of linguistists, regarding the use of several different meanings (eg fireman vs policeman or something like that). When the th form is updated it will appear
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### Re: For Beginners: Plane vs. Hyperplane

PatrickPowers wrote:One of the things that confused me most when getting started with this was the plane. I'm used to planes being flat, being surfaces, and partitioning spaces. In 4D, things are different. Eventually I realized what was confusing me was the distinction between planes and hyperplanes. A plane has dimension 2 while a hyperplane has dimension N-1, with N the number of spatial dimensions. Only in 3D are the plane and the hyperplane one and the same.

A hyperplane is a surface and may be a partition.

A plane is only a partition or surface in 3D. In other-dimensioned spaces it remains useful to specify subspaces such as a plane of rotation.

A hyperplane may have fewer dimensions than a plane. In 2D a plane is 2D, but a hyperplane is 1D. In 1D there are no planes and a hyperplane is zero dimensional. Surfaces and partitions are zero dimensional.

Then came the question of flatness. Is a plane flat? Is a hyperplane flat? There is no real answer. The best I was able to do was to think of a hyperplane as flat from one point of view but having a volume from another point of view, and switch back and forth rapidly. In an N dimensional space this is true of all objects of fewer than N dimensions.

Planes are always flat I think, but in higher spatial dimension they have a sort of skinny flatness, like a line has in 3D. The easiest thing was to think of anything that can be embedded in an N-1 dimensional space as flat.

The 3D Hyperplane in 4D space is equivalent to our 2D Plane in 3D space. In 4D space the 3D Hyperplane is flat. It has zero 4D volume. The sides of a 4D cube (Tesseract) are constructed from 3D Hyperplanes. The walls will have no thickness. The walls will be flat yet be 3D. If you let a moving point particle bump around inside the Tesseract it will eventually bump into every point on every Hyperplane that makes up the Tesseract. If the wall had any thickness the particle would not be able to hit all the points. The animations that are commonly used to explain the Tesseract are extremely misleading and actually detrimental to understanding the Tesseract. They never show the flat sidedness of the Tesseract. A Tesseract is a simple empty box in 4D space. It's walls are made out of 3D planes in 4D space. People on this forum go into talking about all kinds of complicated 4D things and I don't think anybody has grasped the most basic concept of 4D which is the 3D Hyperplane. I think we need to understand the 3D Hyperplane before we can talk about anything else. I think the reason why it seems so complicated is because the Human Brain is just unable to imagine a 3D flat plane. See my Post: Why Humans Will Never Understand 4D Space.
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### Re: For Beginners: Plane vs. Hyperplane

The word 'plane' has the sense of dividing, as well as being two-dimensional. But in 4d, a two-dimensional thing does not divide.

If one keeps relating the words to various senses of three dimensions, you will end up in a terminology confusion quite easily. Using words like 'hyperplane' is not a solution. What happens here is you end up with meanings like the 'AT computer', the 'advanced technology'. This is so antique that it will handle a 32-Megabyte harddrive, and a full 640 K of random access memory.

There are people who study their latin roots. This was the norm when i went to school. What happens here is that without the meaning direct, one can resort to the latin and greek roots, so one says 'tele= far', 'scope = to see', 'phone to hear'. So telephone is to hear-afar. This is what i tried to do with the polygloss. But since there are many new concepts to absorb, it is not a case of take 3d, make 4d. Instead, the same dimension in 3d, serves the role of two different dimensions in 4d. Hyper- means over or above. Hyperspace carries with it, that you can move something out of ordinary space, and move it to a second place. We are hyperspace to two dimensions. We freely lift pieces on the chess-board and place them somewhere else. This is the sort of actions, even geometrically, that hyperspace supposes. But in 4d space, the terrix (or 4-space), is just ordinary space, not hyperspace.

The edge of a knife or sword, in four dimensions, is not a one-dimensional thing, but two-dimensions. The way to think here is that the edge makes a sweep in time, to divide a thing in two. So to divide into two pieces is a plane, or N-1 dimensions, and motion in time produces length. So N-1 is produced by sweeping (ie a 1d travel), an edge that gives N-1 like that, ie N-2.

One of the joys of the polygloss, is that in writing it, I watched a large number of modelers, here and elsewhere, and thought here and there, this needs a word, or that is an interesting idiom. The higher dimensions become harder if you try to use the wrong impressions from three dimensions there.
While the standard art has taken the meanings of words to closely resemble the dimensionality of the present, I have chosen instead to follow the notion that the common words relate generally to solid space, and that it is easier to generate words for the dimensions.
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### Re: For Beginners: Plane vs. Hyperplane

Plane comes from Latin planus, ultimately from Proto-Indo-European *pleh₂- meaning flat. The meaning of "flat" is understood in 3D as having little or no height variations. In geometry, it is taken to be perfectly uniform (i.e. zero height), and thus divides 3D space into two equal parts.

Hyperplane consists of the prefix hyper- which is derived from Greek huper, from the PIE word *upér which means above. Literally it means "above plane" but in this case we are taking it as an analogue of a 2D plane. A hyperplane is therefore understood as a generalization of the standard 2D plane that divides some higher-dimensional space into two equal parts. Even an unbounded line can be counted as a 1-hyperplane since it divides 2D space into two equal parts. But then you run into problems when you describe a 2D plane in 4D. In this case it is still a plane, since when can imagine a 3D hyperplane that contains this plane embedded within the 4D space. In this case you would call them 2-hyperplanes and 3-hyperplanes if we are dealing with 4D space.

Both of these words have native PIE etymons, but not directly into Germanic and English (instead borrowing from Latin and Greek, which are not Germanic languages).

Basically, the trend is that hyper- is used to generalize any 3D-centric term in geometry, like the terms hypercube and hypersphere. Thus a semantic shift happens when geometry is talked about. We tend to prioritize our observations (we are 3D beings after all) rather than speculating based on generalization. Indeed, many use the term 4-cube or hypercube when talking about the tesseract.

A hyperplane is always flat by the virtue of having a zero value in at least one of its dimensional measures, but it keeps some hypervolume and therefore can be visualized. Flatness here is just a result of at least one parameter being exactly zero. Wendy's analysis does help, but we don't have a general consensus.
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### Re: For Beginners: Plane vs. Hyperplane

Eventually, terms derived from hyper- are ambiguous and prone to misinterpretation, because they are, as you said, 3D-centric, and therefore don't always apply, or apply in multiple ways (and are therefore ambiguous) in higher dimensions. Lately, I've come to prefer Wendy's stance of avoiding the use of hyper- altogether. Just call it an n-plane where n is some integer, and it will be unambiguous. A 2-plane is just a plane that fills 2 dimensions. A 3-plane is a plane that fills 3 dimensions. And so on. An (n-1)-plane is one that fills all but one dimension in the ambient space, and therefore divides that space into two halves. Anything with less than (n-1) dimensions cannot divide n-space.

Back to the original topic, in going from 3D to higher dimensions one ought to understand that generally speaking, lower-dimensional phenomena contain coincidences that in higher dimensions become differentiated. So one ought to be careful not to assume that things that go together in 3D also go together in 4D or higher. For instance, in 3D, a 2-plane divides space, and a 2-plane also serves as a rotational plane. However, in 4D, while a 3-plane divides space, a 3-plane does not serve as a rotational plane; that role continues to be played by a 2-plane. Or conversely, in 4D a 2-plane continues to serve as a rotational plane, but a 2-plane no longer divides space. So here we see something that in 3D coincides (the space-dividing 2-plane and the rotational 2-plane), but in 4D (and higher) becomes differentiated.

Other examples of this can be seen elsewhere, such as in certain classes of polytopes. The alternation of a cube in 3D is a tetrahedron, but in 4D and above, the n-simplex is not the alternation of the n-cube, but differentiates into its own class. Similarly, in 3D and 4D the alternated n-cube is regular (tetrahedron and 16-cell), but in 5D it differentiates into its own series of demicubes. Then in 6D, we see the sequence of lower-dimensional demicubes differentiate into yet another class, the Gosset polytopes, that coincide with various things in lower dimensions but become differentiated in 6D and above, up until the grand tesselation in 9D.
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### Re: For Beginners: Plane vs. Hyperplane

When you see the questions in my feed at Quora, you will soon appreciate the motives behind the polygloss.

Dimension, for example, is used to mean 'parallel universe', or 'co-extant universe'. For example, when characters step into another dimension, they are really going to a different 3d world. Curvature in space is entirely different to the usual pictures that come with descriptions of gravity. The usual horn-thing shown for a black hole, is not 'space-time', but the coordinates are actually two space terms and a potential term. It is, in effect, a snooker table, the balls hit on it would in scale, follow what they would in gravity.

Then there is 'the fourth dimension', which you get a lot of space-time guff. No, we are dealing with a different fourth dimension. The fourth dimension of space-time is rather like the flick-cartoons you could do in the margin of books. You flick through the frames and there is the illusion of motion, but it's a lot of stills, and you really can't talk to the character on page 73. The 4d thing comes in when you consider that the characters exist on all pages of the book, and they have a snaking upwards appearence as you stack one image to another. That's 2d space + 1d time as 'the third dimension'.

As quickfur points out, the attributes of polytopes change with dimension. The hexagon in 2d, stands at the head of a chain that passes through the cuboctahedron, a 20-vertex figure runcinated simplex, and then that thereafter. But there is a different chain that passes through the rhombo-dodecahedron, then a strombic pentachoron, and then strombic simplexes thereafter. The strombic simplex is what you get when you look at an n-cube vertex-on: n forshortened cubes at a point, and the outer surface is n(n-1) rhombotopes.

It is not hard, but you really got to keep a clear mind about what is meant, because the words will muddle you up sooner than you give credit for.
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