ICN5D's Algorithm Explained

[ICN5D]

All right, time to explain how my math system works...

This new thread is to be fully referenced through my previous post, where I list the entire enumeration of all shapes and polynomial formulas, that can be made with my notation ( through 5D, so far )

Probably the best course of action is to pick shapes at random later on, and detail how they come about, what the symbols and numbers mean, and compare it to the binomial expansion method. If I leave something out, ASK QUESTIONS! If I leave out a crucial part of the explanation, I probably didn't notice, since I'm the creator and all too familiar with it. There may be some things I take for granted in knowing about, but may not be very clear to everyone else.

So, having said that, let's continue...




The first shape I want to dissect is the one-dimensional line |, a shape everyone knows very well.

Ultimately, a line is made from a 0D point, represented by { * • n }. This sequence is what I call a polynomial formula, or equation. The point { * • n } is equal to the expression (x+1), where ' x ' is the single point ' * ' , and the ' 1 ' is the nulloid ' n ' , the -1 dimensional cell. The symbol • is the dimensional partition, that separates the groups of n-cells by dimension level. The ' • ' symbol can be made with alt-7, in the numpad. Make sure numlock is on! It cannot be made by 7 outside the numpad. Most of you will already know this

To recap:
0D point = * = { * • n }

* - point
• - dimensional partition
n - nulloid


The line can be made by lacing a point to another point, by means of the taper operator " > ". The taper operator, in polynomial form, is represented by " +[*] " , which means to lace all n-cells to a point, and add the n-cells of the base to the final shape. It can also be made by multiplying with a line, but we'll get to that in a bit. So, let's taper us some points, shall we?


The computation format is arranged in the following manner:

{ BASE SHAPE }
[ OPERATOR / SHAPE ]
-------------------------------
{ FINAL SHAPE }


-The BASE SHAPE is the starting shape we will be applying an operator to, or multiplying with in a cartesian product.

-The OPERATOR is, of course, the action function that we apply to transform the BASE into the FINAL shape. An entire shape all together can also be here as well. This makes cartesian products, and the simplest is the line-product [ | • 2:* • n ], when we extrude something into N+1 to make prisms.

-The FINAL SHAPE is the end result of the computation, the product after we've transformed the BASE by means of the OPERATOR.





Lacing a point to a point:

{ * • n }
+[*]
----------------
{ | • 2:* • n }


Once again, the tapering of a shape is a combination of lacing to a point, and addition of the base to the final shape.

Broken down vertically:

LINE = | = { | • 2:* • n }

1D
| - the whole line, made from " * +[*] " , lacing a point to a point, along N+1

0D
2:* - two separated¹ points, made from the addition of the base point to the new one

-1D
n - the nulloid

¹ The symbol " : " stands for the arrangement of the listed n-cells, as being fully separate, and having no incidence. " 2: " means "two separated". Now, obviously, the 2 points are joined by the line, but when referring to just the set of points as a whole, they do not contact each other. Vertices, and shapes that play the role of vertices, will always be denoted as separate in my math system.



Note how the poly equation of the line is { | • 2:* • n }, which is equal to the expression (x²+2x+1) . Relating the powers of X to the dimension level may be a little confusing, since the power number is one higher than the dimension of the corresponding n-cells. Best way to think of it, is the power number stands for the dimension of the viewing perspective. If we're in 1D looking at a line, we will only see points. We have to actually be in 2D in order to see the entire line, from above in a higher perspective. Same goes for viewing a 2D shape, we will only see 1D lines if in 2D. The 2D level will correspond with X³, where we must be in 3D to see the whole 2D shape. And again, we have to be in 4D to see an ENTIRE 3D shape, or we'll only see 2D scans of the 3D shapes. This really isn't all that important, but it will clear up any potential confusion when comparing numbers to numbers.

I apologize if this is confusing, but it's the best way I can describe it. The poly formulas have the form and function of real polynomials, and if I change it, the system would collapse, especially when enumerating the n-simplices. It's all about the inclusion of the nulloid, and being able to complete the number series and derive Pascal's Triangle out of it. Adding the nulloid offsets the power-to-dimension level by +1, simply put






The next shape I would like to dissect is the triangle |>. It can only be made by tapering a line to a point, along 2D, or mult a triangle by a point.

So, we start with a line as the base, and lace to a point, while adding all n-cells of the line to the n-cells of the final shape:

{ | • 2:* • n }
+[*]
-------------------------
{ |> • 3^| • 3:* • n }


Broken down vertically:

Triangle = |> = { |> • 3^| • 3:* • n }

2D
|> - the whole triangle, made from " | +[*] " , tapering a line to point along 2D

1D
3^| - three attached² lines, made from " 2:* +[*] " lacing 2 points to one point, adding the base-shape line to the final shape

0D
3:* - three separate points, made from adding the base's 2 points to the new one

-1D
n - the nulloid


² The symbol " ^ " stands for " attached ", as in all listed n-cells are mutually joined as a whole. All 1D flat line edges will always be denoted as attached in my math system. It's an easy one, as this is normally the case anyway.

Note how the triangle polynomial formula is { |> • 3^| • 3:* • n }, which is equal to the expression (x³+3x²+3x+1). Notice anything familiar here? We're starting to get the binomial expansion series from (x+1)³. Pretty cool, huh?






The next shape I would like to dissect is the square || . It is made by the cartesian product of two lines, and will be the first example of a full product being applied. So, we start with a line as the base shape, and multiply with a line as the operator, making the square:


{ | • 2:* • n }
[ | • 2:* • n ]
-------------------------
{ || • 4^| • 4:* • n }


Broken down vertically:

Square = || = { || • 4^| • 4:* • n }

2D
|| - the full square, made by combining the two terms " | and | " together, making the square ||

1D
4^| - four attached lines, made by mult " (2:* x |) + (2:* x |) " , making four mutually attached lines

0D
4:* - four separated points, made by mult " (2:*) x (2:*) " , making two separate locations of two separate points

-1D
n - the nulloid


The square { || • 4^| • 4:* • n } is equal to the expression (x³+4x²+4x+1) . Omitting the nulloid, the sequence is closer to (x+2)², which sort of violates the whole point of retaining the nulloid in the first place. But, it's worth mentioning, since I'm claiming this relationship to polynomials. N-cubes compute differently than n-simplices in this method, which will be fleshed out in later posts. N-simplices will strictly follow Pascal's Triangle, but n-cubes have their own pattern, which is really no big deal. It just needs to be clarified , in my opinion.





The next shape I wanted to detail is the circle |O . It is made by the process of lathing a line into N+1, or mult a point with a circle. I'm trying to avoid using the lathe operator for now, as it is still in a developmental stage. I can easily lathe the N-1 faces of any shape, but this new system includes ALL n-cells. Lathing is a rotation around an N-1 bisecting plane, where this plane also bisects the appropriate n-cells. I am in the process of hybridizing the old system with the new one. For higher-D shapes, I'll be using the cartesian product with a circle as a way to cheat the system. Bisecting rotations are much more complex than products or tapering.


{ | • 2:* • n }
[O] [()]
-------------------
{ |O • (O) • n }


Broken down vertically:

Circle = |O = { |O • (O) • n }

2D
|O - the whole circle, made by lathing a line into N+1, | [O] = |O

1D
(O) - the hollow circle or 1D edge of a disk, made by joining the 2 points into a point-torus, 2:* [()] = (O)

0D
no individual points exist, more like an infinite number in a 1D disk's edge, as the circle is an infinite-edged polygon

-1D
n - the nulloid



So, that about wraps it up for 0, 1, 2D. In the next posting, I will dissect the 3D shapes in greater detail. Questions are welcomed!

-- Philip

[Keiji]

The square { || • 4^| • 4:* • n } is equal to the expression (x³+4x²+4x+1) . Omitting the nulloid, the sequence is closer to (x+2)², which sort of violates the whole point of retaining the nulloid in the first place. But, it's worth mentioning, since I'm claiming this relationship to polynomials. N-cubes compute differently than n-simplices in this method, which will be fleshed out in later posts. N-simplices will strictly follow Pascal's Triangle, but n-cubes have their own pattern, which is really no big deal. It just needs to be clarified , in my opinion.

Actually, there's nothing "violating" about this expression. The hypercubes are simply x(x+2)ⁿ+1.

This is making sense so far, other than the lathing. Also, it's pretty much the same as what wendy already outlined with her ##, #*, *#, ** operators (one of wendy's few writings that I have actually been able to understand!), just written differently. I don't remember which is which, but they are comb, prism, tegum and taper. They form a nice language by themselves, with comb removing one from the product of the dimensions, taper adding one, and the other two not changing it. Lathing, spherating, or anything similarly "circular" does not mix naturally with it, as far as I know.

[wendy]

The # hides the product, while the * includes a one in the operator. In essence, the products are surface operators, counting the namon (nulloid), and/or volume into the surface.

So the five regular solids (simplex, cross-polytope, sphere, cube, and cubic), are powers where the multiply means the five products, ie

*pyramid* , <#tegum*> , (crind), [*prism#], #comb#

The brackets are used of the radiant products <sum>, (rss) and [max] radiant operators.

But the #comb# product (think that # is part of a cubic lattice!) is used of torus-like figures: all the torotopes are comb-products of two different kinds.

[Marek14]

I've been thinking about this before, but it seems really strange to me how English lacks some simple mathematical terms.

In Czech, my native language, for example, there are well-used one-word terms for straight line and line segment, and, most importantly, separate terms for circle as an edge and circle as a whole object. English has "disk" which means the whole circle and "circle" which can mean either, based on context, but no simple word for just the curve...

[Klitzing]

All right, time to explain how my math system works...
...
0D
point = * = { * • n }
1D
line = | = { | • 2:* • n }
2D
triangle = |> = { |> • 3^| • 3:* • n }
square = || = { || • 4^| • 4:* • n }
circle = |O = { |O • (O) • n }
...
-- Philip

Very clear, and so far surely elementary. But I suppose, a good base to build on. Esp. in not using that mathematically doubtable lathing any more (except for common sense wordings).

But I would like your new symbols to show the dimensionality directly. Thus what about having better:
circle = |O = { |O • (O) • 0 • n } ?

--- rk

[Klitzing]

I've been thinking about this before, but it seems really strange to me how English lacks some simple mathematical terms.

In Czech, my native language, for example, there are well-used one-word terms for straight line and line segment, and, most importantly, separate terms for circle as an edge and circle as a whole object. English has "disk" which means the whole circle and "circle" which can mean either, based on context, but no simple word for just the curve...

True.
And "sphere" is by no means better, meaning both, the full volume shape and the mere surface.
This is a true loss, esp. when being the common base for higher-D namings: hypersphere...

--- rk

[Marek14]

I've been thinking about this before, but it seems really strange to me how English lacks some simple mathematical terms.

In Czech, my native language, for example, there are well-used one-word terms for straight line and line segment, and, most importantly, separate terms for circle as an edge and circle as a whole object. English has "disk" which means the whole circle and "circle" which can mean either, based on context, but no simple word for just the curve...

True.
And "sphere" is by no means better, meaning both, the full volume shape and the mere surface.
This is a true loss, esp. when being the common base for higher-D namings: hypersphere...

--- rk

Well, to be fair, we don't have a separate word for surface and volume of sphere either... just for circles

Here's something I never found out: Is there an English word for an area between two concentric circles?

[quickfur]

I've been thinking about this before, but it seems really strange to me how English lacks some simple mathematical terms.

In Czech, my native language, for example, there are well-used one-word terms for straight line and line segment, and, most importantly, separate terms for circle as an edge and circle as a whole object. English has "disk" which means the whole circle and "circle" which can mean either, based on context, but no simple word for just the curve...

True.
And "sphere" is by no means better, meaning both, the full volume shape and the mere surface.
This is a true loss, esp. when being the common base for higher-D namings: hypersphere...

--- rk

I believe the general term in English (at least as far as mathematics are concerned) is n-sphere for the surface, and n-ball for the solid. But n-sphere is commonly abused to mean n-ball, so this distinction isn't as clear-cut as it ought to be. (And I've been guilty of this myself. )

[Marek14]

Maybe we could call the surface a "sphere peel"

[ICN5D]

Awesome, I really appreciate the feedback from everyone

Actually, there's nothing "violating" about this expression. The hypercubes are simply x(x+2)n+1.

That's a relief! Also a neat little formula, too. I didn't know that one. Makes more sense now, and I'll use it to compare future dissections.

This is making sense so far, other than the lathing. Also, it's pretty much the same as what wendy already outlined with her ##, #*, *#, ** operators (one of wendy's few writings that I have actually been able to understand!), just written differently. I don't remember which is which, but they are comb, prism, tegum and taper. They form a nice language by themselves, with comb removing one from the product of the dimensions, taper adding one, and the other two not changing it. Lathing, spherating, or anything similarly "circular" does not mix naturally with it, as far as I know.

I remember one of wendy's post about >>> is equal to |>> , and it's neat to see it reflected in my computations. And, of course, the lathe is a terrible thing to attempt at describing. I understand it very well, but translating into familiar terms is the challenge. I'm working on it! It's not that bad, just need to show how it works. Visuals are key, and I've been doing it in my head for 6 years...

So the five regular solids (simplex, cross-polytope, sphere, cube, and cubic), are powers where the multiply means the five products, ie

*pyramid* , <#tegum*> , (crind), [*prism#], #comb#

That clears up your notation, thank you I was also lost, but now I can see how it works.

I've been thinking about this before, but it seems really strange to me how English lacks some simple mathematical terms.

In Czech, my native language, for example, there are well-used one-word terms for straight line and line segment, and, most importantly, separate terms for circle as an edge and circle as a whole object. English has "disk" which means the whole circle and "circle" which can mean either, based on context, but no simple word for just the curve...

Hmm, I guess there were way more mathematicians during the evolution of Czech?

Here's something I never found out: Is there an English word for an area between two concentric circles?

I've been calling that one a " line torus ". It has two forms as a hollow tube or flat washer/gasket. No official name exists, as far as I know.

Very clear, and so far surely elementary. But I suppose, a good base to build on. Esp. in not using that mathematically doubtable lathing any more (except for common sense wordings).

But I would like your new symbols to show the dimensionality directly. Thus what about having better:
circle = |O = { |O • (O) • 0 • n } ?

I used a similar version to that. I agree with at least denoting the absence of the n-cells. I used the infinity symbol in place of a zero in a previous format. But, I don't want any confusion with a zero and a spin operator, as both " 0 " and " O " are rather identical. Is there an alt-code for a zero with a slash through it? Like a " no solution " symbol? I suppose it's a matter of font choice, but I want it to be universal.

I believe the general term in English (at least as far as mathematics are concerned) is n-sphere for the surface, and n-ball for the solid. But n-sphere is commonly abused to mean n-ball, so this distinction isn't as clear-cut as it ought to be. (And I've been guilty of this myself. )

I didn't know that. Making me just as equally guilty, and now a little smarter. And, less confused with the terms now

[Keiji]

Alt+2205 apparently gets you ∅ which is the empty set character.

I suppose it's a matter of font choice, but I want it to be universal.

Unicode is universal, please let's not go back to the bad old days of wingdings...

[quickfur]

Alt+2205 apparently gets you ∅ which is the empty set character.

I suppose it's a matter of font choice, but I want it to be universal.

Unicode is universal, please let's not go back to the bad old days of wingdings...

What about just capital C (for Circle? besides being about 3/4 circular)? Using obscure unicode symbols is all nice and everything, but when something is that difficult to type (and don't forget that on non-windows computers the alt key sequence doesn't work the same way!), you can bet that people won't be using that notation very much. Use an uglier, but easier to type notation, and it'll have much better chances of adoption.

[Keiji]

(and don't forget that on non-windows computers the alt key sequence doesn't work the same way!)

I don't even remember any alt- number codes, so even when I'm at a Windows computer, with a number pad, they are useless to me. I usually just copy and paste, googling the name of the character if I need to.

[Klitzing]

Very clear, and so far surely elementary. But I suppose, a good base to build on. Esp. in not using that mathematically doubtable lathing any more (except for common sense wordings).

But I would like your new symbols to show the dimensionality directly. Thus what about having better:
circle = |O = { |O • (O) • 0 • n } ?

I used a similar version to that. I agree with at least denoting the absence of the n-cells. I used the infinity symbol in place of a zero in a previous format. But, I don't want any confusion with a zero and a spin operator, as both " 0 " and " O " are rather identical. Is there an alt-code for a zero with a slash through it? Like a " no solution " symbol? I suppose it's a matter of font choice, but I want it to be universal.

Yep, there is! It is the norse O-slash. it belongs to ANSI, so any west-european font should work here. Within HTML syntax it just is written as &Oslash; resp. &oslash;. But it well can be given as number code too: &#216; resp. &#248; (both decimal). - And this then shows how it can be entered with any non-norse keyoard: with alt-numblock "0216" (Ø) resp. "0248" (ø). - Well not really the Zero of informatics (that one would be accessible only in some unicode fonts), but quite near.

--- rk

[ICN5D]

Oh, man, I remember wingdings! Talk about illegible...

Hmm, well, as for the alt codes, I consult: http://www.alt-codes.net/


Does everyone feel okay with the • symbol? I like it, as it's easy to distinguish and not interpret as another operator. I also like how its bold nature stands out. And, one doesn't need to type a ton of numbers to get it. It also transfers over to plain ole' notepad, where all of my older posts or in-process posts are kept. I feel that anything notepad can save is worthy. I also noticed that the " Consola " font does a great job at spacing out the notation, making it really easy to distinguish the cluster of terms. Last thing I want is too dense of a string to pick out anything. They're complex enough with 4 and 5D shapes, let alone +6D.

What about just capital C (for Circle? besides being about 3/4 circular)? Using obscure unicode symbols is all nice and everything, but when something is that difficult to type (and don't forget that on non-windows computers the alt key sequence doesn't work the same way!), you can bet that people won't be using that notation very much. Use an uglier, but easier to type notation, and it'll have much better chances of adoption.

As for " does not apply " or " zero n-cells " , I have thought about good ole' hyphen, " - " , since I no longer use it for arrangement symbols. It's open for use, and doesn't need alt codes.

My previous version of the poly formulas was using either (∞) or [∞] to represent the infinities. (∞) would be for infinite points along edge of disk, since (O) is the 1-D curved edge. The [∞] represented the infinite edges/points along a margin of some kind, like the [(O)(O)] for duocylinder margin. Retaining the () or [] allowed me to distinguish apart where the infinities came from, either from a product [], or an n-sphere ().

But, in an attempt to clean up and de-uglify the polynomials, I decided pretty quickly to omit them altogether. That way, one doesn't try to compute with them, as it would have already been symbolized by a real n-cell. That's why you don't see anything at all with " zero ", I felt its complete absence signified rather clearly " does not apply/zero ".

Would you prefer:

{ |O|O • 2+|O(O) • [(O)(O)] • [-] • [-] • n }

or

{ |O|O • 2+|O(O) • [(O)(O)] • n }

for the duocylinder? When you go to compute with the poly formula, you have to remember more rules, and do more stuff mentally, when using the longer, and redundant-symbol top one. That's how I feel about it

Or, take the glome, for example:

{ |OOO • (OOO) • (-) • (-) • (-) • n }

or

{ |OOO • (OOO) • n } ?

How long, dense, and redundant of a sequence would you prefer? How many symbols would you prefer to type/memorize? I like simpler to use/memorize/compute.

[Klitzing]

Would you prefer:

{ |O|O • 2+|O(O) • [(O)(O)] • [-] • [-] • n }

or

{ |O|O • 2+|O(O) • [(O)(O)] • n }

for the duocylinder? When you go to compute with the poly formula, you have to remember more rules, and do more stuff mentally, when using the longer, and redundant-symbol top one. That's how I feel about it

Or, take the glome, for example:

{ |OOO • (OOO) • (-) • (-) • (-) • n }

or

{ |OOO • (OOO) • n } ?

How long, dense, and redundant of a sequence would you prefer? How many symbols would you prefer to type/memorize? I like simpler to use/memorize/compute.

You could even omit all that representational stuff of typable / non-typable things, and still show the full dimensionality by writing e.g.
{ |OOO • (OOO) • • • • n }.

--- rk

[ICN5D]

You could even omit all that representational stuff of typable / non-typable things, and still show the full dimensionality by writing e.g.
{ |OOO • (OOO) • • • • n }.

I just may be more receptive to that method. I'm still on the fence with it, just as you are with my lathe computation ! Before I make any decisions, we'll have to wait for the planet to turn around, so everyone else can chime in. Right now, there's only six people in the entire world talking about it, and we're all over the place! Thank you, internet.

I'll be bringing my giant telescope over to a friend's place this evening, to check out some outer space. Afterwards, I'll be back on here.

[Keiji]

You could even omit all that representational stuff of typable / non-typable things, and still show the full dimensionality by writing e.g.
{ |OOO • (OOO) • • • • n }.

I think this is the best method so far.

But what really bugs me is what has done all along; that | isn't fitting to be used where you're using it.

Instead, | (pipe) really belongs with the { and }... as a replacement for •. The reason is, because it's the same height as { and }, so it makes a more natural separator. The different height also makes it look like it's trying to be a separator even when you use it as not a separator - which is one of the biggest reasons why I find your notation as it stands so illegible. So I (capital i) can be used where you're currently using | (pipe).

Try:

{ IOOO | (OOO) | | | | n }

or

{ IOIO | 2+IO(O) | [(O)(O)] | | | n }

or

{ II | 4^I | 4:* | n }

[wendy]

One can always use the letter Ø which is the danish letter o-slash. It's on shift+altgra+L.

[ICN5D]

But what really bugs me is what has done all along; that | isn't fitting to be used where you're using it.

Instead, | (pipe) really belongs with the { and }... as a replacement for •. The reason is, because it's the same height as { and }, so it makes a more natural separator. The different height also makes it look like it's trying to be a separator even when you use it as not a separator - which is one of the biggest reasons why I find your notation as it stands so illegible. So I (capital i) can be used where you're currently using | (pipe).

Big noted on that. You have a valid point. So, I am totally cool with making | a I, but, you wouldn't find both slightly similar? Such is the case with using 0 and O together, to others it may be a little confusing. Someone might ask " Why does one symbol mean two things? " . To the trained eye, there's clearly a difference between a shorter and thicker I versus the taller and thinner | . So, what would you say about { II • 4^II • 4:* • n }?

Although, carrying the remaining zero cells down to 0D looks a little cleaner in { IOOO | (OOO) | | | | n } vs { IOOO • (OOO) • • • • n }. It's just that I don't feel it's understandable with similar looking symbols. I feel that the partition should stand out in some ways, meaning something important, like a barrier or boundary zone of some kind. Scanning the keyboard doesn't give me too many options, but maybe some letter could work. I'm using the : for an arrangement, otherwise it could be


{ II : 4^II : 4-* : n } , where hyphen is now " separated "

{ IOOO : (OOO) : : : : n }

or

{ II - 4^II - 4:* - n }

{ IOOO - (OOO) - - - - n }

to play with some examples.

One can always use the letter Ø which is the danish letter o-slash. It's on shift+altgra+L.

Hmm, when I press shift+altgra+L, it comes up as "[list][\list]". Something to do with the website software. I like the idea of having o-slash without alt-codes, but I can't use it.

[Keiji]

So, what would you say about { II • 4^II • 4:* • n }?

I suppose it's the best compromise of those in your post

[wendy]

Using dots to show removed nodes is what we already do to show the relation of the surtope to the polytope. Klitzing's incmat diagrans show this very elegantly. In the present case, it shows the relation between the slice and the whole figure.

[ICN5D]

So, what would you say about { II • 4^II • 4:* • n }?

I suppose it's the best compromise of those in your post

Right on. I should probably get around to changing them all in the previous posts, but it might not matter. I guess for now on I'll use " I " in future posts.

Using dots to show removed nodes is what we already do to show the relation of the surtope to the polytope. Klitzing's incmat diagrans show this very elegantly. In the present case, it shows the relation between the slice and the whole figure.

Oh, you're right, I didn't notice that. Well, then I guess it's a good thing it's widely used. Just like quickfur pointed out, a fancier looking language, but harder to replicate will have less acceptance.

[ICN5D]

Analysis of the Third Dimension



IOO - SPHERE
IO> - CONE
IIO - CYLINDER
I>> - TETRAHEDRON
I>I - TRIANGLE PRISM
II> - SQUARE PYRAMID
III - CUBE



IOO - SPHERE

IO • (O) •-• n
+[O]+[(O)]
-----------------------
IOO • (OO) •-•-• n


IOO = sphere
•
(OO) = glomohedrix , surface of sphere
•
- = zero edges
•
- = zero vertices





IO> - CONE

IO • (O) •-• n
+[*]
-----------------------------------
IO> • {IO + I^(O)} • (O) • * • n

IO> = cone , from IO +[*] , lacing a circle to a point
•
IO = circle, from adding base to final
I^(O) = glomolatrix pyramid³, made from (O) +[*] , lacing a 1D curved line to single point, aka "anchored line torus"
•
(O) = glomolatrix , edge of disk from base shape
•
* = point


³ The glomolatrix pyramid is the curved 2-cell of the cone. I've been calling it the line torus, anchored line torus, or the attached line torus. The symbol I^(O) can also be written (O)> , but I prefer the first one. When we taper, spin or extrude the cone, the symbols for the new curved cell revert back to the first format, as in IO^(O) , I>^(O) , or II^(O). So, in an effort to maintain simplicity, I choose to represent (O)> as I^(O) . Now, of course, it's quite possible that I don't need to represent it apart from a hollow tube I(O). And, with enough convincing, I just may accept the change. But, for the time being, I feel it's necessary to tell them apart.





IOI - CYLINDER

IO • (O) •-• n
[I • 2:*]
------------------------------------
IOI • {2:IO + I(O)} • 2:(O) •-• n


IOI = cylinder , from IO x I , circle times line
•
2:IO = two separate circles, made from IO x 2:* , playing the role of vertices on a line
I(O) = hollow tube , or glomolatrix prism, made from I x (O) , line times edge of disk
•
2:(O) = two separate disk edges, made from (O) x 2:* , two disk edges times two points
•
- = zero separate vertices





I>> - TETRAHEDRON

I> • 3^I • 3:* • n
+[*]
----------------------
I>> • 4^I> • 6^I • 4:* • n


I>> = tetrahedron , from I> +[*] ,lacing a triangle to a point
•
4^|> = four joined triangles, made from I> + (3^I +[*]) , triangle plus lacing three lines to a point
•
6^I = six joined lines, made from 3^I + (3:* +[*]) , three lines plus lacing three points to a point
•
4:* = four separate points , made from 3:* + * , three points plus one point






I>I - TRIANGLE PRISM , triangle times line

I> • 3^I • 3:* • n
[ I • 2:*]
-------------------------------
I>I • {2:I> + 3^II} • 9^I • 6:* • n


I>I = triangle prism, made from I> x I , triangle times line
•
2:I> = two separate triangles , from I> x 2:*, triangle times 2 points , playing the role of vertices on a line
3^II = three joined squares , from 3:* x I , 3 points times a line
•
9^I = nine joined lines , made from (3^I x 2:*) + (3:* x I) , 3 lines times 2 points plus 3 points times a line
•
6:* = six separate vertices , made from 3:* x 2:* , three points times two points






II> - SQUARE PYRAMID

II • 4^I • 4:* • n
+[*]
-----------------------------
II> • {II + 4^I>} • 8^I • 5:* • n


II> = square pyramid , made from II +[*] , lacing a square to a point
•
II = one square , from adding base to final
4^I> = four joined triangles , made from 4^I +[*] , lacing 4 lines to a point
•
8^I = eight joined lines , made from 4^I + (4:* +[*]) , 4 lines plus lacing 4 points to a point
•
5:* = five points , made from 4:* + * , four points plus another






III - CUBE

II • 4^I • 4:* • n
[ I • 2:*]
-----------------------
III • 6^II • 12^I • 8:* • n

III = cube , from II x I , square times line
•
6^II = six joined squares , from (II x 2:*) + (4^I x I) , a square times 2 points plus 4 lines times a line
•
12^I = twelve joined lines, from (4^I x 2:*) + (4:* x I), 4 lines times 2 points plus 4 points times a line
•
8:* = eight points , from 4:* x 2:* , four points times two points




As you can tell, I've been playing around with representing the full dimensionality of the shapes like the sphere {IOO•(OO)•-•-•n}

-Philip

[ICN5D]

Whoa, almost forgot about this. Here's a rough draft of the basic 4D shapes that can be created with this "dimensionometry" or "transformetrics" or whatever I end up calling it. I will most certainly be making updates to this list, when I have more time to elaborate. You might notice a well defined threshold where I decide to simplify things dramatically. That's when I realized how long the list was , and I at least wanted to put something out there. So, here it is:





Term Glossary

Code: Select all

0-D
n
* - POINT

-----------------


1-D
X
| - LINE
(O) - HOLLOW CIRCLE / GLOMOLATRIX

----------------------------------
2-D
XY
|O - CIRCLE
|> - TRIANGLE
|| - SQUARE
|(O) - LINE TORUS
|^(O) - ANCHORED LINE TORUS
(OO) - HOLLOW SPHERE / GLOMOHEDRIX
(O)(O) - HOLLOW TORUS / TORIHEDRIX
[(O)(O)] - DUOCYLINDER MARGIN

Complex manifold examples:

(O>) - CONIHEDRIX / HOLLOW CONE
(>>) - TETRAHEDRIX / HOLLOW TETRAHEDRON

----------------------------------------

3-D
XYZ
|OO - SPHERE
|O> - CONE
||O - CYLINDER
|>> - TETRAHEDRON
|>| - TRIANGLE PRISM
||> - SQUARE PYRAMID
||| - CUBE
|O(O) - TORUS
||(O) - SQUARE TORUS
|>(O) - TRIANGLE TORUS
(OOO) - HOLLOW GLOME / GLOMOCHORIX
(OO)(O) - HOLLOW SPHERITORUS
(O)(OO) - HOLLOW TORISPHERE
(O)(O)(O) - HOLLOW DITORUS
|(OO) - LINE TORISPHERE
|(O)(O) - LINE DITORUS
|[(O)(O)] - LINE TIGER
[(O)(O)(O)] - TRIOCYLINDER MARGIN
[(OO)(O)] - CYLSPHERINDER MARGIN
[[(O)(O)](O)] - CYLTORINDER MARGIN

----------------------------------

4-D
XYZW
|OOO - GLOME
|OO> - SPHONE
|OO| - SPHERINDER
|O>> - DICONE
|O>| - CONINDER
|O|O - DUOCYLINDER
||O> - CYLINDRONE
|>>> - PENTACHORON
|>>| - TETRAHEDRINDER
|>|O - CYLTRIANGLINDER
|>|> - TRIANGLE PRISM PYRAMID
|>|| - TRIANGLE DIPRISM
||>> - DIPYRAMID
||>| - PYRAMID PRISM
|||O - CUBINDER
|||> - HEMDODECACHORON / CUBE PYRAMID
|||| - TESSERACT
|>[|>] - DUOTRIANGLINDER
|OO(O) - SPHERITORUS
|O(OO) - TORISPHERE
|O(O)(O) - DITORUS
|O[(O)(O)] - TIGER
|O>(O) - CONE TORUS
||O(O) - CYLINDER TORUS / TORINDER
|>>(O) - TETRAHEDRON TORUS
|>|(O) - TRIANGLE PRISM TORUS
||>(O) - SQUARE PYRAMID TORUS
|||(O) - CUBE TORUS
||(O)(O) - SQUARE DITORUS
|>(O)(O) - TRIANGLE DITORUS
||(OO) - SQUARE TORISPHERE
|>(OO) - TRIANGLE TORISPHERE
(OOOO) - HOLLOW PENTASPHERE / GLOMOTERIX
(OOO)(O) - HOLLOW GLOMITORUS
(O)(OOO) - HOLLOW TORIGLOME
(OO)(OO) - HOLLOW SPHERITORISPHERE
(OO)(O)(O) - HOLLOW SPHERIC DITORUS
(O)(OO)(O) - HOLLOW TORISPHERIC TORUS
(O)(O)(OO) - HOLLOW TORIC TORISPHERE
(O)(O)(O)(O) - HOLLOW TRITORUS
|(OO)(O) - LINE TORISPHERIC TORUS
|(O)(OO) - LINE TORIC TORISPHERE
|(O)(O)(O) - LINE TRITORUS
|(OOO) - LINE TORIGLOME

------------------------------------

5-D
XYZWV
|OOOO - PENTASPHERE
|OOO> - GLONE
|OOO| - GLOMINDER
|OO>> - DISPHONE
|OO>| - SPHONINDER
|OO|O - CYLSPHERINDER
|OO|> - SPHERINDRONE
|OO|| - CUBSPHERINDER
|O>>> - TRICONE
|O>>| - DICONINDER
|O>|O - CYLCONINDER
|O>|> - CONINDER PYRAMID
|O>|| - CONE DIPRISM
||OO> - DUOCYLINDRONE
||O>> - DICYLINDRONE
||O>| - CYLINDRONE PRISM
|>>>> - HEXATERON
|>>>| - PENTACHORINDER
|>>|O - CYLTETRAHEDRINDER
|>>|> - TETRAHEDRINDER PYRAMID
|>>|| - TETRAHEDRON DIPRISM
|>|OO - DUOCYLTRIANGLINDER
|>|O> - CYLTRIANGLINDRONE
|>|O| - CYLTRIANDYINDER
|>|>> - TRIANGLE PRISM DIPYRAMID
|>|>| - TRIANGLIE PRISM PYRAMID PRISM
|>||> - TRIANGLE DIPRISM PYRAMID
|>||| - TRIANGLE TRIPRISM
||>>> - SQUARE-TRIPYRAMID
||>>| - SQUARE DIPYRAMID PRISM
||>|O - CYLHEMOCTAHEDRINDER
||>|> - SQUARE PYRAMID PRISM PYRAMID
||>|| - SQUARE PYRAMID DIPRISM
|O|O| - DUOCYLDYINDER
|||O> - CUBINDRONE
|||>> - CUBE DIPYRAMID
|||>| - CUBE PYRAMID PRISM
||||O - TESSERINDER
||||> - TESSERACT PYRAMID
||||| - PENTERACT
|OO[|>] - SPHENTRIANGLINDER
|O>[|>] - CONTRIANGLINDER
||>[|>] - HEMOCTAHEDROTRIANGLINDER
|>>[|>] - TETRAHEDROTRIANGLINDER
|>[|>]| - DUOTRIANGLINDYINDER
|>[|>]> - DUOTRIANGLINDRIC PYRAMID
|OOO(O) - GLOMITORUS
|O(OOO) - TORIGLOME
|OO(OO) - SPHERITORISPHERE
|OO(O)(O) - SPHERIC DITORUS
|O(OO)(O) - TORISPHERIC TORUS
|O(O)(OO) - TORIC TORISPHERE
|O(O)(O)(O) - TRITORUS
|O[(OO)(O)] - CYLSPHERINTIGROID
|O[[(O)(O)](O)] - CYLTORINTIGROID
|O[(O)(O)](O) - TIGRITORUS
|OO[(O)(O)] - SPHERIC TIGER
|O(O)|O - CYLTORINDER
|O|O(O) - DUOCYLINDRITORUS

Inflated complex manifold:

|O>(O>) - DUOCONTERIX

N-Cell Descriptions


GLOME - IOOO

IOOO • (OOO) •-•-•-•n

3D surtopes
(OOO) - the glomochorix, curved 3-surface of the 4-ball





IOO> - SPHONE

IOO • (OO)
+[*]
--------------------------------
IOO> • {IOO + I^(OO)} • (OO) •-• * - n


3D surtopes
IOO - the starting sphere at the base
I^(OO) = (OO)+[*] - a tapering line torisphere, curved and pointed 3-surface lacing to the vertex

2D surtopes
(OO) - the glomohedrix, 2-surface of the base sphere

0D surtopes
* - the vertex at the top, above the sphere along 4D





IOOI - SPHERINDER

IOO • (OO) •-•-• n
[I • 2:* • n]
------------------------------
IOOI • {2:IOO + I(OO)} • 2:(OO) •-•-• n


3D surtopes
2:IOO = IOO x 2:* , two separate spheres, at the ends of the prism
I(OO) = I x (OO) , a line torisphere, a 3D hollow tube that curves into 4D, lacing the two spheres together

2D surtopes
2:(OO) = (OO) x 2:* , the two parallel surfaces of the sphere end-caps, playing the role of sharp edges




IO>> - DICONE

IO> • {I^(O) + IO} • (O) • * • n
+[*]
---------------------------------------------------------
IO>> • {2^IO> + I>^(O)} • {2^I^(O) + IO} • {(O) : I} • 2:* • n


3D surtopes
2^IO> - two cones attached by the circle-base
I>^(O) = I^(O)+[*] , an attached triangle torus, made by lacing the curved 2-cell of a cone to a point

2D surtopes
2^I^(O) - two joined and tapering line torii, the two curved 2-cells of the cone-cells

1D surtopes
(O) - edge of a disk, at the base
I - a line, the orthogonal linear vertex, above the circle in 3 and 4D

0D surtopes
2:* - the two end points of the line vertex



IO>I - CONINDER

IO> • {I^(O) + IO} • (O) • * • n
[I • 2:* • n]
---------------------------------------------------------------------------
IO>I • {2:IO> + IOI + II^(O)} • {2:IO + 2:I^(O) + I(O)} • {2:(O) + I} • 2:* • n


3D surtopes
2:IO> - two separate cones, at the ends of the prism
IOI - a cylinder on the side, lacing two circles together
II^(O) - a square torus, fixed on one side, laces the hollow tube to the line

2D surtopes
2:IO - two separate circles
2:I^(O) - two separate, tapering line torii, the displaced 2-cells of the cone end-caps
I(O) - line torus, laces the two circles together

1D surtopes
2:(O) - two separated disk edges
I - a line, the parallel linear vertex, above the cylinder-cell

0D surtopes
2:* - two points



IOIO - DUOCYLINDER

IO • (O) •-• n
[IO • (O) •-• n]
------------------------
IOIO • 2+IO(O) • [(O)(O)] •-•-• n


3D surtopes
2+IO(O) - two orthogonally bound torii, the rolling surfaces of the duocylinder

2D surtopes
[(O)(O)] - a duoring, the duocylinder margin, edge between rolling sides







IIO> - CYLINDRONE

IIO • {2:IO + I(O)} • 2:(O) •-• n
+[*]
-----------------------------------------------------------------
IIO> • {IIO + 2^IO> + I>^(O)} • {2:IO + I(O) + 2:I^(O)} • 2:(O) • * • n

3D surtopes
IIO - the cylinder at the base
2^IO> - two cones, attached at the vertex
I>^(O) - triangle torus, attached on a face

2D surtopes
2:IO - two sep circles
I(O) - line torus, hollow tube
2:I^(O) - two sep tapering line torii

1D surtopes
2:(O) - two sep disk edges

0D surtopes
* - the vertex, above the cylinder along 4D




I>>> - PENTACHORON

I>> • 4^I> • 6^I • 4:* • n
+[*]
---------------------------------
I>>> • 5^I>> • 10^I> • 10^I • 5:* • n

3D surtopes
5^I>> - five tetrahedra, attached by a vertex

2D surtopes
10^I> - ten triangles

1D surtopes
10^I - ten lines

0D surtopes
5:* - five vertices




I>>I - TETRAHEDRINDER

I>> • 4^I> • 6^I • 4:* • n
[I • 2:* • n]
---------------------------------------------------
I>>I • {2:I>> + 4^I>I} • {8.I> + 6^II} • 16^I • 8:* • n

3D surtopes
2:I>> - two sep tetrahedra
4^I>I - four triangle prisms, joined by an edge

2D surtopes
8.I> - eight triangles in a mixed separated/attached configuration
6^II - six squares

1D surtopes
16^I - sixteen edges

0D surtopes
8:* - eight vertices




I>IO - CYLTRIANGLINDER

I> • 3^I • 3:* • n
[IO • (O) • n]
------------------------------------------------
I>IO • {3^IOI + I>(O)} • {3:IO + 3^I(O)} • 3:(O) •-• n

3D surtopes
3^IOI - three joined cylinders
I>(O) - a triangle torus

2D surtopes
3:IO - three sep. circles, playing the role of 2D vertices
3^I(O) - three joined hollow tubes

1D surtopes
3:(O) - three sep. disk edges






I>I> - TRIANGLE PRISM PYRAMID

I>I • {2:I> + 3^II} • 9^I • 6:* • n
+[*]
----------------------------------------------------------
I>I> • {I>I + 2^I>> + 3^II>} • {11^I> + 3^II} • 15^I • 7:* • n

3D surtopes
I>I - triangle prism, below the 4D vertex
2^I>> - two tetrahedra, joined by a vertex
3^II> - three square pyramids, joined by a vertex

2D surtopes
11^I> - eleven triangles
3^II - three squares

1D surtopes
15^I - fifteen edges

0D surtopes
7:* - seven vertices



I>II - TRIANGLE DIPRISM

I> • 3^I • 3:* • n
[II • 4^I • 4:* • n]
-----------------------------------------------------
I>II • {4^I>I + 3^III} • {4:I> + 15^II} • 24^I • 12:* • n

3D surtopes
4^I>I - four triangle prisms
3^III - three cubes

2D surtopes
4:I> - four triangles
15^II - fifteen squares

1D surtopes
24^I - twenty four edges

0D surtopes
12:* - twelve vertices




II>> - DIPYRAMID

II> • {II + 4^I>} • 8^I • 5:* • n
+[*]
--------------------------------------------------
II>> • {2^II> + 4^I>>} • {12^I> + II} • 13^I • 6:* • n

3D surtopes
2^II> - two square pyramids
4^I>> - four tetrahedra

2D surtopes
12^I> - twelve triangles
II - square

1D surtopes
13^I - thirteen edges

0D surtopes
6:* - six vertices



II>I - PYRAMID PRISM

II> • {II + 4^I>} • 8^I • 5:* • n
[I • 2:* • n]
----------------------------------------------------------
II>I • {III + 2:II> + 4^I>I} • {10^II + 8.I>} • 21^I • 10:* • n

3D surtopes
III - cube
2:II> - two square pyramids
4^I>I - four triangle prisms

2D surtopes
10^II - ten squares
8.I> - eight triangles

1D surtopes
21^I - twenty one edges

0D surtopes
10:* - ten vertices



IIIO - CUBINDER

II • 4^I • 4:* • n
[IO • (O) •-• n]
------------------------------------------------
IIIO • {4^IOI + II(O)} • {4:IO + 4^I(O)} • 4:(O) •-• n

3D surtopes
4^IOI - four joined cylinders
II(O) - square torus

2D surtopes
4:IO - four sep. circles
4^I(O) - four joined line torii

1D surtopes
4:(O) - four sep. disk edges





III> - HEMDODECACHORON / CUBE PYRAMID


III • 6^II • 12^I • 8:* • n
+[*]
--------------------------------------------------
III> • {III + 6^II>} • {6^II + 12^I>} • 20^I • 9:* • n

3D surtopes
III - cube, at the base
6^II>

2D surtopes
6^II - six squares
12^I> - twelve triangles

1D surtopes
20^I - twenty edges

0D surtopes
9:* - nine vertices



IIII - TESSERACT

II • 4^I • 4:* • n
[II • 4^I • 4:* • n]
----------------------------------
IIII • 8^III • 24^II • 32^I • 16:* • n

3D surtopes
8^III - eight cubes

2D surtopes
24^II - twenty four squares

1D surtopes
32^I - thrity two edges

0D surtopes
16:* - sixteen vertices



I>[I>] - DUOTRIANGLINDER

I> • 3^I • 3:* • n
[I> • 3^I • 3:* • n]
-------------------------------------------
I>[I>] • 6^I>I • {6:I> + 9^II} • 18^I • 9:* • n

3D surtopes
6^I>I - six triangle prisms

2D surtopes
6:I> - six sep. triangles , playing the role of vertices
9^II - nine squares

1D surtopes
18^I - eighteen edges

0D surtopes
9:* - nine vertices

[ICN5D]

Here's something I tried out today. It's a cartesian product matrix, showing how the n-cells get combined when we multiply two distinct 3D shapes together.



This is the (triangle prism,cone)-prism

IO> * I>I = IO>I[I>] , 6D Conindric Trianglinder

Code: Select all

 
          IO>   •   I^(O)   +   IO   •   (O)   •   *   
-------------------------------------------------------
I>I  |  IO>I[I>]   I>II^(O)    I>IIO    I>I(O)    I>I   
•    |
3^II |  3^IO>II    3^III^(O)  3^IIIO    3^II(O)   3^II                                 
+    |
2:I> | 2:IO>[I>]   2:I>I(O)   2:I>IO    2:I>(O)   2:I>                                                     
•    |
9^I  |  9^IO>I     9^II^(O)    9^IIO    9^I(O)     9^I                                               
•    |
6:*  |   6:IO>     6:I^(O)     6:IO      6:(O)     6:*                                               


Making a huge polynomial string:


IO>I[I>] • {2:IO>[I>] + 3^IO>II + I>IIO + I>II^(O)} • {2:I>IO + 3^IIIO + 9^IO>I + 3:I>I(O) + 3^III^(O)} • {I>I + 6:IO> + 9^IIO + 12.II(O) + 2:I>(O)} • {2:I> + 3^II + 6:IO + 15.I(O)} • {9^I + 6:(O)} • 6:* • n



Divided up into dimension levels:

6D Rotope
IO>I[I>] - conindric trianglinder

5D Surtopes
2:IO>[I>] - two parallel sep. contrianglinders, has the projection angle of: IO>[I>] || IO>[I>]

3^IO>II - three joined cone diprisms, has the projection angle of : IO>II || IO>I

I>IIO - cyltriandyinder , has the projection of I>IIO || I>II

I>II^(O) - triangle diprism torus , joined on a curved 4-cell

4D Surtopes
2:I>IO - two parallel sep. cyltrianglinders

3^IIIO - three joined cubinders

9^IO>I - nine joined coninders

3:I>I(O) - three sep. triangle torinders

3^III^(O) - three joined and tapering cube torii

3D Surtopes
I>I - triangle prism, playing the role of a vertex from a cone

6:IO> - six sep. cones, playing the role of vertices from a triangle prism

9^IIO - nine joined cylinders, playing the role of edges from a triangle prism

12.II(O) - twelve mixed arrgmt square torii

2:I>(O) - two sep. triangle torii

2D Surtopes
2:I> - two sep. triangles

3^II - three joined squares

6:IO - six sep. circles

15.I(O) - fifteen mixed arrgmt line torii

1D Surtopes
9^I - nine edges

6:(O) - six sep. disk edges

0D Surtopes
6:* - six vertices

[ICN5D]

I've decided to update my notation, and clean up / get rid of some symbols, and change the way I define a duocylinder margin.

• Originally, I used I^(O) to describe the curved 2-surface of a cone. This has been changed to (O)> , which infers the pyramid of a disk's edge. In fact, any use of the carat is no longer needed, with this update.

• The numerous symbols I used to denote whether certain surfaces were connected or not, i.e. , " . " , " : " and " ^ " have all been simplified to " : ". So, when you see 4:I> , this can mean four attached triangles, four separate triangles, or a mix of attached/unattached. In general, n:M , means n-number of m-shapes, regardless of connectivity.

• The original symbol for cartesian product of two disk edges, [(O)(O)], which represent the duocylinder margin, is now (O)[(O)] , which infers the product in the same way as any other shape. If I>[IO] means product of triangle and circle, then (O)[(O)] will mean product of two disk edges. Spherating this with a circle becomes IO(O)[(O)] for a tiger, which differs from a ditorus IO(O)(O). Still not as precise as toratopic notation, but is now more intuitive.

This simpler syntax really helps with the frames of a cylconinder IO>IO , which is a product of cone * circle :

Cone : IO> • { IO + (O)> } • (O) • * • n

Circle : IO • (O) •-• n

combining both together as cartesian product of frame surfaces,

Code: Select all

      |      IO>          IO          (O)>        (O)       *   
      |
------|---------------------------------------------------------------
      |
IO    |    IO>[IO]      IO[IO]      (O)>[IO]     IO(O)      IO       
      |
(O)   |    IO>(O)       IO(O)       (O)>[(O)]   (O)[(O)]    (O)

equals,

IO>[IO] • { IO[IO] + IO>(O) + (O)>[IO] } • { 2:IO(O) + (O)>[(O)] } • { IO + (O)[(O)] } • (O) •-• n

Breaking down, and describing each frame surface, we have:

5-frame
IO>[IO] - cylconinder, equal to IO>IO

4-frames
IO[IO] - duocylinder, equal to IOIO
IO>(O) - cone torus
(O)>[IO] - product of disk-edge pyramid and a circle, once defined as IOI^(O) , a cylinder torus with one end closed off, also circle || torus

3-frames
2:IO(O) - two toruses
(O)>[(O)] - the cylconinder margin, a 3-surface embedded in 5D, as a disk-edge pyramid torus in 5D, rather than 4D ***

2-frames
IO - circle
(O)[(O)] - duocylinder margin

1-frame
(O) - disk edge

0-frame
- no vertices

-1 frame
n - null-frame

The duocylinder IOIO at the base connects to the circle IO at the vertex, by the surfaces of cone torus IO>(O) and (disk-edge pyramid, circle)-prism (O)>[IO] . The cone torus, (disk-edge pyramid, circle)-prism, duocylinder margin, and circle, are all mutually attached by a single surface of (O)>[(O)] , the margin of cylconinder. It's a beautiful object.

*** On the topic of the cylconinder margin, the biggest difference between (O)>(O) and (O)>[(O)] , is that (O)>(O) is 3-surface embedded in 4D, and (O)>[(O)] is 3-surface embedded in 5D. Just like how (O)(O) is 2-surface of torus in 3D, and (O)[(O)] is 2-surface of torus in 4D. The product of a disk edge [(O)] infers one extra dimension of embedding, versus a fiber bundle over circle (O) .