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 separated

^{1}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 (x

^{2}+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 X

^{3}, 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 attached

^{2}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 (x

^{3}+3x

^{2}+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 (x

^{3}+4x

^{2}+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