Aale de Winkel's notation

[Aale de Winkel]

( administrator's note: this thread was split off from http://tetraspace.alkaline.org/forum/viewtopic.php?t=16 )

The notation is throughout the encyclopedia in the articles. but here it is:
Positions within an n-dimensional hypercube order m are given by [[sub]j[/sub]k] (j = 0 .. n-1, k = 0 .. m-1)
(note that the hypercube is discrete space, in continuous space regularly used is x[sub]k[/sub] k = 1 .. n. if time is denoted as the zeroth coordinate taken imaginairy and multiplied with the speed of light we have
x[sub]μ[/sub] == (ict, x[sub]k[/sub]), in most texts c is scaled to 1)
Einstein summation rule regularly used here so
x[sub]μ[/sub] x[sup]μ[/sup] = 0 <=> -t[sup]2[/sup] + [sub]i=1[/sub]∑[sup]n[/sup] x[sub]i[/sub][sup]2[/sup] = 0
pe describing the temperal expansion of the light sphere.

aside from position in the hypercube direction i simularly denote as <[sub]j[/sub]k> (j = 0 .. n-1, k = 0 .. m-1)
this also holds in continues spaces. perhaps though j ranges from 1 to n, and the values 'k' is not limited.
also the hypercube is odular m so k = -1 is equivalent to m-1

when not fixed to a definite value the j and k ranges through all possibilities when combined seperated by a ',' of course the pre-subscripts may not be the same.

hypercube samples.
[[sub]j[/sub]0] the zeroth hypercube position (all coordinates 0)
[[sub]j[/sub]0] <[sub]0[/sub]1 , [sub]k[/sub]0> <[sub]1[/sub]1 , [sub]l[/sub]0>: the hypercubes front plane
[[sub]j[/sub]0] <[sub]k[/sub]1>: the hypercube main n-agonal (ie the line from [[sub]j[/sub]0] to [[sub]j[/sub]m-1])

as said this is used throughout the encyclopedias articles and works fine in pinpointing exactly the hypercube qualifications.
In the encyclopedias article on 'agonals' you find a complete runthrough of samples in a cube! (at the bottom of the page)

[Aale de Winkel]

I think an ending of -idth sounds just a little bit better to my ears for some reason, maybe because of its similarity to width. So, maybe "tetridth" instead of "tetradth"? But what would be the analogous terms for wide and thin - would they be "tetride" and "tetrin"?

I merely used the "dth" of width. Wheter tetridth or tetradth sounds better
I don't know, does anyone know some language specialist. developping
the tetronian language is not my cup of tea, I'm not even a native english
speaker.
Perhaps you'll best start a for others readonly "tetronian language" topic.
and decide wheter it is tetridth or tetradth.

Wheter terms for dx[sub]4[/sub] < δ and dx[sub]4[/sub] > δ
are needed (what δ anyway) I don't know.

[alkaline]

(note that the hypercube is discrete space, in continuous space regularly used is x[sub]k[/sub] k = 1 .. n. if time is denoted as the zeroth coordinate taken imaginairy and multiplied with the speed of light we have
x[sub]μ[/sub] == (ict, x[sub]k[/sub]), in most texts c is scaled to 1)
Einstein summation rule regularly used here so
x[sub]μ[/sub] x[sup]μ[/sup] = 0 <=> -t[sup]2[/sup] + [sub]i=1[/sub]∑[sup]n[/sup] x[sub]i[/sub][sup]2[/sup] = 0

pe describing the temperal expansion of the light sphere.

what do you mean by zeroth coordinate? is this procedure of using imaginary and zeroth coordinates, and multiplying by the speed of light done commonly in relativity or something? I don't know what the μ means when it is the x's subscript. Also, i don't know what "ict" means, or where the speed of light (c) appears in the equation. You define x[sub]μ[/sub], but what is x[sup]μ[/sup]? It would greatly help if you explained these things. (i would assume most other people visiting this site don't know them either).

[alkaline]

Positions within an n-dimensional hypercube order m are given by [[sub]j[/sub]k] (j = 0 .. n-1, k = 0 .. m-1)
...
direction i simularly denote as <[sub]j[/sub]k> (j = 0 .. n-1, k = 0,1)
this also holds in continues spaces. perhaps though j ranges from 1 to n, and the values 'k' is not limited.

when not fixed to a definite value the j and k ranges through all possibilities when combined seperated by a ',' of course the pre-subscripts may not be the same.

hypercube samples.
[[sub]j[/sub]0] the zeroth hypercube position (all coordinates 0)
[[sub]j[/sub]0] <[sub]0[/sub]1 , [sub]k[/sub]0> <[sub]1[/sub]1 , [sub]l[/sub]0>: the hypercubes front plane
[[sub]j[/sub]0] <[sub]k[/sub]>: the hypercube main n-agonal (ie the line from [[sub]j[/sub]0] to [[sub]j[/sub]m-1])

as said this is used throughout the encyclopedias articles and works fine in pinpointing exactly the hypercube qualifications.
In the encyclopedias article on 'agonals' you find a complete runthrough of samples in a cube! (at the bottom of the page)

so position is [[sub]j[/sub]k] and direction is <[sub]j[/sub]k>. I don't quite understand what putting them next to each other means though - for example, [[sub]j[/sub]k]<[sub]j[/sub]k>.

Even with your decriptions, it's hard to picture what these things are denoting. Images that depicted these things would be a really big help. Do you think it would be possible to create an image of a cube with the points, lines, and faces labelled with your notation? If you can't do that soon, it would help to at least get examples of notation by seeing the notation for the following objects: a point, a line, a square, a cube, and a hypercube; also, the notation for the two endpoints of a line, the four corners of a square, etc.

I went the the r-agonals page again, and it doesn't look quite the same as the notation you are using here.

[Aale de Winkel]

I see the terms row, column, etc as being distinct from n-agonals - they refer to parallel lines of data verses the edges/interior lines of an n-cube. The two sets of terms are used in different contexts - except, it seems, for your work which seems to be n-dimensional sets of numbers that add up to certain values. But, even then, i would see it referring to different things - you would say something has three rows, but would you say it has three monagonals? As you mention, the term pillar has been used in the context of magic cubes for the sequence of row, column, etc - but has it been used in any other contexts?

the terms row, column and pillar depict monagonal directions:

The monagonals within a hypercube can in my notation be depicted by

[[sub]j[/sub]0,[sub]k[/sub]q] <[sub]j[/sub]1,[sub]k[/sub]0>

the regular terms around can so be identified by the 'j'
j = 0: row
j = 1: column
j = 2: pillar
j = 3: 3-row (an foggy term I don't use)

aside from the hypercube use of "pillar" I do think "pillars" are the name of the ornamental (yeh "pillars") standing with some buildings.

[Aale de Winkel]

so position is [[sub]j[/sub]k] and direction is <[sub]j[/sub]k>. I don't quite understand what putting them next to each other means though - for example, [[sub]j[/sub]k]<[sub]j[/sub]k>.

putting positions and vectors together one can depict a certain line, plane, cube etc. within a bigger hypercube.
As you might know from some "vector algebra course"(?) a line is depicted by a position and a directional vector.
a plane by a position and 2 vectors etc.
common pre subscripts in juxtaposed point and vectors oght t be the same value.

see the just uploaded commaent on 'row, column and pillar".

I doubt that I can put this into pictures, I'll look into it.

[alkaline]

The monagonals within a hypercube can in my notation be depicted by

[[sub]j[/sub]0,[sub]k[/sub]q] <[sub]j[/sub]1,[sub]k[/sub]0>

the regular terms around can so be identified by the 'j'
j = 0: row
j = 1: column
j = 2: pillar
j = 3: 3-row (an foggy term I don't use)

are you sure you meant to use the letter 'q'? you haven't used it anywhere else and you didn't define it. Maybe you meant the number '1', since it is close to 'q' on the keyboard.

so, if i have this straight, here are how each of the letters is used:

n = dimension of the space
j = dimension of the point or direction: can go as high as n-1

m = "order" (size of hypercube)
k = coordinate value/distance from the origin along an axis: can go as high as m-1 for positions, or unlimited for direction since k is modulo m

l = i'm not sure

i'm also not sure what <[sub]4[/sub]1,[sub]l[/sub]0> means - does this go from one down to zero? why not zero up to one? what do the first and second terms mean here?

when you use k's and j's, is it the highest value they can take, or does it represent all possible values they can take?

[Aale de Winkel]

[[sub]j[/sub]0,[sub]k[/sub]q] <[sub]j[/sub]1,[sub]k[/sub]0>

The actual used letter here does not matter, the equal used presubscipt must be used the same while j != k
so in a cube with j chosen 0 k can still be 1 or 2 while q ranges from 0 to m-1.
In this way you see the formula stands for the rows emerging from the left side square.
This way the presubscripts allow me to make compound statements for any dimension.

John R. Hendricks decided upon using 'n' to stand for the dimension of the hypercube while using 'm' for its order.
The rest of the letters are freely chosen, but needs to be used consistently to make sense.

The vector <[sub]4[/sub]1,[sub]l[/sub]0> thus stands for the tetra-space vector <0,0,0,1>,
or the tetra-time space 5-vector <0,0,0,0,1>

[Aale de Winkel]

what do you mean by zeroth coordinate? is this procedure of using imaginary and zeroth coordinates, and multiplying by the speed of light done commonly in relativity or something? I don't know what the μ means when it is the x's subscript. Also, i don't know what "ict" means, or where the speed of light (c) appears in the equation. You define x[sub]μ[/sub], but what is x[sup]μ[/sup]? It would greatly help if you explained these things. (i would assume most other people visiting this site don't know them either).

counting spatial coordinate from 1 till n (for tetraspace n = 4) one can change these vectors into n+1 vectors.
and use the 0'th position to hold time multiplying this with the (constant) speed of light 'c' into a length, 'i' stands for sqrt(-1) which makes the formulae more convenient.

x[sup]μ[/sup] = T[sup]μ ν[/sup] x[sub]ν[/sub].
In Euclidian (n+1)-spaces the tensor T should be the identity, in Riemanian
(ie curved) spaces the tensor includes the space curvature.
but locally this should also remain the same.
It simply is so that for me the direction I'm standing in (ie the vertical) is quite another direction then your vertical given the sphere we are standing on.

Locally though there is no difference between x[sup]μ[/sup] and x[sub]μ[/sub].
The Einstein summation condition simply says that equal sub and superscrips are summed over. Thus simply dispensing the summing symbol.

Hope this suffices, if more is needed I have to dig into my books since most of it I've forgotten over the passed 20 years.
The reader is excused for not understanding most of this, the mathematical inclined might find it a starting point for setting up tetra-space relativity (go right ahead)

[alkaline]

The vector <[sub]4[/sub]1,[sub]l[/sub]0> thus stands for the tetra-space vector <0,0,0,1>,
or the tetra-time space 5-vector <0,0,0,0,1>

How would you denote <1,0,0,0> and <1,1,1,1>? For some positional vectors, how do you write [0,0,0,0], [1,0,0,0], [0,0,0,1], and [1,1,1,1]? Here are my guesses:

<1,0,0,0> = <[sub]0[/sub]1,[sub]l[/sub]0>
<1,1,1,1> = <[sub]0[/sub]1,[sub]l[/sub]1>
[0,0,0,0] = [[sub]j[/sub]0]
[1,0,0,0] = [[sub]0[/sub]1]
[0,0,0,1] = [[sub]4[/sub]1]
[1,1,1,1] = [[sub]j[/sub]1]

[Aale de Winkel]

You've got it except

<1,1,1,1> = <[sub]j[/sub]1> (but yours is equivelent to this)
and using 0 and 4 is having 5 subscripts which is 1 too manny
in the context of the hypercube the presubscript range is 0 to n-1
in the tetraspae context probably best range is 1 .. 4 (1 .. n (for n-space)
this way presubscript 0 can be reserved for the use of the temporal coordinate

[alkaline]

that was an error in the display - the bottom of the last line got clipped.

So:

[1,0,0,0] = [[sub]1[/sub]1]
[0,1,0,0] = [[sub]2[/sub]1]
[0,0,1,0] = [[sub]3[/sub]1]
[0,0,0,1] = [[sub]4[/sub]1]

but how do you do these?

[1,0,1,0]
[0,1,0,1]
[0,1,1,1]
[1,1,1,0]

[Aale de Winkel]

[1,0,1,0]
[0,1,0,1]
[0,1,1,1]
[1,1,1,0]

simply as is or
[[sub]1,3[/sub]1,[sub]2,4[/sub]0]
[[sub]1,3[/sub]0,[sub]2,4[/sub]1]
[[sub]1[/sub]0,[sub]j[/sub]1]
[[sub]4[/sub]0,[sub]j[/sub]1]

the ',' subscript is new, it simply never occured to me
nor saw any reason for these before, but perhaps there is

[Oren]

Oh, the frustration.
on the one hand, I'm a supergenius and can readily understand all the concepts I've encountered here. My brain is hungry for more.
On the other, I've only got a high school education and I have no idea what most of these mathematical notations mean.

Can someone guide me to a tutorial or something?

[alkaline]

ahh, i think i finally understand. A concrete index subscript assigns the number only to that index; a variable index subscript assigns that number to all indices that haven't been assigned by a concrete index. If an index hasn't assigned a value, its default is zero.

Thus, the tetraspace vector <[sub]4[/sub]1,[sub]l[/sub]0> = <0,0,0,1> because the [sub]4[/sub]1 assigns the value at the fourth index 1, and [sub]l[/sub]0 assigns the value at the indices l=1,2,3 zero. If i understand the system correctly, wouldn't <[sub]4[/sub]1> work also?

Sidenote: how do you specify the dimension of a vector in this system?

[Aale de Winkel]

If i understand the system correctly, wouldn't <[sub]4[/sub]1> work also?

Sidenote: how do you specify the dimension of a vector in this system?

under the assumption that nonspecified default to 0 yes <[sub]4[/sub]1> specifies the tetra-direction.

normally the dimension is fixed by the discussed hypercube, specifications of subhypercubes are using the dimensional vectors of the discussed hypercube.

If one needs to specify the dimension though one might hang a subscript after the closing bracket so: <[sub]4[/sub]1>[sub]4[/sub]

but note: the hypercube is a discrete lattice space, for continues tetraspace positions and also vectors are not limited to integers so:
x[sub]k[/sub] and v[sub]k[/sub] to specify points vectors lines etc. etc.

[alkaline]

Oh, the frustration.
on the one hand, I'm a supergenius and can readily understand all the concepts I've encountered here. My brain is hungry for more.
On the other, I've only got a high school education and I have no idea what most of these mathematical notations mean.

Can someone guide me to a tutorial or something?

well, in space there are points and there are vectors. A vector is a combination of a direction and a length, and can be represented by an arrow. Vectors are location independent. You can represent a vector in the same way you represent a point, with it understood that the vector specified is the arrow drawn from the origin to this point. Here are some examples in realmspace.

[0,0,0] or [[sub]j[/sub]0] = the point at the origin
<0,0,0> or <[sub]j[/sub]0>= a null vector: points nowhere, and has no length

[1,0,0] or [[sub]1[/sub]1] or [[sub]1[/sub]1,[sub]j[/sub]0] = a point a distance 1 away from the origin along the x axis
<1,0,0> or <[sub]1[/sub]1> or <[sub]1[/sub]1,[sub]j[/sub]0> = a vector that points in the positive direction of the x axis: goes from [0,0,0] to [1,0,0]

To assign a vector to a location, using Aale's notation you could do this:

[[sub]1[/sub]1]<[sub]2[/sub]1> = a vector that anchored on [1,0,0] and points in the direction of <0,1,0>, so it's end-point is [1,1,0].
.