Wendy's product notation

Higher-dimensional geometry (previously "Polyshapes").

Postby wendy » Tue Aug 08, 2006 8:29 am

There are a number of base products, based on the w,x,y,z axies, as follows.

[] = prism ie max() maximum
() = spheric ie rss() root-sum-square
<> = tegum ie sum() sum of

When one applies these to unit lengths, at the centre, one gets

() 2d = circle, 3d = sphere, 4d = glome
[] 2d = square, 3d = cube, 4d = tesseract
<> 2d = rhomb, 3d = octahedron, 4d = 16choron

These products can be freely applied to the axies, as follows.

<(x,y),z> circular tegum ("bi-pyramid")
<(w,x),(y,z)> = bicircular tegum
[(w,x),(y,z)] = bicurcular prism "aka duocylinder"
(w,x,y,z) = glome

Sometimes, you might want values to be different, eg

(x,y,z) can be a oblate ellipsoid if x=y>z.
[x,y,z] = squat square prism, x=y>z
<x,y,z> = squat square tegum x=y>z

In the last, if x=y=sqrt(2), z=1, the result will fit 6 to a vertex at the squat end, and thus tile space.

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Postby bo198214 » Tue Aug 08, 2006 3:30 pm

wendy wrote:There are a number of base products, based on the w,x,y,z axies, as follows.

[] = prism ie max() maximum
() = spheric ie rss() root-sum-square
<> = tegum ie sum() sum of


Hey, I am in love with Wendy! This is what I really desperately try to introduce in my "bo's new notation" thread (which should anyway renamed to "bo's notation enhancements" and now perhaps to "bo's and wendy's notation enhancements" :) ).
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Postby Keiji » Tue Aug 08, 2006 4:44 pm

Right. *randomly decides to write out possibilities*

1D
[x] = line

2D
[xy] = square
(xy) = circle

There is no <s>spoon</s> triangle...

3D
[xyz] = cube
[(xy)z] = cylinder
(xyz) = sphere
([xy]z) = ?
<xyz> = octahedron
<(xy)z> = circular tegum

There is no <s>spoon</s> torus...

*shot for the matrix references*

4D
[xyzw] = tesseract
[(xyz)w] = spherinder
[(xy)zw] = cubinder
[(xy)(zw)] = duocylinder
[<xyz>w] = octahedral prism
[<(xy)z>w] = circular tegal prism
[([xy]z)w] = ? prism
(xyzw) = glome
([xyz]w) = ?
([(xy)z]w) = ?
(<xyz>w) = ?
(<(xy)z>w) = ?

Can someone fill in the question marks? :|

I don't like this notation. There are no torii, and there are no triangular shapes. So bleh! :P
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Postby Marek14 » Tue Aug 08, 2006 6:17 pm

Rob wrote:Right. *randomly decides to write out possibilities*

1D
[x] = line

2D
[xy] = square
(xy) = circle

There is no <s>spoon</s> triangle...

3D
[xyz] = cube
[(xy)z] = cylinder
(xyz) = sphere
([xy]z) = ?

This is my "dome" or others' "crind".
<xyz> = octahedron
<(xy)z> = circular tegum

There is no <s>spoon</s> torus...

*shot for the matrix references*

4D
[xyzw] = tesseract
[(xyz)w] = spherinder
[(xy)zw] = cubinder
[(xy)(zw)] = duocylinder
[<xyz>w] = octahedral prism
[<(xy)z>w] = circular tegal prism
[([xy]z)w] = ? prism

I called this "dominder"
(xyzw) = glome
([xyz]w) = ?

I'm no longer sure what were the precise names, but I think that this one is tridome
([(xy)z]w) = ?

Spheridome
(<xyz>w) = ?
(<(xy)z>w) = ?

Well, these were not in my graphotope system, as I only developed it for sphere and prism products. So I never had names for these.

Can someone fill in the question marks? :|

As you can see, I can - most of them, at least.

I don't like this notation. There are no torii, and there are no triangular shapes. So bleh! :P
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Postby bo198214 » Tue Aug 08, 2006 6:20 pm

Hm, now the confusion gets started.
The usage of [] and <> of wendy is quite different to the usage of () in RNS, at least what I can see from the examples.
So wendy can you point out *exactly* how your products are defined?!
I.e. how do we come from such an expression to the implicit equation?

For example:
((xy)z) would be a torus
([xy]z), [(xy)z], <(xy),z> .... would be in my interpretation also be a torus but with one rectangular circle. For wendy the last one is a bipyramid.
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Postby Marek14 » Tue Aug 08, 2006 7:12 pm

bo198214 wrote:Hm, now the confusion gets started.
The usage of [] and <> of wendy is quite different to the usage of () in RNS, at least what I can see from the examples.
So wendy can you point out *exactly* how your products are defined?!
I.e. how do we come from such an expression to the implicit equation?

For example:
((xy)z) would be a torus
([xy]z), [(xy)z], <(xy),z> .... would be in my interpretation also be a torus but with one rectangular circle. For wendy the last one is a bipyramid.


You must realize that these products do NOT include spheration. () is a spheric product, i.e. it makes a^2+b^2=r^2 from a^2=r^2 and b^2=r^2. [] makes max(a^2,b^2)=r^2, similarly. Spheration and RNS notation was only developed later, and the use of () by the two is not related.
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Postby bo198214 » Tue Aug 08, 2006 7:20 pm

So may I summarize that in RNS notation a parenthesis is treated like this:
A=(A<sub>1</sub>...A<sub>n</sub>) -> A=sqrt(A<sub>1</sub><sup>2</sup> + ... + A<sub>n</sub><sup>2</sup>) - r
and in wendys notation like this:
A=(A<sub>1</sub>...A<sub>n</sub>) -> A=sqrt(A<sub>1</sub><sup>2</sup> + ... + A<sub>n</sub><sup>2</sup>)

where the shape is in RNS retrieved by A=0 and in wendys product by A=r? So that we have a sequence of parameters r<sub>i</sub> in RNS and only one parameter r in Wendy's?
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Postby wendy » Wed Aug 09, 2006 7:43 am

The functions rss(), sum(), and max() are all perfectly clearly defined.

A given figure is to be thought of as a radial function, such that it is zero at the centre, and one at the surface. The centre is arbitary, but for the "right" instances, one should use the "real centre" if it exists.

For example, a line of length 2 can be defined as, a centre at 0, and the radial function being equal to "abs(x)", so that it is 1 at x equal -1 and +1 respectively.

a circle would then be rss(x,y). rss is "root-sum-square", so you square what's in the brackets, add them all up, and take the square root. People who play with sinosoidal waves might be familiar with "root-mean-square" or rms.

So, the rss(3,4) = 5, because 3*3+4*4 = 5*5.

The max() is the maximum. Since we are already getting the absolute elsewhere, (by the radial function), we have

max(x,y,z) would give the face that ray through x,y,z would pass, and the distance to that face. So max(0.1, 0.3, 0.9) is 0.9, and this is 9/10 of the way to the surface: ie the surface is hit at (1/9, 1/3, 1).

The sum() is simply the sum. as before we have absolutes, so the absolutes is not less than the smallest.

So if you apply this to the unit-distance from the point (0,0,0), you get a notional sphere of unit radius, a notional cube, and a notional octahedron.

All of these products are individually "open", so

rss(rss(x,y),z) = rss(x,y,z)
sum(sum(x,y),z) = sum(x,y,z)
max(max(x,y),z) = max(x,y,z)

These products are _not_ inter-related, for example,

max(sum(1,7), sum(3,6)) = 9
max(1,7,3,6) = 7
sum(max(1,7), max(3,6)) = 13

and likewise with the polytopes.

What a radiant product defines is the radiant function of the product.

One notes that the "torus product" is not a radiant function. That is, the torus-product a##b does not give a radial statement based on a and b. One notes that the torus-product is a pondering product (ie it looses a dimension), and hence can not be radiant.

The product [(x,y),z]

The radiant value of this is max(sqrt(x*x + y*y),z). The surface of the result is when the radiant function is unity.

So the surface of this figure is when max(sqrt(x**x+y**y), abs(z)) =1

One sees that the thing is bounded by the planes z=1, z=-1 (since this makes the second term 1). It is also bounded by the infinite-cylinder x**x+y**y=1 (over all z), so the intersection of these is a 2-unit cylinder.

And so on.

Note that the rototope (x,y,z) can equally be got by the intersection of three cylinders of the same size, at right angles to each other. This gives a rotational rhombic dodecahedron, bent so that the octaghedron-edges align with the sphere, and the cube prodrudes outwards by a measure of sqrt(1.5).

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Postby Keiji » Wed Aug 09, 2006 8:18 am

Hey I was thinking about the ([xy]z) when I was in bed last night, and it is indeed the crind... I figured out that (ab) where b is just a letter is a growing and shrinking copy of a, moving in the axis b as it does so, following the cosine curve (sine works too, but I prefer cosine because it is at the maximum value at the origin).

So:

([xy]z) = square crind (intersection of two cylinders)
([xyz]w) = cubic crind (intersection of three spherinders - it is a spherinder not a cubinder because cubinders have TWO straight axes so if you intersect two perpendicular cubinders you have already used up all dimensions, and three is impossible)
([(xy)z]w) = cylindrical crind
([xy][zw]) = square duocrind
(<xyz>w) = octahedral crind
(<(xy)z>w) = circular tegal crind
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Postby bo198214 » Wed Aug 09, 2006 8:20 am

@wendy Yes I understood that from what Marek said, a simple yes or no on my previous question had been completely sufficed ;) And had also a somewhat more interactive touch.
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Postby wendy » Thu Aug 10, 2006 7:56 am

There is also a general pyramid product, which will produce triangles from points, etc. The notation for this is **. The product is elongating, that is, it adds a dimension for each application.

So we have

line = point ** point
triangle = point ** point ** point
tetrahedron = point ** point ** point ** point

cone = circle ** point

The product becomes "proper" in five dimensions. This means that there are pyramid products that are not a cumulation of peaks of n-1 dimensions. For example, in 5d

circle ** circle

The product can be constructed thus.

For a given line, triangle, tetrahedron, ..., a point in the interior is a certain fraction of each height. So a triangle with corners x,y,z can be 33%x + 11%y + 56%z. This becomes more obvious when you look at the triangle in the possitive octant, formed by the plane x+y+z+... =1

Perpendicular to each point x, y, z, we place further (mutually orthogonal) spaces X, Y, Z, ... and inscribe figures centred around x, y, z. At any internal point, we have a prism that is 33%X , 11%Y, 56%Z.

To illistrate that a tetrahedron is a line ** line, we note that there are two elements of the product, so the "altitude" is a 2-vertex simplex: a line. Let's place this line running from z=1 to z=-1.

The two bases are as specified in the product, ie line, line. These are both perpendicular to each other and to the altitude. So the first line runs in x=+2 to x=-2, at z=1. The second runs y=+1 to y=-1, running through z=-1 . At any point in the altitude, the cross section is a prism-product of these two: viz

at z 0.5, this is 0.75 * 1 + 0.25 * (-1), so the section here is a rectangle, 75% of x, and 25% of y, or one that is 1.5 * 0.5 wide.

Because the product adds a dimension of stretching, for each application, the highest sum of dimensions in 4d, can only be three dimensions, eg

point * 3d -> 3d-figure pyramid
line * 2d-f -> 2d figure - line-prism
= (2d pyramid) pyramid.

And so it goes.

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Postby PWrong » Thu Aug 10, 2006 8:07 am

Isn't ** exactly the same as tapering (->)?

For example, in 5d

circle ** circle

How can that be a 5D shape?
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Postby moonlord » Thu Aug 10, 2006 8:22 am

I believe she meant disk ** disk because circle ** circle would be three dimensional...
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Postby wendy » Fri Aug 11, 2006 7:59 am

I am not sure. There is a description of tapering a circle to a line, by way of ellipses. This produces a kind of circular roof in three dimensions.

The pyramid product itself does not work this way: each of the bases tapers from 100% at the base to 0% at the apex.

The dimension of a pyramid is the sum of dimensions of the bases, plus the number of pyramid-signs used. So a circle ** circle is 2 +1 +2 = 5.

The torus-product has a negative dimensionality, so each of the ## used, reduces the dimesnion by one, so a circle ## circle = 2 -1 +2 = 3.

The other two products (prism = *#) and tegum (#*) have a zero-dimensional effect, so

circle *# circle = bi-circular prism (duocylinder) = 2 +0 +2 = 4
circle #* circle = bi-circular tegum = 2 +0 +2 = 4

While one might imagine that # and * stand for +1/2 and -1/2 respectively, this is not the case.

The symbols mean as follows

*. volume [no change]
#. surface [reduces dimension by one]
.* drawing [increases dimension by one]
.# extension [no change]

Drawing could safely be thought of as what happens as one might drag out some chewing-gum. Basically, you have a blob, and you stretch it. This adds the extra dimension.

The word tegum means, "covering", the product is a drawing of surface.

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Postby Keiji » Mon Jun 18, 2007 6:43 am

Wendy (or someone else), can you tell me what the following would be?

[<xy><zw>]
[(xy)<zw>]
[(xy)(zw)] duocylinder
<[xy][zw]>
<[xy](zw)>
<(xy)(zw)>
([xy][zw]) duocrind
([xy]<zw>)
(<xy><zw>)

(apart from the two already filled in)
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Postby wendy » Mon Jun 18, 2007 10:54 am

Hi

[<xy><zw>] tesseract, literally, "rhombus-rhombus duoprism"
[(xy)<zw>] circle-square prism
[(xy)(zw)] duocylinder [correct]
<xy> 16choron
<xy> square-circle tegum.
<xy> bi-circular tegum
([xy][zw]) duocrind [the name is not known by me]
([xy]<zw>) bi-square rotation
(<xy><zw>) ditto,

The thing is that a square [xy] is also a rhomb <xy>, the first gives a cube in 3d [xyz], while the second gives an octahedron <xyz>. It is just that [] and <> cross at two dimensions, as they all do at one dimension: ie (x) = [x] = <x> = x.

So something like <xy> gives <<xy><zw>> = <wxyz>.

The last three are the same, being square-square, square-rhombus and rhombus-rhombus spherates. The particular difference is the orientation and size of the three elements.

The surface equation is something like

rss(max(x,y),max(w,z)) = 1.

() rss = root-sum-square, = sqrt(sum of suares) (x,y,z) gives a sphere.
[] max = maximum value [x,y,z] gives a cube
<> sum = sum of values <x> gives an octahedron

For a line in xy, and and a line in wz, this gives a line-segment by way of maximums, these become the axis of an ellipse in the plane (hedrix) formed by these two lines.


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Postby Keiji » Mon Jun 18, 2007 3:31 pm

phpBB's HTML parser messes up < and > when they aren't meant to be HTML (oh, if only we could use htmlentities in phpBB!). Turn off HTML when using < and >.

[<xy><zw>] tesseract, literally, "rhombus-rhombus duoprism" yeah, I should have noticed that!
[(xy)<zw>] circle-square prism also should have noticed the one, it's what I call the narrow cubinder
[(xy)(zw)] duocylinder [correct]
<[xy][zw]> 16choron the <xyzw> is the hexadecachoron, so this would be the wide hexadecachoron like <[xy]z> is the wide octahedron
<[xy](zw)> square-circle tegum what kind of shape is this? any info on it?
<(xy)(zw)> bi-circular tegum see above
([xy][zw]) duocrind [the name is not known by me] duocrind = crind of a two squares, as the ordinary crind is the crind of a square and a line: ([xy]z)
([xy]<zw>) bi-square rotation Hmm... narrow duocrind
(<xy><zw>) ditto doubly-narrow duocrind


wendy wrote:The thing is that a square [xy] is also a rhomb <xy>


On my wiki, I differentiate between [xy] and <xy>, because <xy> is [xy] rotated 45 degrees and then stretched by a scale factor of √2/2. This leads to pairs of shapes like the octahedron <xyz> and wide octahedron <[xy]z>, or the crind ([xy]z) and narrow crind (<xy>z).
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