Klitzing wrote:- ...| is described by abs(... - an xn) + abs(... + an xn) = a0
- ...> is described by abs(... - an xn) + abs(...) = a0
- ...O is described by sqrt((...)2 + an2 xn2) = a0
where the "..." within the equations refers to the left part of the respective equation, corresponding to the previous symbol, etc.
Coefficients moreover have to be chosen appropriate to the demands, e.g.
- when spins should derive true circular shapes (i.e. not ellipses), then (I think) an ought to be equal to the circumradius of "...", right?
- And in order to have taperings of non-vanishing heights, then (I think) an ought to be chosen larger than the circumradius of "...", right?
Did I get it?
--- rk
Very close, the extrude is right, and most of the taper and rotate.
- [...]I : abs([...] - an xn) + abs([...] + an xn) = a0
- [...]> : abs([...] - an xn) + [...] = a0
- [...]O : is a bit more different. As Marek points out, we need to establish direction, and thus the stationary plane. This is highly dependent on the equation, since the notation is a bit ambiguous. This minor obstacle can be overcome with quickfur's suggestion of a subscript axis for the spin operator, Om. This will single out the moving axis 'm' , telling us what the stationary plane is. I'll detail this one further below.
To clarify, a
0 is the circumradius for the entire overall shape. The value and usage of a
n with x
n is only to scale the shape to a better proportion. Without it, we get rectangular prism for a cube, tall skinny pyramids/cones, etc. I found a nice relation to what value a
n should be. It has to do with how many repeated dimension terms there are, with respect to others.
• For example, the unit cube is:
abs(abs(x - y) + abs(x + y) - 2z) + abs(abs(x - y) + abs(x + y) + 2z) = a
There are two separate groups of X and Y, associated with only one Z. The circumradius 'a' is proportioned to those extra X's and Y's differently than Z. In order to get unit cube, we have to scale Z by a multiple of 2. This will balance out the Z extension, and make a perfect cube.
• However, with a unit cylinder:
abs(sqrt(x^2 + y^2) - z) + abs(sqrt(x^2 + y^2) + z) = a
The X, Y, and Z dimensions are proportioned equally, needing no scaling to get unit cylinder.
• For tapered pyramids, the right triangle :
abs(abs(x) - 2y) + abs(x) = a
Circumradius 'a' is proportioned to 2 X's, and only one Y. Scaling Y by 2-fold will make a nicer looking triangle. I guess it's really only cosmetic outside the usage with a unit cube.
• For square pyramid,
abs(abs(x - y) + abs(x + y) -3z) + abs(x - y) + abs(x + y) = a
I found scaling Z by 3 instead of 2 makes a better looking shape. Again, this is more of a cosmetic thing.
Klitzing wrote:From this algebraic description it should be somehow deducable
- that ||O (spinning square, i.e. cylinder) = |O| (extruded disc, i.e. cylinder again)
- and that |>O (spinning triangle, i.e. cone) = |O> (tapered disc, i.e. cone again)
But I cannot see that directly from their algebraic descriptions, even when assuming different coefficients.
(But that might be more my problem with wrangling algebraics right ...
)
Might that be even a more general fact, i.e.
- would O and | generally commute?
- Resp. would O and > generally commute?
That should at least be (dis)provable by means of these algebraic forms.
(But at least | and > should not come out commutable in any case.)
--- rk
Oh, yes! The commutative operators show themselves very nicely in the algebraic definition. But, they can commute
only after some combinations of extrude or taper operators. They can be a little tricky. And, of course, the taper + extrude does not commute , [...]I> =/= [...]>I
• IOI = IIO
• IO> = I>O
• IIIO = IIOI = IOII
• IIO> = IOI> = II>O
• IO>I = I>OI =/= IOI> =/= I>IO
I will explain the bisecting rotate here as well. Simply put, a bisecting rotation will single out an axis to become the rotating axis into the higher dimension. The axis, which cuts as a line segment (through the shape), becomes a circular parameter. If rotating an n-sphere parameter, we simply fatten it up into an n+1 sphere.
•
So, for the IO and OI relationship:1) Starting with line segment, I
2) Bisecting Rotate into 2D, around stationary 0-plane makes circle, IO. Since no axis is stationary, the moving axis will be X. Line segment |x| becomes "circle-segment" sqrt(x^2 + y^2).
3) Extrude into 3D, making cylinder IOI. Extrusions are the simplest cartesian product, where [...]I is the ([...] , line) prism, as infinite shapes [...] stacked with a line segment.
|sqrt(x^2 + y^2) - z| + |sqrt(x^2 + y^2) + z| = a
Conversely, the second way to build a cylinder would be to interchange the spin and extrude operators:
1) Start with line segment
2) Extrude into 2D, making square, as infinite lines stacked within line segment, II
3) Bisecting rotate square into 3D, around stationary 1-plane Y. Either X or Y will work. This will turn x into sqrt(x^2 + z^2), within the square equation:
|sqrt(x^2 + z^2) - y| + |sqrt(x^2 + z^2) + y| = a
which is of course equal to the other orientation
|sqrt(x^2 + y^2) - z| + |sqrt(x^2 + y^2) + z| = a
•
Now for cone, and the commuting O> and >O : 1) Start with line segment I
2) Taper to point along 2D, making right triangle I> :
3) Bisecting rotate into 3D, around stationary 1-plane Y. If using stationary 1-plane X, we get something like a self-intersecting triangle torus, with no major diameter. I call these strange entities cyclotopes, as they are artifacts of the bisecting rotation. Again, as with cylinder, we turn |x| into sqrt(x^2 + z^2):
|sqrt(x^2 + z^2) - 2y| + sqrt(x^2 + z^2) = a
which is equal to the other orientation
|sqrt(x^2 + y^2) - 2z| + sqrt(x^2 + y^2) = a
And, the other way to build cone:1) Starting with line segment, I
2) Bisecting Rotate into 2D, around stationary 0-plane makes circle, IO. Line segment |x| becomes "circle-segment" sqrt(x^2 + y^2).
3) Taper the circle to point along 3D, making cone IO>. This will use the standard taper transformation along axis n, [...]>n : |[...] - xn| + [...] = a
|sqrt(x^2 + y^2) - 2z| + sqrt(x^2 + y^2) = a
•
For rotating cylinder into 4D, we use the same approach as replacing x
n with sqrt(x
n^2 + x
n+1^2). If the moving axis is in an n-sphere parameter, we simply add another dimension within, and fatten to an N+1 sphere.
Cylinder IOI or IIO :
|sqrt(x^2 + y^2) - z| + |sqrt(x^2 + y^2) + z| = a
1) Bisecting rotate into W, around stationary 2-plane XY, will leave Z as axis in motion. X and Y will remain unchanged, but
Z will become
sqrt(z^2 + w^2), making duocylinder IOIO, infinite circles stacked within circle-segment:
|sqrt(x^2 + y^2) - sqrt(z^2 + w^2)| + |sqrt(x^2 + y^2) + sqrt(z^2 + w^2)| = a
2) Bisecting rotate into W, around stationary 2-plane XZ or YZ, will leave Y or X as the axis in motion, respectively. Since X or Y are already inside a circle parameter, sqrt(x^2 + y^2) becomes sqrt(x^2 + y^2 + w^2), making the spherinder IOOI :
|sqrt(x^2 + y^2 + w^2) - z| + |sqrt(x^2 + y^2 + w^2) + z| = a
which is another orientation of
|sqrt(x^2 + y^2 + z^2) - w| + |sqrt(x^2 + y^2 + z^2) + w| = a
Marek14 wrote:What would be the string for 6D cartesian product of two cones?
That one would be IO>[IO>] , the duoconinder. The commuting O> within the cones will hold true, as well. Defined algebraically, as infinite cones XYZ stacked within a 'cone-segment' WVU:
||√(x2 + y2) - 2z| + √(x2 + y2) - |√(w2 + v2) - 2u| - √(w2 + v2)| + ||√(x2 + y2) - 2z| + √(x2 + y2) + |√(w2 + v2) - 2u| + √(w2 + v2)| = a
Marek14 wrote:Toratopes (at least the basic ones) could be added into this system by a ring operator (R), which would signify a rotation around an external line/plane/hyperplane. The equation could be found by putting the previous shape shifted to -an and a reflected copy to +an and applying the spin operator to it.
Yes, that's my fiber bundle operator, A(B) , shape A stretched over surface of shape B. It's a modification of the bisecting rotate, since it acts on only one axis, leaving the others alone. But, instead of turning an axis into a circle-segment, we replace with the whole circle (or whichever shape) along with its circumradius. So, for triangle torus I>(O),
Start with triangle in XY plane,
non-bisecting rotate around 1-plane Y, into Z, will turn
|x| into
|(sqrt(x^2 + z^2) - b)| . The absolute values have to be present around the hollow circle parameter for non-bisecting rotate, if originally there for the transformed axis.
||(sqrt(x^2 + z^2) - b)| - 2y| + |(sqrt(x^2 + z^2) - b)| = a
and non-bisecting rotate around 1-plane X, into Z, will turn
2y into
2(sqrt(y^2 + z^2) - b)||x| - 2(sqrt(y^2 + z^2) - b)| + |x| = a
as the two orientations of a triangle torus I>(O).
For toratopes, the spin operators are within the (), as in (O) for over a circle, (OO) for over sphere, (O)(O) as over a torus, etc. The (O) symbol is the standard non-bisecting rotate, but can be expanded for the surface of other shapes. The symbol (>) is the 1-surface of triangle I>, and (I) is the 1-surface of square II. Simply remove the first extrusion, and place all remaining symbols within the () operator, to denote just the surface of the shape.
I use [(O)(O)] for the duocylinder margin for now, as the product of the 1-surface of two circles. Logically, it may have a better symbol, as in ([O][O]) , or (O)[(O)] , etc. For the 3-manifolds, I use (O)[(O)(O)] for surface of tiger, (O)(O)(O) for surface of ditorus, [(O)(O)](O) for cyltorinder (torus of duocylinder) margin, [(OO)(O)] for 3-frame of cylspherinder IOOIO , (OOO) for surface of glome, etc. I define the cyltrianglintigroid as IO[(>)(O)] , circle over the 2-frame of cyltrianglinder I>IO.
These operators do not commute for closed toratopes, but only a certain way for opens. For example, IO(O)IO = IOIO(O) , cyltorinder is equal to duocylinder torus, since IOIO = IO[IO] . The whole circle product commutes and cannot be split up.
IOO / (III) - SPHERE
IO(O) / ((II)I) - TORUS
IO[(O)(O)] / ((II)(II)) - TIGER
IO(O)(O) / (((II)I)I) - DITORUS
IO(OO) / ((III)I) - TORISPHERE
IOO(O) / ((II)II) - SPHERITORUS
IOOO / (IIII) - GLOME
IOOOO / (IIIII) - PENTASPHERE
IOOO(O) / ((II)III) - GLOMITORUS
IOO[(O)(O)] / ((II)(II)I) - SPHERITIGER
IOO(OO) / ((III)II) - SPHERISPHERE
IOO(O)(O) / (((II)I)II) -SPHERIDITORUS
IO[(OO)(O)] / ((III)(II)) - CYLSPHERINTIGROID
IO[(O)(O)](O) / (((II)I)(II)) - TIGER TORUS
IO(OOO) / ((IIII)I) - TORIGLOME
IO(OO)(O) / (((II)II)I) - TORISPHERITORUS
IO(O)[(O)(O)] / (((II)(II))I) - TORATIGER
IO(O)(OO) / (((III)I)I) - DITORISPHERE
IO(O)(O)(O) / ((((II)I)I)I) - TRITORUS
It gets a bit more ambiguous in 8D, with shapes like (((III)II)((II)I)) and (((II)II)((III)I)) being equal to my symbol IO[(OO)(O)][(OO)(O)] . There would need to be some modification to denote the difference.