Multidimensional Geometry

 
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Philip



Joined: 15 Jan 2015
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PostPosted: Mon Jan 19, 2015 1:52 am    Post subject: Multidimensional Geometry Reply with quote

Hey everyone! This is my first post here, so it's going to be a big one. I've spent the last 7 years developing and learning a few notations that define geometric shapes, with any number of dimensions. Last year I learned the math behind it all, and began rendering these wild things. The more visually impressive types would be the hypertoric varieties. A large library of 3D exploration functions, defined implicitly, are here:

http://hddb.teamikaria.com/forum/viewtopic.php?p=23287#p23287

In their natural dimension, they are all a single toroidal ring. In lower planes, they make multiple copies of an individual hypertorus, spaced out in an array of intercepts. In 3D, all that we will ever see of them are several spheres or tori, that morph and move as one. Shapes in 4D, you will see one or two intercepts. In 5D we'll see 4 at most, 6D can make 8, 7D makes 16, 8D makes 32, etc. Certain types with only 1-sphere-shaped diameters make the most. A nice collection of pics and GIFs are here, on my Imgur account:

http://ninedimensionalbeing.imgur.com/

Here, you'll find wild stuff like,



which is a cycle of four 90 degree turns from the 7D hypertorus notated as ((((II)I)I)((II)I)) . It can be explored with the function :

(sqrt((sqrt((sqrt(x^2 + (y*cos(d) - c*sin(d))^2) - 4.25)^2 + (z*cos(a))^2) - 2)^2 + (z*cos(b))^2) - 1)^2 + (sqrt((sqrt((y*sin(d) + c*cos(d))^2) - 2.5)^2 + (z*((sin(a))*(sin(b))))^2) - 1.25)^2 - 0.4^2 = 0

Parameters a,b, and d are for rotation, and work best when set to the range 0 < a,b,d < 1.57 , for 90 degree turns. Parameter c is for sliding the shape through the 3-plane, and works best when set to -10 < c < 10 . C = 0 is midsection.

For what it looks like you all are doing here, you may find it useful to explore the shape in my favorite go-to 3D plotter:

http://web.monroecc.edu/calcNSF/

This will allow you to dynamically slide and rotate the shape around in higher dimensions, in search of something cool. Once you find a worthy enough structure at the particular angles and depth (as the values of the adjustable parameters), you may then want to rewrite the function to render in the POVray program.

As for the other types of shapes I researched, they won't be as visually fantastic. They will come out as a single, simple 3D shape. But, what's neat about these, are that 90 degree rotations in a higher dimension will transform the 3D shape into a different one. You end up with a single equation that defines multiple 3D shapes, simultaneously. They do make some interesting morphs, though. A good example would be the 5D Spherindrone, notated as IOO>I,



This one can be explored with the implicit function,

abs(abs(sqrt(x^2+(y*cos(a))^2+(z*cos(b))^2) -(y*sin(a))) + abs(sqrt(x^2+(y*cos(a))^2+(z*cos(b))^2) +(y*sin(a))) - 3(z*sin(b))) + abs(sqrt(x^2+(y*cos(a))^2+(z*cos(b))^2) -(y*sin(a))) + abs(sqrt(x^2+(y*cos(a))^2+(z*cos(b))^2) +(y*sin(a))) - 8.5

Parameters a and b are rotation.

I wrote a decent sized list of implicit functions 1D through 4D, here:

http://www.reddit.com/r/hypershape/comments/2p1q3l/the_math_of_shapes_the_shape_of_math/

and 5D, here:

http://www.reddit.com/r/hypershape/comments/2ppwru/geometric_shapes_in_the_fifth_dimension/

The hyperprisms are probably more interesting from a mathematical point of view. It is still possible to combine a prism with a toric shape, and end up with 8 square pyramids that morph and move as one.

Anyways, I thought you all may find this interesting. It's the culmination of 7 years of self-study, outside the mathematical mainstream.
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Last edited by Philip on Mon Jan 19, 2015 8:46 pm; edited 2 times in total
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PostPosted: Mon Jan 19, 2015 4:58 am    Post subject: Reply with quote

Hi Philip,
Welcome to the forum and many thanks for sharing your work and experiences! I can only imagine the hard work and time it took you to come up with this great software . I've installed your applet CalcPlot3D and it's really amusing!
I'll explore it more in depth in the coming days Smile
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Philip



Joined: 15 Jan 2015
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Location: Orlando, FL

PostPosted: Mon Jan 19, 2015 6:15 pm    Post subject: Reply with quote

Oh, sorry, I didn't create CalcPlot3D. That would be Paul Seeburger. Apologies for the misleading word choice! My contribution is writing all the implicit explore functions, exploring the shapes, taking pics and animations. I also invented the notation that looks like "IOOI>", which represents various shapes.
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PostPosted: Tue Jan 20, 2015 1:02 am    Post subject: Reply with quote

Hi,
Sorry, my mistake! Your animations looks great although I didn't fully understand how you extrapolate them from higher dimensional objects.
Also, MathMod may render your equations in real time...
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Philip



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PostPosted: Wed Jan 21, 2015 1:48 am    Post subject: Reply with quote

I can explain that, certainly. But, first, I'd like to know what you mean by 'extrapolate from higher dimensions' . Does it have to do with how I wrote the function? Or, maybe, how do I know what this shape is supposed to be, as in five dimensional?
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PostPosted: Fri Jan 23, 2015 1:03 am    Post subject: Reply with quote

Hi,
Your equations are only functions of 3 parameters so you probably did some projections to get rid of the extra dimentions?
As far as I know, Implicit surfaces with dimensions greater that 3 require complex and time consuming algorithm to be represented as a 3d mesh points.
Ray tracing techniques may be more suitable to visualize them in a 3D space
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Philip



Joined: 15 Jan 2015
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PostPosted: Fri Jan 23, 2015 7:47 pm    Post subject: Reply with quote

Ah, I see what you mean! I'm unaware of this complicated algorithm you mention. Here's how I write a function for a particular shape. I'll use the above posted 5D spherindrone, notated as IOOI>, as an example.

Basically, the sequence IOOI> is a little invention of mine that defines a shape by linear construction operators. It's a blueprint for building it from a 0D point. It can also be used as a guide to procedurally build the implicit equation as well.

Algebraic Definition of Construction Operators I , > , O , M[N]

[...] = arbitrary shape equation/sequence

Extend [...] along axis x_n

    [...]I = |[...] - x_n| + |[...] + x_n| - a


Taper [...] to point along x_n

    [...]> = |[...] - 2x_n| + [...] - a


Bisecting rotate [...]I into x_(n+1) : defined by turning line segment x_n of prism into a circular segment √(x_n + x_(n+1)) , becomes product with circle. If axis x_n is already within an n-sphere, simply add the extra x_(n+1) within the n-sphere.

    [...]IO = |[...] - √(x_n + x_(n+1))| + |[...] + √(x_n + x_(n+1))| - a


Cartesian Product of M and N

    M[N] = |M - N| + |M + N| - a



Building the Spherindrone

* : 0D Point, starting element

I : Line , extend Point along X

|x| = a

IO : Circle , bisecting rotate Line around point, along X into Y

√(x+y) = a

IOO : Sphere , bisecting rotate Circle around axis X, along Y into Z

√(x+y+z) = a

IOOI : Spherinder , extend Sphere along W

|√(x+y+z) - w| + |√(x+y+z) + w| = a

IOOI> : Spherindrone , shrink Spherinder to point while extending along V

||√(x+y+z) -w| + |√(x+y+z) +w| + 4v| + |√(x+y+z) -w| + |√(x+y+z) +w| = a


Here, we end up with a 5D equation. I scale the v-parameter by a factor of 4 to balance out the function, since there are 4 terms of x,y,z and w. This will shorten the height of an otherwise disproportionate tall, skinny pyramid, if using only v.

abs(abs(sqrt(x^2+y^2+z^2) -w) + abs(sqrt(x^2+y^2+z^2) +w) + 4v) + abs(sqrt(x^2+y^2+z^2) -w) + abs(sqrt(x^2+y^2+z^2) +w) = a


The notation sequence IOOI> can be used to parse out the 3D midsections, by using 'i' in place of a symbol. This tells us which dimension to set to 0, making a 3D equation of the 5D equation. One must also rewrite the function using x,y,and z. There are four unique midsections in 3D for a spherindrone:

iiOI> : XY cut , II> sq pyramid

abs(abs(sqrt(0^2+0^2+x^2) -y) + abs(sqrt(0^2+0^2+x^2) +y) + 4z) + abs(sqrt(0^2+0^2+x^2) -y) + abs(sqrt(0^2+0^2+x^2) +y) - 8.5

IOii> : ZW cut , IO> cone

abs(abs(sqrt(x^2+y^2+0^2) -0) + abs(sqrt(x^2+y^2+0^2) +0) + 4z) + abs(sqrt(x^2+y^2+0^2) -0) + abs(sqrt(x^2+y^2+0^2) +0) - 8.5

IOiIi : ZV cut , IOI cylinder
-
abs(abs(sqrt(x^2+y^2+0^2) -z) + abs(sqrt(x^2+y^2+0^2) +z) + 4*0) + abs(sqrt(x^2+y^2+0^2) -z) + abs(sqrt(x^2+y^2+0^2) +z) - 8.5

IOOii : WV cut , IOO sphere
-
abs(abs(sqrt(x^2+y^2+z^2) -0) + abs(sqrt(x^2+y^2+z^2) +0) + 4*0) + abs(sqrt(x^2+y^2+z^2) -0) + abs(sqrt(x^2+y^2+z^2) +0) - 8.5


Now we have four distinct equations for different 3D shapes. These are all using the original 5D equation. Within the 5D equation, three of those dimensions are 'visible, along with two extra open slots. Those open slots are the 4th and 5th dimension, where we can use 'a' and 'b' ,as adjustable parameters, to slide into those higher directions.

Another thing we can do is to set up two separate rotate parameters, to link a visible dimension with an open slot. This will let us tie together the four 3D midsection equations into one single function. This means swapping IiiI> to IOOii, by 90 degree rotation

Using the full 5D equation as a guide, replace:

y -> (y*sin(a))
z -> (z*sin(b))
w -> (y*cos(a))
v -> (z*cos(b))

resulting in:

abs(abs(sqrt(x^2+(y*cos(a))^2+(z*cos(b))^2) -(y*sin(a))) + abs(sqrt(x^2+(y*cos(a))^2+(z*cos(b))^2) +(y*sin(a))) + 4(z*sin(b))) + abs(sqrt(x^2+(y*cos(a))^2+(z*cos(b))^2) -(y*sin(a))) + abs(sqrt(x^2+(y*cos(a))^2+(z*cos(b))^2) +(y*sin(a))) - 8.5


When a,b equals pi/2 = 1.5707 , we see the sine position. When a,b = 0, we see the cosine position. Setting to pi makes a full 180 deg flip.

Using this dual rotate function, we can turn the cross sectioning 3-plane 90 degrees in the 4th and 5th dimensions, which will morph the 3D shapes into different ones, as seen in the above GIF. As a guide for navigation,

[a,b] - 3D Midsection Positions

[0,0] - IOO sphere
[1.57,0] - IOI cylinder
[1.57,1.57] - II> square pyramid
[0,1.57] - IO> cone


This is the method I use to define and plot a multidimensional shape in 3D slices. The hypertoric shapes have another different notation, but uses the same practice of setting extra dimensions to zero, then input rotate parameters. You can also use a rotate + translate parameter to turn and slide in 4 and 5d,

y -> (y*sin(b) + a*cos(b))
z -> (z*sin(d) + c*cos(d))
w -> (y*cos(b) - a*sin(b))
v -> (z*cos(d) - c*sin(d))

'a' slides, 'b' rotates in 4D. 'c' slides, 'd' rotates in 5D. Midsection is when 'a' and 'c' are set to 0. This is the ultimate way to handle a shape, since you can move around to every angle and depth:

abs(abs(sqrt(x^2+(y*sin(b) + a*cos(b))^2+(z*sin(d) + c*cos(d))^2) -(y*cos(b) - a*sin(b))) + abs(sqrt(x^2+(y*sin(b) + a*cos(b))^2+(z*sin(d) + c*cos(d))^2) +(y*cos(b) - a*sin(b))) + 4(z*cos(d) - c*sin(d))) + abs(sqrt(x^2+(y*sin(b) + a*cos(b))^2+(z*sin(d) + c*cos(d))^2) -(y*cos(b) - a*sin(b))) + abs(sqrt(x^2+(y*sin(b) + a*cos(b))^2+(z*sin(d) + c*cos(d))^2) +(y*cos(b) - a*sin(b))) = 8.5

I found CalcPlot3D really excels at dynamic exploration using a function like this. You can move the 3-plane around in real-time, using the four adjustable parameters. But, of course, exploring a 5D shape with 3D is the same as exploring a 3D shape with 1D line segments. There is a lot of missing information from canceling out 2 dimensions, which requires exploration to see everything.

Hope that helps!
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Philip



Joined: 15 Jan 2015
Posts: 9
Location: Orlando, FL

PostPosted: Sat Jan 24, 2015 1:58 am    Post subject: Reply with quote

Here's a few more neat examples in 6D:

IO[IO]>I : 6D Duocylindroninder , prism of the pyramid of the 4D duocylinder. Has infinite 5D duocylindrical pyramids XYZWV stacked within line segment U

XYZWVU

I - 1D Line : Start with line in 1-plane X

|x| = a

IO - 2D Circle : bisecting rotate LINE around stationary point, along X into Y

√(x+y) = a

IOI - 3D Cylinder : extend CIRCLE along line into Z

|√(x+y) - z| + |√(x+y) + z| = a

IOIO = IO[IO] - 4D Duocylinder : bisecting rotate CYLINDER around stationary plane XY, along Z into W

|√(x+y) - √(z+w)| + |√(x+y) + √(z+w)| = a

IO[IO]> - 5D Duocylindrone : shrink DUOCYLINDER to point while extending along V

||√(x+y)-√(z+w)| + |√(x+y)+√(z+w)| + 2v| + |√(x+y)-√(z+w)| + |√(x+y)+√(z+w)| = a

IO[IO]>I - 6D Duocylindroninder : extend DUOCYLINDRONE along line into U

|||√(x+y)-√(z+w)|+|√(x+y)+√(z+w)|+2v| + |√(x+y)-√(z+w)|+|√(x+y)+√(z+w)| - 4u| + |||√(x+y)-√(z+w)|+|√(x+y)+√(z+w)|+2v| + |√(x+y)-√(z+w)|+|√(x+y)+√(z+w)| + 4u| = a






IO>I[IO] = IO>[IOI] : 6D Cubconinder , cartesian product of cone times cylinder. It has infinite cones XYZ stacked within a cylinder-segment WVU.


XYZWVU

I - 1D Line : Start with line in 1-plane X

|x| = a

IO - 2D Circle : bisecting rotate LINE around stationary point, along X into Y

√(x+y) = a

IO> - 3D Cone : shrink CIRCLE to point while extending along Z

|√(x+y) + 2z| + √(x+y) = a

IO>I - 4D Coninder : extend CONE along line into W

||√(x+y)+2z| + √(x+y) - 2w| + ||√(x+y)+2z| + √(x+y) + 2w| = a

IO>IO = IO>[IO] - 5D Cylconinder : bisecting rotate CONINDER around stationary plane XYZ, along W into V

||√(x+y)+2z| + √(x+y) - 2√(w+v)| + ||√(x+y)+2z| + √(x+y) + 2√(w+v)| = a

IO>IOI = IO>[IOI] - Cubconinder : extend CYLCONINDER along line into U

||√(x+y)+2z| + √(x+y) - |√(w+v)-u| - |√(w+v)+u|| + ||√(x+y)+2z| + √(x+y) + |√(w+v)-u| + |√(w+v)+u|| = a



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


IOO>II : 6D Sphone Diprism , cubic prism of the 4D spherical cone. Has infinite 4D sphones XYZW stacked within a square-segment VU

XYZWVU

I - 1D Line : Start with line in 1-plane X

|x| = a

IO - 2D Circle : bisecting rotate LINE around stationary point, along X into Y

√(x+y) = a

IOO - 3D Sphere : bisecting rotate CIRCLE around stationary axis X, along Y into Z

√(x+y+z) = a

IOO> - 4D Sphone : shrink SPHERE to point while extending along W

|√(x+y+z) + 2w| + √(x+y+z) = a

IOO>I - 5D Sphoninder : extend SPHONE along line into V

||√(x+y+z)+2w| + √(x+y+z) - 2v| + ||√(x+y+z)+2w| + √(x+y+z) + 2v| = a

IOO>II - 6D Sphone Diprism : extend SPHONINDER along line into U

||√(x+y+z)+2w| + √(x+y+z) - |v-u| - |v+u|| + ||√(x+y+z)+2w| + √(x+y+z) + |v-u| + |v+u|| = a
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Philip



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PostPosted: Tue Feb 03, 2015 3:15 am    Post subject: Reply with quote

So, what do you think about this technique? Do you think it makes things any easier? Is there anything I can explain better?
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PostPosted: Thu Feb 05, 2015 3:19 am    Post subject: Reply with quote

Hi Philip,
Thanks for the detailed explanation but I'm sorry I still can't understand how it works, especially when you say:
Code:
Basically, the sequence IOOI> is a little invention of mine that defines a shape by linear construction operators. It's a blueprint for building it from a 0D point. It can also be used as a guide to procedurally build the implicit equation as well. 

Parametric surfaces are far more easy to work with in higher dimensions, this is why MathMod support 4d parametric surfaces (K3dSurf support up to 6D, with all the associated rotational plans ).
Do you have some high dimensional parametric examples?
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Philip



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PostPosted: Thu Feb 05, 2015 8:25 pm    Post subject: Reply with quote

I'm afraid I don't have any parametric definitions. I don't yet know how to write and build shapes with it. That's one of the goals this year, as the implicit form is what I've been studying so far. I can explain the notation a little better:

The string of symbols "IOOI>" defines a shape by a series of geometric transformations, along a certain dimension. Each symbol transforms the shape made by all previous symbols to the left, into that corresponding n+1 dimension. The number of symbols is the number of dimensions.


The Five Operators

I - the extend, makes prisms and cubes
> - the taper/scale to a point, makes pyramids and simplices
O - the bisecting rotate, makes spheres and cylinders
(O) - the non-intersecting rotate/fiber bundle, makes toroidal rings
m[n] - the cartesian product, makes the (m,n)- prism

The algebraic definitions of each operator are very simple. In order to construct an equation for a multiple-term, multidimensional shape, we nest the algebraic functions into a combination sequence of extensions, taperings, and rotations. The first symbols are the inner-most nested. The last symbol is the outer-most structure of the equation. To illustrate from 1D to 3D:



Extend Shape M along Axis x_n

mI : |m - x_n| + |m + x_n| = a


Taper Shape M along Axis x_n

m> : |m + 2x_n| + m = a


Bisecting Rotate M-prism into K-space, mI = |m - x_n| + |m + x_n| = a

mIO : |m - √(x_n+x_k)| + |m + √(x_n+x_k)| = a


Non-Intersecting Rotate M-prism into k-space, mI = |m - x_n| + |m + x_n| = a

mI(O) : |m - (√(x_n+x_k)-a)| + |m + (√(x_n+x_k)-a)| = b


Cartesian Product with Shape M and N

m[n] : |m - n| + |m + n| = a




The First Dimension

I - LINE : |x| = a


The Second Dimension

IO - CIRCLE : √(x+y) = a

I> - TRIANGLE : ||x|+2y| + |x| = a

II - SQUARE : |x-y| + |x+y| = a


The Third Dimension

IOO - SPHERE : √(x+y+z) = a

IO(O) - TORUS : (√(x+y) - a) + z = b

IO> - CONE : |√(x+y) +2z| + √(x+y) = a

IOI - CYLINDER : |√(x+y) - z| + |√(x+y) + z| = a

I>> - TETRAHEDRON : |||x|+2y|+|x| + 2z| + ||x|+2y|+|x| = a

I>I - TRIANGLE PRISM : |||x|+2y|+|x| - z| + |||x|+2y|+|x| + z| = a

II> - SQUARE PYRAMID : ||x-y|+|x+y| + 3z| + |x-y|+|x+y| = a

III - UNIT CUBE : ||x-y|+|x+y| - 2z| +||x-y|+|x+y| + 2z| = a

In 3D, there are 2 equals, since the IO and O> pairings can commute. The cone can be expressed as IO> and I>O, as taper of circle = spin of triangle. The cylinder can be expressed as IOI and IIO, as extrude of circle = spin of square. There are many more like this in 4D and beyond.

The 4D list is detailed here. To make a 4D shape, take any 3D shape and append a fourth operator. To write the 4D implicit equation, take the whole 3D equation and replace with M, in the respective 4th operator function.

As a generalization,

N-Cubes:

I - line, 1-cube
II - square, 2-cube
III - cube, 3-cube
IIII - tesseract, 4-cube
IIIII - penteract, 5-cube

N-Simplices:

I - line, 1-simplex
I> - triangle, 2-simplex
I>> - tetrahedron, 3-simplex
I>>> - pentachoron, 4-simplex
I>>>> - hexateron, 5-simplex

N-Spheres:

I - line, 0-sphere
IO - circle, 1-sphere
IOO - sphere, 2-sphere
IOOO - glome, 3-sphere
IOOOO - 4-sphere
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Philip



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Location: Orlando, FL

PostPosted: Sun Sep 20, 2015 10:35 pm    Post subject: Reply with quote

I wanted to share some animations I made recently.

I rendered a new 8D hypertorus with a more advanced explore function than I've ever used. It's a modified form of the rotate/translate function that I wrote to turn the tesseract so that it will pass through a 3-plane vertex-first.

Starting with any 3D cross section equation, we can replace x,y,z and two missing dimensions, labelled as zero, for a general 3D hyperplane:

x = (x*sin(b) + a*cos(b))

y = (y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))

z = (z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))

[xyzT] = ((z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))*sin(t))

[xyzt] = ((z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))*cos(t))

where,

a : slide along the 4th axis w

b : rotate sliding direction on plane 1

c : rotate sliding direction on plane 2

d : rotate sliding direction on plane 3

t : rotates the whole function to another distinct coordinate plane. Does not add a fourth plane of rotation to 'a' . It will make rotate morphs, but best if set to 0 or pi/2 , then using a,b,c,d .

I put this function into a hypertorus equation, and found some really amazing and eye-catching things happen. Especially when we get to 7D and over.

3D Cross-Section Equation, makes 32 torus intercepts in 4x2x4 brick array

(sqrt((sqrt((sqrt(x^2 + 0^2) -10)^2 + 0^2) -5)^2 + (sqrt(y^2 + 0^2) -5)^2) -2.5)^2 + (sqrt((sqrt(z^2 + 0^2) -5)^2 + 0^2) -2.5)^2 = 1



First Animation:



(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + ((z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))*sin(t))^2) -10)^2) -5)^2 + (sqrt((y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))^2) -5)^2) -2.5)^2 + (sqrt((sqrt((z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))^2) -5)^2 + ((z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))*cos(t))^2) -2.5)^2 = 1

a=5 , c=pi/4 , d=0 , t=0 , animate 0 < b < 2pi . Shows a momentary 2x1x4 square of 8 tori in two flip-flopped positions, with a very complex topology change






Second Animation:



(sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + ((z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))*sin(t))^2) -10)^2) -5)^2 + (sqrt((y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))^2) -5)^2) -2.5)^2 + (sqrt((sqrt((z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))^2 + ((z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))*cos(t))^2) -5)^2) -2.5)^2 = 1

analogous to above equation at a=5 , c=pi/4 , d=pi/2 , t=0 , animate 0 < b < 2pi . Shows a momentary 1x1x4x[R1 pair] column of 8 tori in two flip-flopped positions
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nextstep
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Joined: 06 Jan 2007
Posts: 538

PostPosted: Wed Sep 23, 2015 9:05 am    Post subject: Reply with quote

Hi,
very impressive....!
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Abderrahman
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