Slicing toratopes with hyperplanes

Discussion of shapes with curves and holes in various dimensions.

Slicing toratopes with hyperplanes

Postby Marek14 » Mon Jul 24, 2006 12:37 pm

Previously, I have derived how do the toratopes look when sliced with coordinate hyperplane. Today I tried to envision how they look when sliced with arbitrary hyperplane paralel to the coordinate ones.

For 3D torus, there are two ways to slice it - if we have lying it "flat" in xy plane, then in z direction we first encounter a circle which subsequently splits in two circles, which get fixed distance from each other before going back and merging (as all toratopes have mirror symmetry along coordinate hyperplanes, I will only include the first part of slicing from now on)
When we slice the torus from x or y direction, however, we encounter a single point that grows into a series of cassinoid curves (http://astronomy.swin.edu.au/~pbourke/surfaces/egg/ is a good illustration), passing from the connected ones via the lemniscate to the disconnected ones, settling in two circles for the midway cut.

If we take these cuts and rotate them in various ways, we obtain hyperplane cuts of various 4D toratopes.

(31): One of its midcuts are two concentric spheres - and cuts are the "circle -> two circles" cuts of torus rotated around an axis passing through the center. The other midcut is a torus; the cuts are obtained by rotating cassonoid ovals around the "vertical" line passing through the centre (the vertical line is, say, the one that doesn't intersect the separated ovals - the horizontal line intersects them) This shape grows from a point into a disc, then a dimples appear in the middle, finally puncturing the disc and transforming it into a torus.

((21)1): There are three midcuts.
One looking as two toruses based on the same circle (cocircular?). The cut sequence looks like a single torus separating in an "inner" and "outer" tube which drift apart.
The second one looks like two torii with identical inner diameters, but different outer ones. The cuts here look like cassinoid ovals rotated around distand vertical line. It starts as a circle, transforms in a flat torus, then, via a groove in the middle of the surface, separates in two torii.
The third one looks like two torii lying separated from each other. I'm not sure how the cuts look here. I suspect that simple rotation of cassinoid ovals won't get me far here.

(211): First kind of cut starts as circle that "blows up" to become a torus. The second kind are cassinoid ovals rotated around the horizontal axis. In the language of above-mentioned page, egg transforms to melon, then peanut, then splits to eventually end up as two separated spheres.

(22): Tiger has only one kind of cut, and it's another type of rotated cassinoid ovals: this time they are rotated along a distant horizontal line. They start as a circle, transforming in a tall torus, which splits in two torii, one atop another.
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Postby bo198214 » Mon Jul 24, 2006 6:10 pm

why not write a program?
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Postby Marek14 » Mon Jul 24, 2006 9:29 pm

bo198214 wrote:why not write a program?


Not sure where it would be best...
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Postby bo198214 » Mon Jul 24, 2006 10:15 pm

at the place where you live ;)
What do you mean by "where"?
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Postby Marek14 » Tue Jul 25, 2006 5:36 am

bo198214 wrote:at the place where you live ;)
What do you mean by "where"?


I meant what to write the program in.
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Postby bo198214 » Tue Jul 25, 2006 9:23 am

You are not the great decision maker, arent you?
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Postby Marek14 » Tue Jul 25, 2006 12:26 pm

bo198214 wrote:You are not the great decision maker, arent you?


No.
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Postby Marek14 » Fri Jul 28, 2006 11:38 am

Well, I finally found a way to slice the tetratorus in x or y direction. I uploaded a series of images from Mathematica to http://rapidshare.de/files/27316331/tetratorus.zip.html (No place to host them...)

I'd like to ask for help with 5D toruses. They can be visualised as a 2D array of 3D shapes, so I want to ask if there's a way to make sure Mathematica 3D output will have always the same absolute scale, and if there is a way to automatically save pictures.[/url]
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Postby bo198214 » Fri Jul 28, 2006 12:04 pm

Wow nice, but why not make an animated gif (then I dont need to use the slideshow option in my viewer ;) ).
Unfortunately I am a mapler, so I can not help you with this particular question.

What is anyway the general coordinate formula for an general n-dimensional torus? I looked at wikipedia and found that the n-dimensional torus is defined as the crossproduct of n circles, which is equivalent to identifiying the opposite faces of of an n-dimensional cube.
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Postby Keiji » Fri Jul 28, 2006 12:11 pm

Marek14 wrote:Well, I finally found a way to slice the tetratorus in x or y direction. I uploaded a series of images from Mathematica to http://rapidshare.de/files/27316331/tetratorus.zip.html (No place to host them...)


Shouldn't the tetratorus change from two torii to a torus perpendicualr to that? :|

Interesting pictures anyway. I can make better ones if you give me the cartesian equation :P
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Postby Marek14 » Fri Jul 28, 2006 6:33 pm

Rob wrote:
Marek14 wrote:Well, I finally found a way to slice the tetratorus in x or y direction. I uploaded a series of images from Mathematica to http://rapidshare.de/files/27316331/tetratorus.zip.html (No place to host them...)


Shouldn't the tetratorus change from two torii to a torus perpendicualr to that? :|


Nope. This is close to what I have envisioned.

Interesting pictures anyway. I can make better ones if you give me the cartesian equation :P


Well, I finally realized what the general equation is and how to derive it:

1. Take the torus symbol - ((21)1) in our case
2. Expand it to contain ONLY ones: (((11)1)1)
3. Replace each 1 with a variable: (((xy)z)w)
4. Make a square of each variable and add terms within parenthesis:
(((x^2+y^2)+z^2)+w^2)
5. Replace each parenthesis with a term containing square root and one of the radii of the torus:

sqrt[sqrt[sqrt[x^2+y^2]-A)^2+z^2]-B)^2+w^2]-C = 0

Of course, we don't really need the outermost square root:

sqrt[sqrt[x^2+y^2]-A)^2+z^2]-B)^2+w^2 = C^2

Then I just plotted in xzw hyperplane with various values set for y.
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Postby Keiji » Fri Jul 28, 2006 7:30 pm

Well, your equation was mal-formed, but I fixed it, and here are the images:

ImageImageImageImageImageImageImageImage

And now I can see that it is indeed correct, it shouldn't be a perpendicular torus because it would be 4th dimensionally perpendicular and that's what forms the blob you get on the last slice.
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Postby Marek14 » Fri Jul 28, 2006 8:40 pm

Rob wrote:Well, your equation was mal-formed, but I fixed it, and here are the images:

And now I can see that it is indeed correct, it shouldn't be a perpendicular torus because it would be 4th dimensionally perpendicular and that's what forms the blob you get on the last slice.


Great! :) Yes, now I realize that the parentheses didn't match - but the method seems to work for all toratopes including the beasts, so it's all good.

Could you render the singular slices, i.e. those where the overall topology changes via a sharp point? There should be three - between your pictures 2 and 3, 4 and 5, and 7 and 8. I don't remember the exact values for y for them, but they should all be integers. It looks like you used the same diameters as me (4,2 and 1) - the halving ensures that the toratope will never intersect itself.

Generally, one problem with rendering the slices is that the slices are often nested surfaces, like two concentric spheres - but from what I have pondered, it seems that for each toratope, we can find at least one way to slice it without nested surfaces.

I have now pretty much visualized some of the 3D cuts through 5D toratopes - in specific orientation, they require two variables.

Take the tiger#circle, ((22)1). This inocuous-looking shape has a 3D cross-section that looks like four torii lying in the vertices of rectangle. From this situation, if we go in one direction, two pairs of torii (let's say NE with NW and SE with SW) will merge a la the tetratorus pictures. In perpendicular direction, the other two pairs (NE with SE and NW with SW) merge. What if we go a bit in both directions? I have a bit of trouble visualizing this.

The equation for tiger#circle should be:

(Sqrt[(Sqrt[x^2+y^2]-A)^2+(Sqrt[z^2+w^2]-B)^2]-C)^2+v^2=D^2

with A>C+D,B>C+D,C>D, so good values seem to be A=B=4,C=2,D=1
The hyperplane in question to make the cut is xzv
Last edited by Marek14 on Fri Jul 28, 2006 9:12 pm, edited 1 time in total.
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Postby bo198214 » Fri Jul 28, 2006 9:00 pm

*enjoys this thread*
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Postby Marek14 » Fri Jul 28, 2006 9:11 pm

BTW, I have tried to explore the equation for ((22)1) a bit. The results seem to be very interesting, but Mathematica seems to not be the best choice. What did you use for those pictures, Rob?
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Postby Keiji » Fri Jul 28, 2006 9:57 pm

What did you use for those pictures, Rob?


That's a secret :D

Could you render the singular slices, i.e. those where the overall topology changes via a sharp point? There should be three - between your pictures 2 and 3, 4 and 5, and 7 and 8. I don't remember the exact values for y for them, but they should all be integers.


ImageImageImage

I think those slices are interesting. With my parameters (see below), the slices are at 0.7, 1.3 and 2.7 respectively.

It looks like you used the same diameters as me (4,2 and 1) - the halving ensures that the toratope will never intersect itself.


No, I used 2, 1, and 0.3.

Take the tiger#circle, ((22)1)


((22)1) is the circle#tiger. That said, I don't like the definition of spheration anyway.

And isn't it a bit too soon to be going for 5D objects? ;)

If you want, I can make image slices for that as well.
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Postby Marek14 » Sat Jul 29, 2006 7:29 am

Rob wrote:
What did you use for those pictures, Rob?


That's a secret :D

Could you render the singular slices, i.e. those where the overall topology changes via a sharp point? There should be three - between your pictures 2 and 3, 4 and 5, and 7 and 8. I don't remember the exact values for y for them, but they should all be integers.


I think those slices are interesting. With my parameters (see below), the slices are at 0.7, 1.3 and 2.7 respectively.

Yes - if we mark the radii A, B, and C, then the transitions are at B-C, B+C and A-C (I think that final shrink to the point should be A+C, 3.3 in your model)

It looks like you used the same diameters as me (4,2 and 1) - the halving ensures that the toratope will never intersect itself.


No, I used 2, 1, and 0.3.

Take the tiger#circle, ((22)1)


((22)1) is the circle#tiger. That said, I don't like the definition of spheration anyway.


Actually, calling this circle#tiger seems very counterintuitive to me. I mean, how does one build it? By taking a tiger, and replacing each point with a circle. You can't START with a circle.

And isn't it a bit too soon to be going for 5D objects? ;)


Well, slices of 4D toratopes can be produced in form of line of images or an animation. Slices of 5D can be made into an array - I think it's interesting because of all the non-cardinal directions you can move in there. If you want to do some more 4D slices, here are the possibilities:

1. Glome x^2+y^2+z^2+w^2=A^2 - sliced from any direction, midsection is a sphere
2. (31) (Sqrt(x^2+y^2+z^2)-A)^2+w^2=B^2 - sliced from x, y or z (slicing from w produces concentric spheres). Midsection is a torus.
3. ((21)1) (Sqrt((Sqrt(x^2+y^2)-A)^2+z^2)-B)^2+w^2=C^2 - sliced from z (we already did slice from x and y, and slicing from w produces nested torii). Midsection are two concentric torii.
4. (22) (Sqrt(x^2+y^2)-A)^2+(Sqrt(z^2+w^2)-B)^2=C^2 - sliced from any direction. Midsection are two torii atop each other.
5. (211) (Sqrt(x^2+y^2)-A)^2+z^2+w^2=B^2 - sliced once from x or y, and once from z or w. Midsections are two separate spheres in first case, and torus in the second.

If you want, I can make image slices for that as well.


Well, the 5D possibilities are numerous:

1. Petaglome - x^2+y^2+z^2+w^2+v^2=A^2
All of the 3D midsections are spheres.

2. (41) (Sqrt(x^2+y^2+z^2+w^2)-A)^2+v^2=B^2
xyz midsection are two concentric spheres, but xyv one is a torus.

3. ((31)1) (Sqrt((Sqrt(x^2+y^2+z^2)-A)^2+w^2)-B)^2+v^2=C^2
Possible cuts: xyz (four concentric spheres), xyw (two nested torii), xyv (two concentric torii) and xwv (two separated torii)

4. (((21)1)1) (Sqrt((Sqrt((Sqrt(x^2+y^2)-A)^2+z^2)-B)^2+w^2)^2-C+v^2=D^2
Possible cuts: xyz (four nested torii), xyw (two concentric pairs of nested torii), xyv (four concentric torii), xzw (two separated pairs of nested torii), xzv (two pairs of concentric torii), xwv (four torii in a line), and zwv (empty in the middle)

5. ((22)1) (Sqrt((Sqrt(x^2+y^2)-A)^2+(Sqrt(z^2+w^2)-B)^2)-C)^2+v^2=D^2
Possible cuts: xyz (two pairs of nested torii atop each other), xyv (empty in the middle) and xzv (four torii in the corners of rectangle)

6. ((211)1) (Sqrt((Sqrt(x^2+y^2)-A)^2+z^2+w^2)-B)^2+v^2=C^2
Possible cuts: xyz (two nested torii), xyv (two concentric torii), xzw (two separated pairs of concentric spheres), xzv (two separated torii), zwv (empty in the middle)

7. (32) (Sqrt(x^2+y^2+z^2)-A)^2+(Sqrt(w^2+v^2)-B)^2=C^2
Possible cuts: xyz (empty in the middle), xyw (two torii atop each other), xwv (two torii atop each other - different)

8. ((21)2) (Sqrt((Sqrt(x^2+y^2)-A)^2+z^2)-B)^2+(Sqrt(w^2+v^2)-C)^2=D^2
Possible cuts: xyz (empty in the middle), xyw (two pairs of concentric torii atop each other), xzw (two stacks of two torii), xwv (stack of four torii), zwv (empty in the middle, but different from xyz)

9. (311) (Sqrt(x^2+y^2+z^2)-A)^2+w^2+v^2=B^2
Possible cuts: xyz (two concentric spheres), xyw (torus), xwv (two separated spheres)

10. ((21)11) (Sqrt((Sqrt(x^2+y^2)-A)^2+z^2)-B)^2+w^2+v^2=C^2
Possible cuts: xyz (two nested torii), xyw (two concentric torii), xzw (two separated torii), xwv (four spheres in a line), zwv (empty in the middle)

11. (221) (Sqrt(x^2+y^2)-A)^2+(Sqrt(z^2+w^2)-B)^2+v^2=C^2
Possible cuts: xyz (stack of two torii), xyv (empty in the middle), xzv (four spheres in corners of rectangle

12. (2111) (Sqrt(x^2+y^2)-A)^2+z^2+w^2+v^2=B^2
Possible cuts: xyz (torus), xzw (two separated spheres), zwv (empty in the middle)

Funny, isn't it? :D
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Postby Keiji » Sat Jul 29, 2006 8:01 am

Marek14 wrote:
Rob wrote:I think those slices are interesting. With my parameters (see below), the slices are at 0.7, 1.3 and 2.7 respectively.

Yes - if we mark the radii A, B, and C, then the transitions are at B-C, B+C and A-C (I think that final shrink to the point should be A+C, 3.3 in your model)


You got the last two wrong, they go like this:

B-C, B+C, A+B-C, A+B+C

And yes, the last one is where it shrinks to a point.


It looks like you used the same diameters as me (4,2 and 1) - the halving ensures that the toratope will never intersect itself.


No, I used 2, 1, and 0.3.

Take the tiger#circle, ((22)1)


((22)1) is the circle#tiger. That said, I don't like the definition of spheration anyway.


Actually, calling this circle#tiger seems very counterintuitive to me. I mean, how does one build it? By taking a tiger, and replacing each point with a circle. You can't START with a circle.


(x1) is where you replace every point in a circle with x, so,
((22)1) is where you replace every point in a circle with a tiger.

If you want, I can make image slices for that as well.


Well, the 5D possibilities are numerous:

1. Petaglome - x^2+y^2+z^2+w^2+v^2=A^2
All of the 3D midsections are spheres.

2. (41) (Sqrt(x^2+y^2+z^2+w^2)-A)^2+v^2=B^2
xyz midsection are two concentric spheres, but xyv one is a torus.

3. ((31)1) (Sqrt((Sqrt(x^2+y^2+z^2)-A)^2+w^2)-B)^2+v^2=C^2
Possible cuts: xyz (four concentric spheres), xyw (two nested torii), xyv (two concentric torii) and xwv (two separated torii)

4. (((21)1)1) (Sqrt((Sqrt((Sqrt(x^2+y^2)-A)^2+z^2)-B)^2+w^2)^2-C+v^2=D^2
Possible cuts: xyz (four nested torii), xyw (two concentric pairs of nested torii), xyv (four concentric torii), xzw (two separated pairs of nested torii), xzv (two pairs of concentric torii), xwv (four torii in a line), and zwv (empty in the middle)

5. ((22)1) (Sqrt((Sqrt(x^2+y^2)-A)^2+(Sqrt(z^2+w^2)-B)^2)-C)^2+v^2=D^2
Possible cuts: xyz (two pairs of nested torii atop each other), xyv (empty in the middle) and xzv (four torii in the corners of rectangle)

6. ((211)1) (Sqrt((Sqrt(x^2+y^2)-A)^2+z^2+w^2)-B)^2+v^2=C^2
Possible cuts: xyz (two nested torii), xyv (two concentric torii), xzw (two separated pairs of concentric spheres), xzv (two separated torii), zwv (empty in the middle)

7. (32) (Sqrt(x^2+y^2+z^2)-A)^2+(Sqrt(w^2+v^2)-B)^2=C^2
Possible cuts: xyz (empty in the middle), xyw (two torii atop each other), xwv (two torii atop each other - different)

8. ((21)2) (Sqrt((Sqrt(x^2+y^2)-A)^2+z^2)-B)^2+(Sqrt(w^2+v^2)-C)^2=D^2
Possible cuts: xyz (empty in the middle), xyw (two pairs of concentric torii atop each other), xzw (two stacks of two torii), xwv (stack of four torii), zwv (empty in the middle, but different from xyz)

9. (311) (Sqrt(x^2+y^2+z^2)-A)^2+w^2+v^2=B^2
Possible cuts: xyz (two concentric spheres), xyw (torus), xwv (two separated spheres)

10. ((21)11) (Sqrt((Sqrt(x^2+y^2)-A)^2+z^2)-B)^2+w^2+v^2=C^2
Possible cuts: xyz (two nested torii), xyw (two concentric torii), xzw (two separated torii), xwv (four spheres in a line), zwv (empty in the middle)

11. (221) (Sqrt(x^2+y^2)-A)^2+(Sqrt(z^2+w^2)-B)^2+v^2=C^2
Possible cuts: xyz (stack of two torii), xyv (empty in the middle), xzv (four spheres in corners of rectangle

12. (2111) (Sqrt(x^2+y^2)-A)^2+z^2+w^2+v^2=B^2
Possible cuts: xyz (torus), xzw (two separated spheres), zwv (empty in the middle)

Funny, isn't it? :D


12 torii? O_o

This makes me wonder what the sequence is for the number of possible torii... lol.
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Postby PWrong » Sat Jul 29, 2006 9:44 am

Well, I finally realized what the general equation is and how to derive it:

I already posted this here.

((22)1) is the circle#tiger. That said, I don't like the definition of spheration anyway.

A#B isn't valid unless B is a sphere in some dimension. The reason is you need unique axes to align the tiger, but not a sphere.

This makes me wonder what the sequence is for the number of possible torii... lol.

We already found this. 1, 1, 2, 5, 12, 33, 90, 261, 766, 2312, ...
http://tetraspace.alkaline.org/forum/viewtopic.php?t=403&start=30
This mathworld page has the final formula for the number of rotopes and the number of torii.
I also wrote a mathematica notebook to list all the rotopes in dimension n.
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Postby Keiji » Sat Jul 29, 2006 9:53 am

The sequence goes 0,0,1,5,12... if you start at 1D... :|
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Postby PWrong » Sat Jul 29, 2006 10:11 am

Sorry, when I say "torii" I mean all rotopes with brackets around them. So sphere counts, but torinder doesn't.

1D:
1 = line

2D:
2 = circle

3D:
3 = sphere
(21) = torus

4D:
4
(31)
(22)
(211)
((21)1)
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Postby Keiji » Sat Jul 29, 2006 10:34 am

Well, that's a bit silly, how is a line curved, let alone a torus? :roll:
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Postby PWrong » Sat Jul 29, 2006 10:40 am

Well, ok. Let B<sub>n</sub> be the number of bracketed objects.
Then B<sub>n</sub> - 1 is the number of torii. Is that better?
If you want to include objects like torinder, which have brackets in them but are not surrounded by brackets, then we have:
2B<sub>n</sub> - p<sub>n</sub>, where p<sub>n</sub> is the partition function (i.e. the number of rotatopes in nD)

btw, those pictures are great, but they're making the screen really long. Couldn't you split them into two pictures?
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Postby Marek14 » Sat Jul 29, 2006 12:03 pm

PWrong wrote:
Well, I finally realized what the general equation is and how to derive it:

I already posted this here.


I know you posted this. That's why I say that I "realized" it :)


((22)1) is the circle#tiger. That said, I don't like the definition of spheration anyway.

A#B isn't valid unless B is a sphere in some dimension. The reason is you need unique axes to align the tiger, but not a sphere.


Well, it's the shape you get when you replace each point of tiger with a circle - not the other way around. And this operation is not commutative - replacing each point of circle with sphere is something different from replacing each point of sphere with a circle. And, as mentioned, you really cannot replace each point with a tiger since tiger has unique axes. I suspect there's some confusion in definitions here.

This makes me wonder what the sequence is for the number of possible torii... lol.

We already found this. 1, 1, 2, 5, 12, 33, 90, 261, 766, 2312, ...
http://tetraspace.alkaline.org/forum/viewtopic.php?t=403&start=30
This mathworld page has the final formula for the number of rotopes and the number of torii.
I also wrote a mathematica notebook to list all the rotopes in dimension n.
Last edited by Marek14 on Sat Jul 29, 2006 12:52 pm, edited 2 times in total.
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Postby Marek14 » Sat Jul 29, 2006 12:08 pm

Rob wrote:Well, that's a bit silly, how is a line curved, let alone a torus? :roll:


LINE itself is not - but it's border, i.e. two points, are :) The same way as disk is not curved, but a circle is.

No, seriously, I think that this has to do with mid-cuts. It's easiest to say that a toratope midcut is always a set of lower-dimensional toratopes. The mid-cut of torus are two circles, midcut of circle are two points. Simple.

Another point in case is that if we include spheres as toratopes (and we should, as they are the basic building blocks, and occur all the time as midcuts etc), then a line is just 1D sphere.

Finally - is there anything in the definition of toratope that says it must be curved?
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Postby bo198214 » Sat Jul 29, 2006 12:21 pm

Hey Marek and Rob,

give you a jerk! Wouldnt it be a great idea to have a program where you can choose out of a list of 4d toratopes (and where you can put in a shape in CSG notation), and it shows you a 3d slice moving through the shape?
That would be novel and a really appreciable distribution to the 4d community.

PS: Programming language java, then you can put it online everywhere.
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Postby Marek14 » Sat Jul 29, 2006 12:40 pm

bo198214 wrote:Hey Marek and Rob,

give you a jerk! Wouldnt it be a great idea to have a program where you can choose out of a list of 4d toratopes (and where you can put in a shape in CSG notation), and it shows you a 3d slice moving through the shape?
That would be novel and a really appreciable distribution to the 4d community.

PS: Programming language java, then you can put it online everywhere.


Unfortunately, I'm not very good programmer :) Especially not for graphics.
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Postby bo198214 » Sat Jul 29, 2006 1:03 pm

See the global picture, is it worth to write such a program? It is!
Whether you are a good programmer is not that important then anymore.
You can anyway learn a lot by this and *become* a good programmer, especially regarding graphics ;)
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Postby PWrong » Sat Jul 29, 2006 2:01 pm

See the global picture, is it worth to write such a program? It is!

Why don't you do it then? Your building blocks game was pretty good. I'd do it myself, but I can't program in anything except mathematica.
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Postby thigle » Sat Jul 29, 2006 2:02 pm

yep guys common ! bo's right ! do it and become tetraStars!

i don't like shit like rob says "that's a secret what do i do this output in". godam wakeup child ! this is internet - we share ! information restriction is stupid, especially when it ain't worth shit. just what deficit proble does it make to you to answer someone's question about the tools you use ?

if i was your father i would slap you gently and tell you to behave dud ;)
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