RNS and product notations

Discussion of shapes with curves and holes in various dimensions.

Postby moonlord » Tue Aug 08, 2006 12:54 pm

Oh no, you got it wrong :D.

The sheep effect refers to the situation when one in a group does something, and the others follow without thinking. It is also studied in group dynamics, as bo198214 pointed out. Group dynamics study in general how a large group of people/cars/entities will behave in a certain given situation.

Here, me and bo198214 refer to the fact that someone said they're Marek's formulas, then everybody said they're Marek's formulas, and everybody ignored the fact that bo198214 derived them too.

All clear now? :)
"God does not play dice." -- Albert Einstein, early 1900's.
"Not only does God play dice, but... he sometimes throws them where we cannot see them." -- Stephen Hawking, late 1900's.
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Postby PWrong » Tue Aug 08, 2006 1:01 pm

Ok. Well the important thing is, the problem's solved.
Tomorrow I'll go through the 5D and 6D shapes just to make sure.
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Postby Marek14 » Tue Aug 08, 2006 2:24 pm

PWrong wrote:
EDIT: Actually, you use my rules to get the proper form - so the rules were right, I just made a mistake in their application.

True, but I had to assume that the rules apply to objects with more than two nested brackets.


Not really... this was the reason for my Rule 1: Parentheses evaluate from inside out. I probably said it badly, it was supposed to mean that if there are nested parentheses, you evaluate the inner ones first, and then use the result while evaluating the outer ones.
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Postby Keiji » Tue Aug 08, 2006 4:48 pm

How are you supposed to evaluate them from the inside out?

There are no rules regarding what to do with parenthesized cartesian or torus products. So the only way to evaluate nested brackets is to treat the inner brackets as one term, evaluate the outer brackets, then evaluate the inner ones.
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Postby bo198214 » Tue Aug 08, 2006 5:26 pm

Yes but its clear what its meant:

Simply apply the rule (A<sub>1</sub>...A<sub>n</sub>1...1) -> (A<sub>1</sub>x...xA<sub>n</sub> ) # S<sub>n+d-1</sub>
until no more possible and you have the product notation. It does not really matter in what order.
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Postby PWrong » Wed Aug 09, 2006 12:25 pm

Quote:
2. Parentheses of form a1111... with b 1's evaluate to a#(b+1)

I agree with this one.
Example: (211) = 2#4, (3111) = 3#5

I just realised there's two problems here.
First, my example is wrong. (211) = 2#3 by your rule, and (3111) = 3#4. Second, in my list I have (211) = 3#2, not 2#3. However, in the wiki we have (211) = circle # sphere, which seems more correct. So I'll change the list.

Here's the 4D, 5D and 6D rotopes.

4D: 1 + 4 + 4 + 1 = 10
1111 = 1x1x1x1

211 = 2x1x1
22 = 2x2
31 = 3x1
4 = 4

(21)1 = (2#2)x1
(211) = 2#3
(22) = (2x2)#2
(31) = 3#2

((21)1) = 2#2#2

5D: 1 + 6 + 10 + 6 + 1 = 24

11111 = 1x1x1x1x1

2111 =2x1x1x1
221 = 2x2x1
311 = 3x1x1
32 = 3x2
41 = 4x1
5 = 5

(21)11 = (2#2) x1x1
(211)1 = (2#3) x1
(2111) = 2#4
(22)1 = ((2x2)#2) x1
(21)2 = (2#2)x2
(221) = (2x2)#3
(31)1 = 3#2 x1
(311) = 3#3
(32) = (3x2)#2
(41) = 4#2

((21)1)1 = 2#2#2 x1
((21)11) = (2#2)#3
((211)1) = (2#3)#2
((22)1) = ((2x2)#2)#2
((21)2) = ((2#2)x2)#2
((31)1) = ((3#2)#2)

(((21)1)1) = 2#2#2#2


6D: 1 + 10 + 23 + 23 + 8 + 1 = 66
111111

21111 , 2211 , 222 , 3111 , 321 , 33 , 411 , 42 , 51 , 6

(21)111 = 2#2 x1x1x1
(211)11 = 2#3 x1x1
(2111)1 = 2#4 x1
(21111) = 2#5
(22)11 = (2x2)#2 x1x1
(21)21 = 2#2 x2 x1
(21)(21) = (2#2)x(2#2)
(211)2 = (2#3)x2
(221)1 = (2x2)#3 x 1
(2211) = (2x2)#4
(22)2 = (2x2)#2 x 2
(222) = (2x2x2)#3
(31)11 = (3#2) x1x1
(311)1 = (3#3) x1
(3111) = 3#4
(21)3 = (2#2) x3
(31)2 = (3#2) x2
(32)1 = (3x2)#2 x1
(33) = (3x3)#2
(41)1 = 4#2 x 1
(411) = 4#3
(42) = (4x2)#2
(51) = 5#2

((21)1)11 = 2#2#2 x1x1
((21)11)1 = 2#2#3 x1
((21)111) = 2#2#4
((211)1)1 = 2#3#2 x1
((211)11) = 2#3#3
((2111)1) = 2#4#2
((22)1)1 = (2x2)#2 x1
((22)11) = (2x2)#3
((21)2)1 = ((2#2)x2)#2x1
((21)1)2 = 2#2#2 x 2
((21)21) = ((2#2)x2)#3
((21)(21)) = ((2#2)x(2#2))#2
((211)2) = ((2#3)x2)#2
((221)1) = ((2x2)#3)#2
((22)2) = (((2x2)#2) x2) #2
((31)1)1 = 3#2#2 x1
((31)11) = 3#2#3
((311)1) = 3#3#2
((21)3) = ((2#2) x 3)#2
((31)2) = ((3#2) x 2)#2
((32)1) = (3x2)#2#2
((41)1) = 4#2#2

(((21)1)1)1 = 2#2#2#2x1
(((21)1)11) = 2#2#2#3
(((21)11)1) = 2#2#3#2
(((211)1)1) = 2#3#2#2
(((22)1)1) = (2x2)#2#2#2
(((21)2)1) = ((2#2)x2)#2#2
(((21)1)2) = ((2#2#2)x2)#2
(((31)1)1) = 3#2#2#2

((((21)1)1)1) = 2#2#2#2#2
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Postby PWrong » Fri Aug 11, 2006 9:22 am

I'm wondering if I should change the product notation list on the wiki from words to numbers, so instead of "sphere # circle" we have 3#2. I'll put both on and maybe delete the words later.

Also, since objects like the triangular torus can't be uniquely defined with the torus product, I might delete the product notation for those objects. I'll discuss this before I do it though.
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Postby Keiji » Fri Aug 11, 2006 11:08 am

PWrong wrote:Also, since objects like the triangular torus can't be uniquely defined with the torus product, I might delete the product notation for those objects.


Why should they be deleted?

They are perfectly valid objects; it's just that they are ambiguous when expressed in product notation. But then again, product notation doesn't exactly specify parameters for radii of torii... circle # circle could easily mean the self-intersecting type of 3D torus, rather than the "normal" one. ;)

I think we should remove the name-based product notations, and use only the number-based ones, and we should expand out anything that's defined in terms of another rotope (bar the line). Thus, 111 -> 0 becomes (1x1x1) -> 0.
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Postby PWrong » Fri Aug 11, 2006 11:44 am

Why should they be deleted?

I'm not suggesting that we delete the objects, just replace the product notation for them with "none". It's misleading to say that triangular torus is circle#triangle, because it's not.

I think we should remove the name-based product notations, and use only the number-based ones, and we should expand out anything that's defined in terms of another rotope (bar the line). Thus, 111 -> 0 becomes (1x1x1) -> 0.

Done
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Postby Keiji » Fri Aug 11, 2006 11:54 am

One more thing, should we replace the -> with ~? Most operations are only one character and -> does look a bit odd.
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Postby PWrong » Fri Aug 11, 2006 12:36 pm

Good idea
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Postby bo198214 » Fri Aug 11, 2006 12:49 pm

The > is at least mnemonic for tapering.
Its also no commutative operation, so a non-symmetric symbol would be more appropriate, but is no must.
3rd seems not possible to stay with one character symbol for all the here discussed operations.
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Postby wendy » Thu Aug 24, 2006 8:33 am

The pyramid product is indeed associative and communitive.

One can regard, for example, a "taper" of three objects, as a triangle, to which the three objects are orthogonal to, and to each other. The triangle is marked out in % heights (ie perpendicular to wall / maximum perpendicular to wall). These add to one.

For figures X, Y, Z, one has at x,y,z a prism-product xX*yY*zZ, where x+y+z = 1, and xX is X scaled down by a factor of x.

I use <> to represent the tegum-brackets. I suppose you could use {} for the pyramid-product...
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