yeah that’s a cute one. :D When i said above it takes a good 400 pages or so just to reach a point where one can define what one means by “one” i was referring to Alfred North Whitehead and Bertrand Russell ‘s Principia Mathematica, btw. have ya saw Metamath yet? I used to be a formalist myself in terms of philosophy of math, but that seemed to problematic to me eventually… I want math to mean something! So I’ve been a neo platonic idealist (non-dual actually) since maybe 20 years now. puts me in the same camp with Gödel and i can live with that. lmao,. formalism was a way of avoiding paradox btw but these days I embrace paradox and see it as a necessary consequence of human thought. Of course I try to keep it outta my math as well as my code. lmao

Heh, gotta love formal proofs. Check out this proof that 1+1=2.

I have tons of respect for people with graduate degrees in math. They are way crazier than me. :P Crazier in a good way that is.

I think I abused the hell outta the word boundary. Bear in mind I’m not a topologist and I’m tired. haha i read a few real math books on it about 10 or 15 yrs ago…its been that long. Anyway a Möbius strip is counter intuitive to alot of people and that was the point I was trying to make.

Yeah Luke is right vacri. It is hard to be precise and use english lol. Instead of “one sided three dimensional object” I should have said “one sided three dimensional surface” that is essentially what i meant. And further surface here means boundary. But still this lacks true precision, to be totally clear mathematicians resort to Formal languages (in theory that is). In reality human beings don’t care much for complete formality aside from cases where one must be formal such as programming a computer. the buggy nature of most programs tho should show ya the problems humans have with formality. to be totally formal in mathematics tho reaches absurdity, it takes a good 400 pages or so just to reach a point where one can define what one means by “one”.

So with all this in mind a sphere even a solid sphere has 2 boundaries an inside surface and an outside surface. But a Möbius strip as an idealized surface has only one boundary. google it it has all kinds of really cool properties, good to amuse friends family and kids :D

but anyway i was trying to be brief … I know ya can’t tell it. lmao but I was thinking Luke could grok what I was trying to say.

vacri – a hollow sphere is two sided. You have the outer surface, and the inner surface. A solid sphere on the other hand, is one sided – so yeah. But the Möbius strip is way cooler than a plain old sphere.

Starhawk – why would they not consider a sphere a one-sided three-dimensional object?

Yeah i knew ya was only using the Rubik’s cube to introduce the idea and it is a fascinating idea. I love permutation puzzles btw and finite groups …. a personality flaw if ya will ;) With respect to loyds puzzle not only is it 2 dimension and the operation is rather simple it forms a relatively simple permutation group. A rubics cube on the other hand is not only 3-d and the basic operation is far more complex as ya noted and the set of patterns forms a wreath product of two groups and hence the mathematical structure of the permutation group is more complex.

In practical terms this means not only do the pieces making it up have a position on the cube where they “belong” they also have an orientation. People tend to look only at the stickers and not the pieces (ie the corners and the edges). This also includes the center pieces but their orientation is irrelevant on a standard cube as one can’t tell they have been rotated. this isn’t true tho if the cube has pictures on the faces instead of solid colors. lol. Solving a rubics cube btw isn’t even the point of them to me, the point is the beauty of the mathematical theory.

And generalizing the puzzle to nxnxn as well as to m dimensions only increases the beauty. the subject even has a few fundamental unanswered questions floating around as well as several rather fascinating algorithms. i think it is a shame society hasn’t embraced it and similar puzzles as educational devices to teach group theory at an elementary level in HS as an understanding of real algebra would greatly enhance a students understanding of not only mathematics and geometry but also of modern physics.

Starhawk – yeah, I think using something else than hypercube would be a good idea. I just went with the cube, because of the Rubik’s Cube comparison.

Also, I think that Loyd’s Puzzle is easier to solve for many people, because it is a 2D configuration. Each move displaces only a single element. Rubik’s Cube adds a new dimension to the puzzle making it harder to grok.

Travis – I’m amazed that Akismet didn’t snag your post. But I think it still got held in moderation queue because of flood like appearance. :P

So this is a REALLY long article that is REALLY cool, and then you have a REALLY long reply to it…. so i can feel like a badass I am going to post a REALLY long reply as well… but I am just going to copy and paste this whole thing a few times. :D
The whole theory behind it is way creative and I could only immadgen you were thinking about this in the shower or on the toilet because I could not see any other place that you could be soo smart to think of something like this.
:D
So this is a REALLY long article that is REALLY cool, and then you have a REALLY long reply to it…. so i can feel like a badass I am going to post a REALLY long reply as well… but I am just going to copy and paste this whole thing a few times. :D
The whole theory behind it is way creative and I could only immadgen you were thinking about this in the shower or on the toilet because I could not see any other place that you could be soo smart to think of something like this.
:D So this is a REALLY long article that is REALLY cool, and then you have a REALLY long reply to it…. so i can feel like a badass I am going to post a REALLY long reply as well… but I am just going to copy and paste this whole thing a few times. :D
The whole theory behind it is way creative and I could only immadgen you were thinking about this in the shower or on the toilet because I could not see any other place that you could be soo smart to think of something like this.
:D
So this is a REALLY long article that is REALLY cool, and then you have a REALLY long reply to it…. so i can feel like a badass I am going to post a REALLY long reply as well… but I am just going to copy and paste this whole thing a few times. :D
The whole theory behind it is way creative and I could only immadgen you were thinking about this in the shower or on the toilet because I could not see any other place that you could be soo smart to think of something like this.
:D
So this is a REALLY long article that is REALLY cool, and then you have a REALLY long reply to it…. so i can feel like a badass I am going to post a REALLY long reply as well… but I am just going to copy and paste this whole thing a few times. :D
The whole theory behind it is way creative and I could only immadgen you were thinking about this in the shower or on the toilet because I could not see any other place that you could be soo smart to think of something like this.
:D
So this is a REALLY long article that is REALLY cool, and then you have a REALLY long reply to it…. so i can feel like a badass I am going to post a REALLY long reply as well… but I am just going to copy and paste this whole thing a few times. :D
The whole theory behind it is way creative and I could only immadgen you were thinking about this in the shower or on the toilet because I could not see any other place that you could be soo smart to think of something like this.
:D
So this is a REALLY long article that is REALLY cool, and then you have a REALLY long reply to it…. so i can feel like a badass I am going to post a REALLY long reply as well… but I am just going to copy and paste this whole thing a few times. :D
The whole theory behind it is way creative and I could only immadgen you were thinking about this in the shower or on the toilet because I could not see any other place that you could be soo smart to think of something like this.
:D So this is a REALLY long article that is REALLY cool, and then you have a REALLY long reply to it…. so i can feel like a badass I am going to post a REALLY long reply as well… but I am just going to copy and paste this whole thing a few times. :D
The whole theory behind it is way creative and I could only immadgen you were thinking about this in the shower or on the toilet because I could not see any other place that you could be soo smart to think of something like this.
:D

Damn Luke ya know I love that one. lmao.

4-d geometry has always intrigued me esp the claim people can not visualize it. Honestly 3-d geometry can be quite hard to visualize most people would deny the existence of a one sided three dimensional object until ya show them a Möbius strip. And even then some still don’t get it. lol. Even two dimensions has plenty of surprises just how long is the coast of Britain anyway? Mandelbrot says infinite, lmao. My point is 2-d fractals can be unintuitive at first glance and older mathematicians referred to them as pathological monsters.

But playing with computer simulations one can of course develop some intuitive sense of 4-dimensional geometry. A four dimensional cube is really simple to understand, I like to say its no harder than counting to 16. Indeed a binary count to 2 to the nth power is a simple way to generate the coordinates of the vertex points of a n dimensional cube. Of course i like the origin of my coordinate system to be in the center of the cube so a small translation at the end fixes that one. haha. regardless of that tho 4-d geometry is full of surprises and objects more complex than hypercubes can be quite hard to visualize. I’m pretty sure Euclid would have thought we are nuts for even talking about it :D (Altho I read somewhere once Pappus mentioned some Greek mathematicians who had pondered higher dimensional geometry but Pappus considered it a heresy, I forgot where i read it and can’t even say for sure whether it is true. I regret tho the loss of alot of ancient Greek knowledge, I do think tho they knew a hell of alot more than people give them credit for. haha)

But for your puzzle problem we may as well make it fun and make the puzzle a 4 d Regular Polytope that has no 3-d analogue (unlike a hyper cube) and hell while we are at it why not make it a non euclidean object. Of course the hardest part would be realizing it is a projection of a 4-d object and deducing both the type of projection as well as the object itself. After accomplishing that solving it wouldn’t be as hard as one might think just recast the problem as algebraic and forgot all that geometric shit that is close to impossible to visualize. The problem is algebraic anyway, just ask anyone that has tried to solve a Rubik’s cube thru geometric intuition alone. They are seldom if ever successful, It has often amused me how many people can solve Loyd’s 15 puzzle (in a solvable form of course) by i suppose geometric intuitive reasoning yet can not even begin to solve Rubik’s original puzzle. But at an abstract level it is basically the same sort of solution and really trivial at that (see Commutator and Conjugate)

I’m not sure how one could incorporate such an idea into a game tho. Hmmm, i think it is probably “to ambitious for your own good“, but it would be way cool tho :D