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 Post subject: Graphidrons: 2D, 3D, 4D, 5D, 6D symmetries.
PostPosted: Tue Jul 13, 2010 12:45 pm 
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Joined: Fri Nov 04, 2005 12:31 am
Location: Greece, Australia, Thailand, India, Singapore.
I present a video dedicated to the the new puzzles that I call Graphidrons:
http://www.youtube.com/watch?v=jJPj7eMuwrQ


This video features the following list of Graphidrons:

1. Cerberus (a 2D, spatial graph, based on a Great Triambic Icosahedron Face...!).
2. Heron (the simplest 3D structure of all - aka tetrahedron)
3. Hydra (aka Bipartite K3,3 Graph, aka Triangular Double Helix, aka Mobius 3).
4. Olympus (aka Perpendicular Double Helix, aka Mobius 4)
5. Gorgon (the well known Peterson Graph).
6. Chronos (aka K5 Complete Graph, aka 4-Simplex, aka 5-cell, aka Pentachoron, Hyperpyramid).
7. Ananke (same as Chronos, but the corners are colored, not the edges).
8. Pegasus - 5th dimensional! - (aka 5-simplex, aka Hexateron, aka Hexa-5-tope).
9. Argo - 6th dimensional! - (aka 6-simplex, aka Heptapeton, aka Hepta-6-tope).
10. Cosmos (aka Complete Bipartite Graph K4,4, aka Antipode Cube).
11. Hyperion (8-cell, aka Tesseract, aka Hypercube).
12. Pyramus (and actual demo based on the 4D Tetrahedral Prism!).
13. Houlis Cube (16-cell, aka Hexadecachoron)
14. Colossus (24-cell, aka Icositetrachoron).


In the video, there is a summary of most of those, while there is a nice demo
of Pyramus. It works similar to Hyperion, but Pyramus is NOT edge transitive,
a fact which makes it very interesting. I believe this demo makes clear many things.
(the demo takes place from 5:30 to 6:48).

For those not mentioned on a separate thread (e.g. Pegasus and Argo),
please expect a nice video of them in the near future, as they need (and deserve)
to be covered in a more explanatory way. Apologies for the low lights in the video.

Anyway, I hope you enjoy it!

:)


Pantazis

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Educational R&D, Gravity, 4D Symmetry, Puzzle Ninja, Matrix Mech, Alien Technology.


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 Post subject: Re: Graphidrons: 2D, 3D, 4D, 5D, 6D symmetries.
PostPosted: Tue Jul 13, 2010 12:52 pm 
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Joined: Wed Apr 08, 2009 7:46 pm
Location: PA, USA
mind blown.


do you have to keep these on different tables to keep the universe intact?

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It's fully printed in White Strong and Flexible (kind of like me! )


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 Post subject: Re: Graphidrons: 2D, 3D, 4D, 5D, 6D symmetries.
PostPosted: Wed Jul 14, 2010 9:41 am 
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Joined: Fri Jan 29, 2010 2:34 pm
Location: Scotland, UK
Oh dear, now I'm going to have to research some of these online when I have the time. You have some very cool looking puzzles there!

Matt


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 Post subject: Re: Graphidrons: 2D, 3D, 4D, 5D, 6D symmetries.
PostPosted: Wed Jul 14, 2010 2:26 pm 
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Joined: Sat Mar 22, 2003 9:11 am
Location: Marin, CA
I still maintain that the 16-cell is by far the best, although I'm a little curious what it's like to play with the petersen graph (and I'm not sure it makes sense to assign any particular number of dimensions to that one).

Have you considered the demipenteract (no I didn't know that one before, just looking up stuff on wikipedia)? It appears that the only way to get a decent size state space is to go up in number of dimensions, but too high and the puzzle gets too many vertices or too many edges and becomes non-manipulable.


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 Post subject: Re: Graphidrons: 2D, 3D, 4D, 5D, 6D symmetries.
PostPosted: Sun Jul 25, 2010 12:16 pm 
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Joined: Sat Mar 22, 2003 9:11 am
Location: Marin, CA
Going over this again, I'm pretty sure that the Houlis Cube is by far the best of the ones you've proposed thus far, except for maybe the Gorgon, depending how that moves. But there is one worth trying which you apparently haven't - the rectified 5-cell. That has a reasonable balance between simplicity and complexity.

I really wanted to find a more interesting 5-dimensional one, but everything 5-dimensional appears to have way too few or way too many vertices to be really interesting.


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 Post subject: Re: Graphidrons: 2D, 3D, 4D, 5D, 6D symmetries.
PostPosted: Sun Jul 25, 2010 5:40 pm 
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Joined: Mon Mar 30, 2009 5:13 pm
Solving these puzzles in 3D is like rotating a simple 4D object (1x1x1x1) about a plane in 4D space, or rotating a simple 5D object (1x1x1x1x1) in 5D space, etc. Would it would be possible to make a similar 3D projection of a *real* 4D twistypuzzle, i.e., one with at least two 4D shapes that rotate independently? For example, would it be possible to put 2 Chronos together to create a real 2x1x1x1, or maybe even put 16 Chronos together to create a 2x2x2x2?

Just a thought. :D

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 Post subject: Re: Graphidrons: 2D, 3D, 4D, 5D, 6D symmetries.
PostPosted: Sun Jul 25, 2010 10:15 pm 
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Joined: Fri Nov 04, 2005 12:31 am
Location: Greece, Australia, Thailand, India, Singapore.
Yes, it is very possible to have aplenty of new mechanically different designs
(i.e. which require a different solution). For example, the 5-simplex and 6-simplex
are *more* puzzles than Chronos, in the sense that they have more than one
"basic moves" and it took me much more time to become familiar with. Bram is right
when he says that suddenly there can be too many edges, but thank God, there are
also many options. The rectified 5-cell, as well as other, more crazy designs, such as
the 11-cell, are also part of the plan, but I need a lot of time. The Colossus was the
latest puzzle I did before my trip, and I was lucky to have it ready one week before
leaving, so that I could at least test it out!

Regarding creating new puzzles, the combination of this mechanism with twisty parts
is already part of the "hidden words", and it is very possible. But at this super
preliminary stage, making regular polytopes (not necessarily edge-transitive!),
is going to be a priority for some time. Mechanically speaking, we are already
doing some work to produce something safer, stronger and cheaper than the current
ones, which are of excellent quality, but too expensive to produce. Simply put, I can't go
for a car-antenna rampage every time I need new raw materials!!! :P

One thing I know, is that at the IPP I had received some unbelievable comments from
people who *understood* what higher dimensional symmetry is about. But sadly,
only a very small percentage is math inclined to realise and understand the novelty.
I tried to explain the movement to certain people, but they were treating it with too
much respect as they were scared to move it LOL. As for the Colossus (based on
the 24-cell), only Wei Hwa dared to take up the challenge. The rest only awed.
But I can understand it, as I cannot expect people exposed to a new design which is
based on years of work to understand it in their few available minutes. But I have
started some "awareness seminars", and everywhere I go, I am introducing them.

In any case, I have just returned to Australia, so I need to take photos and organise
250 puzzles, catch up with the lectures and tutorials + reseach at the Uni, finalise some
deals with a Toy Museum, work out the final deals with companies which will release the
gravity puzzles in October (sorry, no Aquadron or Krystalledron yet!), update the island
website, and follow up some serious issues with the IPP which is top priority.
(there are more tasks, but those are the ones I can think right now...!)

Please bear with me!!!

:)


Pantazis

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Educational R&D, Gravity, 4D Symmetry, Puzzle Ninja, Matrix Mech, Alien Technology.


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 Post subject: Re: Graphidrons: 2D, 3D, 4D, 5D, 6D symmetries.
PostPosted: Mon Jul 26, 2010 4:22 am 
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Joined: Sat Mar 22, 2003 9:11 am
Location: Marin, CA
My guess is that the 11-cell will prove to be basically non-manipulable, due to edges blocking each other.

The 4d shapes, as Kelvin points out, are much less interesting than the 5d and 6d ones. Unfortunately those are the ones which lack a reasonable middle ground. Maybe the prismatic thing consisting of two 5-cells connected to each other would work.

Also, the costs of production might be easier to keep under control if you put a limit on how much each edge could extend and contract. That might actually make the puzzle more interesting as a puzzle as well.


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 Post subject: Re: Graphidrons: 2D, 3D, 4D, 5D, 6D symmetries.
PostPosted: Mon Jul 26, 2010 7:40 am 
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Joined: Fri Nov 04, 2005 12:31 am
Location: Greece, Australia, Thailand, India, Singapore.
Bram wrote:
My guess is that the 11-cell will prove to be basically non-manipulable, due to edges blocking each other.
The 4d shapes, as Kelvin points out, are much less interesting than the 5d and 6d ones. Unfortunately those are the ones which lack a reasonable middle ground. Maybe the prismatic thing consisting of two 5-cells connected to each other would work.
Also, the costs of production might be easier to keep under control if you put a limit on how much each edge could extend and contract. That might actually make the puzzle more interesting as a puzzle as well.



From the 5D and 6D symmetry designs, I found that there are both manipulable and non-manipulable
state configurations. With an increasing amount of edges, the spacial (knot) placement of those edges
*does* matter for making the structure a "good puzzle challenge" or a "piece of art for decorative purposes".

And the same goes for the 11-cell. But that would require a lot more research than it required for
Pegasus (5D) and Argo (6D). I was not surprised by that, but what I do not know yet is how many
of those states exist for each design. For Chronos (5-cell) a crossing of an edge is what divides the
two "universes" (just like the Cubedron parity!). Of course, this fact does not prevent it from taking
the shape of all 3D projections. However, the case with higher dimensions (or with more complex structures
which have a valency greater than four) changes dramatically! And this part is still work under progress.

Prisms (e.g. Pyramus - the Tetrahedral Prism) have already proved that they are very attractive
(my brother - he is a technician - loved it and solved it without big trouble). But even inverting a
3D tetrahedron (the simplest of all challenges) can be a very cute (and easy) puzzle challenge!

I agree with all the contraction comments, and bandaging in this case could mean that some of them
elongate less, or not at all. Another type of bandaging is the introduction of flexible material on the faces
to prevent movement through them. And last but not least, how about having a non-regular structure? :wink:
(i.e. we are not necessarily restricted in using the same valency for all vertices).

In fact, the more someone thinks about it, the more ideas pop out!

:)


Pantazis

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Educational R&D, Gravity, 4D Symmetry, Puzzle Ninja, Matrix Mech, Alien Technology.


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