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 Post subject: Chronos: Plato's First Shadow!!! (based on the Pentachoron)
PostPosted: Wed Jul 07, 2010 3:26 am 
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Some time ago I mentioned "PAJUPUMA". It stands for "PAntazis JUly PUzzle MAdness". So I am continuing it! LOL

This time I am focusing on the 5-cell, aka 4-simplex, aka Pentachoron: The Chronos Puzzle (prototype)

Attachment:
4D_Chronos.jpg
4D_Chronos.jpg [ 70.03 KiB | Viewed 3236 times ]


This is based on the 4D structure known as Pentachoron (i.e. five tetrahedron spaces). As amazing as it sounds,
*any* of the ten edges can move to the middle of the triangular dipyramid projection. It is a testament to the
four dimensional symmetry that this puzzle has. The first time I played with it, I could not believe my own eyes!!! :-)

Beware though, this is my very first prototype, so it has many imperfections (e.g. some connectors slide LOL)
Video here:

http://www.youtube.com/watch?v=RU6pcmulIYk


This video seems to make it easier to understand the moves based on 4D symmetry, as the design is
much more simpler than the Houlis Cube (which is based on the 16-cell). But watch the video more
carefully... is it *that* easy to move any edge to the middle? Perhaps not. It is a lovely puzzle! ;)

On some other note, this puzzle is dedicated to our lost friend Frank and his fighting spirit. In my very last
correspondence with him, he was delighted for me to add this dedication. If only he was still around to see it alive.
But who knows, he may be watching us, perhaps from the 4th dimension!

:)


Pantazis

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 Post subject: Re: Chronos: Plato's First Shadow!!! (based on the Pentachoron)
PostPosted: Wed Jul 07, 2010 3:43 am 
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I didn't understand ANYTHING about how your Houlis puzzle turned in the video, but this is just fantastic!! I love the turns!! (Well the overlaps sorry :wink: )

Really a pure puzzle :shock:

:D Greg :D

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 Post subject: Re: Chronos: Plato's First Shadow!!! (based on the Pentachoron)
PostPosted: Wed Jul 07, 2010 10:20 am 
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My brain got a little confused, I'll be reading more of that book Kelvin suggested.

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 Post subject: Re: Chronos: Plato's First Shadow!!! (based on the Pentachoron)
PostPosted: Wed Jul 07, 2010 10:54 am 
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This is a great puzzle to show your 4d concept.

How long till I can buy one :lol:

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 Post subject: Re: Chronos: Plato's First Shadow!!! (based on the Pentachoron)
PostPosted: Wed Jul 07, 2010 11:11 am 
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Now that I understand the movement, how do you know when this one is solved?

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 Post subject: Re: Chronos: Plato's First Shadow!!! (based on the Pentachoron)
PostPosted: Wed Jul 07, 2010 11:34 am 
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I envisioned the default position of the pentachoron puzzle to be with 4 vertices on the outside forming a tetrahedron, with the fifth in the middle (vertex-first projection?). You seem to have an edge in the center which gives a triangular dipyramid (edge-first projection?). Is this arbitrary, or is there a practical reason for the choice?

Oh, and another cool puzzle btw :D I think I need to look over some things about 3D projections of the 4D versions of the platonic solids, been a while since I learned about them. Which of the 6 (5,8,16,24,120,600-cell) are you planning to make? I assume the smaller ones are easier to make! :lol: Keep these amazing puzzles coming! :D

Matt


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 Post subject: Re: Chronos: Plato's First Shadow!!! (based on the Pentachoron)
PostPosted: Wed Jul 07, 2010 11:46 am 
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bobthegiraffemonkey wrote:
I envisioned the default position of the pentachoron puzzle to be with 4 vertices on the outside forming a tetrahedron, with the fifth in the middle (vertex-first projection?). You seem to have an edge in the center which gives a triangular dipyramid (edge-first projection?). Is this arbitrary, or is there a practical reason for the choice?


Excellent question! (which is the ones I love to answer!) :)

Being an object which as time tends to zero, 3D tends to 4D, it always has "inside" and "outside" edges.
By minimizing the external edges, we always get the triangular dipyramid. A similar situation applies
to the other designs, but not always!
(A good counterexample is the hypercube, which needs to have the outside cube with maximised edges)

In other words, I could choose for the pentachoron to have another color configuration *and* starting shape,
and baptise it as another puzzle! (I just love the 4D symmetry of regular polychorons!)


bobthegiraffemonkey wrote:
Oh, and another cool puzzle btw :D I think I need to look over some things about 3D projections of the 4D versions of the platonic solids, been a while since I learned about them. Which of the 6 (5,8,16,24,120,600-cell) are you planning to make? I assume the smaller ones are easier to make! :lol: Keep these amazing puzzles coming! :D

Matt


I have made many different ones, and I will be revealing one by one each day.
Some will be simpler than the ones I am showing now. But the last one, is the Colossus,
the puzzle which is the most symmetric object that had ever existed, and that *will* ever exist!
(Yes, this sounds arrogant, but once explained, you will see what I mean!!! Hint: Triality + regular polychoron).

;)


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 Post subject: Re: Chronos: Plato's First Shadow!!! (based on the Pentachoron)
PostPosted: Wed Jul 07, 2010 2:31 pm 
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kastellorizo wrote:
bobthegiraffemonkey wrote:
I envisioned the default position of the pentachoron puzzle to be with 4 vertices on the outside forming a tetrahedron, with the fifth in the middle (vertex-first projection?). You seem to have an edge in the center which gives a triangular dipyramid (edge-first projection?). Is this arbitrary, or is there a practical reason for the choice?


Excellent question! (which is the ones I love to answer!) :)

Being an object which as time tends to zero, 3D tends to 4D, it always has "inside" and "outside" edges.
By minimizing the external edges, we always get the triangular dipyramid. A similar situation applies
to the other designs, but not always!
(A good counterexample is the hypercube, which needs to have the outside cube with maximised edges)

In other words, I could choose for the pentachoron to have another color configuration *and* starting shape,
and baptise it as another puzzle! (I just love the 4D symmetry of regular polychorons!)


So you made the choice to minimize the external edges on these puzzles when possible? In other words, it means that for the pentachoron with minimised external edges the result is the dipyramid, but for the hypercube it isn't possible to minimise external edges so you have take a different approach? Just to clarify.


kastellorizo wrote:
bobthegiraffemonkey wrote:
Oh, and another cool puzzle btw :D I think I need to look over some things about 3D projections of the 4D versions of the platonic solids, been a while since I learned about them. Which of the 6 (5,8,16,24,120,600-cell) are you planning to make? I assume the smaller ones are easier to make! :lol: Keep these amazing puzzles coming! :D

Matt


I have made many different ones, and I will be revealing one by one each day.
Some will be simpler than the ones I am showing now. But the last one, is the Colossus,
the puzzle which is the most symmetric object that had ever existed, and that *will* ever exist!
(Yes, this sounds arrogant, but once explained, you will see what I mean!!! Hint: Triality + regular polychoron).

;)


Pantazis


Sounds great! Any chance of giving us an idea of how many of these you have planned? I'm also curious about the simpler ones. Far as I know, the pentachoron is the simplest 4D shape, but no doubt you have something cool and unexpected planned :D . I don't know what the Colossus is (I feel I should understand the clue better but I don't) but I can't wait to see it. I know about shapes being self-dual (like the pentachoron here), but I have no idea about extending duality to triality (and I may even be barking up the wrong tree here!). Whatever it is, I just hope the universe survives. Between you, Oskar and Bram, lots of people here are getting worried :lol:

Matt


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 Post subject: Re: Chronos: Plato's First Shadow!!! (based on the Pentachoron)
PostPosted: Wed Jul 07, 2010 2:37 pm 
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So if I understand this right it can become any 3d shape with 5 or less vertices?


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 Post subject: Re: Chronos: Plato's First Shadow!!! (based on the Pentachoron)
PostPosted: Wed Jul 07, 2010 4:00 pm 
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bobthegiraffemonkey wrote:
So you made the choice to minimize the external edges on these puzzles when possible? In other words, it means that for the pentachoron with minimised external edges the result is the dipyramid, but for the hypercube it isn't possible to minimise external edges so you have take a different approach? Just to clarify.


Every structure which expresses a polytope can be made by minimised or maximised or combined edges
(like the Houlis Cube). The resulting shapes I have chosen are not the only ones, they are just those which
(for the moment) I think are most suitable. :)



bobthegiraffemonkey wrote:
Sounds great! Any chance of giving us an idea of how many of these you have planned? I'm also curious about the simpler ones. Far as I know, the pentachoron is the simplest 4D shape, but no doubt you have something cool and unexpected planned :D . I don't know what the Colossus is (I feel I should understand the clue better but I don't) but I can't wait to see it. I know about shapes being self-dual (like the pentachoron here), but I have no idea about extending duality to triality (and I may even be barking up the wrong tree here!). Whatever it is, I just hope the universe survives. Between you, Oskar and Bram, lots of people here are getting worried :lol:

Matt


Some of them are using 3D symmetry, but they are interesting objects in graph theory.
And you are 100% right about the Pentachoron. ;)


Beans wrote:
So if I understand this right it can become any 3d shape with 5 or less vertices?


Yes, and this is just another nice property that a simplex structure has.

:)


Pantazis

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 Post subject: Re: Chronos: Plato's First Shadow!!! (based on the Pentachoron)
PostPosted: Wed Jul 07, 2010 8:22 pm 
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First of all, wow! congratulations Pantazis, you have clearly succeeded in what several others have failed in the past. Surely tons of great animations exist, but I never liked those miserable attempts to smootly recreate polychoron rotations in our limited 3d world in real-time (for instance: http://www.youtube.com/watch?v=hDlKV9Usy9U). This is the very first time I see it so smoothly and in real time! Okay, yes, as you said, some of the connectors slided out of place, but I simply can't wait to try first-hand the final version of these puzzles, when all the "imperfections" of your prototypes would have been fixed... Yeah, lots of pressure! :lol:

Could there be a way to synchronize the contraction/expansion of the edges? For example, it would be really amazing if one could perform different rotations of a 8-cell smoothly and keep all sets of four edges roughly the same length, as in this animation (the "type" of rotation changes every minute, the first three types of rotation are very interesting and I particularly like the transition between types of rotation). Would that need intelligent communication between parts? hydraulics? or very tiny connection gears? Am I crazy? :lol:

I hope the next one will be 8-cell.

Skarabajo.

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 Post subject: Re: Chronos: Plato's First Shadow!!! (based on the Pentachoron)
PostPosted: Wed Jul 07, 2010 9:19 pm 
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Neat! I wasn't too taken in with the tesseract one, but this looks very cool. I would tend to think of it more like a sculpture with moving parts rather than a puzzle, but that's certainly not a bad thing...

Many of you in this thread will probably recognize my avatar as a 24-cell... How awesome is it that in 4 dimensions a new sixth platonic solid shows up, and then from 5 dimensions on up there are only 3! I'll give you a hint -- it is way awesome. Any plans to make a puzzle based on that? (I assume you have no designs on the 120-cell and 600-cell...)


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 Post subject: Re: Chronos: Plato's First Shadow!!! (based on the Pentachoron)
PostPosted: Wed Jul 07, 2010 10:47 pm 
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I like this one much more than the "cube" this is much easier to understand.

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 Post subject: Re: Chronos: Plato's First Shadow!!! (based on the Pentachoron)
PostPosted: Thu Jul 08, 2010 3:07 am 
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Iranon wrote:
Many of you in this thread will probably recognize my avatar as a 24-cell... How awesome is it that in 4 dimensions a new sixth platonic solid shows up, and then from 5 dimensions on up there are only 3! I'll give you a hint -- it is way awesome. Any plans to make a puzzle, based on that?

I have a feeling that's his "Colossus" puzzle, based on a hint he gave previously that this puzzle has exactly 600 parts, with about 10 parts per edge, as well as the supersymmety.

Here's the animation:

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 Post subject: Re: Chronos: Plato's First Shadow!!! (based on the Pentachoron)
PostPosted: Thu Jul 08, 2010 10:11 am 
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Iranon wrote:
Many of you in this thread will probably recognize my avatar as a 24-cell... How awesome is it that in 4 dimensions a new sixth platonic solid shows up, and then from 5 dimensions on up there are only 3! I'll give you a hint -- it is way awesome.


Interesting... here is a nice link I found which covers this nicely.

http://local.wasp.uwa.edu.au/~pbourke/geometry/platonic4d/

So... what breaks the equivalent of the dodecahedron and icosahedron in 5D or greater? It's probably the same thing that keeps the equivalent of a 24-Cell from existing in 3D but I'm not sure what that is either.

Let's look at this as going to 3D from 2D or back again. We have the equilateral triangle which is the 2D equivalent of the tetrahedron. We have the square which is the 2D equivalent of the cube. We have a pentagon which is the 2D equivalent of the dodecahedron. But we also have hexagons, septagons, octagons, etc. I see why they don't have 3D equivalents so I guess its similiar to that.

Carl

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 Post subject: Re: Chronos: Plato's First Shadow!!! (based on the Pentachoron)
PostPosted: Fri Jul 09, 2010 1:14 am 
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Iranon wrote:
Many of you in this thread will probably recognize my avatar as a 24-cell... How awesome is it that in 4 dimensions a new sixth platonic solid shows up, and then from 5 dimensions on up there are only 3! I'll give you a hint -- it is way awesome. Any plans to make a puzzle based on that?


Exactly. It was impossible to not appreciate the amazing properties of the 24-cell, and the puzzle
(which surely serves as an artistic object too), is a solid proof of concept that "it's alive"!.

As for some "ancient tips", allow me to add some interesting info from Plato's Cave:
http://www.philosophy-religion.org/hand ... ealism.htm

And for a good laugh, here is the Legend of the Pyramid (it is about the Pentachoron):
http://eusebeia.dyndns.org/4d/pento.html

Reading more about 4D, just makes you hungry to learn even more!!!

8-)


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 Post subject: Re: Chronos: Plato's First Shadow!!! (based on the Pentachoron)
PostPosted: Fri Jul 09, 2010 1:29 am 
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so the pentachoron puzzle you made does have a tetrahedral shape.
If not, I have to say, I'm a little upset...

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 Post subject: Re: Chronos: Plato's First Shadow!!! (based on the Pentachoron)
PostPosted: Fri Jul 09, 2010 6:23 pm 
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elijah wrote:
so the pentachoron puzzle you made does have a tetrahedral shape.
If not, I have to say, I'm a little upset...


But of course it can get many more shapes! The tetrahedron one, is one of the most
beautiful one and it will be used to name a different variation of the Pentachoron.

:)


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 Post subject: Re: Chronos: Plato's First Shadow!!! (based on the Pentachor
PostPosted: Tue Jul 27, 2010 2:17 pm 
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Here is a better video using the competition version of the puzzle:

http://www.youtube.com/watch?v=B0-0ILYz0ug

I go through eight (of the ten in total) colors in the middle of the triangular dipyramid,
and I also show other possible shapes, for example, the tetrahedron (with a corner in the center)
as well as the K5 complete graph form.

:)


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 Post subject: Re: Chronos: Plato's First Shadow!!! (based on the Pentachor
PostPosted: Tue Jul 27, 2010 8:31 pm 
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I like this new version a lot. Can you enlighten us as to what you intend the goal of the puzzle to be? I love the idea of using the telescoping edges as the mechanism, but I can't see any particular solution that stands out.

-π (Eitan)

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 Post subject: Re: Chronos: Plato's First Shadow!!! (based on the Pentachor
PostPosted: Wed Jul 28, 2010 2:04 am 
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pirsquared wrote:
I like this new version a lot. Can you enlighten us as to what you intend the goal of the puzzle to be? I love the idea of using the telescoping edges as the mechanism, but I can't see any particular solution that stands out.

-π (Eitan)


Oh yes!!! I should add this to the video too!

The puzzle is to place *all* the ten different colors in the middle (by using the 4D symmetry move)
but during this process, without going through the same color twice! If that sounds easy, think again,
as when related to the edge in the middle, there is not one, but two types of edges. The six ones
at the top and bottom can exchange directly with the middle one, but the three at the central row
need more than that! So there can be "road blocks" if not careful.

By definition, *more puzzles* can be defined based on the very same structure, by just using a different
3D projection. I have an "easier to show" puzzle idea based on two colors for the Pentachoron.

So it can be regarded as many 3D puzzles in one!

:)


Pantazis

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