Well, many of you have been asking me to place this here ASAP,
so I decided to show it earlier.



Where can I start and where can I finish with this puzzle's properties?



Its symmetric properties (self-dual, triality, regular polytope, edge transitivity,
vertex transitivity, etc, etc etc) can fit easily an entire book.



From the photos, I need to explain a bit, and I will do it as simpler as I can:

The 24-cell projection(s) I made, consists of eight (8) octahedrons.
There is one outside octahedron (in the photos and videos it is colored purple) and there
is an inside (yellow) octahedron. Their six corners are of the purple octahedron
use six other octahedrons (blue, red, white, black, brown, green) to connect with the
six corners of the yellow octahedron.



Now, the purpose of this puzzle: By manipulating the edges, you may exchange the inside
or outside octahedrons with any of the other colored ones. Sounds impossible? It is not!
(it is just too tedious!!! LOL)



It is simply put, the most perfect structure. Take away or add something,
and the perfect symmetry is lost forever!



A nice video (with no solves!) on a very beautiful background:

http://www.youtube.com/watch?v=Drs98RH1sic

A (bad) video, demonstrating how to solve it. I am only *one* move away,
as it is extremely tedious to solve. (When I find a LOT of time, I may do a better one)

http://www.youtube.com/watch?v=-DSMxd7TfvA

This video also demonstrates (towards the end) how flat they can be made when needed!
And all of my designs will be coming with me to Japan (I am flying in 3-4 hours).

Have I said before how much I love mathematics?




Pantazis



PS. My good friend Siva gave me the honor of pressing the submit button for this milestone puzzle.
(I am at his house).

Let me be the first one to say it: Wow!!!

Kudos!

Skarabajo.

AMAZING .

Pantazis, you have to give us time to first understand the other models. Do you want our brains to explode?

I'm now beginning to comprehend the movement of the first two, now comes this monster and confuses me even more

Like someone already said in a previous topic, it would be epic to see a collaboration of you and Oskar.
Two "gangsters" of the universe, breaking the laws of physics. It would be really something ...

I think I want that book as much, if not more, then I want the puzzle. I love symmetry and I know the tetrahedron is self-dual. But I'm not sure how to apply that in 4D. I don't have a clue what triality is. "regular polytope", finally a term I'm comfortable with in 4D. And edge transitivity and vertex transitivity fall back into the no clue catagory. I want to be educated so I can have a chance of being able to fully appreciate these puzzles.

In 3D to find the dual of a shape faces become vertexs and edges stay edges.

Does it make sense to say a diamond is the dual of a square in 2D? Turning vertexs into edges and vice veras?

In 3D we have:
vertexs (which are 0D)
edges (which are 1D)
faces (which are 2D)
volumes (which are 3D)

Going to 4D we have:
vertexs (which are 0D)
edges (which are 1D)
faces (which are 2D and exist between cells)
cells (which are 3D)
Not sure what the next terms is but a 4D equivalent to volume:

Which terms are interchanged for duality for 4D? for triality in 4D? Does triality have a meaning in 3D? Are these even the correct questions or am I out in left field?

Carl

P.S. Just found this site:

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

which answers the duality question for 4D and it appears I was correct about my understanding of duality in 2D as well. Still not sure about triality.

While the 24-cell is very cool, I have to say that as a puzzle experience of the ones you've posted so far the 16-cell is the clear winner.

Also, it looks like your joints could use a lot more work, but you've said these are drafts, so I assume you're working on that.

I'm with bram, but that doesn't make this any less amazing.

The monster is alive!!

This looks stunning, and I'm sure the final version will be a masterpiece. The only problem seems to be that the complexity makes movement difficult so it would require a lot of effort and patience to solve (though I have no doubt it would be worthwhile).

One question though. I understand duality, I understand it being edge-transitive and whatever, and I (think) I pretty much understand the structure. I have no idea what triality means exactly. I know it's some sort of extension of duality, but that's about it. Can you try and explain this please? I'm sure many others will appreciate a description too.

Matt

Holy... Wow. Just Wow.

At the rate you're going, eventually you're gonna do this:

Amazing.This can function the came as the 5 and 16 cell, in the same way a 5x5 can function like a 3x3, right?

A nice link, although it needs some reading:
http://members.home.nl/fg.marcelis/24-cell.htm

Basically, triality is to three vector spaces, what duality is to two.

Regarding which puzzle is the best, that depends on what we want.
For example, Chronos seems nice as an intro to the 4D puzzles,
but it has some interesting twists! The Houlis Cube, is a cube, and that is
what makes it attractive to cubers, but I also like it a lot.
Now, Colossus *is* tedious, but it does provide with some serious challenge,
something like solving a 7x7x7 when compared to a 2x2x2.

But there are still a few more candidates to come. Then we can decide better!

The bigger puzzles work nicer than expected, but they still need to be handled
with care. It will be a huge challenge to make a mass produced one which is
strong enough to escape the wrath of a toddler!!!




Pantazis

I love these pantazis, but they are not my favorites, why must you tempt me so...? I want to see the tesseract!

I love these 4D puzzles. This one's my favourite, just plain and simply due to symmetry. It does somewhat remind me of a Skewb Diamond.

Nice concept, very original - I can imagine people would have some fun toying around with it, but I don't see it as a serious puzzle that will keep people engaged for very long.

Congratulations with your 4D-puzzles! I hope it will be for sale at some point (a complete 4D-Platonics set!!). Several links have been posted already. Here's another one. A very nice introduction in 4D, you can download a movie with a lot of wonderful animations.

http://www.dimensions-math.org/

enjoy!

Thanks all for the nice words, especially Kelvin, I appreciate it.

Now, some terrifying news!!! Wei Hwa took the Colossus from my hands and he changed its shape
many times, and manipulated it to no return!!!

But people do not panic! The Universe is still safe! It is just a little bit different now...!

The Colossus now, does not have the purple outside and the yellow inside, but now it is the
yellow OUTSIDE and the purple INSIDE! (Before that, on the process, it was the brown color
outside and the red color inside).

Well, if there was ONE person in this planet who could do it (he also anticipated the points where
he should not put pressure to avoid damaging), it is HIM!

Photos will be put later... now time to exchange some puzzles!




Pantazis

Pantazis, do you think you could get a picture of the shadow of the Colossus?

(This is a joke. If you don't get it, then shame on you.)

I've been thinking the same thing

A tesseract would indeed be an absolutely incredible puzzle. Please put that as your next priority

But begging aside, building this is a feat like no other. I applaud you, good sir.

Wish granted!(I actually wanted to publicize this more than anyone! LOL)

As for the shadow joke, I did not get it, but my shadow did!!!




Pantazis

It's the name of an old video game for the PS2. Or something to do with the fact the puzzle is the shadow of a 4d object.

O thank god. At first i thought this was a link to the corrupt a wish thread


These puzzles are really cool. but i find it hard to quite understand them.

As promised, here is Wei Hwa solving the puzzle... To exchange the outside purple octahedron
with the yellow one, he first had to use the "intermediate position" where the outside octahedron
was brown color...! Photos:



Once upon a time, there was an object representing the 24-cell projections,
where the outside octahedron was purple, and the inside, yellow!



Then, Wei Hwa decided to change the colors of the most symmetric object in existence!
In the middle, Tom Cutrofello playing with the 16-cell symmetry Houlis-Cube.



The colors are already moving... now the brown is outside, the red inside,
while the rest connect their corners!



Full transformation... The yellow octahedron on the outside, and the purple, inside!
He has turned the object INSIDE-OUT!!!



Confirmation of the colors... as stated any color can be exchanged with the others.
Total number of octahedron states (including 4D and 3D symmetry rotations) is 192.


Now... try to think of how to turn inside-out *ourselves*!!! (*gulp*)




Pantazis

I don't mean to rain on your parade, but isn't a sphere more symmetrical?

Still, this is a pretty impressive. I can't wait for these puzzles to be available for buying.

ok ok, fair enough, I need to underline that we *need* to focus on an object with a *finite* number of morphisms.
On a sphere, vertices and edges could be regarded to be infinitely small... we don't want that here! LOL

More specifically, a real world puzzle can never use the symmetry you described, unless it was a
one-color ball with no cuts, i.e. not a puzzle. This is not the case with the Colossus which is real and moving
using all symmetric properties (almost) to the max!

So please allow me to correct the previous word "object" and substitute it by either the word "puzzle",
or the more detailed description "object with finite morphisms".




Pantazis

That statement is true UNTIL I get SolidWorks.

LOL I must admit, I love your statement!




Pantazis

Good luck making infinitely thin slices?

Considering Kastellorizo just made a puzzle with 4-D symmetry in this 3-D world, maybe cutting slices isn't the only way to get this done?

You could try twisting a brown ball of clay into separate colours of the spectrum: now that *would* be the ultimate twistypuzzle with infinite symmetry. In fact you can see scrambled and solved pictures of this puzzle here.