Thread: "4D-interactive puzzles in MagicTile"

 

It is 4D, in that the puzzle only fits together in a truly regular w=
ay in=20
4 dimensions.  But the puzzle faces are still 2D (the {4,4|4}=20
has 16 2D faces).  This is unlike MC4D, which has 3D faces fitt=
ing=20
together in 4D.  Still, you can see how some of the square faces of =
the=20
{4,4|4} are being warped due to the 4D -> 3D projection.  As in M=
C4D,=20
shift+left drag to do some 4D rotations that will affect this warping.IV>












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From: "Andrey" <andreyastrelin@yahoo.com>
Date: Tue, 06 Mar 2012 16:03:52 -0000
Subject: Re: 4D-interactive puzzles in MagicTile

Roice,
New puzzles are beautiful! It's very nice to see these sceletons of our f=
amiliar 4D bodies and wander around them :)
I solved only one puzzle so far - bitruncated simplex (my favorite polyto=
pe :) ) in F:0:0:1 slicing. It was not very easy, and one of main problems =
is that twising rotation depends on side of face that you click: if you see=
one side, face twists clockwise, but if you go to the other side and make =
the same click, it twists counterclockwise. I think that it's not difficult=
to fix.
This puzzle has a kind of "global non-orientability": if you have wrong o=
riented edge, you can't reorient it by moving around the vertex, you have t=
o make a loop around a tube. I guess that same effects will be in 5x5 and 7=
x7 duoprisms (that are actually just another implementations of planar tori=
:) )

Thank you for these puzzles!

Andrey
=20=20


--- In 4D_Cubing@yahoogroups.com, Roice Nelson wrote:
>
> Recently, I learned that in addition to the regular skew polyhedra that a=
re
> infinite (Melinda's IRPs), there are regular skew polyhedra which are
> finite. They close back around on themselves in 4 dimensions. Here was =
an
> awesome opportunity to make MagicTile genuinely applicable to the 4D_Cubi=
ng
> group :D
>=20
> To start, I've included 4D skew polyhedra associated with some duoprisms,
> the runcinated 5-cell, and the bitruncated 5-cell, though there are addit=
ional
> possibilities for
> the future. These should be pleasant to solve in skew mode, and I'm sure
> many interesting observations can be made of the shapes included so far.
> For instance, check out how the surface of the {4,6|3} relates to 3
> intersecting spheres.
>=20
> I just uploaded some
> picsist>of...
>=20
> - {4,4|4} duoprism, aka hypercube
> - {4,4|5} duoprism
> - {4,4|7} duoprism
> - {4,6|3} runcinated 5-cell
> - {6,4|3} bitruncated 5-cell
>=20
> Although the duoprisms are easily seen as having genus 1, can you figure
> out the genus of the other two? Not trivial to figure out by sight! I
> won't give away the answer, but will say both have the same genus.
>=20
> I changed the 4D viewing controls slightly from MC4D, so that the 4D came=
ra
> distance could be altered with the mouse (I didn't want to rely on a
> slider). Things will mostly be familiar though, and details are in the
> Help menu.
>=20
> This feels like a great plateau, so now I'm going to take a break from
> MagicTile for real :)
>=20
> Enjoy! >
> Roice
>

---

From: Roice Nelson <roice3@gmail.com>
Date: Tue, 6 Mar 2012 11:51:51 -0600
Subject: Re: [MC4D] Re: 4D-interactive puzzles in MagicTile

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Thanks for your thoughts and positive feedback Andrey :)

That's a really cool observation about edge orientation behavior!
Also, about the duoprisms being planar tori, I wanted to mention that the
skew polyhedra are only a different view of the traditional puzzles in
MagicTile. For the duoprisms, you can switch to the {4,4} planar tori view
by setting the option "Show as Skew" to false. Changing this setting for
the bitruncated simplex will show its associated hyperbolic tiling too.

About the twisting direction relationship to the side you click on, I bet
it can be changed without too much effort like you say, so I will plan to
do this relatively soon.

aside: I did mentally debate whether the current behavior was ok or not,
and the same thing happens on all of the IRP puzzles as well. Besides
easier coding, one advantage is that it gives a visual clue of the two
different halves the skew polyhedra divide space into. A similar sort of
thing came up before with non-orientable tilings of the plane - mirrored
faces were twisting in an opposite sense. That had advantage (visual clues
to the topology), but the same disadvantage of making solving difficult. We
settled on making all faces always twist CCW when left-clicked, and it
sounds like I should take the same path here. I would like to make solving
as pleasant as possible for the skews.

Cheers,
Roice

On Tue, Mar 6, 2012 at 10:03 AM, Andrey wrote:

> Roice,
> New puzzles are beautiful! It's very nice to see these sceletons of our
> familiar 4D bodies and wander around them :)
> I solved only one puzzle so far - bitruncated simplex (my favorite
> polytope :) ) in F:0:0:1 slicing. It was not very easy, and one of main
> problems is that twising rotation depends on side of face that you click:
> if you see one side, face twists clockwise, but if you go to the other side
> and make the same click, it twists counterclockwise. I think that it's not
> difficult to fix.
> This puzzle has a kind of "global non-orientability": if you have wrong
> oriented edge, you can't reorient it by moving around the vertex, you have
> to make a loop around a tube. I guess that same effects will be in 5x5 and
> 7x7 duoprisms (that are actually just another implementations of planar
> tori :) )
>
> Thank you for these puzzles!
>
> Andrey
>

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---

From: Roice Nelson <roice3@gmail.com>
Date: Tue, 6 Mar 2012 20:13:55 -0600
Subject: Re: [MC4D] Re: 4D-interactive puzzles in MagicTile

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Hi Andrey,

I went ahead and made your suggested change this evening. So skew puzzles
will twist CCW on a left click regardless of the side you click. It feels
like a definite improvement for both the IRPs and the 4D skew polyhedra, so
thanks again for suggesting that... link to
download

seeya,
Roice



On Tue, Mar 6, 2012 at 10:03 AM, Andrey wrote:

> Roice,
> New puzzles are beautiful! It's very nice to see these sceletons of our
> familiar 4D bodies and wander around them :)
> I solved only one puzzle so far - bitruncated simplex (my favorite
> polytope :) ) in F:0:0:1 slicing. It was not very easy, and one of main
> problems is that twising rotation depends on side of face that you click:
> if you see one side, face twists clockwise, but if you go to the other side
> and make the same click, it twists counterclockwise. I think that it's not
> difficult to fix.
> This puzzle has a kind of "global non-orientability": if you have wrong
> oriented edge, you can't reorient it by moving around the vertex, you have
> to make a loop around a tube. I guess that same effects will be in 5x5 and
> 7x7 duoprisms (that are actually just another implementations of planar
> tori :) )
>
> Thank you for these puzzles!
>
> Andrey

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Hi Andrey,

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