I optimized the Four Fives (or Eight Ones) pattern a little bit, it now has 21 moves.

Four Fives (or Eight Ones):
D2 WF2 WR2 D2 WF2 MU' WR2 VU F2 NR2 F2 NU' ND F2 NL2 F2 ND' F2 NR2 NL2 F2 (21 btm)

Two additional variations on number 5 = 1 + 4 on 4 faces of a 5x5x5 cube:

1.4

14

No doubt that these could be optimized as well!

Good work!! But i don't see any snake in the snakes pattern. I see interleaved rubberbands

Per

You are right: interleaved rubberbands or tangled lines would be more correct.

Hi!

I improved the first variation.

1.4

It is made up of 3 parts:
1. N2F2 N4F2 N2R2 N4R2 TD2 N2F2 N4F2 N2R2 N4R2 TD2 ("wide edges")
2. MR- N2D MR N2D2 MR- N2D MR (centers)
3. F2 L2 N2R2 MU- N2R2 MU L2 F2 (final centers)

Reduced from 39 to 25 turns. Better is surely possible ...

Per

Edit:

So, i could not resist also optimising the second one a bit:

14

Again made from 3 parts

1. WF2 TU WR2 TU2 WR2 TU WF2 ("wide edges")
2. N2R2 N4R2 TD N2F2 N4F2 TD2 N2F2 N4F2 TD N2R2 N4R2 (more "wide edges")
3. F2 L2 N2R2 MU- N2R2 MU L2 F2

Reduced from 36 to 26 turns. I can see cancellation opportunities, but getting lazy

Hi Per,

By cancelling/combining a few turns, both patterns are now 23 moves long:

1.4 (23 moves)
3. R2 B2 MU NB2 MU' TB2 R2
2. N3F2 ND N3F2 ND2 N3F2 ND N3F2
1. N2F2 N4F2 N2R2 N4R2 TD2 N2F2 N4F2 N2R2 N4R2 TD2

14 (23 moves)
3. R2 B2 MU NB2 MU' TB2 R2
2. N2R2 N4R2 TD N2F2 N4F2 TD2 N2F2 N4F2 TD N2R2 N4R2
1. WR2 TU' WF2 TU2 WF2 TU' WR2

Continuing with fairly simple patterns on 4 faces of a 5x5x5 and noting again that 5 = 2 + 3 and that letters 'S' and 'E' are just the inverses of numbers '2' and '3', we got this nice pattern:

23SE (17 moves)
1. D2 WR2 WF2 D2 WF2 WR2
2. WR2 MU' WR2 MU
3. B2 ND' NB' NR NB2 NR' NB ND TB2

Are you able to make 2 mezons pattern also?? I remeber 2 mezons was quite a challenge on the helicopter cube

Per

Hi.

I expanded the 23SE pattern with 1's on the last 2 faces by going up to size 6x6x6.

23SEand1.

Obviously the 1's are done first

Per

Hi,

A variation of the 23SEand1 pattern can be obtained by replacing '1' with '+', to link opposed faces:
23SE++ (26 moves)

The Endless Knot pattern, shown below on a 21x21x21 virtual cube, symbolizes the never ending quest to find shorter algorithms and interesting patterns

Hi, glazik.

One side of a cube 19x19x19:


It is analogue of the trade mark of Russian firm "Pozitronika":


It is possible to make five-coal unit on a Minx 19x19x19.

This mezon or this mezon?

Hi pytlivyj,

Here below is a variation of the Positronika logo on a 19x19x19 cube: Tangled Rings.
The Cisco logo can also be displayed on the same cube.

By searching for other logos, I found this company logo, which wich can be displayed on a 7x7x7 cube:
Logo (20 moves)

I meant the first one - looks like no shift shaping has been done. But of course it is needed but looking at orbitals

The second one looks horrible tp set up. Avoiding voiding lockups ... hmm ...

Per

Even though those patterns look nice. Without sequences, or at least an outline of how to set it all up. I find that not very interesting. As the cubes increase in size, virtually any pattern by trandforming surface to a square tiling on each face is possible. Just makie sure to retainn a valid cube state. Not really hard

Per

Due to the lack of an on-line 3D cube viewer for N > 7, sequences have been listed in this document.

Three-cycles are used for displaying patterns on 3 or 6 cube faces. This technique can easily be scaled up to bigger cube sizes, by using (long) commutators.

Consider for example the 'Logo' pattern on a 7x7x7 cube:

1. Pattern 1 (Color 1):
[N2F N4F, N3D]
[N3F N5F, N4D N2D]

2. Pattern 2 (Color 2):
[N5D, N3F N5F]
[N4D N6D, N4F N6F]

3. Optimize:
[N5D, N3F N5F] [N3F N5F, N4D N2D] [N4D N6D, N4F N6F] [N4F N2F, N3D]

4. Final Algorithm (20 moves):
N3U' N3F N3B' N3U MU' ND N3B N3F' ND' NU' MF NB' NU MU NB NF N3D NF' MF' N3D'

A shorter algorithm may still be found by other means, of course, but optimizing a product of commutators is generally quite simple and perfectly scalable.

Patterns on cube sizes greater than 7 were mainly designed as test cases for the NxNxN engine of my computer program, so I agree with you that they may present a limited interest to cubers, even though it may not have been done before.

The very good document.
On 24 page an inscription "Happy 2012!" is three times written on the contrary. Why?

Chess Master Skewb of the second order:


Chess Master Skewb of the third order:


Chess Master Skewb of the sixth order:


Two "Ð¥" on a Master Skewb:


Corners on a Crazy Tetrahedron:


The turned edges on a Crazy Tetrahedron:

The commutators of centers used throughout the document are based on [NF, NL], which is a two three-cycles: (f r d)(b l u), in that precise order.
The permuted pieces of the first 3-cycle are located on faces F, R, D and those of the second on the opposed faces B, L, U.
It is a property of this type of commutator that patterns on any of the 3 first faces are just the mirrored versions of those on the 3 opposed faces.
This is the reason why letters/numbers are displayed in the reverse direction on opposed faces.

I was going to complete and show a new stripey pattern on the 7x7x7 cube. After setting up the edges, there is still work to be done on the centers. But i will show the untermediate result after setting up edges which is quite nice also. I don't know what to call this.

Based on the well known 6-x pattern.

Per

Another cascading pattern, based on the well known superflip from 3x3x3.
Obviously it has 3 parts. Why does the order of the 3 parts matter??

Per

Here is a possible explanation - maybe too detailed:

The Superflip algorithm can be written as:
MF U B2 D2 MR' D' MR2 MF U R2 D MR2 U B2 U MF2 (16 s*)

The algorithm of the cascading pattern is built from 3 parts A, B and C, each based on the Superflip algorithm:
Part A (5 edges):
M5F U B2 D2 M5R' D' M5R2 M5F U R2 D M5R2 U B2 U M5F2
Part B (3 edges + centers):
M3F TU TB2 TD2 M3R' TD' M3R2 M3F TU TR2 TD M3R2 TU TB2 TU M3F2
Part C (midges + midge-centers):
MF T3U T3B2 T3D2 MR' T3D' MR2 MF T3U T3R2 T3D MR2 T3U T3B2 T3U MF2

It can be seen that A includes edges of B, C and B includes edges of C.
The complete alg is then given by:
A B C

If parts could be ordered differently to give the same cube position, this would mean that A commutes with B, B with C and C with A.
To check this, it is sufficient to compute these commutators [A, B], [B,C], [C, A] and see if they give the Identity position or not. None of them gives the Identity...

I was thinking much simpler. The base 3x3x3 superflip algorithm has a side effect by rotating face centers. This is why we have to work from the outside and inwards - and not opposite

Check this out.

Per

The number of moves of the 'Four Outlined Hearts (90° version)' algorithm has nearly been cut in half, down to 74 btm.
There is of course still room for improvement

It is composed of 5 parts, each based on a number of 4-cycles:

1-Midges
(F2 MR' F2 U') [SF MR SF', R L] (U F2 MR F2)

2- Edges 1
(F2 VU')5 F2

3- Edges 2
(F' L B D2) (L SD2 SR VD SR' SD2 SR VD' R') (D2 B' L' F)

4- Edge-Centers
N3D [N3D', NF NL NB NR] [N3F N3L N3B N3R, ND] N3D'

5- Midge-Centers
(MF2 TU') (NU2 MF D2 MF ND2 NU2 MF' TD2 MF') (TU MF2)

Four Outlined Hearts (90° version)

I have 2 more easy contributions

Stripey dots 1:

Stripey dots 2:

Did some trivial optimisations. Note the similarity of the sequences for the 2 patterns. Yet, quite different outcome

Per

Interesting patterns!

Looking closely at the algorithms, it can be seen that they are products of a number of commutators [A,B], where all moves B' have been grouped to the right as WU.

Continuing further by grouping all moves A to the left as WF, 'stealthy' patterns can be obtained:

SR WF (N2U' N2F') (N3U' N3F') (N4U' N4F') (N5U' N5F') (N6U' N6F') WU SR'
Six Stealthy Patterns 1

SR WF (N2D N2F') (N3D N3F') (N4D N4F') (N5D N5F') (N6D N6F') WU SR'
Six Stealthy Patterns 2

Here is another pattern on a 7x7x7 cube (the algorithm is certainly too long ):

1- Four Crosses
T3U2 MF2 T3U MF2 T3U2 MR2 T3D' MR2

2- Remove Vertical Midge-Centers
MR2 VD2 MF2 VD' MR2 VD2 MF2 VD

3- Remove Horizontal Midge-Centers
[VB' VL' VF' VR', MU']

4- Add Edge-Centers
[N3B' N3L' N3F' N3R', ND' NU] [N3D N3U', NB' NL' NF' NR']

Complete alg (46 moves):
Four Dotted Diamonds (90° Version)

The 2 first can be conjugated to get more symmetric versions of the pattern.

Per

Following your suggestion, setup moves have been added to obtain more symmetric patterns:
Six Stealthy Patterns

Here is another simple pattern, again on a 7x7x7 cube:
Four Cartwheels (90° Version)

Now i want you to extend the cartwheels (90 degrees) to filled diamonds (90 degrees) by adding some missing centers. Curious how short this can be done

Per

Alright. I got 41 moves for the 90° version of the four diamonds.
This may still be too long, though

Corner-centers have been inserted into edge-centers as an attempt to decrease move count.

Here are the details:
1- Four Crosses
MR2 T3U MR2 T3D2 MF2 T3D' MF2 T3D2

2- Corner-Centers
[N3D, N3B' N3F' N3R N3L N3F' N3B']

3- Edge-Centers
[N3B' N3L' N3F' N3R', ND' NU] [N3D N3U', NB' NL' NF' NR']

Four Diamonds (90° Version)


Another simple pattern can also be obtained by adding corners (hasn't been optimized yet):
1- Four Wide Crosses
TU2 M3F2 TU M3F2 TU2 M3R2 TD' M3R2

2- Edge-Centers
[NB' NL' NF' NR', N3D N3U'] [ND' NU, N3B' N3L' N3F' N3R']

3- Corners
R2 SF2 R2 D SF2 R2 SF2 R2 D' SF2

Sixteen Small Crosses (90° Version)

Hi!

I modified the stealth pattern to get some stripes into the shapes.
Then i added some setup moves to make it more symmetric.

I call the final result symmetric statistics!!

Per

Hi Per,

Amazing pattern indeed!

Here is my last contribution (I'm taking a two-week trip abroad shortly...):
Four Jerusalem Crosses (90° Version)

Four Jerusalem crosses (180° Version).Enjoy

1. (M3R2 M3F2 U2)2 - edges
2. R2 F2 M3U2 F2 M3U2 R2 - more edges
3. MR2 WU MR2 WU- - Centers
4. VR2 VL2 MU VR2 VL2 MU- - final centers

Per

Improved the sequence by 2 turns. New version.

Breakdown:
1. V3U2 MR2 MF2 VU2 MR2 MF2 (centers and middle layer)
2. F2 R2 N3U2 N5U2 R2 N3U2 N5U2 F2 (some edges)
3. M3R2 M3F2 U2 M3R2 M3F2 U2 (final edges)

I would guess this can be further optimized ...

I improved the 16 small crosses (90°).

2 quite easy parts...

Per

24 small crosses.

I see many more possibilities with these small crosses ...

24 small crosses (order 3).

Per

6 Canterbury Crosses.

I was trying to extend Jerusalem crosses to 6 sides. But it was impossible to permute middlemost centers the way i wanted ...

Per

I also improved the Four diamonds 90° version.

Main parts:

1. MR2 D- MR2 D U MR2 U MR2 (edges)
2. N2R2 N6R2 N3D- N2R2 N6R2 N3D N3U N2R2 N6R2 N3U N2R2 N6R2 (centers)
3. M3R2 VD- M3R2 VD VU M3R2 VU M3R2 (more centers)
4. TU2 MD (final adjustments)

N2R2 N6R2 M3R2 combines to WR2

Per

I could not resist trying to improve the Dotted Four Diamonds (90°).
Breakdown:

1. SF SR MR2 VD MR2 VD- VU- MR2 VU- MR2 VU2 SR- SF- MD (centers)
2. MR2 D- MR2 D U MR2 U MR2 (edges)
3. N3R2 N5R2 N2D- N3R2 N5R2 N2D N2U N3R2 N5R2 N2U N3R2 N5R2 (more centers)
4. N2R2 N6R2 N3D- N2R2 N6R2 N3D N3U N2R2 N6R2 N3U N2R2 N6R2 (final centers)

Then i combined turns and adjusted top 3 layers. A modest 3 turns s(h)aved. For now i cannot see a better way ...

Per

Properly packaged (order 2).

Good cancellations from the 3 building blocks ...

Per

I managed to futher optimize the Properly Packaged (Order 2) pattern.
Instead of 10 moves it now requires only 6 moves.

NU2 NF2 NR2 NL2 NB2 ND2 (6 btm)

This version has 3 additional stripes:

Gift-wrapped Cube (Order 3)
R D' NR MR NL' U' NR' MR' NL D NR MR NL' U TR' MR' NL
TF' MF' NB U NF MF NB' D NF' MF' NB U' NF MF NB' D' F
U F R' NF2 MF2 NB2 R F' R' NF2 MF2 NB2 R U'
D B' NR2 MR2 NL2 B L B' NR2 MR2 NL2 B L' D'
WD' WR' WD WR
N3R' N3L N3U N3D' N3R N3L' N3U' N3D (74 btm)

Well, the algorithm isn't optimized yet.
It certainly can be done in a more efficient way.


And these are a few more Diamond patterns for the 5x5x5 cube:

4 Diamonds (Order 2)
(MR2 MF2 D2)2
WD WR2 WD' MD WR2 MD (12 btm)

4 Diamonds (Order 4)
MR2 U' MR2 U' D' MR2 D MR2 U2
(WR2 NU')2 VD' WR2 ND WR2 NU2 (18 btm)

6 Diamonds (Order 3)
TF MR2 TF TB MR2 TF TL MF2 TR TL MF2 TL TU2 TD2
NF' NB NR NL' NF NB' NR' NL (22 btm)

Based on the 5x5x5 solution above, I slightly optimized the 4 Diamonds (Order 4) for the 7x7x7:

4 Diamonds (Order 4)
MR2 U' MR2 U' D' MR2 D MR2 U2
(M3R2 NU')2 ND' M3R2 ND M3R2 NU2
(WR2 N3U')2 M2D' WR2 N3D WR2 N3U2 (27 btm)

The above 6 diamonds (order 3) on the 5x5x5 inspired me to do the same on the 7x7x7.

Final result!

Obvious breakdown!!

Per

My idea was to create a cube for my godchild with his name on it.
Originally I liked to have the letters only in one color, but unfortunately this wasn't possible.
So I had to add a couple of 'pixles' of a second color on half of the letters.

LAURIN Cube
U' F' (WR' NF' NB WR NB' NF) F U
NL' B2 NL' NB' ND' NF ND NL B2 NL' ND' NF' ND NB NL2
R2 NU NR' NU' R2 NU NR NU'
L MU' L' F' NU' F L MU L' F' NU F
R2 NU' MR NU R2 NU' MR' NU
VB U ML MB' U' NB' U VB ML' NB' U' MB'
NL' MD NL MD'
D2 F' (MD NL' MD' NL) F D2
B' NU MB' NU' B NU MB NU'
NL2 B' NL2 B MR' MU' B' NL2 B MU MR VL2 NF2 D' MR' D MR NF2 VR' D' MR D NR ML2
NU' R B' MU' U' NR MU NR' U B R' NR MU' NR' VU
MF MU B2 D (ML MD2 ML' MD2) D' B2 MU' MF' (136 btm)

I wonder how much the algorithm of this non-symmetrical pattern can be optimized.
It would be great to bring it at least below 100 moves.

Interesting Laurin pattern, although i prefer symmetric patterns. Less headache to construct

Ok, some quite easy contributions today.

Small orthogonal bars.

Split longer orthogonal bars.

Per

Edit:
Orthogonal U-rings is another more advanced variety.

I can even add 4 more block turns to make it more striking.
Blocked orthogonal U-rings.

Swapping 2 block turns i get another striking pattern.
Blocked L-rings.

Since 2 moves can be saved on the 'Orthogonal U-Rings', the other algorithms can benefit from that as well:

Orthogonal U-Rings
VF VB VR VL VF' VB' VR' VL'
MR ML MF MB MR' ML' MF' MB' (16 btm)

Blocked Orthogonal U-Rings
VF VB VR VL VF' VB' VR' VL'
MR ML MF MB MR' ML' MF' MB'
T3R2 T3F2 T3U2 T3R2 (20 btm)

Blocked L-Rings
VF VB VR VL VF' VB' VR' VL'
MR ML MF MB MR' ML' MF' MB'
T3R2 T3U2 T3F2 T3R2 (20 btm)

The algorithm of this simple pattern certainly can be further optimized:

6 Sieves
ND' F NR NL NB2 NR' F' NB2 NR F NL' F' NR' ND
R2 ND NL' ND' R2 ND D' NL ND' NL' D NL
NB D NR' D2 NB NR' NB' NR ND' D2 NR ND D' NB'
(NB NF NR NL)4 (56 btm)

Hmm. How the heck did you come up with that breakdown to make the pattern? Not even symmetric parts. If not all the cycles had been clockwise i guess the setup would be a lot easier - if not so much shorter.

Per

New easy contribution.

Orthogonal L-rings and short orthogonal bars.

Trivial breakdown. I can extend this with orthogonal blocks (4 extra outer block turns) - but lazy now ...

Per

I was working in reverse order, so my starting point was part 4 which moves all 24 centers in a very symmetrical way. Parts 1 thru 3 are rearranging a couple of centers to achieve the final pattern. Because they are just doing the rest, these parts don't look symmetrical.


My initial solution actually was more straight forward:
6 Sieves
(NR' NF' TL' NU L' NB' NU NB NU' TL NU' L NF
U2 NR NU NR' U2 NR NU'
R' NU' NR NU NR' R
B' NR NU' NR' NU B CF2)2 (64 btm)

It first moves all the centers in the upper half, twists the enitre cube 180° around the F-B axis and does the same again. If you pause the player in the middle of the algorithm you can see that one single algorithm would be enough to build the upper and lower part of the pattern.


Edit:

I just found a better solution:
6 Sieves
(NF B2 NR2 D' (TR' NB NR NB' R) D NR2 B2 NF' CR)4 (52 btm)

... and this saves another 12 moves:
6 Sieves
F' B2
(NR2 D' (TR' NB NR NB' R) D NR2
NL2 U' (TL' NB NL NB' L) U NL2
NU2 R' (TU' NB NU NB' U) R NU2
ND2 L' (TD' NB ND NB' D) L ND2)
B2 F (40 btm)

I improved the

Gift-wrapped cube (order3).

Other approaches are possible. Interesting task

Per