The New Sudoku Players' Forum

Website · by **denis_berthier** » Sun Nov 29, 2020 12:29 pm

PERFECTLY SYMMETRIC PATTERNS

This thread is about perfectly symmetric patterns.
By this, I mean patterns of givens that enjoy a pure visual symmetry without any deviation from it.
If the givens themselves enjoy some symmetry, that's still better, but it's not a condition for being in this thread.

For systematisation purposes, perfectly symmetric patterns will be grouped into families.
Each family will have one (or a few) root pattern(s) and each of these root patterns will have variants defining the actual members of the family.
Note that neither the root pattern(s) nor the variants need have minimal puzzles:
- the root pattern(s) are there mainly for classification purposes;
- the variants are there for eventually producing actual puzzles, but it is interesting to know that some variants don't have actual puzzles or at least not easily found ones.
[Added for precision]: The root patterns must be visually identifiable shapes (henceforth as continuous as possible) - as vague as this condition may be. This may be very restrictive in comparison to the Patterns Game or the systematic studies mentioned in the references.

In the puzzles part of this forum, I've already given examples of perfectly symmetric puzzles. I'll reserve the next posts to say more about their families.
It is very likely that some perfectly symmetric puzzles have already been studied somewhere in this forum. If so, we can add references to them in this thread. [Edit]: Indeed, much has already been done: see the references.

But first, let me introduce a new ([Edit]: not so new indeed: see the references) family and take it as an example of what I'm interested in:

THE OCTAGON FAMILY
The Octagon family has one root pattern, the pure octagon pattern:

Code: Select all: pure octagon: +-------+-------+-------+ ! . . X ! X X X ! X . . ! ! . X . ! . . . ! . X . ! ! X . . ! . . . ! . . X ! +-------+-------+-------+ ! X . . ! . . . ! . . X ! ! X . . ! . . . ! . . X ! ! X . . ! . . . ! . . X ! +-------+-------+-------+ ! X . . ! . . . ! . . X ! ! . X . ! . . . ! . X . ! ! . . X ! X X X ! X . . ! +-------+-------+-------+ --XXXXX---X-----X-X-------XX-------XX-------XX-------XX-------X-X-----X---XXXXX-- 24 givens

It has all the visual/geometrical symmetries one might wish.
It has a problem however: it doesn't have puzzles - or at least none that can be produced in a reasonable amount of time. [Edit]: no puzzle confirmed in the references.

Variant 1: octagon + centre

Code: Select all: octagon+centre: +-------+-------+-------+ ! . . X ! X X X ! X . . ! ! . X . ! . . . ! . X . ! ! X . . ! . . . ! . . X ! +-------+-------+-------+ ! X . . ! . . . ! . . X ! ! X . . ! . X . ! . . X ! ! X . . ! . . . ! . . X ! +-------+-------+-------+ ! X . . ! . . . ! . . X ! ! . X . ! . . . ! . X . ! ! . . X ! X X X ! X . . ! +-------+-------+-------+ --XXXXX---X-----X-X-------XX-------XX---X---XX-------XX-------X-X-----X---XXXXX-- 25 givens

It has the same symmetries as the pure octagon and the same problem. [Edit]: no puzzle confirmed in the references.

Variant 2: open octagon obtained by deleting the 4 central X

Code: Select all: open octagon: +-------+-------+-------+ ! . . X ! X . X ! X . . ! ! . X . ! . . . ! . X . ! ! X . . ! . . . ! . . X ! +-------+-------+-------+ ! X . . ! . . . ! . . X ! ! . . . ! . . . ! . . . ! ! X . . ! . . . ! . . X ! +-------+-------+-------+ ! X . . ! . . . ! . . X ! ! . X . ! . . . ! . X . ! ! . . X ! X . X ! X . . ! +-------+-------+-------+ --XX-XX---X-----X-X-------XX-------X---------X-------XX-------X-X-----X---XX-XX-- 20 givens

It has the same symmetries as the pure octagon and the same problem. [Edit]: no puzzle confirmed in the references.

Variant 3: open octagon + centre obtained by deleting the 4 central X

Code: Select all: open octagon + centre: +-------+-------+-------+ ! . . X ! X . X ! X . . ! ! . X . ! . . . ! . X . ! ! X . . ! . . . ! . . X ! +-------+-------+-------+ ! X . . ! . . . ! . . X ! ! . . . ! . X . ! . . . ! ! X . . ! . . . ! . . X ! +-------+-------+-------+ ! X . . ! . . . ! . . X ! ! . X . ! . . . ! . X . ! ! . . X ! X . X ! X . . ! +-------+-------+-------+ --XX-XX---X-----X-X-------XX-------X----X----X-------XX-------X-X-----X---XX-XX-- 21 givens

It has the same symmetries as the pure octagon and the same problem. [Edit]: no puzzle confirmed in the references.

At this point, you may be bored by this family. Here are now more interesting variants:

Variant 4: octagon + lozenge

Code: Select all: octagon + lozenge: +-------+-------+-------+ ! . . X ! X X X ! X . . ! ! . X . ! . . . ! . X . ! ! X . . ! . . . ! . . X ! +-------+-------+-------+ ! X . . ! . X . ! . . X ! ! X . . ! X . X ! . . X ! ! X . . ! . X . ! . . X ! +-------+-------+-------+ ! X . . ! . . . ! . . X ! ! . X . ! . . . ! . X . ! ! . . X ! X X X ! X . . ! +-------+-------+-------+ --XXXXX---X-----X-X-------XX---X---XX--X-X--XX---X---XX-------X-X-----X---XXXXX-- 28 givens

This time, I could easily generate 1000 puzzles, most of which are very easy and the hardest of which has SER 8.5:

Code: Select all: ..39521...6.....5.9.......87...6...56..2.3..44...1...23.......1.8.....2...75468.. # 300 FNBTY C28.M/S8.f

Variant 5: open octagon + lozenge

Code: Select all: open octagon + lozenge: +-------+-------+-------+ ! . . X ! X . X ! X . . ! ! . X . ! . . . ! . X . ! ! X . . ! . . . ! . . X ! +-------+-------+-------+ ! X . . ! . X . ! . . X ! ! . . . ! X . X ! . . . ! ! X . . ! . X . ! . . X ! +-------+-------+-------+ ! X . . ! . . . ! . . X ! ! . X . ! . . . ! . X . ! ! . . X ! X . X ! X . . ! +-------+-------+-------+ --XX-XX---X-----X-X-------XX---X---X---X-X---X---X---XX-------X-X-----X---XX-XX-- 24 givens

Again, I could easily generate 1000 puzzles, most of which are relatively easy and the hardest two of which have SER 9.0:

Code: Select all: ..67.53...4.....6.8.......25...6...8...3.1...3...7...91.......7.3.....9...29.85.. # 95732 FNBTXYK C24.m/S8.f ..92.31...6.....4.7.......34...2...6...9.4...1...3...56.......2.1.....8...48.73.. # 95028 FNBTWYK C24.m/S8.f

Variant 6: octagon + square

Code: Select all: octagon + square: +-------+-------+-------+ ! . . X ! X X X ! X . . ! ! . X . ! . . . ! . X . ! ! X . . ! . . . ! . . X ! +-------+-------+-------+ ! X . . ! X . X ! . . X ! ! X . . ! . . . ! . . X ! ! X . . ! X . X ! . . X ! +-------+-------+-------+ ! X . . ! . . . ! . . X ! ! . X . ! . . . ! . X . ! ! . . X ! X X X ! X . . ! +-------+-------+-------+ --XXXXX---X-----X-X-------XX--X-X--XX-------XX--X-X--XX-------X-X-----X---XXXXX-- 28 givens

Again, I could easily generate 1000 puzzles, the hardest 12 of which have SER 7.2, e.g.:

Code: Select all: ..34278...4.....7.6.......37..5.9..21.......94..7.1..83.......4.6.....3...41839.. # 307 FNBTY C28.M/S8.f

Variant 7: open octagon + square

Code: Select all: octagon + square: +-------+-------+-------+ ! . . X ! X . X ! X . . ! ! . X . ! . . . ! . X . ! ! X . . ! . . . ! . . X ! +-------+-------+-------+ ! X . . ! X . X ! . . X ! ! . . . ! . . . ! . . . ! ! X . . ! X . X ! . . X ! +-------+-------+-------+ ! X . . ! . . . ! . . X ! ! . X . ! . . . ! . X . ! ! . . X ! X . X ! X . . ! +-------+-------+-------+ --XX-XX---X-----X-X-------XX--X-X--X---------X--X-X--XX-------X-X-----X---XX-XX-- 24 givens

Again, I could easily generate 1000 puzzles, the hardest of which have SER 9.0:

Code: Select all: ..37.24...7.....9.1.......87..3.4..2.........9..1.5..78.......5.2.....1...69.18.. # 2075 FNBTHXY C24.m/S8.f

TOPICS FOR THIS THREAD (non restrictive)
- Finding general families of perfectly symmetric patterns
- Finding interesting patterns in these families (interesting variants of the root pattern(s)
- Finding interesting puzzles for these variants
- [Added]: Finding references to previous work on the above three topics.

In the present context, "interesting puzzles" may mean any of the following (non restrictive):
- has some noticeable resolution path
- uses some exotic pattern for resolution (sk-loop, J-Excocet, Oddagon, ...)
- highest rating (SER or W or...) for this pattern
- has a close neighbour (Hamming distance ≤ 1 or 2) with SER ≥ 11.7 (but strictly speaking, the neighbour is not a member of the family)
- ...
[Edit]: Following some answers to this introductory post, I re-ordered the above list: in the first list, "high rating" appeared first but it is not the main point - though it remains one point of interest.

LIST OF PERFECTLY SYMMETRIC FAMILIES
This section will be updated every time a new family is introduced.

The OCTAGON family: this post
The DIAGONALS family:
The LOZENGE family:
The SLASHES family:
The WAVES family:

[Edit]: it appears that much systematic work has already been done on fully-symmetric puzzles. As explained in the "TOPICS FOR THE THREAD" section, my purpose here is different. It is also different from the Patterns Game purposes.

[Added references mentioned in subsequent posts]:
REFERENCES
On Sudoku symmetries in general: http://forum.enjoysudoku.com/sudoku-symmetry-formalized-t3667.html
A list of valid and invalid fully symmetrical patterns: http://forum.enjoysudoku.com/fully-symmetrical-invalid-patterns-t33569-53.html
http://forum.enjoysudoku.com/ask-for-patterns-that-they-dont-have-puzzles-2-t31323.html
http://forum.enjoysudoku.com/fully-symmetrical-puzzles-t3721.html#p24915
Beyond the topics of this thread, but nice patterns also: http://forum.enjoysudoku.com/sudokus-with-an-original-rare-shape-t4147.html

Website · by **denis_berthier** » Sun Nov 29, 2020 12:30 pm

THE DIAGONALS FAMILY

The diagonal root pattern

Code: Select all: +-------+-------+-------+ ! X . . ! . . . ! . . X ! ! . X . ! . . . ! . X . ! ! . . X ! . . . ! X . . ! +-------+-------+-------+ ! . . . ! X . X ! . . . ! ! . . . ! . X . ! . . . ! ! . . . ! X . X ! . . . ! +-------+-------+-------+ ! . . X ! . . . ! X . . ! ! . X . ! . . . ! . X . ! ! X . . ! . . . ! . . X ! +-------+-------+-------+ X-------X-X-----X---X---X-----X-X-------X-------X-X-----X---X---X-----X-X-------X 17 givens

This root pattern doesn't easily give any puzzle

Variant 1: diagonal + centres

Code: Select all: +-------+-------+-------+ ! X . . ! . . . ! . . X ! ! . X . ! . X . ! . X . ! ! . . X ! . . . ! X . . ! +-------+-------+-------+ ! . . . ! X . X ! . . . ! ! . X . ! . X . ! . X . ! ! . . . ! X . X ! . . . ! +-------+-------+-------+ ! . . X ! . . . ! X . . ! ! . X . ! . X . ! . X . ! ! X . . ! . . . ! . . X ! +-------+-------+-------+ X-------X-X--X--X---X---X-----X-X----X--X--X----X-X-----X---X---X--X--X-X-------X 21 givens

[Edit]: as mentioned by Mike, it seems that this variant first appeared in the Patterns Game: http://forum.enjoysudoku.com/patterns-game-1-0-t5209-1095.html

This variant easily gave me 1000 puzzles. Their two highest SER are 10 and 9.8:

Code: Select all: 2.......8.3..2..1...6...4.....3.8....1..6..7....4.1.....4...6...7..9..3.8.......2 # 95550 FNBTHWXYKG C21.m/S8.f/M1.43.1 SER 10.0 1.......8.7..5..4...6...3.....7.6....4..9..2....2.5.....3...6...2..7..5.8.......1 # 92244 FNBTHWXYG C21.m/S8.f/M1.40.1 SER 9.8

An SER 8.9 example of this pattern appears here: http://forum.enjoysudoku.com/diagonals-centres-t38366.html

Other interesting variants are very likely to be found.

Website · by **denis_berthier** » Sun Nov 29, 2020 12:33 pm

THE LOZENGE FAMILY

This family has two root patterns:

The lozenge root pattern

Code: Select all: +-------+-------+-------+ ! . . . ! . X . ! . . . ! ! . . . ! X . X ! . . . ! ! . . X ! . . . ! X . . ! +-------+-------+-------+ ! . X . ! . . . ! . X . ! ! X . . ! . . . ! . . X ! ! . X . ! . . . ! . X . ! +-------+-------+-------+ ! . . X ! . . . ! X . . ! ! . . . ! X . X ! . . . ! ! . . . ! . X . ! . . . ! +-------+-------+-------+ ----X-------X-X-----X---X---X-----X-X-------X-X-----X---X---X-----X-X-------X---- 16 givens

As it has less than 17 givens, this pattern has no chance of having puzzles.

The lozenge tiling root pattern
So named because, if copied horizontally and vertically, it would generate a semi-regular tiling of the plane (with lozenges of two different sizes).

Code: Select all: +-------+-------+-------+ ! X . . ! . X . ! . . X ! ! . X . ! X . X ! . X . ! ! . . X ! . . . ! X . . ! +-------+-------+-------+ ! . X . ! . . . ! . X . ! ! X . . ! . . . ! . . X ! ! . X . ! . . . ! . X . ! +-------+-------+-------+ ! . . X ! . . . ! X . . ! ! . X . ! X . X ! . X . ! ! X . . ! . X . ! . . X ! +-------+-------+-------+ X---X---X-X-X-X-X---X---X---X-----X-X-------X-X-----X---X---X---X-X-X-X-X---X---X 24 givens

This pattern easily gives puzzles.

Variant of the lozenge tiling pattern: the lozenge tiling with inner small lozenge

Code: Select all: +-------+-------+-------+ ! X . . ! . X . ! . . X ! ! . X . ! X . X ! . X . ! ! . . X ! . . . ! X . . ! +-------+-------+-------+ ! . X . ! . X . ! . X . ! ! X . . ! X . X ! . . X ! ! . X . ! . X . ! . X . ! +-------+-------+-------+ ! . . X ! . . . ! X . . ! ! . X . ! X . X ! . X . ! ! X . . ! . X . ! . . X ! +-------+-------+-------+ X---X---X-X-X-X-X---X---X---X--X--X-X--X-X--X-X--X--X---X---X---X-X-X-X-X---X---X 28 givens

This pattern easily gives puzzles, but it takes much more time than e.g. the lozenge tiling, the octagon+lozenge or the octagon+square patterns. Its puzzles are harder in the mean than those of the above patterns.
I have proposed a puzzle with this pattern here: http://forum.enjoysudoku.com/lozenges-8-3-t38439.html, according to the criteria that it allows a solution with simple bivalue-chains and long oddagons (an exotic pattern).
In the same thread, Mike has given a beautiful puzzle with the same pattern AND symmetry of the givens.

Website · by **denis_berthier** » Sun Nov 29, 2020 12:34 pm

THE SLASHES FAMILY

The slashes root pattern

Code: Select all: +-------+-------+-------+ ! . . X ! . . X ! . . X ! ! . X . ! . X . ! . X . ! ! X . . ! X . . ! X . . ! +-------+-------+-------+ ! . . X ! . . X ! . . X ! ! . X . ! . X . ! . X . ! ! X . . ! X . . ! X . . ! +-------+-------+-------+ ! . . X ! . . X ! . . X ! ! . X . ! . X . ! . X . ! ! X . . ! X . . ! X . . ! +-------+-------+-------+ --X--X--X-X--X--X-X--X--X----X--X--X-X--X--X-X--X--X----X--X--X-X--X--X-X--X--X-- 27 givens

This root pattern has puzzles. One of them (SER=9.3, W=gW=B=11) was given in the Puzzles part of this forum: http://forum.enjoysudoku.com/slashes-t38406.html

Website · by **denis_berthier** » Sun Nov 29, 2020 12:35 pm

THE WAVES FAMILY

The waves root pattern

Code: Select all: +-------+-------+-------+ ! . . X ! X . . ! . . X ! ! . X . ! . X . ! . X . ! ! X . . ! . . X ! X . . ! +-------+-------+-------+ ! . . X ! X . . ! . . X ! ! . X . ! . X . ! . X . ! ! X . . ! . . X ! X . . ! +-------+-------+-------+ ! . . X ! X . . ! . . X ! ! . X . ! . X . ! . X . ! ! X . . ! . . X ! X . . ! +-------+-------+-------+ --XX----X-X--X--X-X----XX----XX----X-X--X--X-X----XX----XX----X-X--X--X-X----XX-- 27 givens

Three puzzles of this pattern were given in the Puzzles part of this forum http://forum.enjoysudoku.com/waves-4-4-t38429.html, http://forum.enjoysudoku.com/waves-7-2-t38427.html and http://forum.enjoysudoku.com/waves-9-3-t38428.html

by **Serg** » Sun Nov 29, 2020 2:00 pm

Hi, Denis!
It looks strange that you didn't give precise definition of symmetries present in your examples. For example, "Octagon" has vertical, horisontal, diagonal, antidiagonal, cental and quarter-turn symmetries. Such patterns were discussed in the thread Fully symmetrical invalid patterns in 2017. Two lists of "basic" patterns were published there - 25 invalid patterns hot having valid puzzles and 49 valid patterns having valid puzzles. If any fully symmetrical pattern is subset of some "basic" invalid pattern, it has no valid puzzles. If any fully symmetrical pattern is superset of "basic" valid patterns, it has at least one valid puzzle.

Serg

by **Afmob** » Sun Nov 29, 2020 2:04 pm

An important reference would be: http://forum.enjoysudoku.com/ask-for-patterns-that-they-dont-have-puzzles-2-t31323.html

Variants 1-3 of the octogon family are subsets of the "magic pattern" and therefore have no valid puzzles.

Edit: As Serg has written, I would suggest to go through the links provided which will answer some of your questions.

by **JPF** » Sun Nov 29, 2020 2:22 pm

You could have a look at this thread about Fully symmetrical puzzles too...

JPF

Website · by **denis_berthier** » Sun Nov 29, 2020 3:24 pm

Hi, Serg, Afmob, JPF,

Thanks for the references. I'll see how to use all this material.

by **m_b_metcalf** » Sun Nov 29, 2020 4:28 pm

denis_berthier wrote:In the present context, "interesting puzzles" may mean any of the following (non restrictive):
- highest rating (SER or W or...) for this pattern

Denis,
This is what the Pattern Game is all about. It contains some highly symmetrical patterns with high ratings, for instance Game 385. You could check on all the played patterns (obtainable on demand with a

Code: Select all: ::: patterns :::

post).

Regards,

Mike

P.S. Here some higher ratings of two of your patternS

Code: Select all: . . 1 2 . 3 4 . . . 5 . . . . . 6 . 7 . . . . . . . 8 8 . . . 3 . . . 9 . . . 7 . 4 . . . 3 . . . 9 . . . 2 2 . . . . . . . 4 Symmetry of givens! . 6 . . . . . 1 . . . 7 9 . 8 5 . . Minimal, ED=9.1/9.1/9.0 . . 1 2 . 3 4 . . . 5 . . . . . 1 . 6 . . . . . . . 7 3 . . . 1 . . . 6 . . . 5 . 8 . . . 4 . . . 3 . . . 8 8 . . . . . . . 2 . 4 . . . . . 9 . . . 7 9 . 6 3 . . Minimal, ED=9.2/9.2/2.6 . . 1 2 . 3 4 . . . 3 . . . . . 5 . 6 . . . . . . . 7 2 . . 1 . 4 . . 8 . . . . . . . . . 5 . . 6 . 9 . . 2 7 . . . . . . . 3 . 8 . . . . . 6 . . . 9 5 . 6 7 . . Minimal, ED=9.1/2.0/2.0

Website · by **denis_berthier** » Sun Nov 29, 2020 4:53 pm

m_b_metcalf wrote:
denis_berthier wrote:In the present context, "interesting puzzles" may mean any of the following (non restrictive):
- highest rating (SER or W or...) for this pattern

This is what the Pattern Game is all about. It contains some highly symmetrical patterns with high ratings, for instance Game 385. You could check on all the played patterns (obtainable on demand with a
Code: Select all
::: patterns :::
post).

I see two differences with the patterns game:
- highest rating is only one of the reasons for a puzzle being interesting,
- the pattterns I consider here are much more restricted than in the patterns game.

Thanks for the examples.

Probably, part of this thread will consist of extracting information from other threads.

by **m_b_metcalf** » Sun Nov 29, 2020 5:07 pm

denis_berthier wrote:
This variant easily gave me 1000 puzzles. Their two highest SER are 10 and 9.8:
Code: Select all
2.......8.3..2..1...6...4.....3.8....1..6..7....4.1.....4...6...7..9..3.8.......2 # 95550 FNBTHWXYKG C21.m/S8.f/M1.43.1 SER 10.0 1.......8.7..5..4...6...3.....7.6....4..9..2....2.5.....3...6...2..7..5.8.......1 # 92244 FNBTHWXYG C21.m/S8.f/M1.40.1 SER 9.8

An SER 8.9 example of this pattern appears here: http://forum.enjoysudoku.com/diagonals-centres-t38366.html

And a whole list of them if you scroll down to the results of this very early Patterns Game.

Regards,

Mike

Website · by **denis_berthier** » Mon Nov 30, 2020 5:27 am

Serg wrote:It looks strange that you didn't give precise definition of symmetries present in your examples. For example, "Octagon" has vertical, horisontal, diagonal, antidiagonal, cental and quarter-turn symmetries.

I didn't want to be too precise about symmetries allowed in this thread. I've added "visual" to the conditions, but this remains very vague (on purpose). In my view, it includes all the algebraic Sudoku symmetries (restricted to patterns), e.g. permutations of floors, as will be clear from the "Waves" family, the symmetry of which is not "geometrical".

Serg wrote:Such patterns were discussed in the thread Fully symmetrical invalid patterns in 2017. Two lists of "basic" patterns were published there - 25 invalid patterns hot having valid puzzles and 49 valid patterns having valid puzzles.

This will be of great help for the present thread and I'll use this before publishing more about the other Families.
To be more precise, skipping the beginning of the thread, the main results are here: http://forum.enjoysudoku.com/fully-symmetrical-invalid-patterns-t33569-53.html and in the next 4 or 5 posts. Can you confirm that those are the final lists of valid and invalid fully-symmetrical patterns?

When I started to be interested in perfect symmetries, it was more for fun than anything else. I wasn't aware that you (and other people) had done such systematic work on them. A good opportunity for me to learn new things. Thanks.

Serg wrote:If any fully symmetrical pattern is subset of some "basic" invalid pattern, it has no valid puzzles. If any fully symmetrical pattern is superset of "basic" valid patterns, it has at least one valid puzzle.

When you say "it has valid puzzles", do you mean "not necessarily minimal ones", "not necessarily symmetrically-minimal ones" or merely (as I understand it) valid ones (i.e.single-solution)?
Can you confirm that the above referenced patterns are resp. maximal/minimal-symmetric?

Website · by **denis_berthier** » Mon Nov 30, 2020 5:33 am

m_b_metcalf wrote:You could check on all the played patterns (obtainable on demand with a
Code: Select all
::: patterns :::
post).

I tried this, but it doesn't seem to work (or maybe it takes time for getting an answer?)

It seems difficult to use the Patterns Game thread. It has 46,000 posts and probably several thousand patterns. Is there any way to search it for some specific pattern?

Website · by **denis_berthier** » Mon Nov 30, 2020 5:53 am

Following the above remarks, I've substantially completed my introductory post. In particular, I've stated some differences in purpose with a systematic search for symmetric patterns (a work already done) and with the Patterns Game.
Let me know if there is still anything obscure.

The New Sudoku Players' Forum

Perfectly symmetric patterns

Perfectly symmetric patterns

THE DIAGONALS FAMILY

THE LOZENGE FAMILY

THE SLASHES FAMILY

THE WAVES FAMILY

Re: Perfectly symmetric patterns

Re: Perfectly symmetric patterns

Re: Perfectly symmetric patterns

Re: Perfectly symmetric patterns

Re: Perfectly symmetric patterns

Re: Perfectly symmetric patterns

Re: THE DIAGONALS FAMILY

Re: Perfectly symmetric patterns

Re: Perfectly symmetric patterns

Re: Perfectly symmetric patterns