## Double lozenge tiling with inner patterns 8.4

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### Double lozenge tiling with inner patterns 8.4

Remember the easy double lozenge tiling with alternating inner lozenges and octagons?
(see here: http://forum.enjoysudoku.com/double-lozenge-tiling-with-inner-patterns-3-8-t38480.html

The pattern is as follows:

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 . !    +-------+-------+-------+29 givens`

When the pattern is repeated horizontally and vertically, it gives a semi-regular tiling of the plane (with lozenges of two sizes), each tile having an inner pattern: an octagon for the larger tiles and a lozenge for the smaller tiles:
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 . . 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 . . 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 . . 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 . . 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 . . 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 . . 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 . . 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 . . 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 . . 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 . . 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 . . 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 . . 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 . . 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 . . 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 . . 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 . . 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 . . 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 .`

I realise that, if you use this pattern as wallpaper, you'll soon get headaches.
However, if you look at it as at a stereogram, you can see it appear farther than the screen.
(There are several techniques for looking that way. Mine is merely looking at infinity.)

The second puzzle I'll propose with this pattern is harder, but solvable with (long) z-chains:

Code: Select all
`   +-------+-------+-------+   ! . 3 . ! . 7 . ! . 2 . !   ! 8 . . ! 4 . 9 ! . . 5 !   ! . . 9 ! . . . ! 8 . . !   +-------+-------+-------+   ! . 9 . ! . 8 . ! . 4 . !   ! 7 . . ! 2 . 6 ! . . 8 !   ! . 8 . ! . 5 . ! . 3 . !   +-------+-------+-------+   ! . . 1 ! . . . ! 6 . . !   ! 9 . . ! 7 . 3 ! . . 2 !   ! . 2 . ! . 1 . ! . 9 . !   +-------+-------+-------+.3..7..2.8..4.9..5..9...8...9..8..4.7..2.6..8.8..5..3...1...6..9..7.3..2.2..1..9. # 95097 FNBTHWYK C28.m/S8.f29 givens, SER = 8.4, W = 7, Z = 12`
Last edited by denis_berthier on Mon Dec 14, 2020 8:55 am, edited 1 time in total.
denis_berthier
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### Re: Double lozenge tiling with inner patterns 8.4

There is also more than one pattern that can be seen. The "harder" one has the middle two rows lined up on top of each other and vertical columns are clear but the rest of the pattern is still a bit blurred.

Leren
Leren

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Joined: 03 June 2012

### Re: Double lozenge tiling with inner patterns 8.4

Leren wrote:There is also more than one pattern that can be seen. The "harder" one has the middle two rows lined up on top of each other and vertical columns are clear but the rest of the pattern is still a bit blurred.

Leren

I can't see your second pattern. Do you see it in 3D also?
I've added two flights of 9 lines each. Is it better now?
denis_berthier
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### Re: Double lozenge tiling with inner patterns 8.4

Actually the second pattern is better. In it there are two clear rows, together and the X's above and below them form an octagon, with each fourth column clearly defined. The easy pattern is unchanged.

This often happens with 3D pictures. They often have a second pattern that is not supposed to be seen but it is there anyway. The reason is that when you see the easy pattern you are not really looking at infinity, but at a point further back than the screen. If you relax your eyes even more, and look even further back, the second pattern comes into focus, but part of it is blurry, because it's not what the designer has intended you to see, and some of the X's don't line up.

In fact I think there are now at least three patterns to be seen, with the second and third having minor differences. In fact it is becoming harder for me to see the easy pattern as my eyes get used to "focusing" on the other patterns.

All patterns are in 3D with the clear part of the second and third patterns "further away than the blurry parts of them.

The phenomenon is (at least metaphorically) similar to harmonics. The easy pattern is the major harmonic and the harder patterns are minor harmonics.

Leren
Last edited by Leren on Mon Dec 14, 2020 9:51 am, edited 1 time in total.
Leren

Posts: 4006
Joined: 03 June 2012

### Re: Double lozenge tiling with inner patterns 8.4

Leren wrote:Actually the second pattern is better. In it there are two clear rows, together and the X's above and below them form an octagon, with each fourth column clearly defined. The easy pattern is unchanged.
This often happens with 3D pictures. They often have a second pattern that is not supposed to be seen but it is there anyway. The reason is that when you see the easy pattern you are not really looking at infinity, but at a point further back than the screen. If you relax your eyes even more, and look even further back, the second pattern comes into focus, but part of it is blurry, because it's not what the designer has intended you to see, and some of the X's don't line up.
In fact I think there are now at least three patterns to be seen, with the second and third having minor differences. In fact it is becoming harder for me to see the easy pattern as my eyes get used to "focusing" on the other patterns.

I finally see "something" farther away from the screen, but it's very unstable. I'll try again. I had never thought that there might be something else hidden behind the first hidden 3D level. Fascinating idea.
denis_berthier
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Posts: 1983
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Location: Paris

### Re: Double lozenge tiling with inner patterns 8.4

Once you get the hang of it, the original pattern becomes harder to see. You see a pattern when your left eye sees an X and your right eye sees another X that is not the same one, but since they are the same shape you brain is fooled into thinking they are the same X. The point is, there is more than one way of lining up different sets of X's. With the primary pattern every X lines up with another X, whereas with the second pattern, some of the X's line up and some don't, so you see a partial clear and partial blurry pattern. The blurry part is when only one eye sees an X and the other sees the white background. Try and look at the screen sideways and see what comes up. Hey, this makes a good break from puzzle solving.

Also, if you haven't seen the incredible Marylin Monroe / Albert Einstein image you can view it here.

Leren
Leren

Posts: 4006
Joined: 03 June 2012

### Re: Double lozenge tiling with inner patterns 8.4

Leren wrote:Once you get the hang of it, the original pattern becomes harder to see. You see a pattern when your left eye sees an X and your right eye sees another X that is not the same one, but since they are the same shape you brain is fooled into thinking they are the same X. The point is, there is more than one way of lining up different sets of X's. With the primary pattern every X lines up with another X, whereas with the second pattern, some of the X's line up and some don't, so you see a partial clear and partial blurry pattern. The blurry part is when only one eye sees an X and the other sees the white background. Try and look at the screen sideways and see what comes up. Hey, this makes a good break from puzzle solving.

I made a copy with only two flights of 9 lines. It makes it easier to see the other pattern, but even so, for me, it remains very unstable. There's a strong attraction to the first 3D pattern.

Leren wrote:Also, if you haven't seen the incredible Marylin Monroe / Albert Einstein image you can view it here.

Too creepy!
denis_berthier
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Location: Paris

### Re: Double lozenge tiling with inner patterns 8.4

As nobody has provided a solution yet, here is one of mine, the solution in W7, with Subsets and all the rules for special cases of whips activated (there's an alternative solution in Z12, i.e. using only z-chains of lengths ≤ 12).

Notice that, as a puzzle, this one has nothing noticeable. It's a typical SER 8.4. I proposed it here mainly for the nice stereogram.

Code: Select all
`(solve-sudoku-grid   +-------+-------+-------+   ! . 3 . ! . 7 . ! . 2 . !   ! 8 . . ! 4 . 9 ! . . 5 !   ! . . 9 ! . . . ! 8 . . !   +-------+-------+-------+   ! . 9 . ! . 8 . ! . 4 . !   ! 7 . . ! 2 . 6 ! . . 8 !   ! . 8 . ! . 5 . ! . 3 . !   +-------+-------+-------+   ! . . 1 ! . . . ! 6 . . !   ! 9 . . ! 7 . 3 ! . . 2 !   ! . 2 . ! . 1 . ! . 9 . !   +-------+-------+-------+)`

Hidden Text: Show
***********************************************************************************************
*** SudoRules 20.1.s based on CSP-Rules 2.1.s, config = W+SFin
*** Using CLIPS 6.32-r779
***********************************************************************************************
whip[1]: c8n6{r3 .} ==> r3c9 ≠ 6, r1c9 ≠ 6
swordfish-in-columns: n6{c2 c5 c8}{r3 r8 r2} ==> r8c3 ≠ 6, r3c4 ≠ 6, r3c1 ≠ 6, r2c3 ≠ 6
swordfish-in-rows: n1{r2 r5 r8}{c8 c2 c7} ==> r6c7 ≠ 1, r4c7 ≠ 1, r3c8 ≠ 1, r3c2 ≠ 1, r1c7 ≠ 1
biv-chain-bn[3]: b7n7{r7c2 r9c3} - b7n8{r9c3 r8c3} - b9n8{r8c8 r7c8} ==> r7c8 ≠ 7
whip[1]: c8n7{r3 .} ==> r2c7 ≠ 7, r3c9 ≠ 7
hidden-pairs-in-a-block: b3{r2c8 r3c8}{n6 n7} ==> r2c8 ≠ 1
biv-chain[3]: r7n2{c6 c5} - c5n9{r7 r5} - b5n4{r5c5 r6c6} ==> r7c6 ≠ 4
z-chain[4]: c6n2{r7 r3} - r2n2{c5 c3} - c3n7{r2 r9} - r9n8{c3 .} ==> r7c6 ≠ 8
whip[5]: c4n6{r1 r9} - r8c5{n6 n4} - c6n4{r9 r6} - c6n1{r6 r4} - c6n7{r4 .} ==> r1c4 ≠ 1
whip[5]: c9n6{r4 r6} - c9n9{r6 r1} - c9n1{r1 r3} - b2n1{r3c6 r1c6} - r4c6{n1 .} ==> r4c9 ≠ 7
whip[7]: r7c6{n5 n2} - r3c6{n2 n1} - r3c4{n1 n3} - r3c9{n3 n4} - r1c7{n4 n9} - r5n9{c7 c5} - c5n3{r5 .} ==> r1c6 ≠ 5
t-whip[6]: b2n5{r3c6 r1c4} - c4n6{r1 r9} - c4n8{r9 r7} - r9n8{c6 c3} - c3n7{r9 r2} - b1n2{r2c3 .} ==> r3c1 ≠ 5
whip[7]: r7n2{c6 c5} - c5n9{r7 r5} - r5n3{c5 c3} - r5n4{c3 c2} - r7c2{n4 n7} - c3n7{r9 r2} - r2n2{c3 .} ==> r7c6 ≠ 5
naked-single ==> r7c6 = 2
z-chain[4]: r3c6{n1 n5} - r3c4{n5 n3} - b3n3{r3c9 r2c7} - r2n1{c7 .} ==> r3c1 ≠ 1
whip[6]: r3c6{n5 n1} - r3c4{n1 n3} - b5n3{r4c4 r5c5} - r5n9{c5 c7} - r1c7{n9 n4} - r3c9{n4 .} ==> r1c4 ≠ 5
whip[1]: b2n5{r3c6 .} ==> r3c2 ≠ 5
t-whip[5]: r4c9{n6 n1} - r5n1{c8 c2} - c1n1{r6 r1} - b1n5{r1c1 r1c3} - b4n5{r4c3 .} ==> r4c1 ≠ 6
whip[6]: c2n5{r8 r5} - r5c8{n5 n1} - r5c7{n1 n9} - c5n9{r5 r7} - r7c4{n9 n8} - r7c8{n8 .} ==> r7c1 ≠ 5
t-whip[6]: r3n4{c2 c9} - r3c1{n4 n2} - r2c3{n2 n7} - b7n7{r9c3 r7c2} - r7c9{n7 n3} - r7c1{n3 .} ==> r1c1 ≠ 4
z-chain[4]: c3n7{r9 r2} - b1n2{r2c3 r3c1} - c1n4{r3 r6} - c6n4{r6 .} ==> r9c3 ≠ 4
whip[6]: r5n9{c7 c5} - r7c5{n9 n4} - r7c1{n4 n3} - r7c9{n3 n7} - b6n7{r6c9 r4c7} - c7n2{r4 .} ==> r6c7 ≠ 9
z-chain[7]: b8n5{r9c6 r7c4} - c4n9{r7 r6} - c9n9{r6 r1} - r1c7{n9 n4} - r8c7{n4 n1} - c8n1{r8 r5} - c8n5{r5 .} ==> r9c7 ≠ 5
naked-triplets-in-a-block: b9{r7c9 r9c7 r9c9}{n3 n4 n7} ==> r8c7 ≠ 4
whip[5]: r1c7{n4 n9} - r5n9{c7 c5} - r5n4{c5 c2} - r8n4{c2 c5} - r7c5{n4 .} ==> r1c3 ≠ 4
whip[1]: r1n4{c9 .} ==> r3c9 ≠ 4
naked-pairs-in-a-block: b3{r2c7 r3c9}{n1 n3} ==> r1c9 ≠ 1
naked-triplets-in-a-row: r3{c4 c6 c9}{n3 n5 n1} ==> r3c5 ≠ 3
biv-chain[3]: r2c7{n1 n3} - c5n3{r2 r5} - r5n9{c5 c7} ==> r5c7 ≠ 1
biv-chain-cn[4]: c5n3{r5 r2} - c7n3{r2 r9} - c7n4{r9 r1} - c7n9{r1 r5} ==> r5c5 ≠ 9
singles ==> r6c4 = 9, r5c7 = 9, r1c7 = 4, r1c9 = 9, r7c5 = 9
naked-pairs-in-a-row: r7{c4 c8}{n5 n8} ==> r7c2 ≠ 5
hidden-pairs-in-a-column: c4{n1 n3}{r3 r4} ==> r3c4 ≠ 5
hidden-single-in-a-block ==> r3c6 = 5
finned-x-wing-in-columns: n5{c2 c7}{r8 r5} ==> r5c8 ≠ 5
stte

.
denis_berthier
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