Technic Tessellations

[DrJB]

Some of my favorite parts are below, and I have a relatively large collection of them.

      

I sometimes use them to build 'periodic' structures and, while doing so, and 3D printing some Escher 'lizards' in the background, the idea of Tessellation came up, and I did the two shapes below.

What other 'tessellations' you know of (or you can come up with) using primarily the parts listed above? Of course, you need not restrict yourself to only those.

 

Edited December 27, 2020 by DrJB

[Technic Jim]

There are some similar ideas in this topic:

https://www.eurobricks.com/forum/index.php?/forums/topic/149048-lego-technic-chemistry-lab-the-lego-molecular-system/

[nerdsforprez]

Of which the OP was a main contributor to 

This thread could probably be closed

[aeh5040]

The other thread was mainly about 3D structures.  I think it makes perfect sense to have a different thread for 2D tesselations.

Those are among my favorite parts too, and posts from DrJB have an unfortunate tendency to trigger an expensive spending spree!

As it happens I have made _exactly_ those two before!  I'll see what else I can come up with...

 

One interesting challenge would be to come up with examples of each of the 17 wallpaper symmetry groups.

Edited December 16, 2017 by aeh5040

[aeh5040]

[DrJB]

7 hours ago, aeh5040 said:

The other thread was mainly about 3D structures.  I think it makes perfect sense to have a different thread for 2D tesselations.

Those are among my favorite parts too, and posts from DrJB have an unfortunate tendency to trigger an expensive spending spree!

As it happens I have made _exactly_ those two before!  I'll see what else I can come up with...

 

One interesting challenge would be to come up with examples of each of the 17 wallpaper symmetry groups.

Thank you for the kind words ... and sorry about the added expenses. On the plus side, many of those connectors are now available 'primarily' in monochromatic colors (grey/white/black) and one needs not invest too much. That said, I'm collecting parts to reproduce your kinetic sculpture (beautiful work) but difficult to get the orange connectors, especially over this side of the Atlantic.

And, for 17 groups, another geometric fascination of mine, I found the app below (iOrnament) sometime back. One can literally spend hours on it. I have not attempted to understand the math behind the reason for only 17 though. In the paid version, he extends the 17 groups to projections on a sphere. 

https://itunes.apple.com/us/app/iornament/id534529876?mt=8

[DrJB]

5 hours ago, aeh5040 said:

Beautiful! We have 3 solutions so far.

I'm working on a BL order ...

[Aleh]

Hm, simply beautiful! Any ideas how th keep this vertically? 

[Gnac]

18 hours ago, aeh5040 said:

https://bricksafe.com/files/aeh/pythagoras/tess.jpg

Ahh, the good ol' Olympic Triskelion!

[DrJB]

Here is another one, using the 'same' basic elements as aeh5040 ... though less compact.

 

[aeh5040]

2 hours ago, DrJB said:

Here is another one, using the 'same' basic elements as aeh5040 ... though less compact.

Nice!  And of course the same can be done with 135 deg connectors throughout.

Thinking a bit more about the 17 wallpaper groups project: In the case of the first picture in this thread (the hexagonal lattice), there is a possible ambiguity about which symmetry group to assign it to.  If we consider symmetries at the level or actual parts, so the brown connector is regarded as different from the LBG one, then I think it is p3m1.  If we just look at the overall shape, and don't distinguish between axles holes and pin holes, it is p6m (in particular it has 6 fold symmetry).

If we want to do it right, I suggest ambiguous cases like this should be avoided...

On 12/15/2017 at 8:53 PM, DrJB said:

... while doing so, and 3D printing some Escher 'lizards' in the background, the idea came up ...

That's a delightful image, by the way...    I wouldn't mind seeing them too.

Edited December 18, 2017 by aeh5040

[aeh5040]

Here are p6m and p6:

[DrJB]

Very nice ... I need to go read a bit more about the 17 Groups :)

I tried few more iterations in LDD and most/all figures I came up with have the p6 symmetry. Essentially, all figures can be reduced to a hexagon with some curvy/crazy edges. This for sure has to do with the parts available to us. Then I discovered a NEW (to me) part on Bricklink. By combining the two parts below, we can then do p4 groups. Pushing this further, we can even try tesselations of 2 groups of shapes.

I can't try the combinations below as the part does NOT exist in LDD yet ... but, I just placed an order for few of them. The holidays for sure will include sometime with Lego.

 +  

By combining the above two parts, we essentially have the 4-branches equivalent of . One can try this other part  ... but I have none. Correction: Found it, in fact, I have 50+ ... that I got some time back from the PAB wall.

Edited December 20, 2017 by DrJB

[DrJB]

Found two more, moving away from p6 to p2/p4 (need to check terminology). The first one is rather 'trivial'. I like the second one, it's starting to look like Moorish Zellige (Tiling/Tesselation) with more than one basic shape.

 

 

Edited December 19, 2017 by DrJB

[Aleh]

Looks cool 

[Didumos69]

Some great tessalation already! Nice winter brain training . Here's my first one:

Edited December 19, 2017 by Didumos69

[DrJB]

Beautiful. I knew you'd join us ... sooner or later :)

Here is another one, the so-called 'chicken-foot' pattern. It's often seen on fabrics/curtains/wallpapers ... p4 I think

Edited December 20, 2017 by DrJB
Milan: Converted oversized image into the link. DrJB: Uploaded smaller photo

[zux]

7 hours ago, Didumos69 said:

Some great tessalation already! Nice winter brain training . Here's my first one

While all of posted creation are very good I like this one the most, probably due to color coding.

[Erik Leppen]

7 hours ago, Didumos69 said:

Some great tessalation already! Nice winter brain training . Here's my first one:

 

What's interesting about this one is that it fits. I was looking at it for a bit until I realized it's not an exact fit, but close. This can be proven using some mathematics. For those wanting to know:

A hexagon consists of 6 equilateral triangles, So, for a hexagon, the "radius" (center-to-corner distance) is equal to the side length. The yellow hexagon has sides of length 3, so the radius is 3 too. The blue hexagon has sides of length 8, so the radius is 8. The height of the blue triangle (side-midpoint-to-center distance) can be found using Pythagoras as height^2 + 4^2 = 8^2, hence height = sqrt(64 - 16) = sqrt(48), wuich is about 6,93. So the red rod, whose endpoints are 3 and 6.93 from the hexagon's centers, has mathematical length 3.93; so the piece used is 0.07 too long.

[Didumos69]

Wallpaper group cmm (2*22)

Wallpaper group p4g (4*2)

Wallpaper group p4g (4*2)

Wallpaper group pgg (22×)

On 12/16/2017 at 8:28 PM, aeh5040 said:

One interesting challenge would be to come up with examples of each of the 17 wallpaper symmetry groups

 

13 hours ago, Erik Leppen said:

What's interesting about this one is that it fits. I was looking at it for a bit until I realized it's not an exact fit, but close. This can be proven using some mathematics. For those wanting to know:

Which makes this a pain to model 

Edited December 20, 2017 by Didumos69

[DrJB]

1 hour ago, Erik Leppen said:

What's interesting about this one is that it fits. I was looking at it for a bit until I realized it's not an exact fit, but close. This can be proven using some mathematics. For those wanting to know:

A hexagon consists of 6 equilateral triangles, So, for a hexagon, the "radius" (center-to-corner distance) is equal to the side length. The yellow hexagon has sides of length 3, so the radius is 3 too. The blue hexagon has sides of length 8, so the radius is 8. The height of the blue triangle (side-midpoint-to-center distance) can be found using Pythagoras as height^2 + 4^2 = 8^2, hence height = sqrt(64 - 16) = sqrt(48), wuich is about 6,93. So the red rod, whose endpoints are 3 and 6.93 from the hexagon's centers, has mathematical length 3.93; so the piece used is 0.07 too long.

Yes, I did the math too and, surprisingly, this works very well in LDD, no complaints whatsoever.

[Didumos69]

3 hours ago, Erik Leppen said:

A hexagon consists of 6 equilateral triangles, So, for a hexagon, the "radius" (center-to-corner distance) is equal to the side length. The yellow hexagon has sides of length 3, so the radius is 3 too. The blue hexagon has sides of length 8, so the radius is 8. The height of the blue triangle (side-midpoint-to-center distance) can be found using Pythagoras as height^2 + 4^2 = 8^2, hence height = sqrt(64 - 16) = sqrt(48), wuich is about 6,93. So the red rod, whose endpoints are 3 and 6.93 from the hexagon's centers, has mathematical length 3.93; so the piece used is 0.07 too long.

51 minutes ago, DrJB said:

Yes, I did the math too and, surprisingly, this works very well in LDD, no complaints whatsoever.

This little house is also an almost perfect fit. The width of the house is in fact slightly more than the 5L suggested by the parts. The roof of the house is part of a hexagon with radius 3 and if I'm not mistaken the width of the house equals twice the height of that hexagon. Following @Erik Leppen's method to calculate the height: height^2 + 1.5^2 = 3^2, hence height = sqrt(9 - 2.25) = sqrt(6.75), which gives 2,60. So the house is 5.2L wide.

[Didumos69]

One more: Wallpaper group p2 (2222)

Edited December 20, 2017 by Didumos69

[Aleh]

Nice! Nice stuff to replace wallpapers in the room 

[Didumos69]

Something based on different shapes: Another Wallpaper group pgg (22x)

Edited December 20, 2017 by Didumos69