Technic Tessellations

[Didumos69]

Yet another one: Wallpaper group p4 (442). This one is based on Pythagorean tiling. It uses only 3 different parts .

Edited December 20, 2017 by Didumos69

[DrJB]

Very nice, I like those. Thinking of the wallpaper groups (simplifying the math), the only 'transformations' allowed/possible are:

1. Translations

2. Rotations

3. Symmetries (mirror images) about 1 or more planes

It appears the rotations/translations are relatively easy to achieve. The symmetries however are not so trivial.

Here are three more, I call them 'Variations on a Theme'. The first is symmetric (three planes), and the other two are its 'twisted siblings' (i.e., no longer symmetric). Yet all three of them still obey translation and rotation.

Edited December 20, 2017 by DrJB

[DrJB]

Speaking of variations (of the above), here are two more, that now result in three 'distinct' tiles

 

[Didumos69]

Another Wallpaper group p4g (4*2) (below you see the primitive that I used to repeat):

Edited December 21, 2017 by Didumos69

[DrJB]

Very nice. So the yellow is a translated/rotated version of the red, and that results in two planes of symmetry.

You really have a knack at doing pentagons inscribed inside hexagons 

Edited December 21, 2017 by DrJB

[Aventador2004]

Do you get these ideas online, or think them up?

[Didumos69]

2 hours ago, Aventador2004 said:

Do you get these ideas online, or think them up?

I get the patterns online, mainly from https://en.wikipedia.org/wiki/Pentagonal_tiling and https://en.wikipedia.org/wiki/Wallpaper_group, but translating them to LEGO is my part. I did come up with this pattern myself though (it's a p6m wallpaper):

I've always been intrigued by tiling planes and spheres, so I know quite a few techniques. I ever contributed to the art of cartography with a self-derived tiling-based equal-area map projection.

 

Edited December 21, 2017 by Didumos69

[Aventador2004]

Just now, Didumos69 said:

I've always been intrigued by tiling planes and spheres, so I know quite a few techniques. I ever contributed to the art of cartography with a self-derived tiling-based equal-area map projection.

Interesting, I quick had a look at those, and they look very interesting to do with lego.

[Didumos69]

Just now, Aventador2004 said:

Interesting, I quick had a look at those, and they look very interesting to do with lego.

You might want take a look at the 3D counter part of this thread and a thread about platonic bodies.

[Didumos69]

The most simple Wallpaper p6 I can come up with:

[JonathanM]

@Didumos69 can you remove the bushes from your p4 to get it done with just 2 parts?

Re the geometry, there's really only one basic operation: reflection. All others are just products (compositions) thereof. e.g. a rotation is a product of two reflections that are non-parallel: it makes a rotation of 2 times the angle of intersection about the point of intersection, with direction specified by the order of composition. Translation is the product of two parallel reflections (again two times the distance between the reflection planes). The third basic type of operation is the glide - basically a translation then a reflection in the axis of movement. All other operations are just combinations of these three.

The 1 dimensional analogue of the wallpaper groups are frieze groups - 2D with repetition in just one dimension. There are 7 of those - no rotations allowed, other than by 180 degrees (two reflections at right angles) as all axes must be parallel or perpendicular with the axis of repetition - thus just translation in the axis of repetition, and reflection in the parallel and perpendicular axes (and the corresponding glides).

[Didumos69]

1 hour ago, JonathanM said:

@Didumos69 can you remove the bushes from your p4 to get it done with just 2 parts?

Yes. I also replaced the 6L axle with a 4L axle.

This is the primitive I used to repeat:

1 hour ago, JonathanM said:

Re the geometry, there's really only one basic operation: reflection. All others are just products (compositions) thereof. e.g. a rotation is a product of two reflections that are non-parallel: it makes a rotation of 2 times the angle of intersection about the point of intersection, with direction specified by the order of composition. Translation is the product of two parallel reflections (again two times the distance between the reflection planes). The third basic type of operation is the glide - basically a translation then a reflection in the axis of movement. All other operations are just combinations of these three.

That makes sense.

1 hour ago, JonathanM said:

The 1 dimensional analogue of the wallpaper groups are frieze groups - 2D with repetition in just one dimension. There are 7 of those - no rotations allowed, other than by 180 degrees (two reflections at right angles) as all axes must be parallel or perpendicular with the axis of repetition - thus just translation in the axis of repetition, and reflection in the parallel and perpendicular axes (and the corresponding glides).

I didn't know about frieze groups. Interesting .

Edited December 21, 2017 by Didumos69

[DrJB]

9 hours ago, JonathanM said:

The 1 dimensional analogue of the wallpaper groups are frieze groups - 2D with repetition in just one dimension. There are 7 of those - no rotations allowed, other than by 180 degrees (two reflections at right angles) as all axes must be parallel or perpendicular with the axis of repetition - thus just translation in the axis of repetition, and reflection in the parallel and perpendicular axes (and the corresponding glides).

I believe these apply only to FLAT patterns (i.e, 2-dimensional only). Once you move to the third dimension, no succession of mirror reflections can produce a rotation. It only works in one special case, when the pattern has one plane of symmetry. Translations are commutative (can change the order without changing the outcome). Rotations however are NOT commutative, the specific sequence of rotations CANNOT be altered.

Edited December 22, 2017 by DrJB

[aeh5040]

You guys are on fire!  Some beautiful tesselations there.

To make things more interesting, for the definitive "17 wallpapers" catalogue, I am going to suggest some extra requirements:

1. The symmetry group should be the same regardless of whether you ignore colors, and whether or not you only look at the overall shape or the individual parts.  (So for example, a 3L axle going through a should be avoided, unless you can tell from the rest of the pattern whether 4-fold rotation about this point is intended to be allowed).

2. It should be rigid, not floppy, and reasonably strong. (So should probably be replaced with , while is best avoided).

3. It should be a mathematically exact fit.

Here is a relatively lame example satisfying these requirements, with (unambiguous) symmetry group p4m:

In a different direction, could we do a Penrose tiling, I wonder?  Presumably the pentagon angles would require some approximations.

By the way, to anyone wondering what we are on about, I found this classification table useful: http://en.wikipedia.org/wiki/Wallpaper_group#Guide_to_recognizing_wallpaper_groups

Edited December 23, 2017 by aeh5040

[aeh5040]

On 12/21/2017 at 12:05 PM, JonathanM said:

Re the geometry, there's really only one basic operation: reflection. All others are just products (compositions) thereof. e.g. a rotation is a product of two reflections that are non-parallel: it makes a rotation of 2 times the angle of intersection about the point of intersection, with direction specified by the order of composition. Translation is the product of two parallel reflections (again two times the distance between the reflection planes). The third basic type of operation is the glide - basically a translation then a reflection in the axis of movement. All other operations are just combinations of these three.

The 1 dimensional analogue of the wallpaper groups are frieze groups - 2D with repetition in just one dimension. There are 7 of those - no rotations allowed, other than by 180 degrees (two reflections at right angles) as all axes must be parallel or perpendicular with the axis of repetition - thus just translation in the axis of repetition, and reflection in the parallel and perpendicular axes (and the corresponding glides).

 

On 12/21/2017 at 3:32 PM, DrJB said:

I believe these apply only to FLAT patterns (i.e, 2-dimensional only). Once you move to the third dimension, no succession of mirror reflections can produce a rotation. It only works in one special case, when the pattern has one plane of symmetry. Translations are commutative (can change the order without changing the outcome). Rotations however are NOT commutative, the specific sequence of rotations CANNOT be altered.

Several things are a bit confused or potentially confusing here:

1. JonathanM: it's true that any isometry can be produced by composing reflections, but not all these groups can be generated by reflections.  Indeed, many of them do not contain any reflections.

2. DrJB: JonathanM was referring to Frieze Groups, which are 2-dimensional (with a 1-dimensional subgroup of translations).

3. DrJB: Not quite.  It's always true in any number of dimensions that a rotation is a composition of two reflections.  However, in 3 or more dimensions there are isometries (rigid motions) that are neither reflections, rotations, translations, nor glide-reflections.  E.g. in 3 dimensions there are screw-rotations, and in 4 dimensions there are compositions of two commuting rotations.

4. Lastly, it's true that rotations do not in general commute in 3 or more dimensions.  But in 2 dimensions rotations about the same point do commute.

Edited December 27, 2017 by aeh5040

[DrJB]

2 hours ago, aeh5040 said:

Several things are a bit confused or potentially confusing here ...

Agree, need a bit more clarity here ...

2 hours ago, aeh5040 said:

3. DrJB: Not quite.  In any number of dimensions that a rotation is a composition of two reflections ...

You're right. Intuitively it makes sense. I was thinking of it initially from a mathematical standpoint, and was not sure a rotation matrix can 'always' be written as the product of two reflections matrices. It might very well be that the solution could at times be not unique.

2 hours ago, aeh5040 said:

4. Lastly, it's true that rotations do not in general commute in 3 or more dimensions.  But in 2 dimensions rotations about the same point do commute.

When I mentioned 'commutativity', I has in mind full rotation in 3D space, and there, for sure the rotations do NOT commute i.e., rotation about x then y axes is NOT the same as rotation about y then x. Thinking of rotation matrices again, if we change the order of matrix multiplication, we get a totally different result. Now, all of this is true in 3D space (and beyond if we're comfortable with that). In 2D space, there is only one axis of rotation (Z, normal to the 2D plane). There, when a rigid body is subjected to two rotations, they are both about the same axis, and as such, they do ADD. In 'my' terminology, one cannot speak about commuting rotations in 2D ... again, pure terminology.

PS. Thanks for a trip back in time ... This reminds me of a graduate course in Dynamics many years ago, and of all the subjects, 3D dynamics is often NOT intuitive ... especially when you include gyroscopic effects. 

2 hours ago, aeh5040 said:

So for example, a 3L axle going through a should be avoided ...

 

That's not fair ... what am I going to do with all the parts I ordered from BrickLink? :(

Edited December 23, 2017 by DrJB

[DrJB]

There are multiple good sources on 17 groups on the web. The one I thought was most 'enjoyable' to read/study is an app (iOS/Android) called iOrnament. I got the screenshot below from it. It explains in a very compact form what the 17 groups are ... 

For example, the first square is a pure translation along 2 axes. The second is a translation plus a rotation. The third is a mirror reflection ...

Edited December 23, 2017 by DrJB

[aeh5040]

On 12/22/2017 at 8:18 PM, DrJB said:

There are multiple good sources on 17 groups on the web. The one I thought was most 'enjoyable' to read/study is an app (iOS/Android) called iOrnament. I got the screenshot below from it. It explains in a very compact form what the 17 groups are ...

That app looks great, but sadly it seems to be iOS only :-(

[technicmath]

On 12/21/2017 at 12:30 PM, Didumos69 said:

Another Wallpaper group p4g (4*2) (below you see the primitive that I used to repeat):

As previously mentioned by Erik Leppen, the length of some parts is mathematically too long:

On 12/19/2017 at 8:21 PM, 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.

Below I made another version of this.

In fact, this is an example of the Cairo pentagonal tiling: see https://en.wikipedia.org/wiki/Cairo_pentagonal_tiling. Four sides of the pentagon have length 1 and one side of the pentagon has length sqrt(3)-1=0.732... In the example above this is scaled by 4 thus one side of the pentagon has length 4(sqrt(3)-1)=2.928... but the parts used give a length of 3. Below I scaled by 7. Then one side of the pentagon has length 7(sqrt(3)-1)=5.124... thus parts giving a length of 5 will not bend the structure but some space needs to be given. Here are 2 examples:

The .ldr file can be found at https://bricksafe.com/files/technicmath/cairo-pentagonal-tiling/Cairo pentagonal tiling.ldr

 

I also made a Penrose tiling: see https://en.wikipedia.org/wiki/Penrose_tiling

There are 2 tiles, see below:

These use a rectangular triangle with side 2 and hypotenuse 7 thus the third side is 3*sqrt(5) by the Pythagoras theorem. Thus for example in the last tile, the length between the lowest and upper point is 6*sqrt(5)+6 which is 12 times the golden ratio and the length of the red sides is 12.

Below is an svg picture:

The .ldr file can be found at https://bricksafe.com/files/technicmath/penrose-tilings/Penrose tilings.ldr