Expanding in three dimensions

0:00 Hi, I'm Henry Segerman.

0:03 In a previous video, I talked about these 3d printed expanding gadgets.

0:06 They use rack and pinion mechanisms to transfer force

0:10 around so that everything expands at the same time.

0:13 These all expand in two dimensions,

0:15 getting bigger on the table but not increasing in height above the table.

0:19 So the obvious question is:

0:21 can we make a mechanism that expands into all three dimensions?

0:25 Well, this is where we're heading:

0:26 a design that expands in all three dimensions.

0:29 The story of how I got there involves mechanical engineering and 3D printing,

0:33 of course, but also lots of symmetry

0:35 and even a bit of crystallography and chemistry.

0:39 First, let's take a look at how the two-dimensional grids work.

0:42 There are junctions that each have a gear inside,

0:45 and the junctions are connected to each other by pairs of racks.

0:49 The racks and gears are a bit difficult to see in this design,

0:51 so I made a demo version to show what's going on more clearly.

0:55 Here's the base for one of the junctions.

0:59 Four racks slide into the base.

1:01 Then we put in a gear that meshes with the four racks.

1:05 When any of the racks moves, it turns the gear which moves the other racks.

1:10 Now let's take another copy of the junction

1:12 and put it to the side of the first one.

1:15 Next, we connect the end of this rack to this base with a bolt,

1:20 and connect the end of this rack to this base with a bolt.

1:25 Now, what's going to happen if we move this rack?

1:29 It will rotate this gear,

1:30 which pulls this rack in, which also pulls its base in.

1:35 That pushes this rack against the gear,

1:38 which rotates the gear, which also moves its racks.

1:42 Okay, let's see it.

1:45 So the force transfers along the mechanism from one junction

1:47 to the next through the pairs

1:50 of racks connecting neighbouring junctions together.

1:53 And we could do the same thing with junctions connected to these racks,

1:56 and we build out the two-dimensional grid.

1:59 This is all that's happening in the big grid designs.

2:01 The only difference with this demo mechanism is that I moved things around,

2:04 so you can see everything that's going on from this camera angle.

2:08 So, how do we make this into some sort of three-dimensional grid?

2:12 At each junction, the gear and the four racks lie in a plane.

2:16 So somehow we need to break out of that plane.

2:19 You might imagine twisting the racks around the gear.

2:21 I have some plans to do that, but that's a story for another video.

2:25 For now, I want to explore what happens if

2:27 we keep the racks in the plane of the gear, but rotate neighbouring junctions,

2:31 like this, so that they don't all lie in the same plane.

2:35 The mechanism still works exactly the same as before.

2:38 The only thing that has changed is that the two

2:40 gears used to be in the same plane,

2:42 but now their planes are rotated relative to each other.

2:46 But how does this pattern propagate outwards?

2:49 When everything was in one plane, we made a square grid.

2:52 Do we still get some kind of grid now?

2:55 What's the large scale pattern?

2:57 This all depends on what the angle between neighbouring junctions is.

3:00 There are lots of options, but your first thought might be to try a right angle.

3:05 Let's visualize what a right angle pattern would look like.

3:08 Here I'm using this building system called "Zometool",

3:11 that lets you build all kinds of mathematically interesting shapes.

3:14 I'm only going to use these blue struts,

3:17 which come out of the zomeballs at right angles to each other,

3:20 but there are other colours of strut that come

3:22 out at other angles for building other structures.

3:25 I'll use the zomeballs to represent the junctions and blue

3:28 struts for the pairs of racks going between the junctions.

3:31 Actually, here's a quick aside: How do you think they make these zomeballs?

3:35 They're injection molded- you squirt molten plastic into a metal mold.

3:40 But how does the mold release the ball once the plastic has solidified?

3:44 It's not like you could just crack the mold into two halves and pull them off.

3:47 Parts of the mold have to go into all of the holes.

3:50 And what about the cavity in the middle?

3:52 How could the bit of the mold that makes the cavity possibly get out?

3:56 Well, let's leave that as a mystery.

3:58 For now, here's a zomeball and four struts representing a junction,

4:02 with four racks coming out of it in a plane,

4:05 and the next junction has its four racks,

4:07 also in a plane but angled at 90 degrees.

4:11 If we keep building outwards in this pattern,

4:13 we discover that it links up with itself here,

4:16 making a kind of right-angled hexagon.

4:19 This is good for expanding mechanisms.

4:21 Relatively short closed loops help with transferring the forces around.

4:25 In the two-dimensional square grid we have loops of length four,

4:29 and six is not much longer.

4:31 And of course it's good that the pattern joins up at all.

4:34 If the angles were slightly off of 90 degrees,

4:36 then the pattern would not quite match up.

4:38 It would instead crash into itself, which obviously wouldn't work.

4:42 Although maybe it shouldn't be surprising

4:44 that the 90 degree pattern does match up nicely.

4:47 This right-angled hexagon is made up of the edges of a cube.

4:51 If you stack cubes together,

4:52 you get the nice regular Minecraft tiling of 3D space,

4:55 and our pattern will live on the edges of that tiling.

5:00 If we continue building the pattern outwards,

5:02 we end up making this larger cube shape made out of eight of the smaller

5:06 cubes and now the whole pattern is made by stacking those larger cubes together.

5:12 Let's actually see what the whole infinite pattern looks like.

5:16 You might have already guessed that this structure shows

5:18 up in chemistry as the atomic structure of a crystal.

5:21 This is the arrangement of niobium and oxygen atoms in niobium monoxide.

5:27 According to Wikipedia, niobium monoxide is a grey solid,

5:30 which sounds pretty boring, but more interestingly, it's also a superconductor.

5:36 Niobium monoxide has the chemical formula NbO,

5:39 so this structure is called the "NbO network".

5:43 Okay, I lied.

5:43 What you're seeing isn't actually the whole infinite pattern.

5:46 But getting this video to loop so that it looks like

5:49 we're endlessly traveling through the NbO network was a fun side-quest.

5:53 This was the same kind of camera motion

5:56 control project as my real-life fractal zoom videos.

5:59 The camera moves forward on a slider,

6:01 going from one view of the object to an almost identical view,

6:05 so we can fade from one view to the other,

6:07 and the video loops In the fractal zoom,

6:10 going from one view to the other corresponds to scaling the 3D print.

6:15 But since the print is a fractal, we can move towards it instead of scaling.

6:19 Here, going from one view to the other

6:21 corresponds to moving through the NbO network.

6:23 So the camera has to move quite far

6:26 into the zometool structure without crashing into it.

6:29 I used this probe lens to do it.

6:31 It can get far enough into the structure that we see the same pattern again.

6:34 So we can fade from one view to the other and make the loop.

6:38 I used this grid on my light, which reduces how much the light spreads out.

6:42 So in the view from the camera, we can't see that far into the distance,

6:46 which reduces the amount of stuff popping in as we go around the loop.

6:50 And I moved the light with the camera so that the lighting

6:52 would match between the start and the end of the loop.

6:55 While I had the zometool structure set up,

6:57 I got some other shots moving into it.

6:59 There are some very pretty views that show the different symmetries,

7:02 but I didn't bother trying to get things lined up well enough to make them loop.

7:07 All right then, side-quest completed.

7:09 So this is the pattern we want to recreate with racks and gears.

7:13 But there are more things that need to be worked out.

7:15 For example, which way does the gear rotate in each

7:18 junction when we are collapsing or expanding the mechanism?

7:21 Can we set it up so that they all rotate the same way?

7:24 What would it even mean to rotate the same way

7:27 when the junctions are in all of these different orientations?

7:30 And here's another thing.

7:32 At each junction the four racks are on two

7:34 different layers which racks go on which layer?

7:38 In other words, the expanding rack mechanism has to break

7:40 a lot of the symmetries of the NbO network.

7:43 I did want to keep as much symmetry as possible though,

7:46 both to reduce the number of different parts I had to design,

7:49 and to make the thing simpler to build.

7:51 Let's come back to the large scale symmetry later.

7:53 For now, here's the design I went with for the junctions.

7:57 It looks a bit different to the demo junctions from before.

8:00 The final design has more symmetry than the demo version.

8:03 Both designs have this 180 degree rotation symmetry,

8:07 but the final design has another symmetry

8:09 that swaps the top and the bottom halves.

8:11 But this other symmetry is not a rotation, and it's not a reflection.

8:15 Let me try to explain.

8:17 Here are two identical copies of the junction.

8:20 If I turn this one over, the other side is not the same as what we started with.

8:24 If you look at the racks on this one,

8:26 they kind of slope upwards as you look to the right.

8:29 While on the original they slope downwards.

8:31 This is in fact a mirror image of what we started with.

8:34 So that extra symmetry is not a rotation, but it isn't just a reflection either.

8:39 If you look around the junction from all directions,

8:41 There's just no plane of mirror symmetry.

8:43 So there's no simpler way to look at it.

8:45 This other symmetry is a combination of a rotation and a reflection.

8:49 Any time I've got some kind of symmetry question,

8:51 the book I reach for is "The Symmetries of Things" by John Conway,

8:56 Heidi Burgiel and Chaim Goodman-Strauss.

8:58 And there's a just released, greatly expanded,

9:00 much abridged follow up, "The Magic Theorem".

9:04 These books give a really nice notation to talk about symmetric objects.

9:08 in that notation, the symmetry of the junction is 2x.

9:12 The 2 means that there's an axis you can look along,

9:14 which has two-fold or 180 degree rotational symmetry.

9:19 The x means that you can rotate it from one view to a reflection of that view,

9:23 without going through a view that has mirror symmetry.

9:26 Here, the book points out that a baseball has the same kind

9:29 of symmetry if you pay attention to the stitching along the seam.

9:32 It's got a two-fold rotation axis here.

9:36 and if you flip it over this way, it will look almost the same,

9:39 except that the seams point in the wrong direction.

9:42 So you get to a reflection of the original view.

9:45 And yet, just like the junction,

9:46 the baseball does not have a plane of mirror symmetry itself.

9:50 I made these 3D prints to illustrate

9:52 the different kinds of spherical symmetries for my book,

9:55 "Visualizing Mathematics with 3D printing".

9:57 Here's a 3D print that has a similar kind of symmetry to our junction,

10:01 except that this one has symmetry type 8x instead of 2x.

10:05 It's got an axis with eight-fold rotational symmetry here.

10:08 That's the eight.

10:09 And we get an x because you can rotate from this view to this one,

10:13 which is a reflection of where we started,

10:15 even though again, there are no planes of mirror symmetry.

10:20 So the symmetry of the junction is 2x.

10:23 This affects how we can 3D print and assemble the junction.

10:26 If a gear and a rack are both horizontal as they're printed,

10:29 then they contact each other along layer lines of the printing.

10:33 If they were printed in different orientations,

10:35 then they would roll over each other's layer

10:37 lines and the mechanism would not run as smoothly.

10:40 So we want to print the gears and the racks horizontally,

10:43 and we also want the print orientations to match

10:46 when we've got axles contacting the inside of gears,

10:48 and these rails that slide in grooves on the racks.

10:53 A single junction then has parts needing to be printed in one,

10:57 two, three, four, five, six different orientations.

11:02 These parts all need to be rigidly connected together,

11:04 ideally without using too much hardware.

11:06 An extra bolt here and there, like I used in the demo junctions from before,

11:11 might not seem that big a deal,

11:13 but this junction design is going to get copied many,

11:15 many times to make a bigger structure.

11:17 So weight becomes an issue.

11:19 So I'm pretty happy with this design,

11:20 in which a single bolt connects all of the parts in a junction together,

11:24 and holds the gear in place on its axle.

11:26 There are these little cap parts that the nut and bolt

11:29 sit in, that hold the racks in place on the base.

11:32 They can be fiddly to put in.

11:34 You kind of need to angle the two racks

11:36 up and then press them into place with the cap.

11:39 Okay, so let's build one of those right-angled hexagons.

11:45 So yeah, this seems to work very well.

11:47 I was a little worried, before I built this, that somehow going around

11:52 all these weird corners would cause a problem.

11:54 But it seems to work fine.

11:57 We only need racks to connect together pairs of junctions,

12:00 so I don't need these racks pointing away from the hexagon.

12:03 So let's remove them.

12:09 For these junctions that don't have racks going off in all four directions,

12:13 I've got a version of the cap part that also fills

12:16 in for one of the two racks on its side of the junction.

12:19 We can also add extra racks meshing

12:21 with the gears that don't connect to bases, like this.

12:24 I think this makes it look cooler.

12:26 It makes the expansion visually bigger.

12:29 One of the advantages of having lots of symmetry

12:31 is that it makes the assembly rules easier to remember.

12:34 There are just fewer rules to think about.

12:36 So here are the rules for this design.

12:39 First, every junction looks the same.

12:41 Apart from when we remove the rack going off

12:43 in some direction like we did for the hexagon.

12:46 Second.

12:47 When two junctions connect together,

12:48 the grooves on both racks have to fit onto rails on the two bases.

12:53 And that's it.

12:54 There's only one way to put two junctions

12:56 together so that both racks fit onto rails.

12:59 The two racks have to have the same handedness,

13:01 and they have to be the right way round.

13:03 So there's only one way to build everything outwards.

13:05 The only choice is how much to build and in what shape.

13:09 Well okay.

13:09 Next, let's do a cube.

13:12 This is made from eight overlapping hexagons, one for each corner of the cube.

13:16 By the way, I'm using red, orange and yellow for the colours

13:19 of the junctions in the three different planes,

13:21 but the colours don't mean anything else.

13:23 All of the junctions are exactly the same as each other apart from the colour.

13:27 This time I didn't add on the outside racks,

13:30 which reduces how much it looks like it expands.

13:32 Although if we just build something bigger,

13:34 the boundary of the shape won't matter so much.

13:42 Okay, now we're getting somewhere.

13:43 This is a nice compact Tetris piece that will fit in my luggage,

13:46 but that I can easily expand out to make a much bigger Tetris piece.

13:51 With this much of the structure built out,

13:53 I can show you why the racks are the length that they are.

13:56 It would be great if I could make them

13:57 longer so that the expansion factor would be bigger.

14:00 But here's the problem.

14:02 This rack is at the same height as this one.

14:05 And when we collapse the structure down,

14:07 this one pokes straight into the side of the other one.

14:09 I don't think there's any way to tweak things

14:11 a little bit to get them to avoid each other.

14:13 The symmetry just doesn't allow it.

14:15 So this is as long as the racks can get,

14:17 if I want it to be able to collapse down this far.

14:20 Still, we get a pretty good expansion factor with these racks,

14:23 just over a factor of two.

14:26 All right, so the Tetris piece is bigger

14:27 than the first cube I made in two directions.

14:30 But since we're doing expanding racks in three dimensions,

14:32 we obviously have to make a structure that is big in all three directions.

14:36 So let's make a two by two by two cube.

14:43 So let's see.

14:46 It seems to work pretty well.

14:49 It's got pretty heavy.

14:50 Just the amount of stuff here.

14:53 So lifting it from there requires a lot of force

14:56 just because I'm lifting half of it up like this.

15:00 Somehow it's easier to do from up here.

15:04 But yeah.

15:09 Here are some motion control shots I took to show

15:11 the movement a bit more smoothly than I can do by hand.

15:19 Here I'm using the probe lens again so we can see inside the structure.

15:24 Alright, you can see we actually get a factor of about two in the collapse.

15:29 Here's one of the original cubes and it's about the same

15:32 size as this two by two by two chunk of cubes here.

15:38 So let's see.

15:41 Now that we've got a two by two by two built out,

15:45 I can explain another interesting feature of the NbO network.

15:49 So again, this is the two by two by two cube.

15:52 It's made of eight copies of this.

15:54 I'm actually going to use the zometool instead just

15:57 because it's a little easier to see what's going on.

16:00 So again this is one of the cubes.

16:02 And here are eight of them stacked together.

16:06 And I've got my Rubik's Cube to indicate where those cubes are.

16:10 And so well, of course you've got one of the cubes

16:13 here in the top corner of this structure here.

16:16 And there's seven more.

16:18 Now, what I want to point out is that there's a whole other set of cubes here,

16:22 which are kind of dual to the first set of cubes.

16:25 So there's one here.

16:26 Here, there's sort of half of this cube

16:28 is built out in the zometool structure here.

16:32 And then next door to that cube is this one here,

16:35 this maybe has only a quarter of the cube built out in zometool.

16:39 And then there's a central one right in here.

16:45 Which again is sort of offset from the original

16:48 eight cubes that we thought we were building.

16:51 But yeah, so there's this whole other sort of dual set of cubes,

16:55 sitting inside of this structure and this other set of cubes is going

16:58 to help us understand the large scale symmetry of the pattern later on.

17:03 This two by two by two cube is also going

17:04 to let us see how the pattern tiles out into space.

17:08 So let's pay attention to this junction here.

17:10 And I've got a copy of it here in the same orientation.

17:13 So you can see the two horizontal racks here.

17:16 And it sort of slopes upwards.

17:18 So if I go back two cubes then we should be looking at this one here.

17:23 This junction here is again in the same orientation.

17:26 And that's not true if I just go back one cube.

17:28 So going from here to here, this one is not in the same orientation.

17:33 I have to rotate it like this to get it to match up.

17:37 So what does that mean?

17:38 It means that if I take two steps then I see exactly the same thing.

17:42 Which means that I could take this entire structure, shift it back two cubes,

17:46 and it would match up with itself and it would tile.

17:48 And that's the same in the other two directions as well.

17:51 If I go this way, it's the same.

17:52 If I go this way, it's the same.

17:54 So as long as I take two steps every time,

17:56 I see exactly the same thing, and I can tile out space with the pattern.

18:01 This book, "The Magic Theorem",

18:02 is greatly expanded over the previous "The Symmetries of Things" in that it

18:06 has many more examples and details about

18:08 the symmetries of two-dimensional things- planar patterns,

18:10 patterns on the surface of a sphere, and so on.

18:14 It's also much-abridged,

18:15 in that it doesn't go into the symmetries of higher-dimensional things,

18:18 like the NbO network.

18:21 So for that, let's go to The Symmetries of Things,

18:23 Chapter 22, on the 35 "prime" space groups.

18:28 Let's not get too far into the details because there's a lot going on here,

18:31 and I certainly don't understand all of it.

18:33 But the short version is that these kinds

18:35 of symmetries of repeating three-dimensional patterns,

18:38 are governed by these two checkerboard patterns of cubes.

18:41 Red checkerboarding with green, and yellow checkerboarding with blue.

18:46 I asked Chaim- this Chaim- about the symmetries

18:49 of the NbO network and my expanding racks version,

18:51 and we worked out that the NbO network has symmetry type 8o:2,

18:56 which Conway names the doubled octad group.

18:59 The expanding racks mechanism breaks some of those symmetries.

19:02 It has symmetry type 4++, the bi-positive hybrid tetrad group, apparently.

19:09 You can see the relations between the different symmetry types on this diagram.

19:13 As we go down these curves we're moving to subgroups,

19:16 so we break some of the symmetries as we move from 8o:2 down to 4++.

19:22 By the way, this zometool structure doesn't quite

19:25 have the same symmetries as the NbO network itself,

19:28 if you pay attention to the geometry of the zomeballs and the struts.

19:32 You can rotate the NbO network around one of its

19:34 junctions by 90 degrees and it looks the same.

19:37 But that's not true if you look closely at the zomeballs and the struts.

19:41 So this zometool version of the NbO

19:43 network actually has yet another symmetry type.

19:46 Chaim and I think it is the null doubled tetrad group, 4o:2.

19:51 One of the things I found most

19:53 confusing about the symmetries of this design was,

19:56 what is the transformation that takes you from, say,

19:59 this orange junction to a neighbouring junction, say this red one?

20:03 So here's a copy of this orange junction.

20:06 You can see that this rack is in exactly the same orientation as this one.

20:11 Of course, I've removed these two here from this one

20:13 because they go off the side of the cube.

20:15 So obviously I have to move- I have to translate over here,

20:18 but I also have to rotate to get it to match up with the red.

20:21 So I move forward and I rotate like this.

20:26 And you can see the orange is in exactly the same orientation as the red.

20:29 And so this was like a right handed screw motion.

20:32 As I move forward, it's like I'm tightening a screw

20:36 with a screwdriver in the usual way you would do so.

20:38 So this is a right-handed screw motion.

20:41 Now what if I want to go from this orange one to this yellow one.

20:44 Again I have to translate.

20:46 But now I do this rotation to get these two to match up- to be the same.

20:53 And this is a left-handed screw motion.

20:56 So why is it that sometimes I have to do a right-handed

20:59 screw motion and sometimes I have to do a left-handed screw motion?

21:03 Why isn't the thing to do to just always turn the same direction?

21:08 There's a really nice way to think about this that Chaim and I discovered,

21:11 once we knew that the symmetry type is 4++.

21:14 We looked up other structures that have

21:16 the same symmetry and found this, the mucuboctahedron.

21:20 This is a surface in space made out of squares and hexagons.

21:24 You can also think of the mucuoctahedron as being

21:26 part of the tiling of space by truncated octahedra.

21:30 An ordinary eight-sided die is in the shape of an octahedron.

21:33 If you cut off all of its corners like this, then you

21:36 get a truncated octahedron and truncated

21:39 octahedra can tile together to fill space.

21:43 If you take the two-dimensional faces

21:45 of that tiling and delete some of the hexagons, you get the mucuboctahedron.

21:50 In the tiling by truncated octahedra,

21:52 every edge meets three polygons: two hexagons, and a square.

21:55 Every edge loses one hexagon, so it ends up with only two polygons next to it.

22:00 That means that we end up with a polygonal surface.

22:03 You could even smooth it out a bit and you get this minimal surface,

22:06 which is called the "D surface".

22:08 The D stands for "diamond".

22:10 The surface cuts space into two sides,

22:12 each of which contains a copy of the molecular structure of diamond,

22:15 which is where the name comes from.

22:18 So if the surface has two sides to it,

22:20 shouldn't the mechanism also have two sides?

22:23 But what could that mean?

22:26 Maybe pause the video for a second if you want to think about it.

22:28 The answer is very satisfying.

22:32 Okay, the junctions and their gears

22:34 correspond to squares in the mucuboctahedron.

22:37 As the mechanism collapses or expands, the gears rotate.

22:40 The two sides of the mucuboctahedron tell you which way the gears rotate.

22:44 Let's see how this works.

22:47 Suppose that you're inside of this cube looking

22:49 out at one of the junctions- say this one.

22:53 Now which way is the gear in this junction

22:55 going to rotate as we collapse the mechanism down?

22:59 It's a little difficult to see because the gear is kind of covered up,

23:03 but you can tell by the positions and the movements

23:06 of the racks that this gear is rotating anti-clockwise.

23:09 As we collapse the structure down.

23:11 Let's take a look at some of the other junctions around here.

23:15 So this one here, again as we collapse down the gear rotates anti-clockwise.

23:21 Maybe let's look at one more- as we collapse down,

23:26 the gear rotates anti-clockwise.

23:28 So you could say that this is an anti-clockwise rotating cube.

23:32 Now what's going to happen if you look

23:34 from the perspective of the next door cube over here?

23:37 Let's look at this junction.

23:39 Now what's going to happen, of course,

23:41 is that the gear rotates clockwise as we collapse down.

23:44 Of course, it must rotate the opposite way

23:48 from when we were looking from this side.

23:49 It's the same gear- you're just looking at it from the back side.

23:53 If you look at a clock from behind,

23:55 the hands of that clock are going to rotate in the anti-clockwise direction.

24:00 So, next to this anti-clockwise cube is going to be a clockwise cube.

24:05 And in fact all of its neighbours are going to be clockwise rotating cubes.

24:09 So that means that in a grid of cubes, we get a checkerboard pattern of cubes

24:13 where the gears rotate clockwise or anti-clockwise.

24:17 But there are also all of the cubes of the dual grid,

24:20 and they also have a checkerboard pattern.

24:22 These two checkerboard patterns are the red,

24:24 green, yellow, and blue balls from the book.

24:27 The mucuboctahedron separates the red and yellow

24:30 balls from the green and blue ones, so it separates the clockwise rotating sides

24:34 of the junctions from the anti-clockwise rotating sides,

24:37 in my geared NbO network.

24:39 So why when we move from one junction to the next do

24:42 we sometimes rotate to the left and sometimes rotate to the right?

24:45 Well, you have to rotate to keep the clockwise rotating

24:48 side of the junction on the correct side of the mucuboctahedron.

24:52 So sometimes that means rotating left and sometimes that means rotating right.

24:55 Isn't that cool?

24:57 If you'd like to print some of this expanding NbO network design for yourself,

25:01 I've put a link to the files in the description.

25:04 And one more thing: I finally got around to setting up a Patreon,

25:08 at patreon.com/HenrySegerman.

25:10 There are some posts up already with some behind

25:12 the scenes details about the Alpha Omega d24 video,

25:16 and some work in progress details for the follow up

25:18 to this video on an expanding racks design based on the triamond lattice.

25:22 I've also just put up a post with some extra details about this video.

25:26 in the future.

25:26 I'll plan to post early versions of videos or other updates

25:29 of things I'm working on, So if you'd like to support my work,

25:32 or just see more about what I'm doing, please consider joining.

25:36 Alright, that's it.

25:37 Thanks for watching!