Chapter 16: Periodic and Aperiodic Tilings

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Welcome back to the Deep Dive.

Today we're not just looking at the surface of materials, we're diving deep into the structural language, the underlying blueprint that dictates how atoms arrange themselves.

Exactly.

Whether it's in a perfect repeating order or in these strange, almost forbidden patterns.

And when you study structural knowledge, you're really trying to decode how physical space is filled.

That's the whole goal.

I mean, everything from simple table salt to incredibly complex metallic alloys, it all relies on this knowledge.

You start with a basic repeating unit, sure, but the real challenge is figuring out how those units tessellate.

Or tile the available space.

Right, without leaving gaps or, you know, causing collisions.

Absolutely.

So our mission today is a deep dive source from a really detailed chapter focused entirely on periodic and aperiodic tilings.

You should think of this as a complete roadmap for understanding the structural classification used in material science.

We're bridging abstract geometry, the math of shapes, with the actual physical arrangement of atoms in crystals and quasicrystals.

And we're going to use this chapter as our guide.

We'll start small with simple 2D tilings, then relax the geometric constraints one by one to find new patterns.

And then take that knowledge into 3D.

And then explode that knowledge into 3D crystal stacking sequences.

The goal here is clarity,

understanding the specific language, the Schlafly symbols, the Ramsdell notation, the Zanoff code that lets scientists describe the structure of matter.

We've previously explored foundational concepts like Bravais lattices and the 17 plane groups, which govern symmetry on a flat surface.

Now we're applying those symmetries directly to the physical act of tiling.

Yep.

So let's jump right in.

How do we define a perfect arrangement of shapes in two dimensions?

Okay, so we start by setting the stage for crystalline solids.

And you have to remember, any crystal structure is fundamentally defined by two things.

The Bravais lattice and the basis.

The Bravais lattice, which is a repeating framework, and a basis, the atoms or molecules attached to each lattice point.

So the act of tiling or tessellation is really the perfect geometric description of how that basis gets packed into a plane.

Okay, so to be precise,

the source defines a tiling, which it calls math culti, as a countable family of closed shapes.

These are the tiles too.

And there are two very strict rules.

Right.

One, the union of all the tiles has to completely cover the entire plane.

No gaps.

No gaps allowed.

Right.

And two,

the interiors of those tiles must be pairwise disjoint.

Which is a very formal way of saying they can't overlap.

Exactly.

They can only touch at their edges or their corners.

So if we impose the absolute strictest requirements, we get to what are called the 2D regular tilings.

And there are three rules that have to hold true at the same time.

First, they must be monohedral.

Monohedral, meaning every single tile is the exact same size and shape.

Yep.

Second, they have to be edge to edge.

They share full common boundaries, not just a vertex meeting halfway up an edge or something.

Okay.

And third.

And third, the tiles themselves must be regular polygons.

So all sides and all internal angles within that tile are identical.

So when you try to tile a floor, you know, you might try a square or maybe a triangle.

But if you try a regular pentagon, you immediately run into a problem.

You do.

And applying those three rules, it drastically limits our options.

So how many regular tilings can possibly exist?

Only three.

The equilateral triangle, the square, and the regular hexagon.

That's it.

No matter how hard you try, you just cannot regularly tile a plane with a pentagon, an octagon, or any other regular polygon.

And to handle these structures efficiently, we need a kind of shorthand language.

This is where the Schlafly symbol notation comes in, written as PNEL.

That's right.

And PNEL is easy.

It just identifies the type of regular polygon we're using.

So if PNA no seal, it's a triangle.

If PDS four, it's a square.

PN8 six door for a hexagon.

Exactly.

And then Talia tells you the number of those polygons that meet at every single vertex in the entire tiling.

So let's take the equilateral triangle tiling.

We have PNA do.

And if you count how many triangles meet at any vertex, you get six.

So the Schlafly symbol is $366.

Right.

And for the square tiling, P is four and four squares meet at every corner.

So that gives us 44 to 12.

And for the hexagon tiling, PSX entry six and only three hexagons meet at a vertex.

So that's $63.

But the real power of the source material here is that it doesn't just list these three.

It proves mathematically why they're the only three possible options.

And that proof, which is described in box 16 .2, it revolves around this really critical geometric constraint.

It does.

For any tiles to meet at a vertex without gaps,

the sum of the angles around that meeting point has to be exactly two P radians.

Or 360 degrees.

360 degrees.

Okay.

Let's unpack the reasoning there.

So first you need the formula for the internal angle of any regular belostited polygon.

And that angle is given by $1 P pi.

Right.

And if you're trying to tile, you have total of these polygons meeting at the vertex.

So the total angle occupied by those toll tiles must be tilt -toler times that internal angle.

And we set that total equal to two pi.

So the equation is two toll six, two P pi on both sides.

And you can immediately cancel out tilt -toler on both sides.

Which leaves us with a pretty simple algebraic relationship that has to hold true for the integers $2.

If you rearrange the terms, you eventually get to two dash two, two glues of dollars.

And since peopling dollars have to be integers greater than two, I mean, you can't have two sided polygon or only two tiles meeting.

This relationship really, really limits the possible combinations.

So if you start testing positive integers for $2, what happens?

Only three solutions emerge that satisfy the equation.

First, if two Pt three dollars for a triangle, $2 has to be six.

So you get $3 a square.

Second, if P three dollars a square, $2 must be four, giving us $4.

And third, if Pt two, the hexagon must be three, which gives us $2, $3.

And what if you try pentagon?

Where to express you?

Well, the internal angle of pentagon is 108 degrees.

If you put three of them together, that's 324 degrees.

Not enough.

And four is too many.

Four, you overlap at 432 degrees.

You just can't find an integer taller that satisfies the equation, two double $25 and two two.

Exactly.

So the geometry itself imposes this strict, beautiful limitation.

It's not a matter of opinion or engineering.

It's a geometric necessity.

So we know the three perfect regular monohedral tilings.

The natural next step is to ask,

what if we relax just one of those roles?

We still want the tiles to be edge to edge and we still want them to be regular polygons, but let's drop the requirement that all tiles have to be identical.

Right.

This brings us to the 2D Archimedean tilings, or as they're also called, uniform tilings.

So now we can use a mix of shapes.

Exactly.

A mix of regular polygons, triangles, squares, hexagons, all in the same tiling.

But the critical constraint we retain is uniformity.

So all vertices must be symmetrically equivalent.

Yes.

If you were miniaturized and stood at any corner intersection in the tiling, the sequence and type of polygons surrounding you would have to be identical every single time across the entire infinite plane.

And this expansion takes us from just three solutions to the 11 Kepler tiles.

Which includes the three regular ones plus eight additional more complex arrangements.

So the Schlafly symbol needs an update to handle this.

It's no longer Piccio.

Right.

It now lists the ordered sequence of polygons around any vertex.

You just read the numbers sequentially as you go around the corner.

Let's use the example from the source.

Three dollars, four cut dots, six delaro, four cut, four colors.

How do we visualize that?

It means that every vertex is surrounded by a triangle, that's the three, followed by a square, the four, followed by a hexagon, the six, and then another square, another four.

And because it's uniform, every single vertex in the whole structure has that exact same order.

That's what guarantees the local angle requirement, the 360 degrees is met, and also that the arrangement is globally consistent.

That sounds like a puzzle.

How did we even figure out there are exactly 11 of these tilings?

Well, this takes us to the computational explanation in box 16 .3.

If you just calculate which combinations of regular polygon angles add up to 360 degrees, you actually find 17 different possibilities.

Wait, 17.

So 17 combinations satisfy the math, but only 11 actually work.

Why do the other six fail?

It's because of incompatible boundary conditions.

Local satisfaction at a single vertex isn't enough.

Ah, so it looks good in one spot, but you can't extend it outwards forever.

Exactly.

If you try to extend one of those six combinations, you quickly find that edge lengths don't match up, or that trying to maintain the vertex arrangement here forces a gap or an overlap over there.

So it's the difference between a local arrangement and a global pattern.

All right.

The 11 Kepler tilings are the only ones that pass both the local angle test and the global uniformity test.

It's a really crucial distinction.

It really is.

It shows the power of structural geometry.

The local math has to be consistent with the global space.

And building on this, we move to section 16 .2 .3,

K -uniform regular tilings.

So if Archimedean tilings are one uniform, meaning one type of vertex, then K -uniform tilings relax that constraint even further, allowing for day different types of symmetrically distinct vertices.

Yes.

And this dramatically increases the complexity.

The source focuses mainly on two uniform tilings, which means there are two distinct types of vertices present in the structure.

How many of those are there?

There are 20 distinct types of two uniform tilings, and they're really essential extensions of the one uniform Archimedean tilings.

For a researcher classifying complex packings, allowing two or more vertex environments is often necessary to describe what they're actually seeing.

Okay, so that takes us to section 16 .2 .4, which introduces duality.

This is a fundamental concept in geometry and crystallography, and it leads us to the Laves tilings.

Right.

And we're changing the rules again.

We were focused on the regularity of the tiles.

Now, we relaxed the need for regular tile shapes, but we imposed regularity on the vertices.

On the vertices.

Specifically, on the valence of the number of edges meeting at that vertex.

If the valence is regular, it means the angular distribution of edges around that vertex is perfectly even.

And the conceptual construction of a dual tiling is, it's really elegant.

You start with an Archimedean tiling and you place a new vertex at the center of every existing tile.

And then, if two tiles in the original tiling shared an edge,

you connect the centers of those tiles with a new edge.

And what you get is the dual structure.

It completely flips the relationship.

Faces become vertices, and vertices become faces.

Exactly.

I see.

And for the regular tilings, the dual preserves regularity.

So the dual of the triangle tiling, the 3666Rs, is the hexagon tiling, the 6333Rs.

It's a beautifully symmetric relationship.

And the duals of the 11 Archimedean tilings are the 11 monohedral tilings with regular vertices.

And these are defined as the Laves tilings.

Named after George Ludwig Friedrich Laves, one of the pioneers we'll get to later.

Correct.

So why is this duality so important in material science?

I mean, it seems like a neat mathematical trick, but the source notes, it has deep crystallographic significance.

It does, because it connects structure directly to analytical methods.

In crystallography, we often analyze structures not in real space with the atomic positions, but in reciprocal space.

Using the reciprocal lattice from X -ray diffraction patterns.

Exactly.

And the dual lattice is often identical to the reciprocal lattice.

So the duality is like an analytical shortcut.

By studying the simple geometry of the dual Laves tiling, you're also studying the geometry of the reciprocal lattice, which is where the diffraction data lives.

Precisely.

And this is crucial for understanding what are called Laves phases, which are these complex intermetallic compounds known for their very intricate packing arrangements.

Let's add a different dimension to symmetry now, one that isn't purely geometric.

Color.

Section 16 .3 discusses color tilings.

Right.

And color is used here as a powerful tool to represent internal symmetries that might not be visible in the crystal structure alone.

Like magnetic symmetry.

Exactly.

For instance, you could assign one color to an atom with a spin up state, and another to an atom with a spin down state.

A colored tiling is just a plane tiling where each tile gets a color, but with the constraint that the coloring must remain uniform across the whole plane.

Uniformity is the key constraint again.

The colored arrangement, so the sequence of colors around any given vertex has to be the same everywhere.

It does.

So let's take our simplest regular tiling, the 306 takes of our triangle tiling, and see how many ways we can color it uniformly with just two colors.

Okay.

So if we analyze the possible color sequences around a vertex, say with color one and color two, we might find a sequence like 11, 11, 12.

Meaning five triangles of color one and one of color two meet at every vertex.

At every vertex.

Or you might have another uniform pattern like an alternating sequence 11, 1, 2, 1, 1, 12,

which results in a totally different global pattern.

So we're not just randomly painting tiles.

We're looking for a color pattern that perfectly respects the underlying symmetry of the geometry.

That's it.

Exactly.

And the source details the specific findings for this 306666 set tiling.

It confirms that there are exactly six possible uniform two -colored tilings.

Six distinct patterns of color symmetry that can exist while still respecting the geometry of six triangles meeting at a corner.

Yes.

And what happens if we try to introduce more colors, say five or six?

The geometry becomes too restrictive.

The complexity of the color scheme starts to fight against the constraint of uniformity.

For the 306666 tiling, if you try to use five colors, there's only one uniform five -colored tiling possible.

Just one.

And only one uniform six -colored tiling.

It's a really profound geometric principle.

Increasing the complexity in one aspect, like color here,

severely limits the available options for uniformity in the entire structure.

Okay.

Everything we've discussed so far relies on periodicity on translational symmetry.

You shift the pattern by a unit vector, it looks exactly the same.

Right.

Now we enter the realm of the forbidding structures.

Structures that lack that traditional periodicity.

Quasi -periodic tilings.

And this is perhaps the most exciting part of structural material science because the discovery of quasicrystals, or QCs,

fundamentally changed how we define a solid.

These are alloys that exhibit non -crystallographic symmetries.

Most famously five -fold and ten -fold rotational symmetry.

And since five -fold symmetry is mathematically impossible in a standard 3D periodic lattice, for decades they were just not to exist.

So these structures are not periodic, but they are highly ordered.

What are they built from?

They're built not from a single simple repeating tile, but from two or more distinct types of bricks.

And the defining difference is that the dimensions of these two types of tiles are related by an irrational number.

You can't describe them with a simple integer ratio.

And the famous example of this is the Penrose tiling in 2D.

Correct.

The Penrose tiling uses two specific rhombuses, a thick one and a thin one, which tiled the plane in a way that shows global five -fold symmetry but never actually repeats.

And the irrational number governing them.

It's the golden ratio, tau tau, which is about 1 .618.

So if they lack translational symmetry, why aren't they considered amorphous, like glass?

This is the critical distinction.

Amorphous solids are completely random, and they produce these diffuse fuzzy diffraction patterns.

Quasicrystals, like icosahedral aliqu alloys, produce diffraction patterns with discrete sharp peaks.

And those sharp peaks confirm long -range order.

They confirm long -range order.

But the pattern of those peaks reveals these non -traditional, non -crystallographic symmetries.

So it's long -range order without translation.

This is the point where it gets a little mind -bending.

How do crystallographers describe something that's stable, ordered, but mathematically impossible in 3D periodic space?

They use the concept of higher -dimensional projection.

Since you can't physically fit five -fold symmetry periodically in 3D space,

scientists mathematically model the structure as a perfectly periodic crystal that exists in a higher -dimensional space.

For example, a 6D cubic, correct?

A 6D cubic crystal, exactly.

I can't even visualize 4D.

How does that help?

Well, think of it this way.

Imagine you have a perfect 3D cube,

and it's casting a shadow onto a 2D surface.

If you tilt the cube just right, its shadow will be a simple square grid, perfectly periodic.

But if you tilt the cube by a specific irrational angle, the shadow becomes complex.

It never perfectly repeats, it uses two or more shapes, but it's still highly organized.

That shadow is the quasicrystal.

So the perfect 6D structure is projected onto our 3D space, which is the shadow.

And that projection preserves the necessary irrational golden ratio relationship, which is what generates the five -fold symmetry we observe.

That's a phenomenal analogy.

It shows that the disorder we see in the quasicrystal is just the consequence of looking at a perfect order from a higher dimension.

That's a great way to put it.

Okay, we've mastered tiling the flat plane.

Now let's see what happens when we try to fill a three -dimensional container.

This moves us from tilings to the five perfect geometric solids or regular polyhedra.

The platonic solids.

These are the 3D analogs of the three regular tilings.

And just as the geometric constraints limited us to three tilings in 2D, they limit us to five solids in 3D.

The tetrahedron, cube, octahedron, dodecahedron, and icosahedron.

Right.

And maintaining consistency, the concept of duality persists in 3D.

The cube and the octahedron are duals.

The dodecahedron and the icosahedron are duals.

And the tetrahedron is a special case.

It's self -dual.

It's self -dual.

We need to adapt our Schlieffel symbol again for 3D.

It shifts from P2 to doll.

How do we read this new symbol?

So ball still refers to the face type, a number of sides on the polygon that forms the face.

P233 for triangular faces, P41st for square faces, and so on.

Twaller now represents the number of faces that meet at any single vertex.

So for the cube, which has square faces, P3 faces meet at every corner.

So the cube is $3.

Exactly.

And the octahedron has triangular faces, so P3 dollars.

And four faces meet at every vertex, making it 4 -3.

And the most complex, the icosahedron has triangular faces, so P3 ziles.

But five faces meet at every vertex, so that's 8 -3.

And just like in 2D, the geometrical constraints of space filling prevent any other integer combinations from forming a perfectly regular convex solid.

Beyond these five platonic solids, the source introduces the Archimedean solids.

These are related, but they're semi -regular.

Yeah, the Archimedean solids are still convex, and they still use regular polygons as faces, but they use two or more types of polygons.

There are 13 of these semi -regular solids.

Which, when you add them to the five platonic solids, gives us 18 distinct structures.

It does.

So how do you create one?

Well, the simplest way is truncation.

If you take a platonic solid, say the cube 300, 300, and you just slice off its eight corners uniformly.

So you just chop them off?

You chop them off.

The original square faces might turn into octagons, and the truncation planes themselves become new triangular faces.

Now you have a structure with two types of faces, octagons and triangles.

That's an Archimedean solid.

And table 16 .1 in the source material provides a beautiful summary of these 18 solids.

It lists the number of vertices, edges, and faces.

And it also confirms that profound relationship from Euler.

Value e plus f equals 2, 2.

It holds true for all of them.

And finally, the chapter pushes geometry even further, extending these ideas to one -dimensional regular polytopes.

Which is, well, it's impossible for us to visualize 40 Euclidean space E44 dollars.

But the math remains consistent.

The source mentions that in E43 dollars there are six regular polytopes, structures like the five cell and the massive 120 cell.

So the principle is the same.

The geometric constraints governing perfect order extend to any dimension.

To any dimension.

Now we connect the dots.

We take the simple 2D 3667 regular triangle tiling, the closest packed arrangement possible in a plane, and apply it to the physical structures of real crystals.

This is the heart of 3D close packing.

Yes.

Section 16 .6 shows that the majority of dense metallic and intermetallic structures can be viewed as stacked layers of spheres, where each layer is one of those 366 buck planes.

And the density achieved in these close packed arrangements is about 74 percent, which is the maximum possible packing density.

It is.

Now in 3D stacking we have three possible positions for any layer of atoms, which we define relative to the layer below it.

We call them A, B, and C.

So if layer A is the first one, the spheres in layer B sit in the depressions created by layer A.

Right.

And layer C sits in the remaining depressions that aren't covered by A or B.

This leads directly to the two fundamental close packed structures.

First, the hexagonal close packed structure, or HCP.

This is the simple alternating sequence.

ABBAB.

The third layer, A, sits directly over the first layer.

In crystallographic terms, this corresponds to the 0 .1 planes in the hexagonal lattice.

And the second one is the face -centered cubic, or FCC, structure.

And this is the sequence ABC ABC.

The repeat only happens every third layer.

This structure, which is the densest packing you can get in the cubic system, corresponds to the 111 planes of the cubic lattice.

So the difference in stacking determines the fundamental symmetry of the crystal.

Fundamentally.

And within these stacked spheres, you have voids, or interstitial sites.

And these are essential for understanding how complex ionic or intermetallic compounds form, especially when smaller atoms try to squeeze in.

We have two main types of sites.

First, the tetrahedral interstitial site, designated to gamma.

How is that one formed?

It's formed by four surrounding atoms, three in one close packed plane, and one atom in the plane right next to it.

These four atoms define the vertices of a tetrahedron.

And because the coordination number is low, just four, this site is the smaller of the two voids.

It is.

The second type is the octahedral interstitial site, or you're better, dole.

And that one is larger.

It's larger, and it's coordinated by six surrounding atoms,

three in the plane above and three in the plane below.

Crucially, the octahedral site is formed by two adjacent triangular voids, one from the top layer pointing down and one from the bottom layer pointing up.

Forming the eight faces of the octahedron.

And that larger volume means it can accommodate bigger interstitial atoms, which influences things like solubility and compound stability, and compounds like CDI2O2.

Now we get to the part that can get overwhelming fast.

The notation system is required to describe complex stacking sequences, or polytypes.

If you have hundreds of layers, ABC notation is useless.

It becomes useless very quickly.

That's why we need compact codes.

The most standard global system is the Ramsdell notation.

And it's concise.

It just lists the total number of layers in the repeat period, followed by a letter for the lattice type.

C for cubic, H for hexagonal, R for rhombohedral.

So the simple FCC structure is just 3C, three layers, cubic.

The simple HCP structure is 2H, two layers, hexagonal.

But what about a more complex polytype, like the 6H structure, where the stacking sequence is ABCACB?

That's a great example for demonstrating the other notations.

So for 6H, which is ABCACB, Ramsdell identifies it as six layers repeating, with overall hexagonal symmetry.

Okay, now how does the highly abstract Zadonov notation handle 6H?

Zadonov uses a sequence of integers, right?

The sequence of integers, and one and two donos, representing the number of consecutive positive and negative stacking steps.

It's a true code.

A positive step is A to B, B to C, or C to A.

And a negative step is the other way around.

A to C, C to B, or B to A.

So for the sequence ABCACB, we have A to B positive, B to C positive, C to A positive.

That's three positive steps.

Okay, so the first number is three.

Then you get A to C negative, C to B negative, B to A negative.

That's three negative steps.

So the notation is just 33.

That is extremely compact.

6H becomes 33.

Why would a crystallographer prefer this highly abstract integer sequence?

Because for very long polytypes, I mean, structures that repeat over 100 layers, the Zadonov sequence stays short, it's easy to read, and it's mathematically useful for structural calculations.

And finally, we have the dollar notation.

This notation identifies each layer based on its local coordination.

A layer is dollar for hexagonal, if its nearest neighbors mimic the HCP stacking environment.

And to dollars for cubic if they mimic the FCC environment.

Right.

So for our 6H example, ABCACB, the dollar notation is AVC.

The layers in the middle have hexagonal coordination, while the outer ones are cubic.

And this language is absolutely essential when discussing polytypism, this phenomenon of materials having different crystal structures just based on stacking variations, especially in critical semiconductors like silicon carbide, Xyle.

Xyle exhibits extensive polytypism 3C, 4H, 6H, 15R, and many, many more.

And the importance here is that structure dictates function.

Sigality polytypes are wide band gap semiconductors used in high -power electronics.

And the stacking sequence directly impacts their electronic and optical properties.

Directly.

The source material highlights this link.

The specific polytype determines whether the energy gap is direct or indirect, and what the magnitude of that gap is.

So 3C has an indirect gap, but 2H has a direct gap.

Which is crucial for light emitting applications.

A small structural change, just how the spheres stack, produces a huge change in electrical performance.

Moving beyond basic stacking, section 16 .7 introduces an advanced application of the 30606 -STEC tiling the triangulation of polyhedral faces.

This concept is used to describe the structure of complex spherical assemblies.

You often see it in virology or large molecular structures where units decorate the surface of a polyhedron like an achosahedron.

So you start with the large triangular faces of the platonic solid.

And you successively divide them into smaller equilateral triangles.

This process is called triangulation.

It creates this fine, intricate tiling pattern on the surface of the solid.

And the complexity is quantified by the triangulation number $2.

How is that calculated?

$ defines the total number of molecular units in the new structure.

The formula is 2T equals PF22, where pi -dollar is a base factor, and $5 is related to the subdivision level.

And the source notes that for the achosahedron, complex assemblies can reach triangulation numbers as high as 47, 87, or even 207.

Wow.

That's an astonishing complexity, built entirely from a simple triangular motif.

It is.

It's a structural logic that nature itself uses to build things like spherical viral capsids, which have to be perfectly ordered but also highly complex.

So it demonstrates how geometry scales from simple flat planes to these intricate spherical architectures.

Hell does.

And finally, we close our journey by recognizing the pioneers in section 16 .8.

This whole framework wasn't built overnight.

Not at all.

We begin with Johannes Kepler from 1571 to 1630.

Long before modern crystallography, Kepler made fundamental observations on crystal structure things like the geometry of snowflakes and the theory of close -packed spheres.

And critically, he provided the foundational geometry determining the 11 plane networks.

The 11 ways a plane can be tiled uniformly, which are the Archimedean tilings we talked about.

Next is George Ludwig Friedrich Lavs from 1896 to 1978.

Lavs was a Swiss crystallographer who provided the necessary mathematical rigor.

He derived and proved Kepler's 11 plane networks.

And his work was pivotal in defining sphere packings and establishing the concept of Lav's phases.

Which are those critical, structurally complex intermetallic compounds?

We have to mention Victor Moritz Goldschmidt from 1888 to 1947.

He was a giant in stereochemistry.

He was instrumental in applying X -ray diffraction techniques to analyze mineral and crystal structures.

And that fundamentally changed the methodology used today to determine the fine structure of atoms in solid materials.

It did.

What an incredible journey into the structure of matter.

We started with the simple three regular tilings, proved their necessity through geometry, expanded to the 11 Kepler tilings and their dual Lavs structures.

Introduced symmetry through color.

And then jumped into the complex application of these 2D rules to 3D crystals.

The core takeaway, I think, is the importance of the structural language.

Whether you're dealing with perfectly periodic structures like HDP or FCC,

they're complex polytypes.

Using Ramesthal, Zdanov and all dollar notations.

Or even the almost paradoxical order of quasicrystals.

The precise language of geometry is indispensable.

It's the key to decoding long range order in materials.

And we finished with a truly provocative idea from that quasicrystal section.

We learned that the 5 -fold symmetry of stable 3D quasicrystals can only be rationalized by mathematically projecting them from a perfect periodic crystal existing in a 6D space.

With the two fundamental tiles governed by the irrational golden ratio.

So if the projection of a 6D crystal can produce such unexpected, yet highly stable ordered structures in our 3D world.

What might be the implications of theoretical crystals projected down from 10 or 12 dimensions?

What stable hidden forms of order driven by even more complex irrational relationships might exist in the universe that we currently lack the framework to even begin observing?

It makes you wonder if our known reality is just a shadow of a larger, more structured dimension.

It does.

Something to mull over as you contemplate the incredible order governing the materials all around you.

Thank you for sharing these sources for today's Deep Dive.

My pleasure.

We'll catch you next time.