The Infinite Pattern That Never Repeats

Summary

This video explores the fascinating world of impossible patterns and materials, starting with Johannes Kepler's theories on planetary orbits and his geometric model of the solar system. It delves into Kepler's conjecture on the efficient stacking of cannonballs and his musings on snowflakes' hexagonal patterns. The narrative then transitions to the concept of aperiodic tiling, exemplified by Roger Penrose's discovery of a pattern that can tile a plane without repeating. This leads to the groundbreaking discovery of quasi-crystals, which defied traditional crystallography and were ultimately recognized with a Nobel Prize. The video ponders the unseen possibilities that challenge our perceptions of the impossible.

Takeaways

• 📘 Johannes Kepler is renowned for discovering elliptical planetary orbits but initially proposed a solar system model using nested spheres and Platonic solids.
• 🔍 Kepler's interest in geometry led him to solve practical problems like the efficient stacking of cannonballs, which he determined could be optimally packed in a hexagonal close packing or face-centered cubic arrangement.
• 🌨️ In his pamphlet 'De Nive Sexangula', Kepler pondered the reason behind the consistent six-cornered shape of snowflakes, hinting at a deeper understanding of crystalline structures.
• 🔲 Kepler's conjecture on the most efficient sphere packing was unproven during his lifetime but was confirmed to be correct in 2017, approximately 400 years later.
• 🔷 The concept of regular hexagons tiling a plane periodically with no gaps was understood by Kepler, who also explored the impossibility of regular pentagons achieving the same.
• 🔺 Roger Penrose's work on aperiodic tiling with two shapes, kites and darts, challenged traditional periodic tiling patterns and introduced a new form of non-repeating patterns.
• 🛠️ Penrose's tiling patterns, although not initially believed to have a physical analog, were later found to be mirrored in quasi-crystals, a discovery that earned Dan Shechtman the Nobel Prize in Chemistry.
• 🔬 The discovery of quasi-crystals, materials that display five-fold symmetry and do not fit into traditional crystallography, was a breakthrough that expanded our understanding of solid materials.
• 🔄 The Fibonacci sequence and the golden ratio are naturally embedded within Penrose's tiling patterns, showcasing the interconnectedness of mathematical constants and geometric patterns.
• 🔑 LastPass, a password management tool, was highlighted as a sponsor, emphasizing the importance of secure password practices and the convenience of password management for users.

Q & A

What is Johannes Kepler most famous for?

Johannes Kepler is most famous for figuring out that the shapes of planetary orbits are ellipses.

What was Kepler's initial model of the solar system before discovering elliptical orbits?

Kepler initially invented a model where planets were on nested spheres separated by the Platonic solids.

What are the five Platonic solids?

The five Platonic solids are the tetrahedron, cube, octahedron, dodecahedron, and icosahedron.

What is Kepler's conjecture regarding the stacking of cannonballs?

Kepler's conjecture states that hexagonal close packing and the face-centered cubic arrangement are both equivalently and optimally efficient for stacking cannonballs, occupying about 74 percent of the volume they take up.

What was Kepler's pamphlet 'De Nive Sexangula' about?

Kepler's pamphlet 'De Nive Sexangula' pondered why snowflakes always form six-cornered shapes and speculated about the self-arrangement of water molecules into hexagonal crystals.

What is the significance of hexagons in tiling a plane?

Hexagons can cover a flat surface perfectly with no gaps, a property known as periodic tiling, and they have six-fold symmetry.

What is an aperiodic tiling?

An aperiodic tiling is a set of shapes that can tile the plane without ever repeating the same pattern, such as the Penrose tiling.

How did Roger Penrose simplify the Penrose tiling?

Roger Penrose simplified the Penrose tiling to just two shapes: a thick rhombus and a thin rhombus, with rules enforced by bumps and notches or matching colors.

What is the connection between Penrose tiling and the golden ratio?

The ratio of kites to darts in Penrose tiling approaches the golden ratio, which is an irrational number and provides evidence that the pattern cannot be periodic.

How did the discovery of quasi-crystals challenge the existing understanding of crystals?

Quasi-crystals, which exhibit five-fold symmetry, challenged the existing understanding that crystals must have one of the 14 known unit cells and cannot have long-range order without repeating patterns.

What was Linus Pauling's initial reaction to the concept of quasi-crystals?

Linus Pauling famously remarked, 'There are no quasi-crystals, only quasi-scientists,' expressing skepticism towards the concept of quasi-crystals that defied the laws of nature at the time.

Outlines

00:00

🌌 Johannes Kepler's Contributions to Geometry and Astronomy

The video script begins with a discussion of Johannes Kepler, a renowned scientist who lived and worked in Prague. Kepler is celebrated for his discovery that planetary orbits are elliptical. Prior to this, he proposed a model of the solar system where planets were arranged on nested spheres separated by Platonic solids, which are shapes with identical faces and vertices. Kepler's interest in geometry also led him to solve practical problems, such as the most efficient way to stack cannonballs, which he found to be hexagonal close packing and face-centered cubic arrangement, a conjecture that took 400 years to prove. Kepler's work on the six-cornered snowflake and his fascination with the hexagonal shape of ice crystals laid the groundwork for understanding atomic and molecular structures. Additionally, Kepler's exploration of tiling with regular hexagons and his unsuccessful attempt to create a periodic tiling with regular pentagons showcased his deep engagement with geometric patterns.

05:01

🔗 Aperiodic Tiling and the Quest for Non-Periodic Patterns

The script then delves into the concept of aperiodic tiling, where patterns can be created that never repeat themselves. It discusses the work of Hao Wang and his student Robert Berger, who disproved Wang's conjecture by demonstrating that certain sets of tiles could only tile the plane non-periodically. This discovery led to a quest to find the smallest set of tiles that could achieve this, culminating in Roger Penrose's work that reduced the set to just two unique shapes, which he used to create a pattern with an almost five-fold symmetry. Penrose's pattern, when overlaid with Kepler's pentagon pattern, aligns perfectly, showcasing a fascinating connection between historical and modern geometric studies. The script also explains the concept of Moire patterns, which are interference patterns that occur when two grids are overlaid, and how they demonstrate the non-periodic nature of Penrose's tiling.

10:02

🎲 Penrose Tiling and the Golden Ratio

The third paragraph focuses on Penrose's favorite pattern, which is composed of two shapes known as kites and darts. These shapes have specific angles and are designed to fit together based on continuous curves. The script explains that these tiles can create an aperiodic tiling, meaning the pattern extends infinitely without repeating. It also introduces the concept that there are an uncountably infinite number of different patterns that can be created with kites and darts, yet they all appear the same when observed. The golden ratio, approximately 1.618, is found in the ratio of kites to darts, hinting at the pattern's connection to five-fold symmetry and pentagons. The script further explores the Fibonacci sequence, which is observed in the spacing of lines within the pattern, reinforcing the pattern's non-periodic and intricate nature.

15:05

🔬 Quasi-Crystals: The Physical Manifestation of Aperiodic Tiling

The fourth paragraph explores the application of Penrose's aperiodic tiling in the physical world, specifically in the discovery of quasi-crystals. These materials, once thought to be impossible due to their violation of traditional crystallography rules, were first theorized and then experimentally confirmed by Paul Steinhardt and Dan Shechtman. The script explains how quasi-crystals, unlike traditional crystals, can exhibit five-fold symmetry and do not follow the 14 known unit cells of crystallography. The discovery of quasi-crystals challenged the scientific community's understanding of matter and led to Shechtman receiving the Nobel Prize in Chemistry in 2011. The script concludes by pondering the existence of phenomena that may be overlooked because they defy current understanding.

20:08

🔐 LastPass: Streamlining Digital Security

The final paragraph shifts focus to the sponsor of the video, LastPass, a password management service. It emphasizes the benefits of using LastPass for securing digital accounts by storing an unlimited number of passwords, enabling cross-device synchronization, and facilitating password sharing. The script encourages viewers to improve their digital security and reduce the cognitive load of remembering multiple passwords by adopting a password manager like LastPass. It also acknowledges the support provided by LastPass for the video content.

Keywords

💡Platonic Solids

Platonic solids are geometric shapes where all faces are congruent, regular polygons with the same number of faces meeting at each vertex. In the video, Johannes Kepler used these solids to model the solar system, placing one between each planet's sphere to reflect his belief in the universe's geometric harmony. The script mentions the cube, tetrahedron, octahedron, dodecahedron, and icosahedron as examples, illustrating Kepler's attempt to find cosmic order.

💡Kepler's Conjecture

Kepler's Conjecture, as discussed in the video, refers to his hypothesis that the most efficient way to stack cannonballs or spheres is through hexagonal close packing and face-centered cubic arrangement, which allows for approximately 74% space utilization. This conjecture was significant in the video as it showcases Kepler's practical application of geometry and was only proven correct 400 years later.

💡Hexagonal Close Packing

Hexagonal Close Packing is a method of arranging spheres in a pattern where each sphere is surrounded by six others, maximizing space efficiency. The video explains that Kepler identified this as the optimal way to stack cannonballs on a ship's deck, which is a practical application of geometric principles and a precursor to modern material science.

💡Aperiodic Tiling

Aperiodic tiling, as introduced in the video, is a pattern of shapes that can cover a plane without repeating. The video delves into the concept through the work of Roger Penrose, who developed a set of two tiles that can tile a plane aperiodically, creating an infinitely repeating pattern without any periodicity. This challenges traditional notions of symmetry and order in geometry.

💡Penrose Tiling

Penrose Tiling is a non-periodic tiling of the plane using two shapes, kites and darts, which can cover an area infinitely without repeating. The video highlights this as a significant discovery in the field of geometry, showing how it relates to the golden ratio and five-fold symmetry, and how it contrasts with the periodic patterns found in traditional tiling.

💡Golden Ratio

The Golden Ratio, often denoted by the Greek letter phi (Φ), is an irrational number approximately equal to 1.618. In the video, it is mentioned in the context of Penrose Tiling, where the ratio of kites to darts in the pattern approaches this ratio, suggesting a connection between the geometry of the tiling and the aesthetically pleasing proportions found in nature and art.

💡Fibonacci Sequence

The Fibonacci Sequence is a series of numbers where each number is the sum of the two preceding ones, typically starting with 0 and 1. The video script notes that the ratio of certain elements in Penrose Tiling follows the Fibonacci Sequence, further linking the mathematical pattern to natural phenomena and the golden ratio.

💡Quasi-Crystal

A quasi-crystal, as mentioned in the video, is a state of matter that exhibits ordered structures without translational symmetry, challenging the classical definition of crystals. The discovery of quasi-crystals, like those made from aluminum and manganese, was a significant breakthrough that validated the theoretical models of Penrose Tiling and expanded our understanding of material science.

💡Five-Fold Symmetry

Five-fold symmetry is a type of rotational symmetry where an object looks the same after a 1/5th turn. The video discusses how this symmetry is rare in traditional geometric patterns but is a key feature of Penrose Tiling and quasi-crystals. This concept is central to the video's exploration of patterns that defy conventional geometric expectations.

💡De Nive Quinquangula

De Nive Quinquangula, or 'On the Six-Cornered Snowflake,' is a pamphlet written by Kepler mentioned in the video. It discusses the hexagonal symmetry of snowflakes, which is a precursor to modern understanding of crystallography. The video uses this work to illustrate Kepler's curiosity about the geometric patterns in nature, which later influenced the development of aperiodic tiling and quasi-crystals.

Highlights

Johannes Kepler's model of the solar system with nested spheres and Platonic solids.

Kepler's belief in geometric regularity in the universe, despite his incorrect model.

Kepler's contribution to practical geometry with his solution for efficient cannonball stacking.

The hexagonal close packing and face-centered cubic arrangement, both proven by Kepler to be optimally efficient.

Kepler's conjecture on the hexagonal shape of snowflakes and his speculation on water molecules.

Kepler's understanding of hexagons' ability to tile the plane periodically.

The discovery of aperiodic tilings that can only tile the plane non-periodically.

Roger Penrose's simplification of aperiodic tiling to just two tiles: a thick rhombus and a thin rhombus.

The Penrose pattern's five-fold symmetry and its relation to the golden ratio.

The Fibonacci sequence's appearance in the Penrose pattern's long and short spacings.

The challenge of creating a physical analog for Penrose patterns, leading to the discovery of quasi-crystals.

Paul Steinhardt's work on quasi-crystals and their five-fold symmetry, contradicting traditional crystal structures.

Dan Shechtman's discovery of quasi-crystals in nature, earning him the Nobel Prize in Chemistry.

The potential applications of quasi-crystals in various industries, such as non-stick cookware and ultra-durable steel.

The philosophical question raised by the existence of quasi-crystals: What else exists beyond our current understanding?

Sponsorship by LastPass, highlighting the importance of secure password management.

Transcripts

A portion of this video was sponsored by lastpass

This video is about a pattern people thought was impossible, and a material that wasn't supposed to exist.

The story begins over 400 years ago in Prague

I'm now in Prague and the Czech Republic, which is perhaps my favorite European city that I've visited so far.

I'm going to visit the Kepler museum,

because he's one of the most famous scientists who lived and worked around Prague.

I want to tell you five things about Johannes Kepler that are essential to our story.

Number one: Kepler is most famous for figuring out that the shapes of planetary orbits are ellipses,

but before he came to this realization,

He invented a model of the solar system

He invented a model of the solar system in which the planets were on nested spheres separated by the platonic solids

What are the platonic solids?

Well they are objects where all of the faces are identical and all of the vertices are identical

which means you can rotate them through some angle, and they look the same as they did before

So the cube is an obvious example.

Then you also have the tetrahedron,

the octahedron,

the dodecahedron which has 12 pentagonal sides,

and the icosahedron which has 20 sides.

And that's it. There are just five platonic solids.

Which was convenient for Kepler because in his day they only knew about six planets

So this allowed him to put a unique platonic solid between each of the planetary spheres

Essentially he used them as spacers

He carefully selected the order of the platonic solids

so that the distances between planets would match astronomical observations as closely as possible

He had this deep abiding belief that there was some geometric regularity in the universe

and of course there is. Just not this.

Two: Kepler's attraction to geometry extended to more practical questions like,

How do you stack cannonballs so they take up the least space on a ship's deck?

By 1611 Kepler had an answer:

hexagonal close packing and the face centered cubic arrangement

are both equivalently and optimally efficient,

with cannonballs occupying about 74 percent of the volume they take up

now this might seem like the obvious way to stack spheres:

I mean it is the way that oranges are stacked in the supermarket.

But Kepler hadn't proved it.

He just stated it as fact, which is why this became known as Kepler's conjecture.

Now it turns out he was right, but it took around 400 years to prove it.

The formal proof was only published in the journal Form of Mathematics in 2017.

Three: Kepler published his conjecture in a pamphlet called deniva sexangula,

on the six cornered snowflake.

In which he wondered, there must be a definite cause why,

whenever snow begins to fall its initial formations invariably display the shape of a six cornered starlet

for if it happens by chance why do they not fall just as well with five corners, or with seven

Why always with six?

In Kepler's day, there was no real theory of atoms or molecules or how they self-arrange into crystals

but Kepler seemed to be on the verge of understanding this

I mean he speculates about the smallest natural unit of a liquid like water, essentially a water molecule,

and how these tiny units could stack together mechanically to form the hexagonal crystal

Not unlike the hexagonal close-packed cannonballs

Four: Kepler knew that regular hexagons can cover a flat surface perfectly with no gaps

In mathematical jargon we say the hexagon tiles the plane periodically.

You know that a tiling is periodic if you can duplicate a portion of it

and continue the pattern only through translation, with no rotations or reflections.

Periodic tilings can also have rotational symmetry.

A rhombus pattern has twofold symmetry because if you rotate it 180 degrees, one half turn,

the pattern looks the same as it did before

Equilateral triangles have three-fold symmetry

squares have four-fold symmetry and hexagons have six-fold symmetry

but those are the only symmetries you can have: two, three, four, and six

There is no five-fold symmetry.

Regular pentagons do not tile the plane

But that didn't stop Kepler from trying

See this pattern right here?

He published it in his book harmonics mundi or harmony of the world

it has a certain five-fold symmetry,

but not exactly, and it's not entirely clear how you would continue this pattern to tile the whole plane

There are an infinite number of shapes that can tile the plane periodically

The regular hexagon can only tile the plane periodically

There are also an infinite number of shapes that can tile the plane periodically or non-periodically

For example, isosceles triangles can tile the plane periodically

but if you rotate a pair of triangles, well then the pattern is no longer perfectly periodic

a sphinx tile can join with another rotated 180 degrees and tile the plane periodically

But a different arrangement of these same tiles is non-periodic

This raises the question: are there some tiles that can only tile the plane non-periodically?

Well in 1961 Hao Wang was studying multi-colored square tiles

the rules were, touching edges must be the same color,

and you can't rotate or reflect tiles only slide them around.

Now the question was, if you're given a set of these tiles,

can you tell if they will tile the plane?

Wang's conjecture was that if they can tile the plane, well they can do so periodically

but it turned out Wang's conjecture was false.

His student, Robert Burger, found a set of 20426 tiles

that could tile the plane but only non-periodically

think about that for a second: here we have a finite set of tiles

Okay it's a large number, but it's finite.

And it can tile all the way out to infinity without ever repeating the same pattern

there's no way even to force them to tile periodically

and a set of tiles like this that can only tile the plane non-periodically is called an aperiodic tiling

and mathematicians wanted to know,

Were there aperiodic tilings that required fewer tiles?

Well Robert Burger himself found a set with only 104.

Donald Knuth got the number down to 92,

and then in 1969, you had Raphael Robinson who came up with six tiles.

Just six, that could tile the entire plane without ever repeating

Then along came Roger Penrose, who would ultimately get the number down to two

Penrose started with a pentagon.

He added other pentagons around it and of course noticed the gaps

But this new shape could fit within a larger pentagon which gave Penrose an idea:

What if he took the original pentagons and broke them into smaller pentagons?

Well now some of the gaps start connecting up into rhombus shapes,

Other gaps have three spikes.

But Penrose didn't stop there:

He subdivided the pentagons again

Now some of the gaps are large enough that you can use pentagons to fill in part of them

and the remaining holes you're left with are just rhombuses, stars, and a fraction of a star that penrose calls a justice cap.

You can keep subdividing indefinitely, and you will only ever find these shapes

So with just these pieces, you can tile the plane aperiodically

With an almost five-fold symmetry.

The fifth thing about Johannes Kepler is that if you take his pentagon pattern,

and you overlay it on top of Penrose's,

The two match up perfectly.

Once Penrose had his pattern, he found ways to simplify the tiles.

He distilled the geometry down to just two tiles: a thick rhombus and a thin rhombus.

The rules for how they can come together can be enforced by bumps and notches, or by matching colors

And the rules ensure that these two single tiles can only tile the plane non-periodically

Just two tiles go all the way out to infinity without ever repeating

Now one way to see this is to print up two copies of the same Penrose pattern

and one on a transparency and overlay them on top of each other

Now the resulting interference you get is called a Moire pattern,

where it is dark the patterns are not aligned

you can see there are also some light spots

and that's where the patterns do match up.

And as I rotate around you can see the light spots move in and get smaller

and then at a certain point they move out and get bigger

and what I want to do is try to enlarge one of these bright spots,

and see how big of a matching section I can find

oh yes yes!

It's like all of a sudden everything is illuminated

I love it

So these patterns are perfectly matching up here here here here and here but not along these radial lines

and that is why they look dark.

So what this shows us is that you can't ever match any section perfectly to one beneath it:

There will always be some difference.

So my favorite Penrose pattern is actually made out of these two shapes

which are called kites and darts

And they have these very particular angles

and the way they're meant to match up is based on these two curves.

So you can see there's a curve on each piece

and so you have to connect them so that the curves are continuous

and that's the rule that allows you to build an aperiodic tiling from these two pieces

so uh I laser cut thousands of these pieces

and oh i'm gonna try to put them together and make a huge Penrose tiling

Oh man. Come on

If you stare at a pattern of kites and darts

you'll start to notice all kinds of regularities like stars and suns

but look closer and they don't quite repeat in the way you'd expect them to

these two tiles create an ever-changing pattern that extends out to infinity without repeating

Does this mean there is only one pattern of kites and darts

and every picture that we see is just a portion of that overall singular pattern?

Well the answer is no.

There are actually an uncountably infinite number of different patterns of kites and darts that tile the entire plane

and it gets weirder:

If you were on any of those tilings,

you wouldn't be able to tell which one it is.

I mean you might try to look further and further out gather more and more data, but it's futile.

Because any finite region of one of these tilings appears infinitely many times

in all of the other versions of those tilings

I mean don't get me wrong those tilings are also different in an infinite number of ways,

But it's impossible to tell that unless you could see the whole pattern, which is impossible

There's this kind of paradox to Penrose tilings

where there's an uncountable infinity of different versions,

but just by looking at them, you could never tell them apart

Now what if we count up all the kites and darts in this pattern?

Well I get 440 kites and 272 darts.

Does that ratio ring any bells?

Well if you divide one by the other you get 1.618

That is the golden ratio.

So why does the golden ratio appear in this pattern?

Well as you know it contains a kind of five-fold symmetry

and of all the irrational constants, the golden ratio phi is the most five-ish of the constants.

I mean you can express the golden ratio as 0.5 plus 5 to the power of 0.5 times 0.5

The golden ratio is also heavily associated with pentagons

I mean the ratio of the diagonal to an edge is the golden ratio

And the kite and dart pieces themselves

are actually sections of pentagons,

Same with the rhombuses

So they actually have the golden ratio built right into their construction

The fact that the ratio of kites to darts approaches the golden ratio, an irrational number,

provides evidence that the pattern can't possibly be periodic.

If the pattern were periodic, then the ratio of kites to darts

could be expressed as a ratio of two whole numbers:

The number of kites to darts in each periodic segment.

And it goes deeper:

If you draw on the tiles not curves but these particular straight lines,

well now when you put the pattern together you see something interesting

they all connect up perfectly into straight lines

there are five sets of parallel lines

this is a kind of proof of the five-fold symmetry of the pattern

but it is not perfectly regular

take a look at any one set of parallel lines.

You'll notice there are two different spacings.

Call them long and short

From the bottom we have long short long short long long..

wait that breaks the pattern

These gaps don't follow a periodic pattern either

but count up the number of longs and shorts in any section:

here I get 13 shorts and 21 longs,

and you have the fibonacci sequence

1 1 2 3 5 8 13 21 34 and so on

And the ratio of one Fibonacci number to the previous one approaches the golden ratio

Now the question Penrose faced from other scientists was, could there be a physical analog for these patterns?

Do they occur in nature, perhaps in crystal structure

Penrose thought that was unlikely.

The very nature of a crystal is that it is made up of repeating units

Just as the fundamental symmetries of the shapes that tile the plane had been worked out much earlier,

the basic unit cells that compose all crystals were well established.

There are 14 of them

and no one had ever seen a crystal that failed to fit one of these patterns.

And there was another problem

Crystals are built by putting atoms and molecules together locally,

whereas Penrose tilings -- well, they seem to require some sort of long range coordination.

Take this pattern for example

You could put a dart over here and continue to tile out to infinity. No problems.

Or you could put a kite over here on the other side. Again, no problems.

But, if you place the kite and dart in here simultaneously, well then this pattern will not work

I mean, you can keep tiling for a while

but when you get to somewhere around here, well it's not gonna work.

You can put a dart in there which completes the pattern nicely,

But then you get this really awkward shape there, which is actually the shape of another dart

but if you put that one in there then the lines don't match up.

The pattern doesn't work.

So how could this work as a crystal?

I mean, both of these tiles obey the local rules,

but in the long term, they just don't work.

Paul Steinhardt and his students were using computers to model how atoms come together into condensed matter,

In the early 1980s,

that is essentially solid material at the smallest scales,

and he found that locally, they like to form icosahedrons

but this was known to be the most forbidden shape because it is full of five-fold symmetries.

So the question they posed was, how big can these icosahedrons get?

They thought maybe 10 atoms or 100 atoms,

but inspired by Penrose tilings, they designed a new kind of structure.

A 3D analog of Penrose tilings now known as a quasi-crystal.

And they simulated how x-rays would diffract off such a structure

and they found a pattern with rings of 10 points, reflecting the five-fold symmetry

Just a few hundred kilometers away, completely unaware of their work,

another scientist Dan Shechtman created this flaky material from aluminum and manganese

and when he scattered electrons off his material, this is the picture he got.

It almost perfectly matches the one made by Steinhardt

So if Penrose tilings require long range coordination, then how do you possibly make quasi crystals?

Well i was talking to Paul Steinhardt about this and he told me,

if you just use the matching rules on the edges,

those rules are not strong enough and if you apply them locally you run into problems like this.

You misplace tiles.

But he said if you have rules for the vertices --

the way the vertices can connect with each other,

those rules are strong enough locally so that you never make a mistake and the pattern can go on to infinity

One of the seminal papers on quasi crystals was called De Nive Quinquangula:

on the pentagonal snowflake in a shout out to Kepler.

Now not everyone was delighted at the announcement of quasi crystals,

a material that up until then people thought totally defied the laws of nature.

Double Nobel Prize winner Linus Pauling famously remarked,

"There are no quasi crystals, only quasi scientists"

burn

But uh Shechtman got the last laugh.

He was awarded the Nobel Prize for chemistry in 2011

and quasi crystals have since been grown with beautiful dodecahedral shapes

they are currently being explored for applications from non-stick electrical insulation and cookware, to ultra durable steel.

And the thing about this whole story that fascinates me the most is,

What exists that we just can't perceive because it's considered impossible?

I mean the symmetries of regular geometric shapes seemed so obvious and certain

that no one thought to look beyond them, that is until Penrose.

And what we found are patterns that are both beautiful and counter-intuitive,

and materials that existed all along that we just couldn't see for what they really are

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