In this article we will continue our discussion of crystal structure and growth.  Then we will move into the work of Roger Penrose, pentagonal tilings and quasicrystals.

# Crystal Growth

Crystal growth consists of the addition of new atoms, ions or polymer strings into the characteristic arrangements of a crystalline Bravais lattice.

Crystalline solids are typically formed by cooling and solidification from the liquid state.

An ideal crystalline surface grows by the spreading of single layers or by the lateral advance of the growth steps bounding the layers.

Bravais Lattices:  1 – oblique (monoclinic), 2 – rectangular (orthorhombic), 3 – centered rectangular (orthorhombic), 4 – hexagonal, and 5 – square (tetragonal).

# Crystal Shape Transformations

Variations in temperature and pressure may change one crystal structure into another.

For example, sulfur transforms from orthorhombic to monoclinic at 96 degrees C.

Fluorite will grow from cubical to octahedral and back again due to truncating – the corners being cut off.  Recall that the cube and octahedron are Platonic solid duals of each other.  The ions rotate, expand or contract to become a different lattice.

A cube transforming into an octahedron by cantellation (cutting down edges).

Cubes can also change into dodecahedra.

A cube transforming into a dodecahedron by cantellation.  Credit: Frank Chester

Octahedra can change into icosahedra then dodecahedra.

Credit: Drunvalo Melchizadek – The Ancient Secret of the Flower of Life, Volume I

“Each pattern and crystal, no matter how complex it gets, will turn into one of the five Platonic solids if you truncate it just right, showing the innate nature of the five Platonic solids in crystal structure.”1

# Bravais Lattices

Bravais Lattices are also known as ‘Space Lattices’.

They refer to a three-dimensional lattice where the tiles space without any gaps or holes.  This means they tessellate.

When the discrete points are atoms, ions, or polymer strings of solid matter, the Bravais lattice concept is used to formally define a crystalline arrangement and its finite frontiers.

A crystal is made up of a periodic arrangement of one or more atoms repeated at each lattice point.

3D Crystal structure of water aligning with lattice points.

Diamond crystal lattice

# X-ray Diffraction Photographs

X-ray diffraction photographs are a technique used for determining the atomic and molecular structure of a crystal, in which the crystalline atoms cause a beam of incident X-rays to diffract into many specific directions.

The first ones were taken by Walter Friedrich and Paul Knipping in Munich under direction of Max von Laue in 1912.

Max von Laue (1879-1960) was a German physicist who studied under Max Planck and won the Nobel Prize in Physics in 1914 for his discovery of the diffraction of X-rays by crystals.

# Laue Pattern

A Laue Pattern is a diffraction pattern of a stationary single crystal, obtained by X-ray.

“The Laue method consists of the following: a narrow X-ray beam with a continuous spectrum is directed at a stationary single crystal, which serves as a diffraction grating. The diffraction pattern produced by the crystal is recorded on a photographic film located behind the crystal. In addition to the center spot formed by the undeflected X-ray beam, various additional spots also appear in the Laue pattern. The number and arrangement of these spots depend on the type of crystal and on its orientation relative to the beam.

The X-ray beam strikes the crystallographic planes and is reflected (X-ray diffraction); the direction of the reflected beam, in accordance with the Bragg-Vul’f [Wulff-Bragg] condition, corresponds to the direction of the diffraction maximum, to which all of the atoms on the crystallographic plane contribute. Each set of parallel crystallographic planes accounts for one spot on the Laue pattern.

The Laue pattern makes it possible to determine the direction of the axes of symmetry of a crystal (that is, to perform its “orientation”; this is especially important for unfaceted crystals). Furthermore, it is possible to evaluate the degree of a crystal’s perfection and to determine certain defects, such as block structure, mosaic structure, and the presence of internal deformations (asterism, X-ray topography), from the distribution of intensity in the spots.”2

*Note – Bragg’s Law (Wulff-Bragg’s condition) is a special case of Laue diffraction.  It gives the angles for coherent and incoherent scattering from a crystal lattice.

NaCl [001] Beam Back Scatter Mode & NaCl+Twin [111] Beam Back Scatter Mode

See here that the diffraction pattern of a beryl crystal displays the flower of life pattern.

Now we will move onto the fascinating topic of quasi-crystals.  In order to understand quasi-crystals, we must first discuss pentagonal tilings.

# Pentagonal Tilings

Recall that the regular pentagon does not tile a plane.  This means it cannot be laid flat on a surface, with its shape repeating, with all sides touching and no spaces in between.  It cannot tessellate with itself.

However, when laying it on a plane, it does do many interesting and beautiful things with other shapes filling in the open spaces.

A pentagonal tiling is a tiling of the plane where each individual piece is in the shape of a non-regular pentagon.

There are fifteen known types of pentagons that tile a plane.  These are not regular pentagons.  In these tilings, all the tiles are the same shape and size.  (See Article 54)

# Johannes Kepler

Johannes Kepler (1571-1630), the famous German mathematician, astronomer and astrologer was the first to show, in 1619 in Harmonices Mundi, how a seed pattern can be grown from the center using pentagons, pentagrams and golden triangles.

# Dr. Roger Penrose

Dr. Roger Penrose (1931-present), an Oxford mathematician, physicist and philosopher of science took up the work were Kepler left off.

Interestingly, Dr. Penrose argues that the known laws of physics are inadequate to explain the phenomenon of consciousness and has written books on the connection between fundamental physics and human and animal consciousness.

He is quoted as saying, “I think I would say that the universe has a purpose, it’s not somehow just there by chance … some people, I think, take the view that the universe is just there and it runs along – it’s a bit like it just sort of computes, and we happen somehow by accident to find ourselves in this thing. But I don’t think that’s a very fruitful or helpful way of looking at the universe, I think that there is something much deeper about it.” 3

# Penrose Tilings

Penrose discovered many pentagonal tilings in 1974 that he named Penrose tilings.

Penrose tilings are where two shapes tile to fill the plane with aperiodic ‘non-repeating’ pentagonal elements at all scales.

“These patterns have been found to underlie the nature of most liquids.

They are, for instance, cross-sections through water.”4

Dr. Penrose developed these ideas based on the 1938 article Two Basic Types of Statistical Distribution by Czech geographer, demographer and statistician Jarmir Korcak.

Dr. Penrose wanted to figure out how to lay pentagon-shaped tiles to fully cover a flat surface.

“Penrose noticed that the two triangles within the pentagon [golden triangle and golden gnomon) that have golden proportions can be assembled in pairs, forming all-new symmetrical tiles that can be combined into different patterns.  For example, two acute golden triangles can be combined to form a ‘kite’, while two obtuse triangles with golden proportions can form a ‘dart’.  Furthermore, a kit and dart can be combined to form a rhombus with sides of length phi.”5

This led to non-periodic tiling with remarkable properties.

They lack translational symmetry.

They are self-similar.  That is, the same patterns occur at larger and larger scales (holographic).

There are three main types of Penrose tilings.  These are P1, P2, and P3.

P1 – uses pentagons, pentagrams, a ‘boat’ (roughly 3/5 of a star) and a ‘diamond’ (a thin rhombus).

P2 – uses the ‘kite’ and ‘dart’.

The kite is a quadrilateral with 4 interior angles: 72°, 72°, 72° and 144°.

The dart is a non-convex quadrilateral with four interior angles: 36°, 72°, 36° and 216°.

P3 – uses a pair of rhombuses (rhombs).

The thin rhomb has 4 corners with angles: 36°, 144°, 36° and 144°.

The thick rhomb has 4 angles: 72°, 108°, 72°, and 108°.

In 1984 such patterns were observed in the arrangement of atoms is quasicrystals.

High-magnification microscopic images & x-ray diffraction patterns of quasicrystals reveal dodecahedral symmetries and the appearance of the golden ratio.

Quasi-crystals reveal five fold symmetry and underlying long range order.

Dr. Penrose says, “My interest in the tiles has to do with the ideas of a universe controlled by very simple forces, even though we see complications all over the place.”

He followed conventional rules to make complicated patterns, just as the universe does.

It was his attempt to see how the complicated could be satisfied by very simple rules that reflect what we see in the world.

His pentagonal tilings are multidimensional structures helping to show how the universe can be built as a hologram.

Incidentally, the rhombic triacontahedron (pictured below) is the 3D analog of a Penrose tiling – the building block of a quasicrystal.

# Quasicrystals

Quasicrystals were discovered by Dan Shechtman in 1982.

Dan Shechtman (1941-present) is a Professor of Materials Science at the Technion – Israel Institute of Technology and Iowa State University.

He earned the Nobel Prize in Chemistry in 2011 for work he did in the eighties.  It took 19 years for the scientific community to admit the importance of his work and honor his discovery of groundbreaking pentagonal-symmetry of quasicrystals.

While on sabbatical in 1982 Shechtman discovered a new state of matter – quasicrystals – midway between crystalline and amorphous states.

“In 1982, scientist Dan Shechtman captured an image with a scanning electron microscope that seemed to contradict basic assumptions in the field of crystallography, a branch of chemistry that studies crystalline solids.  Ten bright dots appeared in each circle, revealing a diffraction pattern of ten-fold symmetry.  The prevailing wisdom at the time held that crystals could only possess two-fold, three-fold, four-fold and six-fold rotational symmetry, but Shechtman’s discovery changed all that.  In fact, it was so unbelievable that he was asked to leave his research group in the course of trying to defend his findings.  The battle raged on, and eventually other scientists were forced to re-examine their understanding of the nature of matter.  With the help of Penrose’s tiling mosaics, the scientific world gradually began to accept Shechtman’s findings.”6

Everyone was skeptical about quasicrystals, even Schechtman himself when he first saw images like the above one.  Credit: Dan Shechtman

On the scale of solids they lie somewhere between crystals proper and glass.

In other words, they are something in between a structure with repeating, symmetric units, and one with completely random building blocks.  They are a perfect balance of order and chaos.

Consider a molten glob of aluminum and manganese (Al6Mn).  In a molten state they are in a chaotic disordered state (it was assumed).  If cooled slowly it was assumed it would cool into a glob.

Shechtman cooled it very quickly with liquid nitrogen.

He discovered 10 fold symmetry in the resulting crystals (see image below).

This was not supposed to exist in mainstream crystallography.  There was not supposed to be 5-fold symmetry.

Credit: Dan Shechtman

He published his results in Physical Review Letters in 1984.

A huge scientific debate ensued.

The definition of a crystal was not changed until 1992.

Since the time of Abbe Hauy, a crystal used to be defined as having “a regularly ordered, repeating 3D pattern”.

Due to these new discoveries it is now defined as a solid with “a discrete diffraction diagram”.

Recall that a diffraction diagram is where you hit a crystal with x-rays and capture the x-ray on film in extreme high resolution.  This shows the overall gist of what the crystal looks like.

Professor Shechtman got beautiful geometry:  6 point, 5 point, and 10 point.

What mainstream science has a hard time admitting is that atoms with particles could never make a crystal like this.

They can’t be built out of units this way.

“Similar to microclusters, quasi-crystals appear to not have individual atoms anymore, but rather that the atoms have merged into a unity throughout the entire crystal.”7

A.L. Mackay writes, “Fractal structures with five-fold axes everywhere require that atoms of finite size be abandoned.  This is not a rational assumption to the crystallographers of the world, but the mathematicians are free to explore it.”

Vibration (frequency, wavelength, harmonics) is the key to structure, not particles.

The 1994 Encyclopedia Britannica shows these quasicrystals in relation to platonic solids.  The image below is from the 1998 Encyclopedia Britannica.

The Encyclopedia Britannica states, “Three views of the icosahedral symmetry of quasicrystalline aluminum-manganese. (Top) View is along the fivefold symmetry axis; (centre) rotating by 37.38° reveals the threefold axis, and (bottom) rotating by 58.29° reveals the twofold axis.”

Quasicrystals have been found that are dodecahedra and icosahedra.

Scanning electron microcopy images of quasicrystals. a, Al 65 Cu 20 Fe 15 dodecahedral quasicrystal

Al63Cu24Fe13is Icosahedrite.

Ho-Mg-Zn is a Pentagonal Dodecahedron.

• Al71Ni24Fe5 – from a Khatyrka meteorite – has a diffraction pattern that reveals a ten-fold symmetry.
• TiMn has a 10-fold diffraction pattern.

Shectmanite ‘snowflakes’ form when an aluminum/manganese alloy is cooled rapidly.

5-fold geometric snowflakes.  Credit: Dan Shectman.   From Nelson, David R., Quasicrystals, Scientific American, Vol. 255, No. 2 (August 1986), pp. 42-51

Europium atoms linked with para-quaterphenyl-dicarbonitrile have a cubic/triangular network (seen below).

Scanning tunneling microscopic image of the quasicrystalline network built up with europium atoms linked with para-quaterphenyl-dicarbonitrile. Credit: J. I. Urgel / TUM

Al-Co-Ni has different 5-fold symmetry variants (see bottom left).

Credit: Zhanbing He, Haikun Ma, Hua Li, Xingzhong Li & Xiuliang Ma.  Read more here.

# Qualities of Quasi-crystals

Quasi-crystals are ordered but not periodic – their neat patterns never exactly repeat.

They are very hard.

They have low friction.

They don’t conduct heat very well (good candidate for protective coatings on items ranging from airplanes to non-stick cookware).

# Natural Quasi-crystals

Three natural specimens have been found in 4.5 billion-year-old meteorite from the Khatyrka region in northeastern Russia.

In 2009 the first one was found.8  It was icosahedrite, a quasicrystalline alloy of aluminum, copper and iron (Al63Cu24Fe13).

Icosahedrite diffraction pattern.

In 2015 a second one was found.  It was Al71Ni24Fe5.  It was the first found with decagonal symmetry.

In 2016 the third one was found.9

Credit: Luca Bind et al.

They have also found a quasi-crystal with icosahedral 20-sided symmetry – it resembles flat 10-sided disks stacked in a column.

Credit: Steinhardt et al.

It was previously assumed quasi-crystals only existed in the laboratory.  They were thought to be too fragile and energetically unstable to form in nature.  These meteorite discoveries proved otherwise.

# Conclusion

In this article we have briefly discussed crystal growth, crystal lattices and x-ray diffraction patterns.  We also discussed the discovery of quasi-crystals, which are based upon pentagonal tilings discovered by Sir Roger Penrose.

The important thing is to note how the Platonic solids continually fit into the structure of the atomic, molecular and mineral worlds.

We also continue to see evidence that subatomic particles such as protons and electrons cannot be discrete particles, but part of a larger wave structure.

This information perfectly fits into our theory of all in reality being composed of a fluid-like crystallized Aether that organizes itself into geometric fractal-holographic patterns upon which all physical matter grows – like a crystal.

In the next ten articles we will jump from examining the geometry of the atomic, molecular and mineral worlds to the geometry of plant, insect and animal life.

Something very interesting occurs when we jump from the mineral scale to the plant scale – the first scale containing what is traditionally viewed as “living”.  We begin to exit the realm of pure geometric solids and enter the realm of the golden ratio.

Everyone is well aware of the mind-bendingly large variety of plant, microorganism, insect and animal life.  We are also well aware that none of these life forms (at least most of them) do not exist as pure cubes, octahedra, dodecahedra and so forth, as we have been seeing in the mineral world.

The way this wide variety of life forms grow upon our geometric matrix and still have a nearly infinite variation is to use the golden ratio and Fibonacci sequence.

Of course, it is to be remembered that the Fibonacci sequence comes from the golden ratio and the golden ratio comes from the pentagon and the pentagon comes from the dodecahedron.

So even though we see a wide variety of life forms structured in all kinds of ways, all life is still growing upon a matrix of oscillating Platonic solid geometry, particularly the dodecahedron and icosahedron.

The difference is that the structures are far, far more complex (and much more rapidly changing) than what we have been looking at in the atomic, molecular, and mineral worlds.

1. Melchizadek, Drunvalo, The Ancient Secret of the Flower of Life, Volume 1, Clear Light Trust, 1998
2. Kolpakov, A.V, The Great Soviet Encyclopedia, 1979
3. Morris, Errol, A Brief History of Time, 1991
4. Lundy, Miranda, Sacred Geometry, Wooden Books, 1998
5. Meisner, Gary B., The Golden Ratio: The Divine Beauty of Mathematics, RacePoint Publishing, 2018
6. Wilcock, David, The Divine Cosmos, https://www.divinecosmos.com/start-here/articles/97-the-divine-cosmos-chapter-03-sacred-geometry-in-the-quantum-realm?%20fontstyle=f-larger&%20fontstyle=f-larger&%20fontstyle=f-larger&%20fontstyle=f-smaller
7. Meisner, Gary B., The Golden Proportion: The Divine Beauty of Mathematics, Race Point Publishing, 2018
8. http://www.nature.com/articles/srep09111
9. http://www.dailymail.co.uk/sciencetech/article-4016890/Third-natural-quasicrystal-discovered-Russian-meteorite-hosted-two-examples.html