Return to Free Library

Return to Number Menu

Previous Article                                                                         Next Article


We will now take a look at pentagonal tilings, Penrose tilings and quasicrystals.


Pentagonal Tilings

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

The regular pentagon does not tile on a plane. There will always be gaps, as seen below:

However, non-regular pentagons do many interesting and beautiful things as we shall see.



15 Types of Irregular Convex Pentagon that Tile the Plane

There are 15 known types of pentagons that tile a plane.  These are not regular pentagons.  That means the side lengths are not all equal.  However, in these tilings, all the tiles are the same shape.

The first five were found by Karl Rheinhardt.  Three more were found by Richard Kershner in 1968.  Richard E. James found 1 more.  Marjorie Rice found four more types by 1977.

In 1985 Rolf Stein found the 14th version and in 2015 the 15th version was found by Casey Mann, Jennifer McLoud and David Von Derau.1


Here are the fifteen types:



Nonperiodic Monohedral Pentagonal Tilings

Nonperiodic = not recurring at regular intervals; non-repeating

Monohedral = all tiles are the same size and ‘shape’.  That is, every tile in the tiling is congruent to a fixed subset of the plane.

Nonperiodic monohedral pentagonal tilings can be constructed like the examples below.



Dual Uniform Tilings

There are three isohedral pentagonal tilings generated as duals of the uniform tilings.

Isohedral = A monohedral tiling where for any two tiles there is a symmetry of the tiling that maps one tile to the other.

These isohedral pentagonal tilings are:



Aperiodic Tilings

Aperiodic tilings are non-repeating.



Johannes Kepler (1571-1630)

Kepler was the first known to experiment with aperiodic pentagonal tilings.

In 1619 he published Harmonices Mundi that showed how gaps between pentagons on a place could be filled by using pentagrams, decagons, and golden triangles, as seen below.



Dr. Roger Penrose (1931-present)

Roger Penrose was an Oxford mathematician, physicist and philosopher of science who took up where Kepler left off and discovered aperiodic pentagonal tilings in 1974 now known as Penrose tilings.

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.”2


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.”3


Icosahedral water clusters.  Credit: Martin Chaplin – Water Structure & Science



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.”4

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).

Pentagons – red, dark blue & grey

Pentagrams – green

Boats – light blue

Diamonds – yellow



P2 – uses the ‘kite’ and ‘dart’.

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

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



P3 – uses a pair of rhombuses (rhombs).

The thin rhomb (light blue) has 4 corners with angles: 36°, 144°, 36° and 144°.

The thick rhomb (dark blue) has 4 angles: 72°, 108°, 72°, and 108°.



Penrose Tilings & Quasicrystals

In 1984 Penrose tiling patterns were observed in the arrangement of atoms in quasicrystals.

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

Quasi-crystals reveal fivefold 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 (seen below) is the 3D analog of a Penrose tiling – building block of a quasicrystal.




Quasicrystals were discovered by Dan Schechtman 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 the 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.”5

Everyone was skeptical about quasicrystals, even Schechtman himself when he first saw imagees 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.

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”.

A ‘diffraction diagram’ results from hitting a crystal with x-rays then capturing the x-ray on film in extreme high resolution.  This will show 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 in 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.”6


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, and harmonics) is the key to structure, not particles.


The 1994 Encyclopedia Britannica shows these quasicrystals in relation to platonic solids.

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.


Schectmanite ‘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.

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.7   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.8

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.




In this article we have  discussed the discovery of quasi-crystals, which are based upon pentagonal tilings discovered by Sir Roger Penrose.

We see 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.


  1. Mulcahy, Colm, Martin Gardner at 101 (“It’s as not-so-easy as 3, 4, 5”), Scientific American, 28 October 2015.
  2. Morris, Errol, A Brief History of Time, 1991
  3. Lundy, Miranda, Sacred Geometry, Bloomsbury Publishing, 2001
  4. Meisner, Gary B., The Golden Ratio: The Divine Beauty of Mathematics, RacePoint Publishing, 2018
  5. Wilcock, David, The Divine Cosmos,
  6. Meisner, Gary B., The Golden Ratio: The Divine Beauty of Mathematics, RacePoint Publishing, 2018

Return to Free Library

Return to Number Menu

Previous Article                                                                         Next Article