Hi, and welcome to the fun and exciting crash course in Quasicrystals, that's a tongue twister.

The first thing we're going to talk about is tiling, no not quite that kind of tiling. We're actually going to be making the tiles from various shapes. The idea is to see which tiles fill the plane, without leaving any gaps! and figure out what it is about the Geometry of these shapes that determines which ones will tile the plane and which ones won't.  

One thing we notice about all the shapes that tile the plane is that they have 2 or more lines of symmetry. This is because in order to get shapes to fit together without leaving any gaps they need to be equal and opposite in 2 dimensions, in order to tile a 2 dimensional plane.

Which of these shapes have symmetry? Which ones will tile the plane? Correct me if I'm wrong, but THESE ones don't form patterns. The one's I've circled in green might, but I only need to test one of them because I can already identify them as homogenous topological transformations of one another. . .

homogenous means "the same everywhere" or "equal on all sides" 
topology is a branch of mathematics that deals with the connectedness of points.
this only works because I do the same thing to both sides, symmetrically.

An ordinary arrow will tile the plane, however this arrow has a  broken tail and will leave gaps.

It turns out that all shapes which tile the plane obey several simple geometry rules:
The shape must have two or more lines of symmetry, and The angles must either add up to or divide evenly into 360 degrees in order to complete the circle and fill the plane.

When we expand from 2 dimensions to three, we are no longer tiling a plane but filling in 3d space. This is the geometry behind crystallography, the science of crystals. A crystal tiles the 3d plane in much the same way that the 2d tilings we just used. Only crystals use real atoms as the building blocks, instead of abstract shapes and lines on a piece of paper. 

These are some examples of building block shapes that will tile the 3d plane so to speak, and form crystals. You may want to pause it to note their geometry and symmetry.

The rules for 3d are similar to the ones for 2d, only you are dividing into a sphere instead of a circle.
Which means you are dealing with two angles instead of one.
The Polar Angle must be divide evenly into 180 degrees (or 1pi)
While the Azimuthal angle must divide evenly into 360 degrees (2pi)

There are 7 different Crystal systems with Defining Symmetry
for more information on them I suggest you go to wikipedia and read up!
I also suggest using the Microsoft Paint program or any other graphics program to fool around with shapes. Copy and past shapes to tile the plane. If you find you can do it without rotating the shapes, this means that they have "Translational Symmetry"  

The Penrose tiling (which is like a 2d quasticrystal) does not exhibit translation symmetry, only rotational symmetry`

Now that we've given you the basics on crystals and the geometry of tiling. I want to move on to our main subject: Quasicrystals

Regular crystals have translational symmetry, where the same unit part is repeated over and over again with no rotation. Quasicrystals have rotational symmetry, where the same unit part is repeated over and over, but with a rotation about an angle.

Quasicrystals are structural forms that are both ordered and nonperiodic. They form patterns that fill all the space but lack translational symmetry. The term and the concept were introduced originally to denote a specific arrangement observed in solids which can be said to be in a state intermediary between crystal and glass. Producing Bragg diffraction, they share a defining property with crystals, but differ from them by lacking a simple repeating structure.

Mathematical artefacts known as 'aperiodic tilings' were invented in the early 1960s, but some twenty years later physical experiments gave conclusive evidence of their material existence. Within the field of crystallography and solid state physics the discovery has produced a paradigm shift which is indeed a minor scientific revolution.

[1] It was realized that quasicrystals had been investigated and observed earlier
[2]but until then the prevailing views about atomic structure of matter lead to their being explained away.

An ordering is nonperiodic if it lacks translational symmetry, which means that a shifted copy will never match exactly with its original. The ability to diffract comes from the existence of an indefinitely large number of elements with a regular spacing, a property loosely described as long-range order. Experimentally the aperiodicity is revealed in the unusual symmetry of the diffraction pattern. 

The first officially reported case of what came to be known as quasicrystals was made by Dan Shechtman and coworkers in 1984. Between a mathematical model of a quasicrystal, such as the Penrose tiling, and the corresponding physical systems, the distinction is taken to be evident and usually does not have to be emphasized.


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Notes:

The buildings blocks of matter are the elements from the periodic table. The nucleus has a charge that keeps the corresponding number of electrons in orbit around itself. Think of the electron orbits like balloons that are tied together and repel each other. The electron orbits is what gives shape to atoms. The shapes of these atoms determine how they will fit together with other atoms of various shapes, most of the time they don't fit at all. Atoms that have shapes with translational symmetry form crystals. 
Think of atoms as magnetic spheres with bumper arms which make them stick to other atoms of different sizes and shapes differently. Take oxygen for example which is the most corrosive or reactive element on the periodic table, because it has two strong arms that like to rip other molecules apart. 



 flux-growth technique. As heat treatments indicate this phase is stable at high temperatures above 850 °C close to the melting temperature and metastable in a lower temperature regime up to about 500 °C. We performed compression experiments in these two temperature regimes in two different orientations, with the compression axis parallel as well as inclined by 45 degrees to the tenfold axis. We studied the microstructure of deformed samples by transmission electron microscopy.