Crystal Math


So…when I was first coming up for ideas about this project my first thought was that I wanted to do food because if I can remember anything about my BC Calc class second block it was that I was always hungry. The idea of rock candy somehow came up. So I started looking up the mathematics of rock candy which then led me to this whole universe of geometry, vectors, 3D structure and all in all CRYSTAL MATH…

The idea with the rock candy was to see what the trend of the rate of volume for the crystals would look like. My hypothesis was that the volume would increase exponentially since the growth of sugar crystals depends on the amount of surface area that’s available to stick to.


I took out the rock candy to measure about every twelve hours to get volumes by both integration and by water displacement.

Day .5
Day 1

Day 1.5

Day 2


Day Volume by Integration Volume by Water Disp. (cc)
.5 3.498 3
1 7.35 6
1.5 24.15 17
2 27.35 22


The graph of this data is here:

My hypothesis turned out to be for the most part correct!

As I was making rock candy I also started looking up the properties and structure of sugar…turns out there are 7 basic crystal structures…


There is actually a lot of math that relates to these structures such as vectors and the differences in bonding angles depending on which structure your crystal happens to be…

To see what these vectors look like, here’s the link…

As it turns out, sugar is monoclinic that’s why its crystals form into cube-like shapes.
So if I were to go to one corner of the sugar cube and label three vectors a, b, and c these vectors would ideally stay equal to each other as the crystal grows.

Then I bumped into a few names as I was researching and this one name popped up everywhere…Dmitry Kondrashov.

Dmitry Kondrashov is a professor at the University of Chicago. He studies crystal structure and crystal lattices using x-ray technology. Kondrashov composed a geometric theory of crystal growth in 1998 and I decided to give it a read. The math involved in calculating errors and factors that impact the ideal crystal shape go far beyond my competence of math knowledge however, I still learned quite a bit. There were many components to his theory but adjusting to time and my knowledge of this subject I focused on what Kondrashov mentions to be a “Wulff shape”

The Wulff shape is related to the Wulff construction method and it’s used to determine the equilibrium shape of a crystal that has a fixed volume in a separate phase like vapor or saturated solution, like the sugar solution used to make rock candy. In order to obtain this equilibrium it must abide by the principles of Gibbs thermodynamics by minimizing the total free energy of the crystal’s surface. When two fluids touch the portion of total free energy is proportional to the area of the surface in which the fluids are in contact. Motion of the planes created by the two fluids can be calculated using several equations such as these…the only issue is figuring out where the corners will meet so a system of linear equations would need to be simultaneously calculated as well. Kondrashov’s theory takes into account many of the factors that impact crystal growth such as energy loss, spacing,phase changes, and surface tension among other things. Another interesting Kondrashov states is that in order to concentrate on the shape of the crystal rather than the size, energy is often replaced by surface tension in many of the equations that are part of his geometric algorithm.


Kondrashov briefly discusses the main search in the world of crystallography, the main objective is to find and predict the equilibrium shape of a crystal from its internal structure using math.

All in all, through this project I’ve continued my love for 3D structure, not only are crystals beautiful to look at the amount of calculations and math that go into them is pretty interesting as well.

Thank you for reading!

Recipe for Rock candy….

On a stick

1 cup of water
3.5 cups of sugar
lollipop sticks
glass jars
flavoring (optional)

1. First dip the lollipop sticks in water and coat the area you’d like the crystals to grow in sugar, this gives the 2. crystals something to latch onto.
3. Heat water and sugar and bring to a boil, boil for about ten minutes or until you can’t see the sugar crystals anymore.
4. Let solution cool and add flavoring and food coloring if desired.
5. Pour solution into glass jars take clothespins to dip only the portion you’d like to grow crystals on, into the solution (make sure the lollipop sticks are dry!).
6. Wait for crystals to grow!

“Broken Glass”

half a cup of water
¾ cup of light corn syrup
2 tsp of flavoring
2 cups of sugar
food coloring (blue if you’re a fan of Breaking Bad)

1. Heat sugar, corn syrup, and water and bring to 285 degrees
2. Remove from burner, add food coloring and flavoring
3. Pour into a thin layer on a baking sheet that has been lined with foil and greased with cooking spray.
4. Let sit for at least 30 minutes until brittle
5. Break with a mallet and eat 🙂


Works Cited