A. Why Study Crystals?
What are crystals? The answer to that question is the end result of a great deal of research and study by many people over many years. Crystals can be gem-like solids as seen in quartz crystals. So one might believe that this leads to the study of the regular and semi-regular solids of geometry, but crystals are more than that. Crystals are systematic arrangements of molecules to form physical solids. Crystals are solid state physics. When we look at the gem-like object we are looking at a single crystal, when we look at a slab of marble or a sheet of steel we are looking at many crystals all next to each other, polycrystalline forms. Crystals are beautiful, everyone likes beauty. Why are crystals beautiful? Crystals have symmetry.
There are many topics from mathematics that relate to crystals and many topics of crystallography that are math. How can crystals be represented as three dimensional objects? There are many answers to that question: Miller indices, space groups of symmetries, drawings, projections.
Crystals are the evidence of the atomic structure of matter. There are three laws of crystal morphology:
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1. Crystals grow naturally with plane faces.
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2. The angles between faces of a crystal are characteristic of the crystal.
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3. Law of Rational Positions. When three noncoplanar edges of a crystal are used as coordinate axes, the coordinates of the points where the planes of the crystal faces intersect the axes can be expressed in small integers.
If we postulate that crystals are made of congruent building blocks stacked one upon the other, one next to the other, each of the laws can be derived.
When we were taught the scientific method in elementary school we were told scientists collect data and then interpret it to get theories. So one would expect those laws to have come first and then the theory. Actually the history is more complicated. The scientists had other theories (like alchemy) and were looking at the data from those points of view. The data were available but the interpretation took time.
It seems that postulational systems are a way of thinking that people naturally follow. The laws or postulates are simplifying principles that explain and predict what happens. If the predictions don’t match up it takes time for people to reject the old laws and look for new ones.
B. Growing Crystals.
Another reason to study crystals is to grow them. Growing crystals is an opportunity for students to have a hands on activity. The two references to use are
Crystalls and Crystal Growing
by Alan Holden and Phylis Morrison and
Crystals—A
Handbook for School Teachers
by Elizabeth A. Wood.
I should have started this activity much earlier. I should have set it up in the classroom last April or May to see how my students would react to science apparatus in a math room. I have grown an alum crystal. The process of getting the seed is satisfying by itself.
To get seeds one dissolves the alum in some water and then places the solution on a shallow plate to evaporate, the shallower the plate the better. Examine your seeds with a magnifying glass or even a microscope. Philip and Phylis Morrison in their book
Ring of Truth
suggest watching the seeds form under the microscope as the water evaporates.
Once you have the seed tie it with a thread and suspend it in a supersaturated solution and wait. The more seeds we have growing the better our chances of seeing a big, beautiful crystal. So see if you can inspire your students to try growing crystals independently.
If you are not a science teacher you may not know that Macalester and Bicknell, the chemical supply company on Henry Street in New Haven will not sell chemicals to private individuals. So either see your friendly science teacher or plan far enough ahead to get a school supply purchase order. Meanwhile, grow crystals with the chemicals you have in the house or can get from the supermarket: alum, sugar, salt, borax, epsom salt or what have you.
