By Andrew Freda, posted August 3, 2015 – 

A chemist who understands why a diamond has certain properties, or why nylon or hemoglobin have other properties, because of the different ways their atoms are arranged, may ask questions that a geologist would not think of formulating, unless he had been similarly trained in this way of thinking about the world. —Linus Pauling (“The Place of Chemistry in the Integration of the Sciences,” Main Currents in Modern Thought [1950])

One of my favorite “Geometry and . . .” units that I do with my students involves chemistry. I find that students come to a stronger understanding of some of the important terms and ideas of geometry when they see similar ideas in what they may consider an unrelated field (and the ideas of chemistry that we examine are great fun).

I start by asking simple questions: Is a diamond the hardest stone in the world because it is made of “hard stuff,” and is the graphite in pencils slippery because it is made of “slippery stuff?” Is a diamond expensive because it is made of “rare” atoms, and is the graphite in pencils inexpensive because it made of “easily found” atoms? Students are invariably surprised to learn that diamonds and graphites are both naturally occurring forms of carbon, but the arrangement of the carbon atoms—the geometry of these molecules—accounts for the difference. An enterprising student usually asks if we can make diamonds using pencils, and the answer is yes. In fact, the quality of synthetic diamonds is now so high that even the most experienced jewelers need a mechanical device to decide whether a diamond is natural or synthetic.

Once the class begins to understand the idea that shape—the arrangement of atoms—determines the properties of a substance, we move on to the use of geometric terms in chemistry. In geometry, congruence does not depend on orientation: A triangle is congruent to another of corresponding sides, and angles have the same measure. Not so in chemistry, where orientation can make the difference between an effective drug and an ineffective one. Enantiomers are mirror-image molecules that are congruent but reversed. An example is the drug L-DOPA, which is used to treat Parkinson’s disease; its congruent mirror image, D-DOPA, has no effect (L is left and D is right). I ask students to imagine the level of testing and understanding that scientists need to undertake when looking for new treatments: Not only do they need to find the correct molecule; it must also “face” the correct way!

I make a concerted effort to discuss the geometry of water and snow during our long western Massachusetts winter term. A great resource Snow Crystals, which treats students to fascinating photo galleries of the various types of snowflakes, the basics of how snowflakes are created, and a wonderful argument (based on probability) that no two snowflakes are alike. After this unit, students will never look at snow the same way again!

When we are ready for a challenging three-dimensional problem that requires trigonometry, the class can look at a methane molecule and its special geometry. This is a nice version of the dry textbook problem that asks students to find the angle formed by a segment from a vertex to the center of a tetrahedron then to another vertex.

ANDREW FREDA, afreda@deerfield.edu, teaches at Deerfield Academy in Deerfield, Massachusetts. He has spoken and written on a variety topics, including the importance of visual representations in algebra, teaching Geometry through stand-alone “units,” ways to understand probability through the use of a dice game, the use of fractals in the high school curriculum, and the role of CAS in the mathematics classroom.