American Scientist

Tetrahedron Angles

To the Editors:
In the Q&A with Miguel José Yacamán (First Person, September–October 2024), there is an arithmetic error. In the second question about five-fold symmetry, Yacamán says, “If I have a two-fold symmetry, if I rotate the crystal 180 degrees, all the atoms fall in the same place. When I rotate it by 70.2 degrees, the atoms will not match.” The number should be 72 degrees because 360 divided by 5 equals 72, not 70.2.

William J. Saucier
Madison, WI

Dr. Yacamán responds:
The reader is correct in general, but this specific case is different. In a regular tetrahedron with the crystal structure of gold, the angle between the sides of the tetrahedron is 70.53 degrees. Therefore, if we pack five tetrahedra to form a decahedron (70.53 degrees 5 times), the total is 352.65 degrees rather than 360 degrees. Therefore, from the classic point of view the experimental observation of decahedra is not possible. I apologize for the confusion caused by my effort to avoid more technical data.