By Class Teacher, Jeanette Voss

“The right thing is to proceed from second dimension to third, which brings us to cubes and other three dimensional figures.”—Plato

Geometry teaches flexible thinking. Apart from knowing the traditional concepts of symmetry, triangles, parallel lines, angles, etc., it teaches students to visualize objects and to think about or to reason about their spatial relationships. In the 8th grade geometry block called, Mensuration, Stereometry, and Loci, the class practiced the constructions of geometric forms, built Platonic/and Archimedean solids, and created beautiful depictions of functions in the study of curves, called loci.

Ancient Greek polymaths, Plato and Archimedes, were fascinated by the geometry of nature and studied a combined total of 31 geometric solids and their duals. When we look closely, we can discover that many patterns in nature are based on the mathematics of three-dimensional space. Although archeologists found geometric stone shapes resembling the five solids in Scotland in 1500 B.C.E., it was Plato who assigned these solids the attributes of the four ancient elements of earth, air, water, fire, and a fifth, which he called the “quintessence” or the “cosmos”. Platonic solids are regular polygons, which means that all their faces and sides are equal. Geometric patterns and shapes resembling these solids can be found when we look closely at atomic structures of elements, at crystalline shapes of salts as well as at models of microbes, radiolarians, and viruses.

The class studied the properties of the five Platonic solids by drawing them into their morning lesson books and by constructing them, first with clay, and later with a compass and straight edge on sturdy paper. The latter activity required a high degree of skill and accuracy in the use of these tools. As their studies progressed and the students became accomplished at the construction of the nets, they discovered that four Platonic solids have another Platonic solid as their dual: the hexahedron (cube) “hides” the octahedron; the icosahedron has within itself the shape of the dodecahedron; only the tetrahedron, has itself as a dual.

Archimedes, in turn, enumerated 13 solids and their duals, for a total of 26 geometric solids. While transforming a clay cube to its dual, the octahedron, a number of 8th graders discovered the hexagonal face of an Archimedean solid, the truncated hexadron, while transitioning the cube to the octahedron.

As stated above, Platonic solids are applied to describe formations in molecular theory. They were also useful in the field of cosmogony (theory of the origin of the universe). Johannes Kepler, for example, attempted to use them to explain planetary distances and motions in his Mysterium Cosmographicum by drawing planetary paths inside the Platonic solids. Many artists have created 3D models of his thesis. In the field of crystallography, out of the 31 geometric solids, only three, Plato’s hexahedron and Archimedes’ truncated octahedron and rhombic dodecahedron, can be close packed into crystal structures.

As a growing number of mobiles of Platonic and Archimedean solids dangled about the classroom, the students ended the block with the study of curves, also called loci. Following an idea by Jamie York, the students stretched their minds and their capabilities to imagine lines beginning with the simple idea of going on a “treasure hunt”. They generated curves first in their mind and then on paper, thus finding first the mind-image and later the language of loci. We began with a simple problem: If a treasure is buried 100 feet from a tree on a field, where can it be found? The answer is that it is found in a circle around that tree. Then students progressed to the more technical language of “finding the locus of points equidistant from a line and a point”. The solution to this query is the shape of a parabola, which we constructed by finding the intersections of concentrically expanding circles and regularly spaced lines. This activity allowed the students to visually create a curve that they will get to depict later in high school through mathematical calculations of functions.

The flexibility in thinking that is required to mentally visualize space, shapes, and their relationships to each other was challenging, but I hope that it will have benefitted the class to advance in their ability to stretch their minds, which will be important in the future and not only in geometry. Click here to view more photos.