Tag Archives: geometry

Ovals upon ovals!

A circle has a center c and a radius r. The points on the circle are all a distance of exactly r units from c. What if we start with two fixed center points? The set of points on an ellipse have a fixed sum-distance to two center points c1 and c2 (called foci). An ellipse has kind of an oval shape. Ellipses occur in nature – the Earth’s orbit is not a perfect circle but rather an ellipse, with the Sun being just one of the two foci.

In this activity we investigated how to draw ellipses using pushpins and string. We also asked whether it is possible to have more than two fixed center points. Continue reading Ovals upon ovals!

Multiplying points in the plane

It is easy to add points in the plane “coordinatewise”, that is, by writing (x,y)+(z,w)=(x+z,y+w). This even has a very nice geometric interpretation: completing the parallelogram with first three corners (0,0), (x,y), and (z,w). But it isn’t easy to multiply points because the simple rule (x,y)*(z,w)=(xz,yw) doesn’t allow division. In this activity we looked for something better. Continue reading Multiplying points in the plane

Origami and geometry

A Platonic solid is a polyhedron with the same regular polygon on every side, and also the same number of sides at every vertex. The cube is the most well-known example, and the tetrahedron (pyramid with a triangular base) is another one. In fact, there are only five such figures possible! The other three are called the octahedron (8 sides), dodecahedron (12 sides), and icosahedron (20 sides).

How could these more obscure figures have been discovered in the first place? And how can we know for sure that there aren’t any more examples? We used origami paper to ponder these questions hands-on in this session, which was delivered by our special guest Paul Ellis of Manhattanville College and the Westchester Area Math Circle. Continue reading Origami and geometry