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!
Suppose you have a room, and you need to guard it using ceiling mounted cameras. Assume the cameras can see in 360 degrees. If the room is very simple, such as a square shape, you can get away with just one. But if the room is more complicated, say with nooks and crannies, you may need many cameras. How can you find how many? Continue reading Taking in the whole room
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
It is hard to believe that this was only our first discussion using Geogebra, the all-powerful program for visualizing geometry problems. In fact we used just a tiny fraction of the tool’s capabilities to explore four-sided figures, or quadrilaterals. Continue reading Quadrilaterals
In this session we asked what solids can be made using regular polygons as sides. We investigated this almost entirely by playing around with snap-together polygons. And thanks to our campus 3d printer, we got to handle some of the more obscure shapes too. Continue reading Solids from regular sides
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
You are probably familiar with the hexagonal tiling of the plane:
There is something special about hexagons that makes this successful. In this discussion we asked what would happen if we tried to tile using hexagons together with other polygons. For example, if at every corner you place two hexagons and one pentagon, you will get a polygonal “soccer-ball”. Continue reading Hyperbolic surfaces
Linear transformations have many applications. We looked into questions about how linear transformations move points around in the plane. The activity has more information about the tasks, which include lots of “2 by 2” matrices.
How much fruit is in that slice of fruit? In other words, what is the area of a polygon? If the polygon is drawn on a grid, how can you find the area by counting the grid points?
Continue reading Polygons and Area
When we look at a highway, we often think of it as a straight line. Of course it is not straight—it curves with the earth. Still, if we say that the straight lines on the surface of the earth are the great circles, then we can begin to do geometry. Continue reading Spheres and geometry