geometry

Slices of Hanging Cubes

Hang a cube from one of its vertices. Now, if you slice it horizontally through its center, you get a hexagon. What if you do this with a 4-dimensional cube, i.e., a tesseract? The slice will yield a 3-dimensional object— what does it look like? Answer:...

Continue reading...

Koch Tetrahedron

In Koch Snowflake we saw an interesting fractal snowflake-like object that is obtained when gluing smaller triangles iteratively to the sides of a big triangle. So, what happens if you do something similar to a tetrahedron? That is, suppose you take a regular tetrahedron (all side lengths...

Continue reading...

Hyperbolic Geometry

In the Fun Fact on Spherical Geometry, we saw an example of a space which is curved in such a way that the sum of angles in a triangle is greater than 180 degrees, where the sides of the triangle are “intrinsically” straight lines, or geodesics....

Continue reading...

Volume of a Ball in N Dimensions

The unit ball in Rn is defined as the set of points (x1,…,xn) such that x12 + … + xn2 <= 1. What is the volume of the unit ball in various dimensions? Let’s investigate! The 1-dimensional volume (i.e., length) of the 1-dimensional ball (the interval [-1,1]) is...

Continue reading...

Sphere Eversions

If you take a loop of string in the plane and place an arrow along it pointing clockwise, is it possible to deform the string, keeping it in the plane, so that the arrow points counterclockwise, without causing any kinks in the string? A moment’s...

Continue reading...

Sierpinski-Mazurkiewicz Paradox

If you’ve seen the Banach-Tarski paradox, you know that it is possible to cut a solid 3-dimensional ball into 5 pieces and reassemble the pieces using only rigid motions to form two solid balls each the same size as the original. The construction depends on the Axiom...

Continue reading...

Chords of an Ellipse

Consider N equally spaced on points on the unit circle, with the point P=(1,0) as one of these equally spaced points, and draw (N-1) chords from P to every other point. In Chords of a Unit Circle, we saw that the product of the lengths of...

Continue reading...

Sliding Chords

Take a circle C, and a chord in the circle. Now slide the chord around the circle. As you do this, the midpoint of the curve will trace out a smaller concentric circle. Call the area between the two circles A(C). Now suppose you do the same...

Continue reading...

High-Dimensional Spheres in Cubes

How good is your intuition in high dimensions? Take a square and divide it into its four quadrants. Inscribe a circle in each. Now, draw a circle whose center is at the center of the big square and whose radius is just big enough to touch...

Continue reading...