20 Apr

This recent post described a construction that demonstrated the “Three-D Pythagorean Theorem.” Here we dive a bit deeper into this bit of mathematics that may be less familiar than it deserves. A right tetrahedron has, by definition, a vertex where three right angles meet (there can be only one). An example is shown in the… [Read more]

04 Oct

[Note this post is a bit outdated, and not maintained; the most up-to-date version of this information can be found in my pages on the OEIS wiki.] I’ve recently become interested in a family of interrelated sequences that can be found in the Online Encyclopedia of Integer Sequences (OEIS). These sequences all have to do… [Read more]

30 Aug

In both the Woven SSD and the Woven GSD, you calculate the inset (from the points where the rods meet near their ends) for putting marks on the rods by multiplying their lengths by 2/(3+√5). Where does that strange-looking number come from? The key is that both figures consist of (regular) pentagrams, just interlocked in… [Read more]

09 Mar

If you look again at the diagram of an icosahedron in a cube (at the right), you’ll see that because of its symmetry, all three of the icosahedron vertices nearest the top front cube vertex are the same distance from that cube vertex. That equidistance means that the body diagonal of the cube passes through… [Read more]

09 Mar

As mentioned and illustrated in the post on the Anticos, it’s possible to inscribe an icosahedron in a cube. (In this case, that technically means that given a cube, you can choose two points on each face of the cube such that the convex hull of the resulting set of twelve points is a regular… [Read more]

30 Jul

How does the value of the following improper integral compare to 1? I.e., is it smaller, larger, or exactly equal to 1? (This problem was proposed to Math Horizons Playground by Mehtaab Sawney of Commack High School. And for all of you $\pi$-ists out there, $\tau$ is of course just the radian measure of a… [Read more]

30 Jul

There is no excerpt because this is a protected post.

30 Jul

For definiteness, the unlabeled points do in fact trisect each of the sides of the triangle. Stumped? You can peek at the answer using the password “threedian”.

30 Jul

There is no excerpt because this is a protected post.

30 Jul

What are the next few terms in this sequence of smallish numbers? 1 1 1 3 1 3 1 1 3 1 1 3 1 1 3 1 1 1 3 2 1 1 3 2 1 3 1 1 3 1 1 1 3 2 1 1 3 2 1 1 3 ? ? ?… [Read more]

30 Jul

There is no excerpt because this is a protected post.

11 Apr

This post contains the details of the claim made in “More Spherical Construction” that you can determine the side length of a spherical square from the ratio between the lengths of its diagonals. We’ll do this on a sphere of radius one; everything scales by a factor of the radius for a general sphere. The… [Read more]