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]

10 Apr

This post contains the details of computing the height and radius of 1-cup and 1/4-cup Gengzhi Goblets.

10 Apr

So I had zeroed in on the proof of the Archimedean volume relationship as the source of my giveaway for G4G13. But how to create an interesting variation? The first thing to notice is that the function of h that appears in the proof in the cross-sectional area of of the cone, namely h², could… [Read more]

09 Apr

So it became time to decide on Studio Infinity’s giveaway at the 13th Gathering for Gardner (G4G13). By tradition, at least, it’s considered a plus for giveaways to connect with the number of the conference — 13 in this case. So this mathematical free association starts with the number 13. What thoughts does 13 evoke?… [Read more]