# Classroom Capsules and Notes

Capsules By Courses. We are organizing the capsules into courses, when possible using the same topics as are used in Course Communities. So far we have organized capsules for the following courses:

You may select topics within each course.

## Featured Items

##### Upper Bounds on the Sum of Principal Divisors of an Integer

After a brief historical discussion of the aliquot parts of an integer the authors use elementary arguments to prove a theorem about the sums of its principal divisors.

##### Spherical Triangles of Area $\pi$ and Isosceles Tetrahedra

The tiling of a sphere by spherical triangles is discussed.

##### A Theorem of Frobenius and Its Applications

The Frobenius Theorem states that if $d$ is a divisor of the order of a finite group $G$, then the number of solutions of $x^d = 1$ in $G$ is a multiple of $d$. The article includes a proof and applications of the theorem.

##### Proof without Words: Viviani's Theorem

In an equilateral triangle, the sum of the distances from any interior point to the three sides is equal to the altitude of the triangle.

##### Why Euclidean Area Measure Fails in the Noneuclidean Plane

The author discusses measuring areas of triangles in a non-Euclidean setting.

##### Hidden Group Structure

Certain subsets of the ring of integers mod n with hidden group structure are discussed.