Browse Classroom Capsules and Notes

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Wronskian Harmony

A closed form of the Wronskian for \( sin(kx) \) and for \( e^{kx}, k=1,2,\ldots,n \) is obtained. The derivation depends on trigonometric identities and properties of the determinant....

Demystifying the Projective Plane

The author describes explicitly the projective plane as a surface formed from a square by joining each pair of opposite edges in a definite way.

Definitely \(\sim\) Positively the Pits

The author classifies the quadratic forms defined by simple 2 by 2 matrices and illustrates them with corresponding quadratic surfaces.

Generating Nonpowers by Formula

For each \(n>1\), the authors find a closed formula to generate a subsequence of positive integers that is the complement of the sequence of perfect \(n\)th powers.

A Note on the Five-Circle Theorem

The author shows geometrically that two congruent incircles constructed inside a triangle imply the congruence of the other two incircles.

On the Symmetry Group of the Dodecahedron

The author offers a new proof of the theorem on the symmetry group of the Dodecahedron based only on elementary group theory and easy visualizations.

A Golden Frustum

Using a cone and a frustum with appropriate size, the author divides a cylinder into three parts of equal volume.

The Gaps Between Consecutive Binomial Coefficients

The authors determine both maximum and minimum of the gap between two consecutive binomial coefficients in any row of Pascal's triangle.

Lowering the Odds for "Even Up"

The set of all "even up" card games is modeled by a group of binary strings, which shows a game outcome is determined by luck, not skill. The probability for each outcome is obtained...

Proof without Words: Squares of Triangular Numbers

The author proves visually that the square of the \(n\)th triangular number equals the sum of the cubes of the first \(n\) positive integers.