Suppose we have a region $\Omega \subset \mathbb{R}^n$ that is a polytope, i.e. a non-empty intersection of half-spaces. Visually this would be a polygon in 2D or a polyhedron in 3D – “polytope” is a single term to describe such objects regardless of the number of dimensions. The important feature is that the boundary of … Read More “Polytope Identities from Vector Calculus” »
Any motion of a rigid object is equivalent to a rotation around some axis followed by a translation along that axis. We can therefore represent any motion mathematically using some object that includes information for a single rotation and a single translation. An extremely convenient object for this purpose is the dual quaternion. Part of … Read More “The Exponential and Logarithm for Dual Quaternions” »
Introduction In dynamics, one frequently needs a derivative of a vector in an inertial frame of reference in order to apply Newton’s laws when it is much more convenient to express that vector in a different frame. The Basic Kinematic Equation (also known by many other, less-informative names) relates the derivatives in two frames to … Read More “What is the Basic Kinematic Equation?” »
Every whole number (except 1) can be factorized in one and only one way into a product of prime numbers. For example: $$\displaystyle \begin{aligned} 21 &= 3\times 7 \\ 34 &= 2\times 17 \\ 35 &= 5\times 7 \end{aligned} $$ By looking at the factorizations one can easily tell when two numbers share a common … Read More “Numbers With Nothing In Common” »
You know about electricity, and you know about magnetism, and you’ve probably heard the word “electromagnetism” somewhere before to refer to electricity and magnetism combined in some sense. But why do these two seemingly different phenomena get lumped together into one word, whereas there’s no such thing (yet?) as “electrogravity”? If you know some more … Read More “Magnetism from Relativity” »
Consider a family of polynomials $\{P_0, P_1, P_2, \cdots\}$ where each $P_n$ has degree $n$. The polynomials are orthogonal if there is an inner product such that $(P_i, P_j) = 0$ if and only if $i\neq j$. Now, any given $P_{n+1}$ can be written in terms of all the lower-degree polynomials: $$\displaystyle P_{n+1} = r_{n+1,n+1}xP_n … Read More “Recurrence Relations for Some Orthogonal Polynomials” »
A sultan has granted a commoner a chance to marry one of his $N$ daughters. The commoner will be presented with the daughters one at a time and, when each daughter is presented, the commoner will be told the daughter’s dowry (which is fixed in advance). Upon being presented with a daughter, the commoner must … Read More “The Sultan’s Daughters Problem” »
$\pi$ is the most well-known special number, but there is also $e$. Whereas $\pi$ has the easy interpretation of being the ratio of a circle’s circumference to its diameter, and everyone knows what a circle is, $e$ arises in calculus with which many people have no familiarity. Here’s something interesting… Without calculus, the closest one … Read More “What is e?” »
A previous post used the typical series-expansion-plus-linear-algebra approach for finding finite-difference formulas to derive approximations to the first derivative of any desired order of accuracy. If you’ve ever used that method yourself, you probably know how tedious it is, and that post doesn’t make it look much less so. On top of that, for all … Read More “All Finite-Difference Formulas for All Derivatives” »
Several competitive decks in Vintage Magic: The Gathering are powered by the card Bazaar of Baghdad. Deck construction rules mandate at least 60 total cards in a deck with at most four copies of any given card. The game starts by each player drawing seven cards then performing, if he wishes, a series of “mulligans” … Read More “Mulligans in the Bazaar” »