Dork Scratchings

Here's a neat trick for estimating the distance to objects.  Stick out your thumb and look at it with just one eye open, then with just the other eye open.  The distance to the object is the distance it appears to jump multiplied by 10 .  This works because the distance from your eyes to your thumb on your outstretched arm is about 10 times the distance between your eyes.   I thought I'd have a go at using this to measure the distance to the moon

Here's a nice theorem by Gauss The (2D) shoelace theorem Suppose $\mathbf{x_1},\mathbf{x_2},...,\mathbf{x_n}$ are ordered vertices of a polygon.  Let $x_i, y_i$ be the coordinates of $\mathbf{x_i}$ and define $\mathbf{x_{n+1}}$ to be $\mathbf{x_1}$.   Then the area of the polygon is plus or minus $$ \frac{1}{2}\sum_i{x_iy_{i+1} - x_{i+1}y_i} $$ This formula gives rise to the name of the theorem as shown by the illustration below Proof We're going start off by proving the simple case where the polygon is a triangle.  The first thing to notice is that if you take any two adjacent vertices, .e.g. $\mathbf{x_1}$ and $\mathbf{x_3}$, they form a triangle with the origin.  This has an area equal to half of the parallelpiped formed by the two vertices (treated as vectors from the origin). And the area of the parallelpiped is plus or minus $det(\mathbf{x_1}\vert \mathbf{x_3})$, with the sign being positive if the columns taken in order produce an anti-clockwise motion around the origin.