Charlie Co

After tests, students ponder their class grades while teachers seem to design tests that yield bell-shaped grade histograms. In this two-part series, I delve into the math and Python solution for this intriguing problem of finding the most probable distribution. For this video (Part 1), we first derive the number of ways (permutations) N students can be distributed into a grade distribution, noting that the grade distribution with the most number of permutations would be the most probable one. Next, we realize that when the number of students (N) is large, factorials become large quantities that cannot be represented with 64-bit floating point data types. Thus, we turn to deriving Stirling’s approximation (ln N! = N ln N - N), which involves doing the opposite of the traditional Reimann sum. That is, we approximate the sum of rectangular areas by an integral. Following an integration by parts, ignoring small terms, and grouping, we come to a tractable approximation for the natural log of the number of permutations of a grade distribution: -N ∑ p (ln p) where p is the fraction of students with a given score.

Balancing chemical equations to determine reaction stoichiometry is an important first step in chemistry. Most chemical equation balancing problems in introductory chemistry can be done by inspection. Harder problems such as the combustion of rocket candy (sucrose/sorbitol mixture) with potassium nitrate, as discussed here, benefit from a systematic algebraic approach. In particular, with 5 atomic species (C, O, H, N, K) to balance, I setup then solve the 5 simultaneous linear equations to find the correct stoichiometry. In a future video, I will use regex (regular expressions) and Numpy (np.linalg.solve) to automate the setup and solution of these algebraic equations for chemical reactions of almost unlimited complexity.

Linear regression is typically used to find lines of best fit to data. Here, we transform the equation of a circle to a linear form to fit the center and radius of a circle to data points.

An entirely geometric derivation of the volume formulas for pyramids and cones that offers insight to the origin of \(\frac{1}{3}\) in the volume formulas. This was also my entry to the Khan Academy Junior Breakthrough Challenge.

The rotation matrices are derived here via rotation of the x and y cartesian coordinates and using trigonometric relations to determine the coordinates of a point in the new rotated x' and y' coordinates.

Using the rotation matrices derived earlier, I go through the nitty gritty details of writing Python code, without using image processing packages, to rotate an image of my dad and I pixel by pixel.

Blender models are constructed here to visualize in three dimensions the correspondence in volume of a hemisphere to the volume between a cone and its enclosing cylinder leading to the volume formula for spheres. This landmark discovery was made by Archimedes in the 3rd century BC.

This is by a long margin the toughest math problem I have encountered in my 14.5 years of existence. It is simple initially, but bringing together the geometric concepts with the multiple absolute value inequalities was truly a challenge. I worked and wrote equations non-stop in this 23 minute video to solve a single math problem!

Archimedes Principle states that the buoyant force is equal to the weight of the fluid displaced. It is used here to derive an expression for the density of a floating cylinder given the density of the fluid, the radius of the cylinder, and the highest point of the cylinder relative to the fluid level.

Most students learn of parabolas as \(y=ax^2+bx+c\). However, parabolas have a more general geometric definition. A parabola is the set of points that are equidistant to a point called the focus and a line called the directrix. Here, the equation of a parabola relative to a point and a line, which is defined by its slope and intercept, is derived. Because the line need not be parallel to the x-axis, the resulting parabola can be oriented at any angle relative to the x and y axes. The resulting equation is implemented and plotted using Python.

Animated blender models to illustrate Cavalieri's Principle are constructed step by step in this video using Jacques Lucke incredible Blender add-in Animation Nodes.

A geometric proof of why the centroids of triangles is always located \(\frac{2}{3}\) along any median is explained and illustrated here using a Blender model.

I used to do YouTube/Twitch live streaming of me playing games. Before I made enough videos to earn a nicer real background, I used to have a green screen, which is replaced with a 3D Blender model image of a wall with a medkit, health potions, and a wall of lighted acoustic tiles. This background is in fact visible behind me in this video, and yes! - that background is 100% fake! This video goes into the step by step details of how the acoustic tile wall section was created using Blender. There is another video on my YouTube channel going into details of how the medkit is modeled in Blender.

This is a fun parody video where I used DaVinci Resolve corner pinning feature to do a screen replacement and create video evidence to convince my friends that I'm playing a healthy dose of video games! DaVinci Resolve is an amazing video editing software provided for free by Blackmagic Design that has been used in editing multiple Hollywood blockbusters.

The expression for kinetic energy \(\frac{1}{2}mv^2\) is taught in elementary schools. I dislike memorizing stuff that comes from nowhere, but the traditional derivation of this expression for kinetic energy requires calculus. Here, it is derived using elementary mathematics via conservation of energy.

This video explains why the sum of interior angles of an n-sided polygon is \(180(n-2)\) degrees. We could not find a suitable background and thought it would be a good idea to record it in a big garbage bag with holes for ventilation and cables for the lamps! There was even another Math in a Garbage Bag video on my YouTube channel...

My dad tells me these simulations are far from accurate as they are not calculated from the so called Navier-Stokes equations. But my gosh, it is so easy and fun to make these fluid simulations in Blender.