Links to cool graphs I made on Desmos! Most of these graphs are interactive, so feel free to click and drag. Some of the graph features can only be viewed on the Desmos website, so if you are curious or would like to tinker with the formulas yourself, please click the links in the title. Most graphs will also have more details and explanations on the pages linked.
Phase Portrait Exploration
Graph the phase portrait for a homogenous linear system (i.e. x’ = Ax). Sometimes the trajectories will disappear, this is because the solution involves an imaginary step and Desmos does not natively support complex operations, so workarounds have to be creative.
Bike Exploration
This graph was inspired by an example mentioned in Ben Orlin’s book “The Only Constant is Change.” More details can be found in the link.
Gradient Field
Visualize the gradient field of a 3-dimensional conic. The black graph is the intersection of the 3D surface and the xy-plane. The vector field math was adapted from another Desmos creator’s graph which you can find here.
Change of Basis Application
Shows how reflecting points across an arbitrary line through the origin is easier if we use a non-standard basis for the xy-plane.
Linear Maps and Change of Bases
Represent a point in the xy-plane as both a linear combination of the standard basis and as a linear combination of a dynamic non-standard basis. Drag the orange point around to explore!
Linear Maps
Visualizes some of the infinitely many non-standard bases that span the xy-plane!
Dot Product Exploration 2
This project encodes the surprising and elegant geometric interpretation of the dot product as the product of the length of a vector and some projection onto it.
Simple Dot Product Equality
This graph attempts to visualize this property of the dot product: Consider a vector c and a line L perpendicular to span(c). Now consider vectors a and b along the L. We claim that dot(a, c) = dot(b, c). More clearly, this graph demonstrates the geometric meaning of the dot product as the product of the length of the vector and some projection onto it. Since the projections of a and b onto c are identical, we know that their dot products must be equal too!
Simple “e” Exploration
A cool way to link discrete math to e via combinatorics. e is truly a profound number; it’s the natural language of growth itself!
Simple Hexagonal Circle Packing
Hexagons are truly are remarkable mathematical jewel in the cosmos. They adorn the natural world as chemical bonds, honeybee nests, and even massive planetary storms. I began exploring the nature of hexagonal packing as part of a Stanford CS 109 project; the fruits of that project are in this repo.
Heron’s Formula Exploration
This project demonstrates the utility of Heron’s formula. It shows that in some cases, using Heron’s formula is vastly simpler than computing the area of a triangle via normal means (i.e. A = bh / 2). Most of the utility of this graph is only accessible on the Desmos website. To truly understand how useful Heron’s formula is, it’s best to see first how simple it is to calculate the triangle area using Heron’s formula and then compare it to the mess of algebra needed to compute it the traditional way. However, since embedded Desmos graphs cannot display equations, you have to click on the link in the subtitle to appreciate its full majesty.
Extreme Value Theorem
The Extreme Value theorem is an extremely intuitive result. Simply put, it claims that a real-valued continuous function on a closed
interval attains a minimum and a maximum. We see below that the red function has both a minimum point and and maximum point
on the closed interval between the purple lines.
PROBABILITY
Simple Mean of Random Samples Exploration
This project explores the claim: the expectation of the mean of a random sample from a distribution is the same as the expected value of the population. It’s worth pondering why the histogram resembles a normal distribution.
Simple Argmax Equivalence Exploration
This simple project attempts to visualize the claim that the argmax of a function f(x) is the same as the argmax of the log of that function (since log is monotonic). While this is mathematically and graphically intuitive, I felt it was helpful to see it first-hand.
Color Triangle
Each corner of the triangle represents a “color axis” (between 0 and 255). For example, if you move the point to the vertex marked “G” the RGB value will be (0, 0, 255), so the triangle will turn fully green. The color triangle can’t represent every RGB value, but it’s still fun to explore!