Delta Thoughts
Hello, world! Hello, World ! I’m Adi Mittal, a student at Terman Middle School . I enjoy food, math, running, martial arts, and music of memes. My main intent in starting this blog is to share my thoughts, ideas, and outlooks on cool math. My primary goal is to create content that's interesting and to share my thoughts on the world around me. I hope my ideas and thoughts can appeal to you, just as…
Spinning Coins Coins! As some of you may relate to this, I love to take a coin, and just spin it on a flat surface or table. It's just satisfying, but being put to shame by the so called "Fidget Spinner". I was spinning a coin a few days ago, and was ultimately bored at the time, so I decided to ask myself a simple question: What information from the coin can be taken away from it spinning? This…
This triangle though... How do you... How do you even... This is when you know the problem you are about to be shown, will be annoying. When a friend of mine first introduced this problem, I thought this would be very, VERY simple, to solve. Use some angle properties, use the given similar triangles, and soon enough, a solution will be found. Of course this didn't work. I tried a few other things,…
Answer: Too fast The Earth has a diameter of approximately 12742000 meters. Most people of course wouldn't travel that far, but what if you did? How fast can you get across with nothing but yourself? That's essentially what people have asked in the form of the question: How long will it take to fall through the center of the Earth? SPOLIER WARINING: THIS WARNING IS TO INDICATE ANY WORD FOLLOWING…
I tried... There is just no introduction needed here. The problem at hand is probably one of the hardest, most controversial topic in computer science: Prove that $P = NP$, or otherwise In case this is not clear (or never have heard this problem before), it is to show that all NP-hard problems are P problems, or show that they are not equal. An NP-hard problem is a problem that cannot be solved in…
Okay, you don't make that much... Not much for an introduction this post. Found this problem when looking for interesting problems for myself. Shoutout to Harvard's Problem of the Week (from 2002 to 2004). The problem at hand is: Consider the following game: You flip a coin until you get tails, and the amount of money you win is equal to number of coins you end up flipping (i.e. If you flip a…
System 1 and System 2 Persuasion Tactics and Their Impacts on Secondary School Absenteeism This was done as part of the Advanced Authentic Research program during the 2019-2020 academic school year. Over 7 million students across the United States missed 15+ days of school in the 2015-16 school year (US Department of Education, 2012). These chronically absent who miss 10% of their academic year…
The hidden relationship of quartics and phi This post looks to describe an interesting property intrinsic to any and all quartic functions, and it has to do with the relationship between the functions' inflection points. Below is a Desmos graph, with labeled $f(x)=Ax^4+Bx^3+Cx^2+Dx+E$, the general quartic equation, as well as its 2 inflection points $P$ and $Q$. A third point $R$ is labeled, which…
The utility of right angles If you ever seen a bike at night, you've likely noticed the bright reflector many people use to ensure they're visible while riding. Why are they so effective at creating such visibility? It lies in the construction of the reflector itself. Looking closely at a bicycle reflector, you will notice that they aren't just plain mirrored facets; they have an almost pixelated,…
Yes, the flamethrowing guitar was necessary This post is a collection of essays analyzing scenes, themes, dialogue, and scoring of George Miller's 2015 120-minute action packed car chase of a movie, Mad Max: Fury Road . It is truly one of my favorite, raw action movies I've seen in a long time as it knows exactly what it is: an action movie. It delivers in spades what every action enthusiasts…
Stability amidst the chaos Introduction Let me propose a question to start. Try to solve the following: $\large{x^{x^{x^{x^{.^{\hspace{.07cm}.^{\hspace{.07cm}.}}}}}} = 2}$ An infinite power tower which supposedly equals 2? Seems unlikely, but those familiar with these infinite-operation type problems likely know the strategy to solve this. Notice how there's a copy of our equation stacked on top…
How to guess mathematically Today, I want to talk about a really powerful tool in math and statistics, that on its own may seem very niche, the concept behind it is something really—and I mean really —powerful and is how many other discoveries and tools are made and immortalized. In particular, I want to talk about the Metropolis-Hastings algorithm and Markov chain Monte Carlo methods. If you want…
A powerful tool to reimagine counting Try typing the fraction $\frac{1}{98}$ into your calculator see what you get. Don't have one on hand? Here's a calculator ready and waiting for you. Next try $\frac{100}{9899}$. See if anything stands out to you. Even with the few amount of decimals this displays, you might notice some patterns appearing. $\frac{1}{98}$ expanded as a decimal appears to contain…
Time to flip circles inside out --> Today I want to talk about a type of geometry I think is grossly overlooked, especially when compared to the popularity of its Euclidean brother. In a world where linear transformations are the norm between translations, rotations, and dilations, sometimes it's hard to see anything but them as the workhorse geometric tools. However, there is an additional…
Why grids love squares so much We are all familiar with the idea of a grid. From making up the small pixels on our screen, to the compact city maps of New York, grids pop up everywhere due to the kind nature of the innate squares built into them; grids are extremely space efficient packing in squares above and below each other while still maintaining a sense of order. But, why do we grids love…
Modelling the past 17 months in 17 minutes COVID-19 is one of those events that will likely not just define the way people will interact with each other, but likely entire socieities. I wouldn't even be surprised to see this pop up in an AP US History textbook in a decade just for how long the pandemic has been drawn out for. So it should be no surprise that from the first month of the pandemic, a…
A puzzle to test in the forest If you break a stick at 2 uniformly random points into 3 segments, what's the probability you can form a triangle out of those 3 segments? As with all puzzles, drawing something always helps. Label the ends of the stick to be 0 and 1, and we'll make the first break at point $x$. The key to this puzzle is to use the triangle inequality : no side of a triangle can be…
Monopolize the world with mx+b You're a business tycoon. You have ideas in mind but no products on hand. You need to make a quick buck now and all you have is some needlework on your belt to carry you. If you wanted to maximize profit between making hats and shirts, what combination of the two should you make? Well, let's look at what our goal is. If I make \$15 per shirt, and \$10 per hat, we can…
The ultimate secret to win at pool and laser tag Today's post is one that's been months in the making. It originally started as one that only covers a single problem, but quickly branched off as I delved deep into dozens of papers and videos, just with more and more questions coming up. We're going to be discussing one of the oldest mixed studies of algebra and geometry: dynamical systems .…
Basically, Pixar should hire me Last post, we looked at different types of billiard problems , a class of math problems analyzing how light bounces with different setups of mirrors. Notably, we saw how straight lines make for very simple, easy to compute mirrors, while others like circular ones , can be incredibly frustrating. A large portion of last post's content, though, was made up of…
Flexible? Maybe. Flex-able. Definitely This will be a slightly more theoretical, conceptual post than the others, but these tricks have that mathematical and problem-solving elegance that is too good not to share. We'll go over a shortcut for some certain integration by parts problems, and one that allows us to make educated guesses of antiderivatives to find an answer. First, let's take a loot at…
The end of an era It's a been a little while since I've last posted. As part of my calculus class, we end the year with an exploration into a calculus related topic that we present to the rest of the class. I and my partner chose to explore the origins behind the brachistochrone : the curve of fastest descent for a rolling ball. Below is the related write-up I did as part of this project, and…
See? Video games ARE useful A couple months back, we covered a little bit about some random circle computations and facts I had collected over the months leading into that post. In it, we highlighted and rederived the basic raytracing equation for circles and spheres. In a few words, we used properties of vectors to be able to reduce the problem of where a line intersects a sphere into a quadratic…
Capture the world as it (almost) is Out of all the apps on my phone, the camera is one that I can't see myself without anymore. Between being out with friends, or travelling with family, the camera rarely remains idle as I capture memories forever. Though, there is one particular feature of mobile photography I've come to especially love: the panorama. Even this struggles to capture how well the…
Is this really math anymore? Many who have dipped their toes in math for even a little bit will know that $\sqrt{2}$ is irrational. Like many other well known mathematical constants like $\pi$ and $e$, $\sqrt{2}$ can't be written out as a fraction, and its decimal expansion goes on forever without repeating. It's a novel fact, and comes with a fairly simple proof too. Claim: $\sqrt{2}$ is…
How bees outsmart us all What shape has the largest area to perimeter ratio? Some might have a guess, or an intuitition for what the answer should be, but it's a surprisingly difficult question to pin down as a proof. In fact, a rigorous proof wasn't given until around 1840 (J. Steiner)! Some Observations There are some facts about our ideal shape we can observe quickly. First, it has to be convex…
Why even mathematicians should care about history Pythagorean Theorem : the sum of the squares of the length of the legs of a right triangle is equal to the square of the hypotenuse. The Pythagorean theorem is perhaps the most famous statement in all of mathematics. $a^2 + b^2 = c^2$ is practically drilled into the minds of everyone that went through any form of secondary school education. But let…
No introduction. Just cool. Last post, I summarized the basics of formal logic , starting from a philosophical context before building up to more broad and involved results, coming closer to what just looks like math by the end of it. One of those results which was briefly thrown into the conclusion was the Compactness Theorem . You won't need to know a lot of logic to appreciate the power of this…
Everything you need for discrete and continuous induction. I've been on a kick for mathematical foundations recently. Maybe it's all that logic I've been looking at . One of those key ideas that finds its way into math is the Principle of Mathematical Induction . It's likely even if you've only taken high school math classes, you've encountered induction in one way or another. The importance of…
Your password is not protected perfectly, but it's good enough. Proofs, to me, are the cornerstones of all science. What separates math, physics, chemistry, and all other fields of science is the level of rigor we expect for professionals to conduct themselves. We don't take anything for fact until we are convinced irrefutably to call it so. We can conjecture all we want, but it only holds so much…
A classic fact and its 10000 word Wikipedia spiral. Irrational numbers are a bit strange to think about. In the sense, they were the first "new" type of number to really challenge early and young mathematicians. We have the natural numbers $\mathbb{N}$ like $\{0,1,2,3,\cdots\}$. From there, we then include the negative numbers $\{ \cdots,-2,-1,0,1,2,\cdots \}$ to have the integers $\mathbb{Z}$,…
Where modern number theory begins. It's not unfair to call prime numbers the foundation for not just number theory and mathematics, but many mathematicians' careers. At an early age, around 7 years old in 1st-2nd grade in the US, kids learn multiplication and division. From there, we learn that we can break down numbers into smaller numbers, we call factors or divisors. For example, $2024 = 2…
Actually important online doomsday prepping Previously, I had the opportunity to cap off my high school math career to write and present a topic of my choosing . In that case, I spent a couple weeks looking at the basics of the calculus of variations and its physics-inspired methods. To round off my undergraduate curriculum, I was given a similar opportunity on a much larger scale. Spanning across…