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Author: Shreyas

Small Displacements Revisited: The Fermat Point

Small Displacements Revisited: The Fermat Point

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1. Introduction This post will continue my investigation into the “method” of small geometrical displacements that I discussed in my previous post. I will apply it to a famous problem regarding triangles: the Fermat point. The Fermat point of a triangle is defined as the point in the plane of the triangle that minimizes the sum of the distances to each of the vertices. As we learned in the last post, the method of small displacements can be very useful…

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The “Method” of Small Displacements

The “Method” of Small Displacements

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1. Introduction One of the most important things about calculus and its subfields is the ability to think about “very small” changes in certain variables and pull out information from them. The derivative of a variable with respect to another variable essentially tells us the relation between a very small change in (termed ) and the very small change in (termed ) that results. But how do we define “very small?” Limits are a powerful way of doing so: we…

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The Higher-Dimensional Pythagorean Theorem

The Higher-Dimensional Pythagorean Theorem

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1. Introduction We have all heard of the familiar Pythagorean theorem, relating the side lengths of a right triangle. But, have you heard of De Gua’s Theorem? An interesting generalization of the Pythagorean theorem into 3 dimensions, this theorem is very similar result for “right tetrahedrons” where three of the edges are mutually perpendicular. Suppose, in such a tetrahedron we have     Denote the area of as . Then,     In today’s post, we will investigate whether a…

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The Cardinality of the Real Numbers

The Cardinality of the Real Numbers

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1. Introduction , The natural numbers are perhaps the most fundamental and intuitive set in mathematics. But they have a strange property: they never end (as some of you may have found out by attempting a “name the biggest number” game in childhood). Of course, these numbers may be extended by various logical arguments: what happens if we subtract a larger number from a smaller one, or divide a number into another number that is not its multiple, or take…

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Volume of a n-dimensional Ball: Part 2

Volume of a n-dimensional Ball: Part 2

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1. Introduction The first installment of this post investigated the question of computing the volume of a unit ball (and by extension any ball) of dimensions. The main result we obtained was that the volume can be computed recursively as , where     In this post, we will use this result to prove an explicit formula for . 2. The Formula A quick search online yields the following explicit formula for :     where the function extends the…

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Volume of a n-dimensional Ball: Part 1

Volume of a n-dimensional Ball: Part 1

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1. Introduction This post investigates an interesting question I had been thinking about: we all know that the area of a circle of radius equals , and the volume of a sphere of radius equals . But what if we enter the territory of even higher dimensions? We should be able to say something about the “volume” of an n-dimensional sphere, where “volume” is interpreted as the equivalent measure of space in that many dimensions (e.g. area for 2D, regular…

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The Napoleonic Triangle: A Classic Result

The Napoleonic Triangle: A Classic Result

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Introduction Given any triangle , suppose we take each side and and construct an equilateral triangle on it. You may notice that given a line segment, there are two ways to construct an equilateral triangle on it (one on each side); we will eventually consider both ways, but for now suppose each triangle is constructed on the outside of , i.e. they do not overlap with . We now consider finding the centers of each of these equilateral triangles, as…

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Yet Another Interpretation of Solutions to Quadratic Equations

Yet Another Interpretation of Solutions to Quadratic Equations

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Today’s post will be fairly short and simple, but may have interesting usages in other areas. Given a quadratic equation , it is well known that solutions for are given by the quadratic formula, . There are a multitude of ways to derive this: Completing the square: can be reduced to . At this point, we know easily how to solve equations of the form , and by extension , by simply taking square roots (being wary of the two…

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Basic Derivation of Heron’s Formula

Basic Derivation of Heron’s Formula

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Introduction This post is a basic derivation of the famous Heron’s formula, which allows calculation of the area of any triangle in terms of its side lengths . We refer to the following diagram: As is convention, let sides be opposite to angles . Further, let the altitude length and . Then, by the Pythagorean Theorem applied to triangles and , we have the following two equations:         Subtracting the two equations cancels the and yields ….

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Chicken McNugget Theorem – Bezout’s Identity

Chicken McNugget Theorem – Bezout’s Identity

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Given an unlimited number of 2 and 5 dollar bills, what all dollar values can be created? Clearly all multiples of 2 and 5 can be created, but they can also be combined to form many other values such as 7 (2 + 5), 13 (5 + 4 * 2) and so on. In fact, there are only a finite number of values that cannot be created in such a way by combining 2 and 5 dollar bills, and in…

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