Category Archives: Philosophy

Understanding Geometry – 4

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Aleksej Ivanovič Markuševič’s book, “Remarkable Curves” discusses the properties of ellipses, parabolas, hyperbolas, lemniscates, cycloids, brachistochrone, spirals and catenaries.  Among these “lemniscates” are the ones that I encountered only once before starting undergraduate education (all other curves appeared frequently in physics textbooks) and that too just to calculate the area enclosed by this curve. So I will discuss the properties of lemniscates in this post.

Let’s begin with the well-known curve, ellipse. An ellipse is the locus of points whose sum of distances from two fixed points (called foci) is constant. My favourite fact about ellipses is that we can’t find a general formula for the perimeter of an ellipse, and this little fact leads to the magical world of elliptic integrals. This, in turn, leads to the mysterious elliptic functions, which were discovered as inverse functions of elliptic integrals. Further, these functions are needed in the parameterization of certain curves, now called elliptic curves. For more details about this story, read the paper by Adrian Rice and Ezra Brown, “Why Ellipses are not Elliptic curves“.

Lemniscate is defined as the locus of points whose product of distances from two fixed points F_1 and F_2 (called foci) is constant. Lemniscate means, “with hanging ribbons” in Latin.  If the length of the segment \overline{F_1F_2} is c then for the midpoint of this line segment will lie on the curve if the product constant is c^2/4. In this case we get a figure-eight lying on its side.

Lemniskate_bernoulli2.svg

Lemniscate of Bernoulli; By Kmhkmh (Own work) [CC BY 4.0], via Wikimedia Commons

The attempt to calculate the perimeter of the above curve leads to elliptic integral, hence can’t derive a general formula for its perimeter. Just like an ellipse!

If we equate the value of the constant product not to c^2/4 but to another value, the lemniscate will change its shape. When the constant is less than c^2/4, the lemniscate consists of two ovals, one of which contains inside it the point F_1, and the other the point F_2.

oval

Cassini oval (x^2+y^2)^2−2c^2(x^2−y^2)=a^4−c^4; Source: https://www.encyclopediaofmath.org/legacyimages/common_img/c020700b.gif

When the product constant is greater than c^2/4 but less than c^2/2, the lemniscate has the form of a biscuit. If the constant is close to c^2/4, the “waist” of the biscuit is very narrow and the shape of the curve is very close to the figure-eight shape.

biscuit

Cassini oval (x^2+y^2)^2−2c^2(x^2−y^2)=a^4−c^4; Source: https://www.encyclopediaofmath.org/legacyimages/common_img/c020700b.gif

If the constant differs little from c^2/2, the waist is hardly noticeable, and if the constant is equal or greater than c^2/2 the waist disappears completely, and the lemniscate takes the form of an oval.

oval.gif

Cassini oval (x^2+y^2)^2−2c^2(x^2−y^2)=a^4−c^4; Source: https://www.encyclopediaofmath.org/legacyimages/common_img/c020700a.gif

We can further generalize this whole argument to get lemniscate with an arbitrary number of foci, called polynomial lemniscate.

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Building Mathematics

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Let’s talk about the work of a mathematician. When I was young (before highschool), I used to believe that anyone capable of using mathematics is a mathematician. The reason behind this was that being a mathematician was not a job for people like Brahmagupta, Aryabhatta, Fermat, Ramanujan (the names I knew when I was young). So by that definition, even a shopkeeper was a mathematician. And hence I had no interest in becoming a mathematician.

Then, during highschool, I came to know about the mathematics olympiad and was fascinated by the “easy to state but difficult to solve” problems from geometry, combinatorics, arithmetic and algebra (thanks to AMTIVipul Naik and Sai Krishna Deep) . I practiced many problems in hope to appear for the exam once in my life. But that day never came (due to bad education system of my state) and I switched to physics, just because there was lot of hype about how interesting our nature is (thanks to Walter Lewin). 

In senior school I realised that I can’t do physics, I simply don’t like the thought process behind physics (thanks to Feynman). And luckily, around the same time, came to know what mathematicians do (thanks to Uncle Paul). Mathematicians “create new maths”. They may contribute according to their capabilities, but no contribution is negligible. There are two kinds of mathematicians, one who define new objects (I call them problem creators) and others who simplify the existing theories by adding details (I call them problem solvers). You may wonder that while solving a problem one may create bigger maths problems, and vice versa, but I am talking about the general ideologies. What I am trying to express, is similar to what people want to say by telling that logic is a small branch of mathematics (whereas I love maths just for its logical arguments).

A few months before I had to join college (in 2014), I decided to become a mathematician. Hence I joined a research institute (clearly not the best one in my country, but my concern was just to be able to learn as much maths as possible).  Now I am learning lots of advanced (still old) maths (thanks to Sagar SrivastavaJyotiraditya Singh and my teachers) and trying to make a place for myself, to be able to call myself a mathematician some day.

I find all this very funny. When I was young, I used to think that anyone could become a mathematician and there was nothing special about it. But now I everyday have to prove myself to others so that they give me a chance to become a mathematician. Clearly, I am not a genius like all the people I named above (or even close to them) but I still want to create some new maths either in form of a solution to a problem or foundations of new theory and call myself a mathematician. I don’t want it to end up like my maths olympiad dream.

Magic Cubes

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Last week I attended a talk (by a student) about Magic Squares. I learned a bunch of cool facts about them (like how to devise an algorithm to construct them). Towards the end of the talk, one student from the audience suggested the possibility of Magic Cubes. I got very excited about this idea since it pointed towards the stereotypical mathematical ideology of generalizing the examples in order to see the deeper connections.

I myself don’t know much about Magic Cubes (or even Magic Squares) but would like to quote W. W. Rouse Ball & H. S. M. Coxeter from pp. 217 the book “Mathematical Recreations and Essays” (11th Ed.) :

A Magic Cube of the n^{th} order consists of the consecutive numbers from 1 to n^3, arranged in the form of a cube, so that the sum of the numbers in every row, every column, every file, and in each of the four diagonals (or “diameters “), is the same-namely, \frac{1}{2}(n^3 + 1). This sum occurs in 3n^2 + 4 ways. I do not know of any rule for constructing magic cubes of singly-even order. But such cubes of any odd or doubly-even order can be constructed by a natural extension of the methods already used for squares.

I would like to read about these magic hyper-cubes in future. And if you know something interesting about them, let me know in the comments below.

Under 40

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The age of 40 is considered special in mathematics because it’s an ad-hoc criterion for deciding whether a mathematician is young or old. This idea has been well established by the under-40 rule for Fields Medal, based on Fields‘ desire that:

while it was in recognition of work already done, it was at the same time intended to be an encouragement for further achievement on the part of the recipients and a stimulus to renewed effort on the part of others

Though it must be noted that this criterion doesn’t claim that after 40 mathematicians are not productive (example: Yitang Zhang).  So I wanted to write a bit about the under 40 leading number theorists which I am aware of (in order of decreasing age):

  • Sophie Morel: The area of mathematics in which Morel has already made contributions is called the Langlands program, initiated by Robert Langlands. Langlands brought together two fields, number theory and representation theory. Morel has made significant advances in building that bridge. “It’s an extremely exciting area of mathematics,” Gross says, “and it requires a vast background of knowledge because you have to know both subjects plus algebraic geometry.” [source]
  • Melanie Wood: Profiled at age 17 as “The Girl Who Loved Math” by Discover magazine, Wood has a prodigious list of successes. The main focus of her research is in number theory and algebraic geometry, but it also involves work in probability, additive combinatorics, random groups, and algebraic topology.  [source1, source2]
  • James Maynard:  James is primarily interested in classical number theory, in particular, the distribution of prime numbers. His research focuses on using tools from analytic number theory, particularly sieve methods, to study primes.  He has established the sensational result that the gap between two consecutive primes is no more than 600 infinitely often. [source1, source2]
  • Peter Scholze: Scholze began doing research in the field of arithmetic geometry, which uses geometric tools to understand whole-number solutions to polynomial equations that involve only numbers, variables and exponents. Scholze’s key innovation — a class of fractal structures he calls perfectoid spaces — is only a few years old, but it already has far-reaching ramifications in the field of arithmetic geometry, where number theory and geometry come together. Scholze’s work has a prescient quality, Weinstein said. “He can see the developments before they even begin.” [source]

Intra-mathematical Dependencies

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Recently I completed all of my undergraduate level maths courses, so wanted to sum up my understanding of mathematics in the following dependency diagram:

mat-dependency (1)

I imagine this like a wall, where each topic is a brick. You can bake different bricks at different times (i.e. follow your curriculum to learn these topics), but finally, this is how they should be arranged (in my opinion) to get the best possible understanding of mathematics.

As of now, I have an “elementary” knowledge of Set Theory, Algebra, Analysis, Topology, Geometry, Probability Theory, Combinatorics and Arithmetic. Unfortunately, in India, there are no undergraduate level courses in Mathematical Logic and Category Theory.

This post can be seen as a sequel of my “Mathematical Relations” post.

Understanding Geometry – 3

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In some of my past posts, I have mentioned “hyperbolic curvature“,”hyperbolic trigonometry” and “hyperbolic ideal points“. In this post I will share some artworks, based on hyperbolic geometry, by contemporary artists (from Tumblr):

To explain the mathematics behind the construction of these pictures I will quote Roger Penrose from pp. 34 of  “The Road to Reality“:

Think of any circle in a Euclidean plane. The set of points lying in the interior of this circle is to represent the set of points in the entire hyperbolic plane. Straight lines, according to the hyperbolic geometry are to be represented as segments of Euclidean circles which meet the bounding circle orthogonally — which means at right angles. Now, it turns out that the hyperbolic notion of an angle between any two curves, at their point of intersection, is precisely the same as the Euclidean measure of the angle between the two curves at the intersection point. A representation of this nature is called conformal. For this reason, the particular representation of hyperbolic geometry that Escher used is sometimes referred to as the conformal model of hyperbolic plane.

In the above-quoted paragraph, Penrose refers to Escher’s “Circle Limit” works, explained in detail by Bill Casselman in this article.

Understanding Geometry – 1

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When we think about mathematics, what comes to our mind are the numbers and figures. The  study of numbers is called arithmetic and the study of figures is called geometry (in very crude sense!). In our high school (including olympiad level) and college curriculum we cover various aspects of arithmetic. I am very much satisfied with that treatment, and this is the primary reason for my research interests in arithmetic (a.k.a. number theory).

But, I was always unsatisfied with the treatment given to geometry in our high school curriculum. We were taught some plane Euclidean geometry (with the mention of the existence of non-euclidean geometries), ruler and compass constructions, plane trigonometry (luckily, law of cosines was taught), surface area & volume of 3D objects, 2D coordinate geometry, conic sections and 3D coordinate geometry.  In the name of Euclidean geometry some simple theorems for triangles, quadrilaterals and circles are discussed, like triangle congruence criterias, triangle similarity criterias, Pythagoras theorem, Mid-Point Theorem, Basic Proportionality Theorem, Thales’ Theorem, Ptolemy’s theorem, Brahmagupta theorem etc. are discussed. Ruler-compass constructions are taught as “practical geometry”. Students are asked to cram the formulas of area (including Brahmagupta’s formula and Heron’s formula) and volume without giving any logic (though in earlier curriculum teacher used to give the reasoning). Once coordinate geometry is introduced, students are asked to forget the idea of Euclidean geometry or visualizing 3D space. And to emphasize this, conic sections are introduced only as equations of curves in two dimensional euclidean plane.

The interesting theorems from Euclidean geometry like Ceva’s theorem,  Stewart’s Theorem,  Butterfly theorem, Morley’s theorem (I discussed this last year with high school students), Menelaus’ theorem, Pappus’s theorem,  etc. are never discussed in classroom (I came to know about them while preparing for olympiads). Ruler-compass constructions are taught without mentioning the three fundamental impossibilities of angle trisection, squaring a circle and doubling a cube. The conic sections are taught without discussing the classical treatment of the subject by Apollonius.

In the classic “Geometry and Imagination“, the first chapter on conic sections is followed by discussion of crystallographic groups (Character tables for point groups),  projective geometry (recently I discussed an exciting theorem related to this) and differential geometry (currently I am doing an introductory course on it). So over the next few months I will be posting mostly about geometry (I don’t know how many posts in total…), in an attempt to fill the gap between high-school geometry and college geometry.

I agree with the belief that algebraic and analytic methods make the handling of geometry problems much easier, but in my opinion these methods suppress the visualization of geometric objects. I will end this introductory post with a way to classify geometry by counting the number of ideal points in projective plane:

  • Hyperbolic Geometry (a.k.a. Lobachevsky-Bolyai-Gauss type non-euclidean geometry) which has two ideal points [angle-sum of a triangle is less than 180°].
  • Elliptic Geometry (a.k.a. Riemann type non-euclidean geometry) which has no ideal points. [angle-sum of a triangle is more than 180°]
  • Parabolic Geometry (a.k.a. euclidean geometry)  which has one ideal point. [angle-sum of a triangle is 180°]