Tag Archives: geometry

Triangles and Rectangles


Consider the following question by Bill Sands (asked in 1995):

Are there right triangles with integer sides and area, associated with rectangles having the same perimeter and area?

Try to test your intuition. The solution to this problem is NOT so simple. The solution was published by Richard K. Guy in 1995: http://www.jstor.org/stable/2974502 

If you have a simpler solution, please write it in a comment below. Even I don’t understand much of the solution.


Intra-mathematical Dependencies


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


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 – 2


If you want to brush up your high school geometry knowledge, then KhanAcademy is a good place to start. For example, I learned a new proof of Pythagoras Theorem (there are 4 different proofs on KhanAcademy) which uses scissors-congruence:

In this post, I will share with you few theorems from L. I. Golovina and I. M. Yaglom’s “Induction in Geometry ”  which I learned while trying to prove Midpoint-Polygon Conjecture.

Theorem 1: The sum of interior angles of an n-gon is 2\pi (n-2).

Theorem 2: The number of ways in which a convex n-gon can be divided into triangles by non-intersecting diagonals is given by


Theorem 3: Given a \triangle ABC, with n-1 straight lines CM_1, CM_2, \ldots CM_{n-1} drawn through its vertex C, cutting the triangle into n smaller triangles \triangle ACM_1, \triangle M_1CM_2, \ldots, \triangle M_{n-1}CB. Denote by r_1, r_2, \ldots r_n and \rho_1, \rho_2, \ldots, \rho_n respectively the radii of the inscribed and circumscribed circles of these triangles (all the circumscribed circles are inscribed within the angle C of the triangle) and let r and \rho be the radii of the inscribed and circumscribed circles (respectively) of the \triangle ABC itself. Then

\displaystyle{\frac{r_1}{\rho_1} \cdot\frac{r_2}{\rho_2} \cdots\frac{r_n}{\rho_n} =\frac{r}{\rho} }

Theorem 4: Any convex n-gon which is not a parallelogram can be enclosed by a triangle whose sides lie along three sides of the given n-gon.

Theorem 5 (Levi’s Theorem): Any convex polygon which is not a parallelogram can be covered with three homothetic polygons smaller than the given one.

The above theorem gives a good idea of what “combinatorial geometry” is all about. In this subject, the method of mathematical induction is widely used for proving various theorems. Combinatorial geometry deals with problems, connected with finite configurations of points or figures. In these problems, values are estimated connected with configurations of figures (or points) which are optimal in some sense.

Theorem 6 (Newton’s Theorem): The midpoints of the diagonals of a quadrilateral circumscribed about a circle lie on one straight line passing through the centre of the circle.

Theorem 7 (Simson’s Theorem): Given a \triangle ABC inscribed in the circle S with an arbitrary point P on this circle. Then then feet of the perpendiculars dropped from the point P to the sides of the \triangle ABC are collinear.

We can extend the above idea of Simson’s line to any n-gon inscribed in a circle.

Theorem 8: A 3-dimensional space is divided into \frac{(n+1)(n^2-n+6)}{6} parts by n planes, each three of which intersect and no four of which have a common point.

Theorem 9: Given n spheres in 3-dimesnional space, each four of which intersect. Then all these spheres intersect, i. e. there exists a point belonging to all the spheres.

Theorem 10 (Young’s Theorem): Given n points in the plane such that each pair of them are at a distance of at most 1 from each other. Then all these points can be enclosed in a circle of radius 1/\sqrt{3}.

I won’t be discussing their proofs since the booklet containing the proofs and the detailed discussion is freely available at Mir Books.

Also, I would like to make a passing remark about the existence of a different kind of geometry system, called “finite geometry“. A finite geometry is any geometric system that has only a finite number of points. The familiar Euclidean geometry is not finite because a Euclidean line contains infinitely many points. A geometry based on the graphics displayed on a computer screen, where the pixels are considered to be the points, would be a finite geometry. While there are many systems that could be called finite geometries, attention is mostly paid to the finite projective and affine spaces because of their regularity and simplicity. You can learn more about it here: http://www.ams.org/samplings/feature-column/fcarc-finitegeometries

Understanding Geometry – 1


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°]

Real vs Complex numbers


I want to talk about the algebraic and analytic differences between real and complex numbers. Firstly, let’s have a look at following beautiful explanation by Richard Feynman (from his QED lectures) about similarities between real and complex numbers:


From Chapter 2 of the book “QED – The Strange Theory of Light and Matter” © Richard P. Feynman, 1985.

Before reading this explanation, I used to believe that the need to establish “Fundamental theorem Algebra” (read this beautiful paper by Daniel J. Velleman to learn about proof of this theorem) was only way to motivate study of complex numbers.

The fundamental difference between real and complex numbers is

Real numbers form an ordered field, but complex numbers can’t form an ordered field. [Proof]

Where we define ordered field as follows:

Let \mathbf{F} be a field. Suppose that there is a set \mathcal{P} \subset \mathbf{F} which satisfies the following properties:

  • For each x \in \mathbf{F}, exactly one of the following statements holds: x \in \mathcal{P}, -x \in \mathcal{P}, x =0.
  • For x,y \in \mathcal{P}, xy \in \mathcal{P} and x+y \in \mathcal{P}.

If such a \mathcal{P} exists, then \mathbf{F} is an ordered field. Moreover, we define x \le y \Leftrightarrow y -x \in \mathcal{P} \vee x = y.

Note that, without retaining the vector space structure of complex numbers we CAN establish the order for complex numbers [Proof], but that is useless. I find this consequence pretty interesting, because though \mathbb{R} and \mathbb{C} are isomorphic as additive groups (and as vector spaces over \mathbb{Q}) but not isomorphic as rings (and hence not isomorphic as fields).

Now let’s have a look at the consequence of the difference between the two number systems due to the order structure.

Though both real and complex numbers form a complete field (a property of topological spaces), but only real numbers have least upper bound property.

Where we define least upper bound property as follows:

Let \mathcal{S} be a non-empty set of real numbers.

  • A real number x is called an upper bound for \mathcal{S} if x \geq s for all s\in \mathcal{S}.
  • A real number x is the least upper bound (or supremum) for \mathcal{S} if x is an upper bound for \mathcal{S} and x \leq y for every upper bound y of \mathcal{S} .

The least-upper-bound property states that any non-empty set of real numbers that has an upper bound must have a least upper bound in real numbers.
This least upper bound property is referred to as Dedekind completeness. Therefore, though both \mathbb{R} and \mathbb{C} are complete as a metric space [proof] but only \mathbb{R} is Dedekind complete.

In an arbitrary ordered field one has the notion of Dedekind completeness — every nonempty bounded above subset has a least upper bound — and also the notion of sequential completeness — every Cauchy sequence converges. The main theorem relating these two notions of completeness is as follows [source]:

For an ordered field \mathbf{F}, the following are equivalent:
(i) \mathbf{F} is Dedekind complete.
(ii) \mathbf{F} is sequentially complete and Archimedean.

Where we defined an Archimedean field as an ordered field such that for each element there exists a finite expression 1+1+\ldots+1 whose value is greater than that element, that is, there are no infinite elements.

As remarked earlier, \mathbb{C} is not an ordered field and hence can’t be Archimedean. Therefore, \mathbb{C}  can’t have least-upper-bound property, though it’s complete in topological sense. So, the consequence of all this is:

We can’t use complex numbers for counting.

But still, complex numbers are very important part of modern arithmetic (number-theory), because they enable us to view properties of numbers from a geometric point of view [source].

Dimension clarification


In several of my previous posts I have mentioned the word “dimension”. Recently I realized that dimension can be of two types, as pointed out by Bernhard Riemann in his famous lecture in 1854. Let me quote Donal O’Shea from pp. 99 of his book “The Poincaré Conjecture” :

Continuous spaces can have any dimension, and can even be infinite dimensional. One needs to distinguish between the notion of a space and a space with a geometry. The same space can have different geometries. A geometry is an additional structure on a space. Nowadays, we say that one must distinguish between topology and geometry.

[Here by the term “space(s)” the author means “topological space”]

In mathematics, the word “dimension” can have different meanings. But, broadly speaking, there are only three different ways of defining/thinking about “dimension”:

  • Dimension of Vector Space: It’s the number of elements in basis of the vector space. This is the sense in which the term dimension is used in geometry (while doing calculus) and algebra. For example:
    • A circle is a two dimensional object since we need a two dimensional vector space (aka coordinates) to write it. In general, this is how we define dimension for Euclidean space (which is an affine space, i.e. what is left of a vector space after you’ve forgotten which point is the origin).
    • Dimension of a differentiable manifold is the dimension of its tangent vector space at any point.
    • Dimension of a variety (an algebraic object) is the dimension of tangent vector space at any regular point. Krull dimension is remotely motivated by the idea of dimension of vector spaces.
  • Dimension of Topological Space: It’s the smallest integer that is somehow related to open sets in the given topological space. In contrast to a basis of a vector space, a basis of topological space need not be maximal; indeed, the only maximal base is the topology itself. Moreover, dimension is this case can be defined using  “Lebesgue covering dimension” or in some nice cases using “Inductive dimension“.  This is the sense in which the term dimension is used in topology. For example:
    • A circle is one dimensional object and a disc is two dimensional by topological definition of dimension.
    • Two spaces are said to have same dimension if and only if there exists a continuous bijective map between them. Due to this, a curve and a plane have different dimension even though curves can fill space.  Space-filling curves are special cases of fractal constructions. No differentiable space-filling curve can exist. Roughly speaking, differentiability puts a bound on how fast the curve can turn.
  • Fractal Dimension:  It’s a notion designed to study the complex sets/structures like fractals that allows notions of objects with dimensions other than integers. It’s definition lies in between of that of dimension of vector spaces and topological spaces. It can be defined in various similar ways. Most common way is to define it as “dimension of Hausdorff measure on a metric space” (measure theory enable us to integrate a function without worrying about  its smoothness and the defining property of fractals is that they are NOT smooth). This sense of dimension is used in very specific cases. For example:
    • A curve with fractal dimension very near to 1, say 1.10, behaves quite like an ordinary line, but a curve with fractal dimension 1.9 winds convolutedly through space very nearly like a surface.
      • The fractal dimension of the Koch curve is \frac{\ln 4}{\ln 3} \sim 1.26186, but its topological dimension is 1 (just like the space-filling curves). The Koch curve is continuous everywhere but differentiable nowhere.
      • The fractal dimension of space-filling curves is 2, but their topological dimension is 1. [source]
    • A surface with fractal dimension of 2.1 fills space very much like an ordinary surface, but one with a fractal dimension of 2.9 folds and flows to fill space rather nearly like a volume.

This simple observation has very interesting consequences. For example,  consider the following statement from. pp. 167  of the book “The Poincaré Conjecture” by Donal O’Shea:

… there are infinitely many incompatible ways of doing calculus in four-space. This contrasts with every other dimension…

This leads to a natural question:

Why is it difficult to develop calculus for any \mathbb{R}^n in general?

Actually, if we consider \mathbb{R}^n as a vector space then developing calculus is not a big deal (as done in multivariable calculus).  But, if we consider \mathbb{R}^n as a topological space then it becomes a challenging task due to the lack of required algebraic structure on the space. So, Donal O’Shea is actually pointing to the fact that doing calculus on differentiable manifolds in \mathbb{R}^4 is difficult. And this is because we are considering \mathbb{R}^4 as 4-dimensional topological space.

Now, I will end this post by pointing to the way in which definition of dimension should be seen in my older posts: