You must have seen long-division method to compute decimal representation for fractions. Astonishingly, I never pondered about how one would divide an irrational number to get decimal representation. Firstly, this representation will be approximate. Secondly, we have been doing this in name of “rationalizing the denominator” stating the reason that division by irrationals is not allowed. But, in fact, this is the same problem as faced while analysing division algorithm for Gaussian integers.

Bottom line: Numbers are just symbols. We tend to assign meaning to them as we grow up. Since the set of real numbers, rational numbers and integers form an Euclidean domain, we can write a division algorithm for them. For example, we don’t have special set of symbols for 3 divided by π, but 3 divided by 2 is denoted by 1.5 in decimals.

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 , 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 in general?

Actually, if we consider as a vector space then developing calculus is not a big deal (as done in multivariable calculus). But, if we consider 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 is difficult. And this is because we are considering 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:

Dimension ≡ Dimension of the underlying vector space

Yesterday, I was fortunate enough to attend a lecture delivered by Dr. Ritwik Mukherjee, one of my professors, to motivate the study of algebraic topology. Instead of using the “soft targets” like Möbius strip etc. he used the following profound theorem for motivation:

If is continuous then there exists an such that: .

This is known as Borsuk-Ulam Theorem. To appreciate this theorem, one need to know a fundamental theorem about continuous functions known as Intermediate Value Theorem:

If a continuous function, , with an interval, , as its domain, takes values and at each end of the interval, then it also takes any value between and at some point within the interval.

Here is a video by James Grime illustrating Borsuk-Ulam Theorem in 3D:

Though the implications of the theorem itself are beautiful, following corollary known as Ham sandwich theorem is even more interesting. Here is a video by Marc Chamberland explaining this theorem:

Here is a video by Michael Stevens illustrating Brouwer fixed-point theorem in some interesting cases:

Now the applications of this theorem are numerous, and there is a book dedicated to this theorem: “Fixed Points” by Yu. A. Shashkin. But my favourite application of this fixed point theorem is to the board game called Hex, explained by Marc Chamberland here:

If you come across some other video/article discussing the coolness of “Borsuk-Ulam Theorem” please let me know.

Edit (18 May 2018): Proof of Brouwer’s Fixed Point Theorem by Tai-Danae Bradley:

This semester I am taking a course about protein structures. Here is a quick intro to proteins:

Though I have taken some other biology courses in past years, I found this course very much relatable to mathematics. Proteins are made up of “amino acids”. Though, chemistry allows large number of possible structures for amino acids (considering steric hindrance etc.), nature uses only 20 unique amino acids to make billions of different proteins. In my opinion, these 20 amino acids are “axioms” of protein building just like the 5 axioms of euclidean geometry.

Using just 20 amino acids we can get a large variety of protein structures, just like creating any kind of shape in euclidean space using just 5 axioms. Even more fascinating is the existence of “Quasisymmetry in Icosahedral Viruses”. An awesome article explaining this is available here. Note that, the term “triangulation number” stated in that article was not borrowed from mathematics. It’s a term used to study symmetries in icosahedral viruses and refers to “the square of the distance between 2 adjacent 5-fold vertices.”