PHIbonacci ∞

Nikola Tesla invented the Tesla coil, whilst Charles Darwin discovered evolution. It wouldn’t make sense to say Tesla discovered the Tesla coil, or to think that Darwin invented Evolution.

The difference between inventing and discovering something lies in the core of what is man-made and what is made by nature.

Hence there arises a big question in my mind: Is mathematics a  man-made invention or a discovery of something made by nature?

I’d say both somehow, because mathematics is a very intertwined network of principles.

It could be a discovery, if we combine ideas according to Richard Feynman’s simple words: “The imagination of nature is far, far greater than the imagination of man.”;

Mathematics is about creating worlds, and yet, I suppose, the actual framework of mathematics may be a discovery; Thus the results of math is actually an invention. Still, the underlying principles, say logic, creativity and an incredible amount of intuition, are a discovery in another field: Cognitive Science!

The tangled hierarchy of universes in recursion

by Dywiann Xyara  © 2019

The steps towards perfection

Hey guys! I made a blog about my current project “Mathematical Surrealism”. If you are interested take a look :)

Mandelbrot set

How can you visualize a 4-dimensional graph? Real valued functions of one variable are nicely graphed on a 2D gird (1 number line for the inputs crossed with 1 number line for the outputs) and we can talk about things like slope, area under the curve, etc. But complex functions of one variable would require a 4D grid (2 number lines for the inputs crossed with 2 number lines for the outputs). Although there are some clever ways to visualize complex functions with contour plots, 3D colored plots, two 2D plots, etc., notions like the integral as the area under the curve do not cross over. Unlike with integrals learned in introductory calculus, the complex integral has no geometric realization. Consequently, complex functions are often dealt with completely abstractly.

BUT WHY BOTHER WITH 4D FUNCTIONS? 4D STUFF ISN’T EVEN REAL! Right?

For mathematics a 4th dimension is introduced whenever it is useful or of interest. Often, a higher dimensional framework is useful in solving problems. For example, finding roots of polynomials sometimes requires the use of complex numbers. Also the famous sphere packing problem was solved in higher dimensions first which later gave insights into how to solve the problem in lower dimensions https://www.quantamagazine.org/20160330-sphere-packing-solved-in-higher-dimensions/. But there are more concrete examples too. Well known for this is Einstein’s work.

Oh right, I heard the 4th dimension is time!

Yes, in his theory of relativity, Einstein utilized a 4-dimensional geometry in which he assigned 3 dimensions to represent space (left-right, back-forth, up-down) and then a 4th dimension to represent time. This ended up being useful for describing the universe at a large scale.

But is there a real life 4th dimension? You know, like, is there a strange place I can go, become invisible, travel through time, etc?

In math, a group is a particular collection of elements. That might be a set of integers, the face of a Rubik’s cube–which we’ll simplify to a 2x2 square for now– or anything, so long as they follow 4 specific rules, or axioms.

Axiom 1: All group operations must be closed, or restricted, to only group elements. So in our square, for any operation you do—like turn it one way or the other—you’ll still wind up with an element of the group. Or for integers, if we add 3 and 2, that gives us 1—4 and 5 aren’t members of the group, so we roll around back to 0, similar to how 2 hours past 11 is 1 o’clock.

Axiom 2: If we regroup the order of the elements in an operation, we get the same result. In other words, if we turn our square right two times, then right once, that’s the same as once, then twice. Or for numbers, 1+(1+1) is the same as (1+1)+1.

Axiom 3: For every operation, there’s an element of our ground called the identity. When we apply it to any other element in our group, we still get that element. So for both turning the square and adding integers, our identity here is 0. Not very exciting.

Axiom 4:  Every group element has an element called its inverse, also in the group. When the two are brought together using group’s addition operation, they result in the identity element, 0. So they can be thought of as cancelling each other out. Here 3 and 1 are each other’s inverses, while 2 and 0 are their own worst enemies.

So that’s all well and good, but what’s the point of any of it? Well, when we get beyond these basic rules, some interesting properties emerge. For example, let’s expand our square back into a full-fledged Rubik’s cube. This is still a group that satisfies all of our axioms, though now with considerably more elements, and more operations—we can turn each row and column of each face.

Each position is called a permutation, and the more elements a group has, the more possible permutations there are. A Rubik’s cube has more than 43 quintillion permutations, so trying to solve it randomly isn’t going to work so well. However, using group theory we can analyze the cube and determine a sequence of permutations that will result in a solution. And, in fact, that’s exactly what most solvers do, even using a group theory notation indicating turns.

From the TED-Ed Lesson Group theory 101: How to play a Rubik’s Cube like a piano - Michael Staff

Michael Atiyah (Notices of the AMS, 2010)