Intereting Posts

Primes of the form $a^2+b^2$ : a technical point.
Why is $\cos(x)^2$ written as $\cos^2(x)$?
Degree of $\mathbb{Q}(\zeta_n)$ over $\mathbb{Q}(\zeta_n+\zeta_{n}^{-1})$
limits of integration and derivative
Solutions of $p!q! = r!$
Demonstration: If all vectors of $V$ are eigenvectors of $T$, then there is one $\lambda$ such that $T(v) = \lambda v$ for all $v \in V$.
How to show that $a_n$ : integer $a_{n+1}=\frac{1+a_na_{n-1}}{a_{n-2}}(n\geq3), a_1=a_2=a_3=1$
Limit approach to finding $1+2+3+4+\ldots$
About matrix derivative
Integers of the form $a^2+b^2+c^3+d^3$
Prove $x \equiv a \pmod{p}$ and $x \equiv a \pmod{q}$ then $x \equiv a\pmod{pq}$
Determine y-coordinate of a 3rd point from 2 given points and an x-coordinate.
Solutions to Diophantine Equation $x^2 – D y^2 = m^2$
What is a co-prime?
Definition of vector field along a curve

Are there any properties of the Mandelbrot set that can be analysed without a knowledge of complicated topology?

Considering the fact that the set is based on a quadratic function, are there any interesting properties of the set that can be proved using algebra or relatively simple geometry? If there are, then what are those properties?

- Determine the set $\{w:w=\exp(1/z), 0<|z|<r\}$
- Is there anything special with complex fraction $\left|\frac{z-a}{1-\bar{a}{z}}\right|$?
- Prove that $\sum\limits_{k=0}^{n-1}\dfrac{1}{\cos^2\frac{\pi k}{n}}=n^2$ for odd $n$
- Book recommendations for self-study at the level of 3rd-4th year undergraduate
- Integrating around a dog bone contour
- Formula for calculating residue at a simple pole.

- Proving $|f(z)|$ is constant on the boundary of a domain implies $f$ is a constant function
- Conjecture: Every analytic function on the closed disk is conformally a polynomial.
- Prove that $f$ is a polynomial if one of the coefficients in its Taylor expansion is 0
- Does $|f|$ locally constant imply $f$ constant globally?
- Zoom out fractals? (A question about selfsimilarity)
- What is $iav-\log(v)$? Any series expansion or inequality for it?
- Why $\displaystyle f(z)=\frac{az+b}{cz+d}$, $a,b,c,d \in \mathbb C$, is a linear transformation?
- Writing the roots of a polynomial with varying coefficients as continuous functions?
- Holomorphic function with zero derivative is constant on an open connected set
- Ahlfors “Prove the formula of Gauss”

There are a number of properties about the Mandelbrot set that you can understand without knowing much math beyond basic algebra. I will list and explain a number of them, and provide refrences for further study. I will first, however, attempt to explain why such attempts will often be futile.

Part of what makes the Mandelbrot so incredibly famous is the fact that simple properties of the simplest possible non-linear iterated map:

$$f_c: z \mapsto z^2+c$$

can produce such an incredibly complicated structure that not only requires sophistocated mathematical tools to understand but isn’t even very well understood yet.

The set has two equvalent definitions, both relating to the same thing but in different ways. One is that $M$ consists of those values $c$ such that $f_c^n(0)$ remains bounded as $n \to \infty$, and this definition makes sense, and can be quite useful for a number of things. The other, equivalent, definition is that $M$ consists of those values $c$ such that $J(f_c)$, the Julia set for the map $f_c$

is connected. Even proving that these two defintitions are equivalent, while not “difficult” per se, requires some elementary, but not neccessarily obvious, results from complex analysis. A lot of properties of $M$ can only be derived easily from this second condition, (or from the first by proving it is equivalent to the second, but that doesn’t really count) and connectedness is, at it’s heart, a topological definition. Many characterizations come from the proof that a Julia set is either connected or contains nothing but disconnected points, and this dichotomy requires more than simple algebra as well. So without this you won’t get the true beauty of this beast, and without the knowledge of geometric measure theory a number of incredible resuts (e.g. the Hausdorff dimension of the boundary of the set is 2) don’t even make sense.

So ultimately yes, the Mandelbrot set stems from a very simple problem, but it requires a sophistocated set of tools to fully understand.

That being said, there are a number of results that are certainly accessible without topology or complex analysis. The first of which was referenced in the comments, that $M$ is bounded by the circle of radius 2.

This is quite easy to prove, although not quite “trivial”. You should definitely be able to manage it by yourself if you haven’t already. Simply think of where $f_c$ sends $0$ after the first iteration.

But don’t stop there! There are plenty of other cool geometrical things on $M$ that don’t require particularly complicated machinery. For example: if you read a little about bifurcation diagrams, specifically for $x^2 +c$ with $c \in \Bbb{R}$, you’ll understand a little bit about chaos theory, and understand this diagram:

But this is the exact same function as generates the Mandelbrot set. So if you look at the negative real axis of M you should see a striking similarity:

And this knowledge of at least some of the structure of the Mandelbrot set comes only from the study of one very simple dynamic system, which you can analyse using only algebra and perhaps a little calculus. Fully understanding bifurcation theory does require a lot of work, but that isn’t necessary to understand this one simple connection.

But in the end, understanding $M$ requires a lot of prerequisites, while observing fascinating properties of it requires nothing but a computer and curiosity. Perhaps fascination with some aspect of the fractal will lead you to learn challenging math in order to develop a deeper understanding.

A book for further reading geared towards students with nothing more than high school level math education would be The Computational Beauty of Nature by Flake which discusses a lot of topics tangential and related to the study of fractals.

Do you know about Misiurewicz Points? They’re polynomials roots on the M-set boundary. Here is a Question with some discussion of them along with a plot and some simple Mathematica code for calculating them.

- Tensor product of a module with an ideal is isomorphic to their standard product
- How to prove: $2^\frac{3}{2}<\pi$ without writing the explicit values of $\sqrt{2}$ and $\pi$
- How would one go about proving that the rationals are not the countable intersection of open sets?
- Why are the fundamental theorems of calculus usually associated to the Riemann Integral?
- Showing independence of random variables
- Stuck on proving uniform convergence
- graph is dense in $\mathbb{R}^2$
- Proving $\pi(\frac1A+\frac1B+\frac1C)\ge(\sin\frac A2+\sin\frac B2+\sin\frac C2)(\frac 1{\sin\frac A2}+\frac 1{\sin\frac B2}+\frac 1{\sin\frac C2})$
- Is a left invertible element of a ring necessarily right invertible?
- Calculate sum of an infinite series
- Finite sum of reciprocal odd integers
- What is the most efficient method to evaluate this indefinite integral?
- Nowhere continuous function limit
- Contraction of compact sets
- Are projective limits always subobjects of a product and dualy, inductive limits quotient objects of a coproduct?