Fractal Dimensions

The YouTube channel 3Blue1Brown contains some excellent videos. The video linked to here on Fractal Dimensions is no exception. Beginning with self-similar shapes, like the Koch snowflake and Sierpinski Triangle, the commentator calculates their fractal dimension and then goes on to look at non-self-similar shapes such as the coastlines of England and Norway. The screenshot below displays the fractal dimensions for the coastlines of these two countries.

What emerges is that the fractal dimension is a measure of the roughness of a shape and that these shapes really exist, unlike mathematically perfect shapes like lines, circles, cubes and so on. These latter shapes have dimensions of 1, 2 and 3 corresponding to the one, two and three dimensions that they inhabit. A fractal shape like the one shown below however, can be considered to be halfway between a line and a two dimensional shape like a square. It has a fractal dimension of 1.5:

Here is the video:

Double Factorial

On Day 24771, my tweet was:

24771: 3×23×359; member of OEIS A259359: numbers n such that n!!-8 is prime (useful as a means of generating probable primes) #mathematics. 

Prior to this I'd been perplexed until I read the following in Wikipedia:

The double factorial should not be confused with the factorial function iterated twice, which is written as (n!)! and not n!!

This of course is quite confusing. The Wikipedia goes on to say that:

The sequence of double factorials for even n = 0, 2, 4, 6, 8,... starts as
1, 2, 8, 48, 384, 3840, 46080, 645120,... (sequence A000165 in the OEIS) 

The sequence of double factorials for odd n = 1, 3, 5, 7, 9,... starts as
1, 3, 15, 105, 945, 10395, 135135,... (sequence A001147 in the OEIS) 

Perhaps the alternative term semifactorial is more appropriate and odd factorial is sometimes used for the double factorial of an odd number. Interestingly, the Wikipedia article mentions that double factorials also arise in expressing the volume of a hypersphere. The 4-dimensional hypersphere, or 3-sphere, was the subject of my previous post. The formula for the volume of an n-dimensional hypersphere is:

 

which reduces to the familiar formula for the volume of a sphere when n=3.

3-Sphere

I've been currently reading "Reality in Not What It Seems: The Journey to Quantum Gravity" by Carlo Rovelli (translated from the Italian in which it was originally written). It's a fascinating book but early on, Rovelli makes mention of the 3-Sphere and says:

Einstein’s idea is that space could be a 3-sphere: something with a finite volume (the sum of the volume of the two balls), but without borders. The 3-sphere is the solution which Einstein proposes in his work of 1917 to the problem of the border of the universe.

Well, this was a new concept to me and naturally I needed to investigate. I was fortunate to stumble across a question in Quora titled: How can one visualize a 3-sphere? All three answers to the question were illuminating in different ways. Of course, the 3-sphere is a specific type of n-sphere: there is a 1-sphere, a 2-sphere as well as a 4-sphere, 5 sphere and so on. It helps to start with the 1-sphere and 2-sphere before tackling the 3-sphere.

In the diagram shown below a 1-sphere (circle) existing in 2 dimensions is represented in 1 dimension as two lines joined together at their ends:

In the next diagram, a 2-sphere (sphere) existing in 3 dimensions is represented in 2 dimensions as two circles joined together at their boundaries.

This gives us some idea of how to represent a 3-sphere, or hypersphere, existing in 4 dimensions. It can be envisioned as two spheres joined across their boundaries. This is shown in the diagram below:

The following YouTube video does a very good job of explaining the mathematics underlying the 3-sphere in terms of the Hopf fibration:

During this video, the lecturer (Niles Johnson) mentions Cartesian products and Quaternions. I've heard of both but should try to reacquaint myself with them.

ADDENDUM: added on January 11th 2019

I was reminded of the 3-sphere by a recently read article that begins:

Our universe has antimatter partner on the other side of the Big Bang

In a CPT-symmetric universe, time would run backwards
from the Big Bang and antimatter would dominate

The two three dimensional spheres that make up the fourth dimensional 3-sphere seem perfectly suited to accommodate the matter-antimatter pair of universes. If one accepts that time is an illusion (see this blog post of mine title The Illusion of Time) then these two structures are eternally existent in a timeless 4-D Universe.

Abundant Numbers

Today's numbered day is 24762 and one of its claims to fame, according the OEIS, is its membership in A228964: smallest sets of 7 consecutive abundant numbers in arithmetic progression. The initial abundant number is listed. The abundant numbers forming this arithmetic sequence are 24762 | 24768 | 24774 | 24780 | 24786 | 24792 | 24798. The next abundant number is 24800 which breaks the pattern.

It might be appropriate in this post to remind myself what constitutes an abundant number and to list some interesting facts about them. To begin, a definition from Wikipedia:

In number theory, an abundant number or excessive number is a number for which the sum of its proper divisors is greater than the number itself. The integer 12 is the first abundant number. Its proper divisors are 1, 2, 3, 4 and 6 for a total of 16. The amount by which the sum exceeds the number is the abundance. The number 12 has an abundance of 4, for example. 

Definition: a number \(n\) for which the sum of divisors \(\sigma(n)>2n \), or, equivalently, the sum of proper divisors (or aliquot sum) \( \text{s}(n)>n\). 

Abundance is the value \( \sigma(n)-2n\) (or \( \text{s}(n)-n\)).

Some of the interesting facts about abundant numbers are:

• the smallest odd abundant number is 945
• The smallest abundant number not divisible by 2 or by 3 is 5391411025
• infinitely many even and odd abundant numbers exist
• every integer greater than 20161 can be written as the sum of two abundant numbers 
• every multiple (beyond 1) of a perfect number is abundant
• every multiple of an abundant number is abundant
• an abundant number with abundance 1 is called a quasiperfect number, although none have yet been found

ADDENDUM: Friday, June 12th 2020

I came across some further interesting facts about abundant numbers on this site. Here is what was mentioned:

• There are at least 10000 pairs of known consecutive abundant integers.
  See A096399 and this file by T. D. Noe.
  
• The triple 171078830, 171078831, 171078832 was apparently found by Laurent Hodges and Michael Reid in 1995.
  
• There are at least 1000 triples of consecutive abundant numbers.
  See A096536 and this file by Donovan Johnson.
  
• The starting term of the smallest consecutive 4-tuple of abundant numbers is at most:
  141363708067871564084949719820472453374 (39 digits)
  by Bruno Mishutka, November 1st 2007. See A094628.

Feigenbaum Constants

The Feigenbaum constants are right up there with the big ones but it was only after watching a Numberphile video that I understood the significance of the delta 𝛅 constant. Here is the video that I watched:

The Feigenbaum constant 𝛿 is defined by Wikipedia as follows:

It's easier to see with a specific example of f(x), an old friend from Chaos Theory namely ax(1-x) and a table of values for a:

I created a spreadsheet in Google Sheets to illustrate the bifurcation process. Below is an example of the first bifurcation when a=3.3:

Below is an example of the second bifurcation, taken when a=3.5:

And so on it goes.

The Beal Conjecture

I was led to the Beal Conjecture by following the answers to this question in Quora:

What are three distinct positive integers a, b and c such that \(a^3+b^3=c^4\)

Of course, we know from Fermat's Last Theorem that there are no solutions to \(a^3+b^3=c^3\) but what if c is raised to the fourth power as it is here in this question? It turns out that there are infinitely many solutions and one nice, easy solution generator was proposed in one of the answers and that's shown below:

It's clear that all solutions will involve numbers that have a common factor because \(a = kc \) and \( b = lc \). No pair of numbers can be coprime. Another answer however, referred me to the Beal Conjecture. Below is an excerpt from Wikipedia:

The article goes on to say that:

Fermat's Last Theorem established that \( A^n + B^n = C^n \) has no solutions for n > 2 for positive integers A, B, and C. If any solutions had existed to Fermat's Last Theorem, then by dividing out every common factor, there would also exist solutions with A, B, and C coprime. Hence, Fermat's Last Theorem can be seen as a special case of the Beal conjecture restricted to x = y = z.

I should say a little about Andy Beal whose name adheres to the conjecture:

(In 1993) Andy Beal was working on Fermat's Last Theorem when he began to look at similar equations with independent exponents. He constructed several algorithms to generate solution sets but the very nature of the algorithms he was able to construct required a common factor in the bases. He began to suspect that co-prime bases might be impossible and set out to test his hypothesis by computer. Andy Beal and a colleague programmed 15 computers and after thousands of cumulative hours of operation had checked all variable values through 99. Many solutions were found: all had a common factor in the bases. While certainly not conclusive, Andy Beal now had sufficient reason to share his discovery with the world. In the fall of 1994, Andy Beal wrote letters about his work to approximately 50 scholarly mathematics periodicals and number theorists. By offering a cash prize for the proof or disproof of this important number theory relationship, Andy Beal hopes to inspire young minds to think about the equation, think about winning the offered prize, and in the process become more interested in the wonderful study of mathematics. Information regarding the $1,000,000 cash prize that is held in trust by the American Mathematics Society can be obtained at the University of North Texas web site: http://www.math.unt.edu/~mauldin/beal.html.

Daniel Andrew "Andy" Beal is an American banker, businessman, investor, poker player, and amateur mathematician. He is a Dallas-based businessman who accumulated wealth in real estate and banking. Wikipedia

Born: November 29, 1952 (age 64), Lansing, Michigan, United States

Net worth: 10.5 billion USD (2017) Forbes

Education: Baylor University, Michigan State University

Organizations founded: Beal Bank, Beal Aerospace

SageMath

I was looking at a comparison of numerical analysis software on Wikipedia and came across SageMath that was free and that had a version available for OS X. Oddly, there is no version available for Windows. Some of its expensive proprietary competitors are Mathematica, Maple and MATLAB. A version of Mathematica is free on the Raspberry Pi and I have the Pixel OS installed as a virtual machine but the licence does not apply to this scenario and so it is unfortunately missing.

Once the dmg file was downloaded I installed it and opened it in terminal and then activated the notebook, from where worksheets can be created. Here is a screenshot of the notebook in Chrome:

The username is admin and my password is the usual. Here is what the introductory page to the notebook says about Sage:

Welcome! 

Sage is a different approach to mathematics software. 

The Sage Notebook 

With the Sage Notebook anyone can create, collaborate on, and publish interactive worksheets. In a worksheet, one can write code using Sage, Python, and other software included in Sage. 

General and Advanced Pure and Applied Mathematics 

Use Sage for studying calculus, elementary to very advanced number theory, cryptography, commutative algebra, group theory, graph theory, numerical and exact linear algebra, and more. 

Use an Open Source Alternative 

By using Sage you help to support a viable open source alternative to Magma, Maple, Mathematica, and MATLAB. Sage includes many high-quality open source math packages. 

Use Most Mathematics Software from Within Sage 

Sage makes it easy for you to use most mathematics software together. Sage includes GAP, GP/PARI, Maxima, and Singular, and dozens of other open packages. 

Use a Mainstream Programming Language 

You work with Sage using the highly regarded scripting language Python. You can write programs that combine serious mathematics with anything else.

Of course, I'll have to master Python if I want to create any serious worksheets. At least it's set up now on my laptop so it's there is I want to play with it. 

2017: A New Year

As the new year begins, it seems appropriate to look at the mathematical character of the number that will identify it: 2017. As a start, this number is prime, in fact the 306th prime. It's nearest neighbours are 2011 and 2027. Thus there will not be another prime year for a decade. Working through the Online Encyclopaedia of Integer Sequences (OEIS), I was made aware initially that the number was linked to one circle and 23 ellipses, each with the major axis equal to the diameter of the circle and major/minor axes coincident with the x and y axes.

The equation all twenty four shapes have in common is \(ax^2+bxy+ cy^2 = 2017\).

In the case of the circle, \(a=1, b=0, c=1\) and so \(x^2 + y^2 = 2017\). The integer solutions are \(x=9\) and \(y=44\). This is the point A on the circle c in the diagram below.

For the ellipses, \(a=1, b=0\) and so \(x^2 + cy^2 = 2017\). The values of \(c\) for which there are integer solutions to \(x\) and \(y\) are 2, 3, 7, 14, 21, 24, 27, 31, 33, 42, 46, 56, 66, 81, 84, 87, 88, 93, 112, 232, 253, 462 and 1848. I've plotted the cases of \(c\)=2, 3, 7 and 14 in the diagram below. The associated points are (37, 18), (17, 24), (15, 16) and (1, 12).

 

For the equation \(ax^2+bxy+ cy^2 = 2017\), when \(a\) is not equal to 1 but \(b\) is still 0, the following \((a, b, c)\) values give integer solutions to \(x\) and \(y\): (4, 0, 9); (2, 0, 41); (5, 0, 17); (2, 0, 65); (2, 0, 95); (8, 0, 65). All these ellipses lie inside the circle \(x^2+y^2=2107\) and have been graphed below (with part of the surrounding circle visible):

If \(xy\) terms are allowed, then there is another whole series of ellipses of the form \(ax^2+bxy+cy^2 = 2017\) where the following values of (a, b, c) yield integer solutions for x and y: (1, 1, 1); (9, 6, 1849); (1, 26, 1); (1, 8, -8); (1, 10, 1); (2, 1, 3); (1, 1, 8); (1, 1, 22); (4, 1, 4); (4, -1, 4); (4, -3, 5); (2, -1, 10); (2, 1, 12); (2, -1, 12); (1, 20, 1); (1, 31, 1); (8, 8, 97); (37, 4, 37); (28, 12, 57); (57, 18, 193). A few of these I've plotted below (with the circle included for comparison):

2017 can also be written as the sum of three (not distinct) cubes, namely \(7^3+7^3+11^3\). Thus it seems that 2107 is unusual in that it can be linked in 2-D space to a large numbers of ellipses. Of course, ellipses are the orbits followed by celestial bodies within the solar system trapped by the gravitational forces of larger bodies such as the Sun and planets.

Consider one of the ellipses above, say \(4x^2+9y^2=2017\) and a general point \((x, y)\) situated on it. The only integer values of \(x\) and \(y\) that satisfy this equation are (±22, ±3) as shown below: