My Math Forum  

Go Back   My Math Forum > College Math Forum > Number Theory

Number Theory Number Theory Math Forum


Reply
 
LinkBack Thread Tools Display Modes
June 18th, 2019, 08:37 AM   #1
Newbie
 
Joined: Jun 2019
From: Sweden

Posts: 8
Thanks: 0

Symmetries when expanding Thue Morse sequence in layers of rings

By generating Thue Morses sequence in rings and study the natural numbers N (including 0) represented by radial binary combinations some geometrical properties emerges, such as:

* All odd integers will be arranged in a specific geometric order
* Even integers will be arranged in a specific geometric order

* The one complement will allays be an opposite radial combination where the two radial combinations together constitute a diagonal.

* The two complement for a radial combination can be found by symmetry.


The presentation got a lot of graphics so I can´t post it here but I put it up on my blog if someone cares to take a look.

Link:

Some symmetrical properties constructing the natural numbers using Thue Morse sequence
Tudde is offline  
 
June 19th, 2019, 01:49 AM   #2
Newbie
 
Joined: Jun 2019
From: Sweden

Posts: 8
Thanks: 0

Example of symmetris

An example of symmetries regarding odd numbers in a n=4 ring system (or universe - in the terms used in set theory). The two complement to an odd number represented by a binary radial combination will always be perpendicular.




Sets of even numbers twos complement got similar symmetric proprieties.
Tudde is offline  
June 19th, 2019, 08:10 PM   #3
Newbie
 
Joined: Jun 2019
From: Sweden

Posts: 8
Thanks: 0

Finding the two's complement for even integers

The two´s complement for the set of even integers represented by binary radial combinations in a n=4 ring system colored light blue (the sets are more formally defined in the presentation - see link above) can be found by rotating Pi/2^2=Pi/4 within the set - illustrated by the animation below.



The two´s complement for the set of even integers represented by binary radial combinations colored pink can be found by rotating Pi/2^3=Pi/8 within the set - illustrated by the animation below:



The two´s complement for the set of even integers represented by binary radial combinations colored yellow can be found by rotating Pi/2^4=Pi/16 , that is to say Pi/2*n within the set - illustrated below:



The binary radial combination represented by orange color only consist of one element and will have it self as twos complement

And lastly - the twos complement for the number 0 will result in a carry outside the system – sort of hitting the infinity wall for this universe and a loophole to the next universe (from n=4 to n=5 in the case above).


Above symmetries can be found in ring system n=0 ..... n=5 and can probably proven for all n by induction by generating the Thue Morse sequence in a L-system similar to the way that it can be proven that a given diagonal will consist of a radial combination and it's one complement (see the presentation)
Tudde is offline  
June 27th, 2019, 11:46 PM   #4
Newbie
 
Joined: Jun 2019
From: Sweden

Posts: 8
Thanks: 0

If we sum the radial combinations in each set defined as above in a system where n=4 we see that starting from the set of odd numbers going anticlockwise the sums adds up to 256, 128, 64, 32, 16 and 0 – see picture below. The general patterns for the sums in a n-system of the sets starting from the odd numbers going anticlockwise seems to be 2^(2n), (2^(2n-1), 2^(2n-2)…. 2^n and lastly 0 (the zero can alternatively be ignored if we exclude it from N). This can probably be proved as a general rule for all n by induction since its true for n=0 to n=5

Tudde is offline  
June 29th, 2019, 02:15 AM   #5
Newbie
 
Joined: Jun 2019
From: Sweden

Posts: 8
Thanks: 0

The sets of radial combinations also shows how “close” they are to be an odd number as shown below for a system where n=4. It can probably also be proven for an arbitrary n by some sort of induction proof since it’s s true for n=0 to n=5
Tudde is offline  
July 3rd, 2019, 01:13 AM   #6
Newbie
 
Joined: Jun 2019
From: Sweden

Posts: 8
Thanks: 0

The system can also be used to do arithmetic. As in the animated example below where 9 is added to 12 giving the result 21. Subtraction can be done by finding the two complements according to the rules established above.
Tudde is offline  
July 5th, 2019, 12:29 AM   #7
Newbie
 
Joined: Jun 2019
From: Sweden

Posts: 8
Thanks: 0

The system can also be used as an alternative way to express boolean truth tables. The radial combinations marked red represent the truthtable for the NOT-function:

Tudde is offline  
July 17th, 2019, 05:47 PM   #8
Newbie
 
Joined: Jun 2019
From: Sweden

Posts: 8
Thanks: 0

The diagonal will consist of two radial combinations that are the ones complement to each other. This is true for all radial combinations in a n=5 ring system and can be proved for all n ring systems by induction, as show below:

Tudde is offline  
Reply

  My Math Forum > College Math Forum > Number Theory

Tags
expanding, layers, morse, rings, sequence, symmetries, thue



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
question regarding the symmetries of the triangle goodfeeling Abstract Algebra 4 November 15th, 2012 10:30 PM
Linear symmetries of the pentagon. Math4dummy Algebra 1 February 5th, 2012 05:44 PM
errata for Morse & Feshbach - Methods of Theoretical Physics becko Physics 8 June 5th, 2011 06:36 AM
Half symmetries jk22 Algebra 6 August 14th, 2010 07:27 AM
Expanding Sequence A128335 Infinity Computer Science 15 July 5th, 2007 06:33 AM





Copyright © 2019 My Math Forum. All rights reserved.