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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 
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. 
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 Lsystem 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) 
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 nsystem of the sets starting from the odd numbers going anticlockwise seems to be 2^(2n), (2^(2n1), 2^(2n2)…. 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 
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 
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. 
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 NOTfunction: 
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: 

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