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June 18th, 2019, 08:37 AM  #1 
Newbie Joined: Jun 2019 From: Sweden Posts: 10 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: 10 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: 10 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: 10 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: 10 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: 10 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: 10 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: 10 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: 
July 21st, 2019, 05:34 AM  #9 
Senior Member Joined: Aug 2008 From: Blacksburg VA USA Posts: 354 Thanks: 7 Math Focus: primes of course 
This looks interesting, unfortunately I personally have no time at the moment as I am in the midst of a renovation and move. A bit surprised no one has commented ...

July 21st, 2019, 08:38 PM  #10 
Newbie Joined: Jun 2019 From: Sweden Posts: 10 Thanks: 0 
@billymac00 Thanks for your comment! Hope you will get some spare time to check it out! And good luck with your renovation! Perhaps no one has commented because it's an odd way of looking at numbers. Normally you line them up on the number line and perhaps this circle approach, expanding the Thue Morse sequence, seems a bit awkward. But I do think the generated symmetries and the relations between the numbers are quite intriguing and I would very much appropriate if someone cared to take a look. But I also understand that people got other things to do and/or are into some other areas of math or thinks this is to odd. Generating the ring system is quite simple by expanding the Morse Thue sequence starting from the inner circle filling neighboring cells clockwise with 0 and a 1 by applying the Lsystem (0 → 01), (1 → 10) as shown in the animation below The inner circle represent 2^0, the first ring 2^1, the next ring 2^2 and so on. Binary radial combinations can then be represented by the combinations of ones and zeros generating the numbers 0 to 31 in binary in the example above. At first the numbers might seem to be rather randomly displaced but, as shown above, there are some very interesting patterns and symmetries hiding in the way the natural numbers are arranged. Last edited by Tudde; July 21st, 2019 at 08:40 PM. 

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