1

Are these nodes related to graph theory? Or they can provide us an idea to use the graph theory or network theory as a tool to unravel the secrets of nature? We will show some methodologies that we applied to generate new binary sequences by using these nodes. We are then able to formulate a new mathematical structure from these nodes. We believe that this structure will provide a new Boolean Simplification method which also be able to applyto DNA research and give us a glimpse to the origin of life.

‘Luo Pan’ or Chinese compass

2

Now, let us follow a methodology (3 steps) and we will see some amazing properties of ‘Luo Pan’. Let us startfrom a single node which we named symbol number 1. Later on, we noticed that we can use tree diagram in discrete mathematics to extract the properties of these nodes.

Step 1

Step 2

Step 3

Symbol number 1

Symbol number 2

3

Symbol number 2

Symbol number 3

Step 1

Step 2

Step 3

4

Symbol number 3

Symbol number 4

Step 1Step 2

Step 3

5Symbol number 4

Symbol number 5

We assume by now you will know how to proceed without showing the steps.

6

Symbol number 5

Symbol number 6

7Symbol number 6

Symbol number 7

8

Symbol number 7

Symbol number 8

9As we can see the patterns will return to the Single node/symbol number 1. Signifying that there is a periodicity.

Symbol number 8

Symbol number 1

10

Summarizing the sequences into circularform.

Symbol number 1

Symbol number 2

Symbol number 3

Symbol number 4

Symbol number 5

Symbol number 6

Symbol number 7

Symbol number 8

11

We can see that the nodes will keep on increasing. Showing a sign of production of materials.

12

The nodes started to merged. Showing a tendency of simplification.

13

Further merging of nodes.

14

2 into 1 ?

15

Further simplificationwill return to ‘1’ !

16

The nodes in these symbols are only allowed to colour as either white or black.

Let us employ discrete mathematics to make an analytical analysis and see what we can find.

Let us start with the symbol number 1as shown below.

or

Symbol number 1

We will get either white or black. 2 possibilities

17

We suspect that this is the ‘1’ that mentioned by ‘Lao Tze’. A famous ancient Chinese philosopher.

18

Can we define in this manner?

19

Next, we go to symbol number 2. There 4 possibilities as shown below.

or

Symbol number 2

or or

Can we define in this manner?

20

Next, we go to symbol number 3.

Can be colour as white or black

Can be colour as white or blackCan be colour as white or black

To find out the possibilities of the symbol number 3 , we realized that we can change the zero and one in the binary sequence of 000, 001, 010,011, 100, 101, 110 and 111 to black and white. As shown.

21This binary sequence can be used to represent the 23 = 8 possibilities

of the symbol number 3.

22

We found that these combinations are Thue-Morse sequence. We defined black as 0 and white as 1.

They are 00, 01, 11, 10.

AA’B B’BB’

23

Next, we can see that these combinations are binary sequence. Again, we defined black as 0 and white as 1.

They are 00, 01, 10, 11 .

A A’B B’ B B’

24

We can see that fractal Tai Ji diagram can be related to Thue-Morse sequence.

25

Binary sequence will be generated if we just rotate the small Tai Ji symbol in the right side (dark side) of the bigger Tai Ji symbol.

26

Next, we come to symbol number 4. We redrawn the nodes and put them into a triangular formfor clarity purpose.

If every node can only be coloured as black or white. How many possibilities are there for this symbol?

27

1 1 1 1 1 1

Symbol number 4 Symbol number 4without lines connected.

We will colour the nodes with either black or white in clock-wise direction.

We have straighten the nodesso that we can relate them to binary sequence which will help usin colouring.

28

0 0 0 0 0 0

Symbol number 4 Symbol number 4without lines connected.

We will colour the nodes with either black or white in clock-wise direction.

We have straighten the nodesso that we can relate them to binary sequence which will help usin colouring.

29

0 0 0 0 0 00 0 0 0 0 10 0 0 0 1 00 0 0 0 1 10 0 0 1 0 00 0 0 1 0 10 0 0 1 1 00 0 0 1 1 10 0 1 0 0 00 0 1 0 0 10 0 1 0 1 00 0 1 0 1 10 0 1 1 0 0

0 0 1 1 0 10 0 1 1 1 00 0 1 1 1 1

30

0 1 0 0 0 00 1 0 0 0 10 1 0 0 1 00 1 0 0 1 10 1 0 1 0 00 1 0 1 0 10 1 0 1 1 00 1 0 1 1 10 1 1 0 0 0

1 1 1 1 1 1

There will be 26 = 64 possibilities.

31

32

33

Extracted from Page 24 of the book “A new kind of Science” by Professor Stephen Wolfram, also the Chief Executive Officer of “MATHEMATICA” software company.

If Professor Stephen Wolfram was able to simulate a lot of living

and non-living beings in nature by using his cellular automaton, we believe that we will also able to make it and probably other discoveries will also be made through using the combinations of the symbol number 1 until number 8.

34

Reference to the page 25 of “A new kind of science”

Professor Stephen Wolfram’scellular automaton and his simulated results.

35

Professor Stephen Wolfram’scellular automaton and his simulated results.

36

Professor Stephen Wolfram’scellular automaton and his simulated results.

37

We have found out these two symbols will give us a completeness new sequence by using our methodologies which in turn can be proven to be lead to Thue-Morse sequence through a simple rule. Please refer to the next slides.