“All things come out of the one, and the one out of all things”
Herakleitos (535 – 475BC)
Herakleitos (535 – 475BC)
Doubling
An example of one of many number patterns within the Jia Xian's triangle is the numbers across the triangle add up to the numbers that double as shown below and is the same sequence as revealed in the Enneagram. See the Enneagram diagram below from the website A New Understanding.
1 = 1
1 + 1 = 2
1 + 2 +1 = 4
1 + 3 + 3 + 1 = 8
1 + 4 + 6 + 4 + 1 = 16
1 + 5 +10 +10+ 5 + 1 = 32
1 + 6 +15+20+15+ 6 + 1 = 64
An example of one of many number patterns within the Jia Xian's triangle is the numbers across the triangle add up to the numbers that double as shown below and is the same sequence as revealed in the Enneagram. See the Enneagram diagram below from the website A New Understanding.
1 = 1
1 + 1 = 2
1 + 2 +1 = 4
1 + 3 + 3 + 1 = 8
1 + 4 + 6 + 4 + 1 = 16
1 + 5 +10 +10+ 5 + 1 = 32
1 + 6 +15+20+15+ 6 + 1 = 64
Sierpinski Fractals
A fascinating, but until recently, hidden fractal pattern emerges when ∑ addition is applied to the Jai Xian triangle. This is very similar to the Sierpinski Fractal pattern that is already well know using odd numbers. (See both fractal patterns below for comparison).
Notice all the numbers become 9 on the 27th row and column and 2 + 7 = 9 . Beyond the 27th row and column the fractal pattern repeats itself but doubles just as the Sierpinski Fractal pattern does.
Note: the lower triangle is lying on its side.
Below is an excellent ed.ted.com talk on Jia Xian triangle that also includes Sierpinski Fractal Triangle and doubling.
FIBONACCI
Fibonacci, or more correctly Leonardo da Pisa, was born in Pisa in 1175AD and introduced the Latinspeaking world to the decimal number (accounting) system from the Arabs. Fibonacci is best known for a simple series of numbers, introduced in Liber abaci and later named the Fibonacci numbers in his honour.
Fibonacci Sequence
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987………..
One of the interesting aspect of the Fibonacci sequence numbers is their direct relationship with Jia Xian triangle as they are generated by adding the diagonal numbers in the triangle as shown below:
Fibonacci, or more correctly Leonardo da Pisa, was born in Pisa in 1175AD and introduced the Latinspeaking world to the decimal number (accounting) system from the Arabs. Fibonacci is best known for a simple series of numbers, introduced in Liber abaci and later named the Fibonacci numbers in his honour.
Fibonacci Sequence
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987………..
One of the interesting aspect of the Fibonacci sequence numbers is their direct relationship with Jia Xian triangle as they are generated by adding the diagonal numbers in the triangle as shown below:
By applying sigma summation to the Fibonacci numbers and placing then in 12 pairs to create a cycle of 24 the paired numbers when added all become 9 !! See the chart below and also the 2nd video below.
HARMONIC FREQUENCIES
Another fascinating relationship Jia Xian triangle has is with harmonic frequencies in particular music. By taking the doubled numbers that were generated from the triangle and multiplying these by 3 and repeating this it generates a series of numbers that are the frequencies of harmonics within music and standing wave patterns. Part of this sequence is shown below showing the number trebling across the table and doubling downwards. The notes that are generated increase by an octave downwards and continue across through all the notes eventually returning to C. The numbers that are generated will reoccur often as we explore other aspects of mathematics namely spherical geometry.
C G D A E B .....
1 = 1 x 3 = 3 9 27 81 243 .....
1 + 1 = 2 x 3 = 6 18 54 162 486.....
1 + 2 +1 = 4 x 3 = 12 36 108 324 973.....
1 + 3 + 3 + 1 = 8 x 3 = 24 72 216 648 1944.....
1 +4 + 6 + 4 + 1 = 16 x 3 = 48 144 432 1296 3888.....
1 + 5 + 10 + 10 + 5 + 1 = 32 x 3 = 96 288 864 2592 7776.....
1 + 6 + 15 + 20 + 15 +6 + 1 = 64 x 3 = 192 576 1728 5184 15552 .....
When the Harmonic Frequency numbers are converted to sigma numbers they change the 1st column into a series of 6 repeating numbers and the 2nd column numbers into a oscillating series of 3’s and 6’s, and then in all the other columns the numbers become 9,s!! We saw this phenomena when we converted the Jia Xian triangle numbers, they also ∑ to 9!! Something interesting appears to be going on here.
1 = 1 3 9 9 9 9
1 + 1 = 2 6 9 9 9 9
1 + 2 +1 = 4 3 9 9 9 9
1 + 3 + 3 + 1 = 8 6 9 9 9 9
1 + 4 + 6 + 4 + 1 = 7 3 9 9 9 9
1 + 5+10+10+ 5 + 1 = 5 6 9 9 9 9
1 + 6 + 15+20+15+6 + 1 = 1 3 9 9 9 9
SOLFEGGIO FREQUENCIES
The Sofeggio harmonics frequencies also abound with nines. There are nine of them and the cycles per second for each of the nine frequencies ∑ to 3, 6 or 9 and when all added together ∑ to 9. The numbers within each set of 3 numbers also cycle 1 through to 9 starting at the one in each of the following numbers 174, 741 and 417. Also there are 3 sets of numbers that share the same numbers but in a different order. Another example is 852, 285 and 528. They form equilateral triangles with each other. See the diagram below.
Another fascinating relationship Jia Xian triangle has is with harmonic frequencies in particular music. By taking the doubled numbers that were generated from the triangle and multiplying these by 3 and repeating this it generates a series of numbers that are the frequencies of harmonics within music and standing wave patterns. Part of this sequence is shown below showing the number trebling across the table and doubling downwards. The notes that are generated increase by an octave downwards and continue across through all the notes eventually returning to C. The numbers that are generated will reoccur often as we explore other aspects of mathematics namely spherical geometry.
C G D A E B .....
1 = 1 x 3 = 3 9 27 81 243 .....
1 + 1 = 2 x 3 = 6 18 54 162 486.....
1 + 2 +1 = 4 x 3 = 12 36 108 324 973.....
1 + 3 + 3 + 1 = 8 x 3 = 24 72 216 648 1944.....
1 +4 + 6 + 4 + 1 = 16 x 3 = 48 144 432 1296 3888.....
1 + 5 + 10 + 10 + 5 + 1 = 32 x 3 = 96 288 864 2592 7776.....
1 + 6 + 15 + 20 + 15 +6 + 1 = 64 x 3 = 192 576 1728 5184 15552 .....
When the Harmonic Frequency numbers are converted to sigma numbers they change the 1st column into a series of 6 repeating numbers and the 2nd column numbers into a oscillating series of 3’s and 6’s, and then in all the other columns the numbers become 9,s!! We saw this phenomena when we converted the Jia Xian triangle numbers, they also ∑ to 9!! Something interesting appears to be going on here.
1 = 1 3 9 9 9 9
1 + 1 = 2 6 9 9 9 9
1 + 2 +1 = 4 3 9 9 9 9
1 + 3 + 3 + 1 = 8 6 9 9 9 9
1 + 4 + 6 + 4 + 1 = 7 3 9 9 9 9
1 + 5+10+10+ 5 + 1 = 5 6 9 9 9 9
1 + 6 + 15+20+15+6 + 1 = 1 3 9 9 9 9
SOLFEGGIO FREQUENCIES
The Sofeggio harmonics frequencies also abound with nines. There are nine of them and the cycles per second for each of the nine frequencies ∑ to 3, 6 or 9 and when all added together ∑ to 9. The numbers within each set of 3 numbers also cycle 1 through to 9 starting at the one in each of the following numbers 174, 741 and 417. Also there are 3 sets of numbers that share the same numbers but in a different order. Another example is 852, 285 and 528. They form equilateral triangles with each other. See the diagram below.
PRIMES
In turn the prime numbers are associated with a Harmonic Number by ± 1. But not all Harmonic Numbers are associated with a Prime Numbers. Why is this? It's all based on the formula Primes = 6n ± 1 which was discovered by Leibniz a famous German mathematician . We will come across him again shortly.
All Harmonic Numbers greater than 6 are divisible by 6. Why?
Because: 1 + 2 + 3 = 6
1 x 2 x 3 = 6
Prime numbers, are also related to the cycle of 24!! The primes are in blue and in the second table are shown in their sigma numbers. Note the primes that have been converted to their sigma numbers cluster either side of the lines which sigma to 3's, 6's or 9's shown in red!!
SPHERICAL GEOMETRY
The sigma's of numbers within spherical geometry both 2D and 3D are intimately related to 9. This is due to the use of 360 degrees in a circle, which of course sigma's to 9. So a deeper question is why do we divide circles into 360 degrees. Below is an excellent video that summarizes 3 possible reasons for this or maybe a combination of them all. But there may be an answer closer to home. More on that later. The summation to nine by the angles generated by halving the angles within a circle are seen in the example below and is also covered in the video Number 9 code. This halving is of course the opposite of doubling which we have already considered.
The sigma's of numbers within spherical geometry both 2D and 3D are intimately related to 9. This is due to the use of 360 degrees in a circle, which of course sigma's to 9. So a deeper question is why do we divide circles into 360 degrees. Below is an excellent video that summarizes 3 possible reasons for this or maybe a combination of them all. But there may be an answer closer to home. More on that later. The summation to nine by the angles generated by halving the angles within a circle are seen in the example below and is also covered in the video Number 9 code. This halving is of course the opposite of doubling which we have already considered.
Before we move on below are two excellent video presentations one is on why 360 degrees and the other on aspects of the Number 9 code some of which are not covered in this study.


Time
As summarized in a section in the video above time is also related to 360 and therefore 9 since the Earth rotates through 360 degrees in a day which equals 24 hours ( ∑'s to 6)
In a day there are:
1440 (= 9) minutes
8640 (= 9) Seconds
In a week there are:
10,080 (= 9) minutes
In a year there are:
525,600 (= 9) minutes
We discovered earlier that the cycle of 24 is a significant aspect of the Sigma code revealing the underlying patterns we consider in the Fibonacci sequence and prime numbers. Here it is again much closer to home.
Internal Angles of Regular Polygons
The internal angles of polygons have the same property of the circles 360 degrees and all sigma to 9 as shown below.
Also a curious aspect of these internal angle numbers is when the angles of paired polygons are added they all add to 1080 which ∑ to 9.
As summarized in a section in the video above time is also related to 360 and therefore 9 since the Earth rotates through 360 degrees in a day which equals 24 hours ( ∑'s to 6)
In a day there are:
1440 (= 9) minutes
8640 (= 9) Seconds
In a week there are:
10,080 (= 9) minutes
In a year there are:
525,600 (= 9) minutes
We discovered earlier that the cycle of 24 is a significant aspect of the Sigma code revealing the underlying patterns we consider in the Fibonacci sequence and prime numbers. Here it is again much closer to home.
Internal Angles of Regular Polygons
The internal angles of polygons have the same property of the circles 360 degrees and all sigma to 9 as shown below.
Also a curious aspect of these internal angle numbers is when the angles of paired polygons are added they all add to 1080 which ∑ to 9.
If we look at the internal of angles within polygons we fine the same affinity toward the summation to 9. An example of this is the pentagram shown with its internal angle of 36, 72 and 108. 36 +72 + 72 = 180
Note: That the angles in the pentagram above all relate to the Harmonic Frequency numbers which we looked at early. See these highlighted in red below. This relations of harmonic frequencies and angles continues when we look at the Platonic Solids.
C G D A E B .....
1 = 1 x 3 = 3 9 27 81 243 .....
1 + 1 = 2 x 3 = 6 18 54 162 486.....
1 + 2 +1 = 4 x 3 = 12 36 108 324 973.....
1 + 3 + 3 + 1 = 8 x 3 = 24 72 216 648 1944.....
1+4 +6 + 4 + 1 = 16 x 3 = 48 144 432 1296 3888.....
1 + 5 + 10 + 10 + 5 + 1 = 32 x 3 = 96 288 864 2592 7776.....
1 + 6 + 15 + 20 + 15 +6 + 1 = 64 x 3 = 192 576 1728 5184 15552 .....
C G D A E B .....
1 = 1 x 3 = 3 9 27 81 243 .....
1 + 1 = 2 x 3 = 6 18 54 162 486.....
1 + 2 +1 = 4 x 3 = 12 36 108 324 973.....
1 + 3 + 3 + 1 = 8 x 3 = 24 72 216 648 1944.....
1+4 +6 + 4 + 1 = 16 x 3 = 48 144 432 1296 3888.....
1 + 5 + 10 + 10 + 5 + 1 = 32 x 3 = 96 288 864 2592 7776.....
1 + 6 + 15 + 20 + 15 +6 + 1 = 64 x 3 = 192 576 1728 5184 15552 .....
Platonic and Archimedean Geometry
The following table show how the internal angles and the sum of their faces of the Platonic solids. All of these ∑ to 9!!
The following table show how the internal angles and the sum of their faces of the Platonic solids. All of these ∑ to 9!!
A further example of angles, this time the Sum of the Face Angle in the table above, relating to the Harmonic Frequency numbers is highlighted in red below. This time with a factor of 10.
C G D A E B .....
1 = 1 x 3 = 3 9 27 81 243 .....
1 + 1 = 2 x 3 = 6 18 54 162 486.....
1 + 2 +1 = 4 x 3 = 12 36 108 324 973.....
1 + 3 + 3 + 1 = 8 x 3 = 24 72 216 648 1944.....
1 +4 +6 + 4 + 1 = 16 x 3 = 48 144 432 1296 3888.....
1 + 5 + 10 + 10 + 5 + 1 = 32 x 3 = 96 288 864 2592 7776.....
1 + 6 + 15 + 20 + 15 +6 + 1 = 64 x 3 = 192 576 1728 5184 15552 .....
The 5 Platonic and the 13 Archimedean* solids are all derived from the tetrahedron. In the Number 9 Code video above on Fibonacci tetrahedral numbers were shown being derived from the numbers in Jia Xian triangle .
See the relationship to the tetrahedral shown below
*Note that 5 + 13 = 18 = 9
5 &13 are both Fibonacci and Prime numbers!
C G D A E B .....
1 = 1 x 3 = 3 9 27 81 243 .....
1 + 1 = 2 x 3 = 6 18 54 162 486.....
1 + 2 +1 = 4 x 3 = 12 36 108 324 973.....
1 + 3 + 3 + 1 = 8 x 3 = 24 72 216 648 1944.....
1 +4 +6 + 4 + 1 = 16 x 3 = 48 144 432 1296 3888.....
1 + 5 + 10 + 10 + 5 + 1 = 32 x 3 = 96 288 864 2592 7776.....
1 + 6 + 15 + 20 + 15 +6 + 1 = 64 x 3 = 192 576 1728 5184 15552 .....
The 5 Platonic and the 13 Archimedean* solids are all derived from the tetrahedron. In the Number 9 Code video above on Fibonacci tetrahedral numbers were shown being derived from the numbers in Jia Xian triangle .
See the relationship to the tetrahedral shown below
*Note that 5 + 13 = 18 = 9
5 &13 are both Fibonacci and Prime numbers!
Catalan Solids
In mathematics, a Catalan Solid or Archimedean duals are dual polyhedron to an Archimedean solid. The Catalan solids are named for the Belgian mathematician, Eugène Catalan, who first defined them in 1865.
Note: all the sigma numbers of the Archimedean edge ( E ) and vertices (C) numbers and the sigma’s of all the Catalan face (F)and edge (E) numbers are either 3, 6 or a 9 !