Definitions. It’s all about definitions. I prefer to *discover a* *definition*.

Lets go!

- Assume that center of symmetry is angle of 90 degrees.
- Convert angle from degree to radian and calculate:
- sin(90)
- cos(90)
- tan(90)

- If
*90**nine*and*zero*take**two**numbers from:*sin()**–*after decimal point*cos()**–*after decimal point– before decimal point and maybe after decimal point – i.e. when degree will be build of higher amount of numbers than amount numbers before decimal point*tan()*

- If number will be build of three numbers i.e.
you take three numbers*911* - Sum all taken numbers.
- The symmetry means the sum of numbers will be perfomed on degrees:
and*89*and that sums will be equal. The length of symmetry will be equal*91*– one number before and after 90 degrees.*1* - When the sums will be equal at numbers
and*88**92***2**– two numbers before and after 90 degrees.

We will use some C++ code to show it.

Definitions:

const double PI = std::acos(-1); double degreeToRadian(double degree) { return degree * PI / 180.0; } double radianToDegree(double radians) { return radians * 180 / PI; } double _sin(double x) { return std::sin(degreeToRadian(x)); } double _cos(double x) { return std::cos(degreeToRadian(x)); } double _tan(double x) { return std::tan(degreeToRadian(x)); }

Now we can calculate 89 and 91 degress:

int main() { std::cout.precision(20); std::cout << "sin(89) = " << _sin(89) << std::endl; std::cout << "cos(89) = " << _cos(89) << std::endl; std::cout << "tan(89) = " << _tan(89) << std::endl; std::cout << "sin(91) = " << _sin(91) << std::endl; std::cout << "cos(91) = " << _cos(91) << std::endl; std::cout << "tan(91) = " << _tan(91) << std::endl; return 0; }

Result will be:

`sin(89) = 0.`

`99`

`984769515639126958 cos(89) = 0.`

`01`

`7452406437283376345 tan(89) =`

`57`

`.289961630759876243 sin(91) = 0.`

`99`

`984769515639126958 cos(91) = -0.`

`01`

`7452406437283476959 tan(91) = -`

`57`

`.289961630759549394`

So what now? We can see:

sin(89) = sin(91)

abs(cos(89)) = abs(cos(91))

abs(tan(89)) = abs(tan(91))

And as we define above we take numbers from trigonometry functions from degree ** 89** and degree

**.**

*91*9 + 9 + 0 + 1 + 5 + 7 =

**31**

9 + 9 + 0 + 1 + 5 + 7 =

**31**

Symmetry exists because two sums are equal.

When the symmetry is calculated between multipliers of 90 degrees then symmetry should exists.

But what if I tell you the secret if the symmetry exists when the *center* is available on 102 degress?

Let us check it.

int main() { std::cout.precision(20); std::cout << "sin(101) = " << _sin(101) << std::endl; std::cout << "cos(101) = " << _cos(101) << std::endl; std::cout << "tan(101) = " << _tan(101) << std::endl; std::cout << "sin(103) = " << _sin(103) << std::endl; std::cout << "cos(103) = " << _cos(103) << std::endl; std::cout << "tan(103) = " << _tan(103) << std::endl; return 0; }

You will get:

sin(101) = 0.98162718344766397571 cos(101) = -0.19080899537654480436 tan(101) = -5.1445540159703107008 sin(103) = 0.97437006478523524589 cos(103) = -0.22495105434386480914 tan(103) = -4.3314758742841590333

You will find:

sin(101) != sin(103)

cos(101) != cos(103)

tan(101) != tan(101)

So where is that symmetry?

As we are following the previous definitions:

- from
*sin(101)*take: “9”, “8”, “1”:**after**decimal separator - from
*cos(101)*take: “1”, “9”, “0”:**after**decimal separator - from
*tan(101)*take “5”, “1”, “4”:**before**decimal separator and after

and

- from
*sin(103)*take: “9”, “7”, “4”:**after**decimal separator

from*cos(103)*take: “2”, “2”, “4”:**after**decimal separator

from*tan(103)*take “4”, “3”, “3”:**before**decimal separator and after

From *_numbers(101)* you will build a sum:

9 + 8 + 1 +

1 + 9 + 0 +

5 + 1 + 4 = **38**

And from *_numbers(103) *you will build a sum:

9 + 7 + 4 +

2 + 2 + 4 +

4 + 3 + 3 = **38**

So it means:

*sum*(*_numbers(101)*) = *sum*(*_numbers(10*3*)***)**

So it is fact:

When angle number is build of 4 numbers (i.e. “1337”) or any other quantity of numbers – using the same amount of numbers to calculate symmetry from *sum(_numbers(x))* will create real symmetry results!

Also we can observe that symmetries are not only at multipliers of 90 degress but at other degree values with different symmetry lengths.

Pretty clever, huh?

I am open for suggestions how it can be possible.

At incoming **Part 3.** I will introduce c++ code and result text files for larger range of numbers.

Code for this part can be found at Github.