pae's more research

I tried to researach more on relationships between numbers and shapes, geometry, and there are!!

Pythagoreans were the first kind that discovered the relationship which they adored numbers, but back then they knew only just positive whole numbers
In Pythagoreans number and symbolism method, they classified numbers into 2 types: masculine and feminine. In which odd numbers are masculine and evens are feminie. The reason why odds are masculine because they believed that odd numbers are stronger than evens. Here are why:

- when divided, even numbers have nothing in the center (unlike odd)

- odd+even = odd

- 2 evens can never produce an odd, while 2 odds can produce aneven

Also the Pythagoreans especially liked number 10 which they introduced Tetractys, a set of four things

Gnomons (related to previous project, maybe?)

For Pythagoreans, gnomons were the odd integers/ the masculine numbers. BY starting with the monad, 1, the sum of it and any consecutive number of gnomonss is a square number

1 + 3 = 4

1 + 3 + 5 = 9

1 + 3 + 5 + 7 = 16

Pythagoreans famous example of connection between numbers and shapes is Pythagora's theorem, where right triangle is created due to ralation of adjacent sides to their hypotenuse.

a squared*b squared = c squared. Also they discovered a method of finding all the triples of whole numbers which could form the sides of such right triangles known as Pythagorean triads or triples

Later, they discovered that whole numbers and their ratios semmed mysteriously connected to ' pleasing intervals of sound' from a vibrating string. Within these ratios. 1/2, 3/4, 2/3, etc, they can produce pleasing resonaces of chords that are appealing to human ear because they seem to be the fundemental importance of sound. By using these ratios, perfect chords, pitches, and notes of sound are created.

They believed that what is pleasing to the ear is also pleasing to the eye and the mind as well.

Due to all grammatical or logical interconnection of these unexpected discoveries, Pythagoreans and his folloowers infer that ' All is number'

The invention of number was simply because of a need to keep account of sheep, leading to further kind of numbers which we called mathematics. Because of mathematics and numbers, they allowed people to explore new horizons: to fly, to walk on the moon, to go into deep space, etc. Then some questions aroused from these facts.

If the language of number was a human invention to faciliate human affairs (count sheep), why should this mathematicsl language be one which the Univeres apparently not only understands, but utself speaks?

Example of involvement of numbers in art

The painting concerns a remarkable equation discovered by Swiss mathematical genius, Leonhard Euler. Euler found an unexpected connection between pi and integers from 1 to infinity. It was unexpected because pi is not defined as a number but interm of circles
From Pythagorean theories above, Euler's equation is like a mathematical yinyang sign. It finds a perfect balance between a digital world on the left ( integers) and an undivided continuum on the right (circle's seamless curve)

In the painting, the artist wanted to reflect the thought that Euler's small but powerful equation seemed to conjure up. For instance, a relationship between numbers and circles or geometries then applied to the form of landscape: mountains and sea whcih they in one way opposite but also similar in a way (connected) where mountains are static and sea is dynamic

The artist tried to show that things which appear to be opposite from one point of view can also be connected from another valid point of view. Also representing a symphonic of sea becoming mountains, and the other way around, of sea and mountains precipitating out of togetherness into separate entities.

Why base 10 number is used?

There are many number-base systems were invented such as base2, base12, or base16. Base 2 number uses only 0 and 1 integer to run all the system and mainly use with computer programs. Apparently, base 12 system seems to be very practical because it can be divided by manny numbers while base 10 have less. Also base 12 is somehow used in today's world. Days and nights are measure in 12- hour block with subdivided into 60 minutes and minutes into 60 seconds. All divisible by 12

Therefore, the only main rean we use base 10 number system is because we have 10 fingers
Duedecimal multiplication system

Pythagoreans represented numbers by Pattern of dots. Result of arranging pebbles into patterns

Therefore 9 pebbles can be arranged in 3 row*3 columns

O O O

O O O

O O O

and 10 pebbles can be arranged in 4 rows

O

O O

O O O

O O O O

square numbers can all be subcivided by a diagonal line into 2 triangular numbers, therefore square numbers is always the sum of 2 triangular numbers

Common number patterns

1.Arithmetic sequences --> made by adding same value each time

1, 3, 5, 7, 9, 11 where comon difference is 2

2. Geometric sequences --> made by multiplying same value each time

2, 4, 5, 16, 32 multiplying by 2

Special sequences

1. Triangular numbers --> generated from a pattern of dots which form a triangle

1, 3, 6, 10, 15, 21

2. Squared numbers --> next number is made by squaring by its position number

( example 2nd number is 2 squared)

1, 4, 9, 16, 25, 36

3. Cube numbers --> next number is made by 3 times of its position number

1, 8, 27, 64, 125

4. Fibonacci numbers --> adding the 2 numbers infromt together

0, 1, 1, 2, 3, 5, 8, ...