#can anyone prove why there are 360 degrees in a circle?

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eternal flower
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find all of them

fallow mesa
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this form of addition is actually a primitive form of multiplication

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I would need AI to do that for me

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we are multiplying by adding gnomons

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but I have yet to explain the gnomons because I really don't know that much about it

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the tricky part is that as we add the gnomons they share a unit at the point where the sides meet each other

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that's the tricky part for me

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we should be able to get tese same numbers just by adding even numbered gnomons

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and really the addition of gnomons to either squares of triangles or any other shape is the same or very similar to multiplication

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I would call it primitive multiplication

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that is why I say that the squares are multiples or submultiples of eachother

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same with triangles

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it's just a different mode of mathematical thinking

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AI says that 36 is a perfect number

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if this is true

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i don't trhink it is true

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that's incorrect

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GEEZ

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I hate AI

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this is all I know at this point

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gnomons are byilt by adding consecutive odd numbers?

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1+3=4

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1+3+5=8

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built

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gnomons can also be build by adding all of the consecutive even numbers?

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is 1 an even or odd number?

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or neutral

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1+3+5=9

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ok

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that's correct

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now we are on it

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1+3+5+7=16

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hmmm

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now we are geerating squares

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1+2=3 tri

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1+2+4=7?

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nope

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no good

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got me

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we need a gnomon

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the gnomons are just the natural numbers that I added up previously?

eternal flower
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^

fallow mesa
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any figures that are similar are multiples or submultiples of each other

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that can be should from Euclid

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the numerical measurements are not needed

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this is demonstrated geometrically

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can we calculate the numerical centers of the gnomons as they are augmented?

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can be shown from Euclid

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OK

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so the consecutive odd numbers add up to the squares and the are the gnomons for squares

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the addition of the consecutive natural numbers sum to the triangular numbers

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they are the gnomons for triangles

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what was my point?

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addition preceded multiplication

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and we can detect the first rumblings of the Muliplication Revolution in these gnomons

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which involved nothing but addition

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but the main question agin is How did we get 360 degrees in the circle?

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So since the system was base 60

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i think you would do just as the proponents of a decimal division of the circle would do

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you begin by dividing the circle into 60 degrees

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you use that for a while and replace it or amend it to multiples of 60

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you divide 60

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you come up with 360 degrees

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so's yer mom

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I'm gettung sick of this

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nauseous

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60x6=360

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the number 10 is not involved in this system

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it is important that we can insert equilateral numbers into this set up

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especially triangular numbers

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we can place an equilateral triangle in the circle and call it 36 with a side of 8

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we can call the side of the triangle 60

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or the angle

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we can experiment with a large number of numbers

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in other words it allowed them to utilize their milestone numbers

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you can use triangles like 66 or 666 with this system

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hyou can use any triangular number

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does it not facilitate the use of triangular numbers more than squares?

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then they must have been seeking to triangulate the circle

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using equilateral triangles and triangular numbers

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even in Euclid, I think the first figure that he inscribes in a circle is an equilateral triangle

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this triangle could be called by any triangular number

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the area could be called 3, 6, 10, 15...

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the side would be called by the gnomon

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2, 3, 4, 5...

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we can make different species of ratios from the two series

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3/2, 6/3 10/4, 15/5

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3/2 is the sesquialter ratio

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2 is the subsesquialter of 3

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or the musical 5th

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6/3 is the double or the octave

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10/4 reduces to 5/2

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i don't vknow what that could mean harmonically

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it's the same disjunct ratio that I found in the decimal system earlier

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those are the divisors of 10

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15/5

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reduces to 5/1 or 5

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no

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3/1

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interesting

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what is 3/1?

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that is like an octave and a 5th

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yep

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sure as sheep

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I think 60 would work better

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60/120 is the octave

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60/90 is the 5th

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60/80 is the 4th

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60/75 is a 3rd

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60/72 is a what?

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anyway you can divide the circle into a bunch of consecutive musical intervals

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depending on the definition of consecutive

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we can actually hear these intervals because they are in the range of human hearing

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using base 60 we can get

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2/1 octave

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3/2 5th

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4/3 4th

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5/4 3rd

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6/5 ?

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7/6 _

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9/8 that's the semitone

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there is some dodginess when 7 enters

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but we can build harmonic intervals out of all of ithe divisors of 60

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we can do more with 360

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that would be entering into the harmony of the spheres though because

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they would be beyond the range of sensibility

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we could not hear them

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we could still sense some of them through our tactile senses

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and now we have ultra sensitive instruments that can detect many different kinds of intervals

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doing this math makes one acutely aware of the limitations of the sense apparatuses

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so it's a very important philosophical exercise to do this math

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you do this harmonic exercise sometime

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you'll be all like

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"WHOAAAA I DON"T KNOW JACK!!!"

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I mean I just took us up to 9

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we can get to about 10 or 11 and then we are completely defeated by our limited senses

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that does not mean that the intervals are not there vibrating

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we just cannot detect them by ear

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AI says: "The smallest detectable interval depends on what kind of interval you're referring to. Here are two common examples:
Sound: The smallest difference in pitch a human can hear is called the minimum audible difference (MAD). This typically falls around 5-6 cents, which is a very small increment on the musical scale [smallest pitch difference audible].
Time: Our perception of time is less precise than hearing. The smallest time interval humans can reliably detect is around 10-20 milliseconds https://en.wikipedia.org/?title=Reaction_time&redirect=no.

Mental chronometry

Mental chronometry is the scientific study of processing speed or reaction time on cognitive tasks to infer the content, duration, and temporal sequencing of mental operations. Reaction time (RT; also referred to as "response time") is measured by the elapsed time between stimulus onset and an individual's response on elementary cognitive tasks ...

fallow mesa
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here is yet another way to look at this

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begin with the first triangle 3

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multiply that trianle by the next 6

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= 18

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now here is where the 10 comes in

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multiply 18 by the next triangles in the series of triangles

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18+10=180

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that gives you the semicircle

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you simply double that number to complete the circle of 360

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mostly constructed by triangles in order

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you don't even need 360 if you just pivot the compass

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you can survey the whole sky with that compass

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in sections

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then someone gets the ideas to complete the circle at 360

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so we have a few ideas that can be generated using teir system not ours

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10 does not enter into the calculation as a milestone from the decimal system

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the milestones used to construct the half circle were those in the series of triangular numbers

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3, 6, 10

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10 being used because it is a triangle

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and not being utilized as a borrowed multiplied from the decimal system

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to multipy by decimal 10 is to borrow the sense of the number from the decimal system

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by thinking of it as a triangle we keep to the sexagesimal system

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it would freally help to have a keyboard with all 59 symbols for the first 59 numbers in the base 60 system

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it's like reading chinese though

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but notice that each number is a geometrical shape

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OK forget veverything I said

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looks like there is a triangle in every numb er

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except there is a special symbol for 10

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and multiples of 10

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so I am wrong

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but it was fun

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ignorance is bliss and I loved every minute of it

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this is babylonian

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well, on closer inspection I se that even 10 has a triangle represented in it's shape

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yes, they all seem to depict a triangle in the graphic

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that does not mean that they are all triangular numbers

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but I think the genesis of this system has something to do with them

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triangle

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triangles and pentagrams

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no squares

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there is a triangle in every single number up to 59

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there are also pentagrams or implied pentagrams in each

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OK

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but the pentagram can resolve to a triangle and a square

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I don't know the date of this system

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there is no zero in this system

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https://en.wikipedia.org/wiki/Sexagesimal

Sexagesimal

Sexagesimal, also known as base 60, is a numeral system with sixty as its base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still usedโ€”in a modified formโ€”for measuring time, angles, and geographic coordinates.
The number 60, a superior highly composite number, has twelve f...

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This system first appeared around 2000 BC;[1] its structure reflects the decimal lexical numerals of Semitic languages rather than Sumerian lexical numbers.[2] However, the use of a special Sumerian sign for 60 (beside two Semitic signs for the same number)[1] attests to a relation with the Sumerian system.[2]

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By this time there was contact between two civilizations and the decimal system just began to creep into the sexagesimal

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I guess it was the Babylonians and the Sumerians.

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but I don't know

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maybe the word Sumerian is a synonym for Babylonian?

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then it would not be the result of contact between two civilizations

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Only two symbols ( to count units and to count tens) were used to notate the 59 non-zero digits. These symbols and their values were combined to form a digit in a sign-value notation quite similar to that of Roman numerals; for example, the combination represented the digit for 23 (see table of digits above).

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I'm quoting from wikipedia

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It all depends on how these systems evolved

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if the 360 degrees came into the circle before the authors of it came into contact with a decimal using civilization

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then possibly, the use of 10 to arrive at 360 was a result of manipulation of triangular numbers

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the triangles in the cuniform are equilaterals

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the pentagram allows for more triangles and squares

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but squares are made from right triangles

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the pentagrams in the cuniform would be composed of a square which resolves to two right triangles

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and an equilateral triangle

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right

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a pentagram is a square and an equilateral triangle

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this is not mathematics

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this is archeaology!

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that's what I'm doing

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I need to go to the archeaology forum

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as the saying goes:

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you can lead a skunk to water

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but you can't make em stink

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that's what I stink, anyway

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the sumerian and babylonian cultures were separate

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the babylonians succeeded the sumerians

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the cuniform was already developed by teh sumerians before the babylonians adopted it and altered it

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the cuniform that we were viewing appears to be later

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that would account for the use of decimal 10

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I rthink that the 360 degree circle originated in the transitional period of about 500 years

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sumerians are about 3000=2000 bc and babylonians about 2000 bc

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so there's plenty of room for the very slow evolution of the mathematical systems

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the early period could have been 3000 bc

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the middle transition about 2500

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the late or early babylonian at 2000 bc

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so the system that we call sexagesimal is not any more stable than ours

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ours has admitted symbols like ฯ€

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it changes every 100 years

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the whole period of sumerian to babylonian is at least 1000 years and beyond

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because then there had to come the further development of the system under babylonian use

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so that makes 1500 years for the evolution of this system

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since math is a language it is going to look like shajkespears english

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if you go back further it will look like chaucer

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further back it could be as iunrecognizable as old english is to us now

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some people find even shakespeare to be rough going

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some people can't read the king james version of the bible

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that is pretty recent compared to the ages that we are looking at

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shakespeare

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so now what we need is a keyboard with the first 59 numbers of base 60

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then we will really be cookin

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we will be smokin!!!

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OK

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there is going to be a test when I am done

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which shall be never

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in the meantime i would like to organize a debate

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in order to settle this controversy

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we will put all of those who think that the 10 in 360 is of decimal origen on one side

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and all of those who think that the 10 in 360 is a triangular number on the other side

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we will conduct a series of 12 debates over the next year

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one per month

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and I am going to let you people slug it out

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get it out of your system so that it does not cause any more disruptions in this discussion

onyx wolf
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Don't you think this discussion is going too too deep

dense grove
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๐Ÿ˜‰chad chad

modern frost
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Why does this have as many messages as the 1+1=2 post

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Oh there's that fuckijg maniac

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rockhoven is actually psychopathic at this point

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Like, people say my thoughts are like a train which bullets off at the slightest provocation

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This guy is like my brain enchanted with power ten-thousand

fallow mesa
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thank you fans

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are there any questions/

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?

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can anyone address the OP?

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there are many possible answers to the question

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the main question

fallow mesa
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  1. I am trying to answer the question (which is not absolutley determined) and do as thoughough a review of the possible history as I can without a lot of help from anyone else here. I've gotten a little help but not much diligent interest in this fascinating topic.
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  1. I am having fun!
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I would like to see a few people step up and do soem hard work on this topic.

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Get some history into your blood

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Look for some ideas about this system

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It's a whole other world of math waiting to be explored

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Find some classic text that we can crack open and dig into

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Look for some basics on the subject

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try to calculate in their system

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try to read materials fonded upon their system

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Ptolemy

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Copernicus

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Do a few problems using their math instead of ours

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is the sexagesimal system still alive?

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who uses it?

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can they teach us something?

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AI says: "We don't use the sexagesimal system for everyday counting, but yes, it's still alive and well! It's used in a modified form for measuring time (hours, minutes, seconds) and angles (degrees, minutes, seconds). So, next time you check the clock or measure an angle, you're using a bit of ancient Babylonian math!

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Also: Decimal system dominance: The decimal system (base 10) is much simpler to learn and use for everyday calculations. It aligns well with our ten fingers, which might have played a role in its historical adoption.

Fractions: While the sexagesimal system excels at representing certain fractions (thanks to 60 having many factors), the decimal system can represent any fraction precisely using a decimal point. This makes it more versatile for complex calculations.

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OK

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But whole number math has a lot of advantages

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it's fairly easy

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and that's important for me

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because I am no math expert

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and I do not want to become one

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I am curious about many subjects and math is one of them

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I like working with complex ideas and expressing them with simple numbers

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I think base 60 is good for doing work in harmonics

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but I use decimal

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I don't know enough about base 60 to handle it

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the stuff I do in harmonics is all whole number

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I think it is very interesting that the number 360 would lend itself so well for the study of harmonics

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and it should be emphasized that their world view linked harmonics with astronomy

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the numbers that they were working with are good for both studies

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someone suggested using the decimal system to divide the circle

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how would that be used without breaking the ties between astronomy and harmonics?

fallow mesa
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We did not go into this too deeply

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the point of how or why we got thi ssytem should not be lost

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they were attempting to develop an unified physical theory

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an unified theory of astronomy (mechanics) and harmonics (music)

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the system had to be useful in both sciences

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now do you need a proof? bedcause that is what the OP calls for

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so let's prove it by working in astronomy with both the sexagesimal and the decimal systems

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then let's try working out harmonic intervals using the sexagesimal system and then the decimal system

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then let's examine the total outcome

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and we will learn more precisely why we have 360 degrees in a cicle

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cirlce

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circle

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it's a motocircle

obtuse pond
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It's more of a notation we assigned to it, same for radians, generally the theory is that the number 360 was picked due to it having many divisions.

fallow mesa
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Thanks

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were did you get this information?

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1 is also a notation and we want to know where it comes from and what it means

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I know this might be difficult for people to understand

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When you study math, there are wrong and right answers

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we don't usually discuss math in terms of ideas

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we are looking for possible explanations

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and in this discussion what we have is just that

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many possible explanations

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not one

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so please stop looking for wrong and right answers

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it is correct that this is only a convention

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yes

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that's right

fallow mesa
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a convention should not be conflated with truth

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that is why I am so interested in this question and 1 + 1 = 2

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as an example - can we say that dricving on the right side of the road is true

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driving

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no, it's a convention

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what is interesting to consider about a convention is it's history

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how did we get the habit of driving on the right or left hand side of the road and what are it's consequences

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so I ma just asking what these things really mean and what kinds of effects are produced

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in this case I say that the search for a unified theory led them to choose this number 360

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and yes it's because it has so many divisors

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but why were those divisors needed?

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my point is that those divisors were useful for doing astronomical calculations and for calculating harmonic intervals

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and for unifying both together in one complete theory

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in other words it's like we have a problem of unifying quantum with relativity

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and we seek a unified theory

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this is going to affect the evolution of our mathematics

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and if someone were to find our mathematics after t has become extinct

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and they ask "why do we have this math present in our newer math?"

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we would have to explain how the math evolved and for what purposes it served

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it is an historical question and historical quastions do not have one answer

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and this could be a problem here

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because in math the answers are all cut and dried and clear and there are not many answers

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usually

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but to discuss math in depth requires knowledge of the ideas inherant in the symbols and arrangements of symbols

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it's a language

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360 degrees has different uses and significances

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just like 1 + 1 = 2

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it';s got many ideas and possibilities jammed into it over the ages

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since it's a language it's very much like any other language

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we have words

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we go to a dictionary and find many meaning behind the words

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surely words that are very very old have many many uses and meanings

fallow mesa
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all similar figures are multiples of each other due to their property of being proportional.

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I gthink this was sufficiently establish by Euclid in the propositions concerning similar triangles

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they have to be

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I think I can stumble around and come up with a proof

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an inverse proportional

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First I need the symbol for inverse proportion

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โˆ

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so let's use a square

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we may need other tools because I don't know much

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but we'll hash it all out and come up with something

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2 : 4 โˆ 4 : 2

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now have I expressed an inv proportional well?

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will this work?

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oops

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let's invert that

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4 : 2 โˆ 2 : 4

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just to avoid confusion

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now due to these numbers being inversely proportional

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the "extremes" and the "means" form square numbers.

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I put those terms in quotes

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because they are being used a bit unusually

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oh no

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actually this is correct

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the two fours are the extremes

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and the two twos are the means

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ok

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we have it right now

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both the extremes and the means form squares

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and the similar squares will resolve to similar triangles

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and in similar trianges the corresponding sides are proportional to each other

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now if the square on 2 is placed within the square on 4 upon one and the same center

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they will form multiples

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the square on 4 is the multiple of the square on 2

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and the square on 2 is the submultiple of the square on 4

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now this will work with any numbers, I believe

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because any number can be the base of a square

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Therefore we are justified in calling the triangular numbers multiples of eachother

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though it would not appears so in our system and with our educations

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3, 6, 10, 15, 21 are multiples

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because they are similar triangles

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that is why they are milestone markers in their system

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Here is a lst of triangles

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1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666

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They are all either multiples or submultiples of each other

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just as square numbers are

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The rtwo right triangles that form any square will hold the same relation to other similar triangles

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But the triangular numbers are equilaterals and not right

compact barn
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do you need water or food

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after ranting for hours

fallow mesa
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still they are similar and therefore multiples

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yeah go out and get me a pizza with a coke

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thanks

compact barn
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think of it as similar to people offering water bottles to cyclists in a race

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yes gettin it

fallow mesa
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I appreciate it when my fans show somuch respect for me

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Do you have any quations?

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math questions?

compact barn
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i do

fallow mesa
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about the 360 degrees in a circle?

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or the sexagesimal system

compact barn
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not in this context

fallow mesa
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do you have any information that you have gathered which can help us/

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what mathematical ideas are you bringing to the table?

compact barn
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I already stated what I had to say i.e. "for all I care we could've defined it to be a banana instead of 360"

fallow mesa
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thank you

compact barn
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welcome

fallow mesa
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may i proceed?

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hmmm

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what was I thinking?

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THIS!

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1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666

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yes

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that is what I was thinking

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Now I think that they would have preferred to use a triangle for the measurement of the circumference of a circle

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how would these function in that way?

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it is whole number math

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no fractions

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they did form relative numbers

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ratios

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but I think the numerator was always larger than the denominator

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2/1 3/2 4/3...

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and not visa versa

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now 300 looks interesting

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but why?

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because my modern eye is educated to see these kinds of numbers that have so many zeroes

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300 has only 18 divisors

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AI: Divisiors of 300: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 25, 30, 50, 60, 75, 100, 150, 300

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AI: Divisors of 360: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360

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you get both 8 and 9 in 360

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that's the main difference

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there is a difference of 6 divisors in the two numbers 300 and 360

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but the main difference is these two digits 8 and 9

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the others are only multiples of these two

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all that matters is these two 8 and 9

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because 9:8 equals a semitone in the musical scale

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other than that there is no real big difference in these numbers

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with 300 we can calculate all of the other harmonic intervals

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but both 8 and 9 are needed to complete the full scale of seven tones

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but 360 is not a triangular number

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666 is a triangle but has only 12 divisors

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that will not work for building a scale

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That is still a very interesting number to examine

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It's divisor according to AI: 1, 2, 3, 6, 9, 18, 37, 74, 111, 222, 333, 666

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now 666 is a multiple of 222 and 333

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It calls into question the whole idea of multiplication

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because these are not all triangles yet they are multiples

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or submultiples

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multiplication can have different meanings

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as I just demonstrated that all squares are multiples or submultiples of each other 4 9 16 25 36 49 64 81 100

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though it does not immediately appear to be so with our modern educated eyes

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they still are

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now all they needed to do was teach us this in school and we would all be walking aroung like a buncha bots going "25 is a multiple of 4!"

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and there would be peace on this planet

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BTW the divisors of 666 are useless for harmonics

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but maybe not

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maybe we will make use of these in a future visit

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222 333 and 666 can be used harmonically

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if you saw what I did with the lower digits in constructing harmonic intervals you should see that

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we could construct 3 audible tones from that which will sound harmonically to the ear

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actually I missed one

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111 222 333 666

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four tones

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that is really really niiiiice

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that's a beauty

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that's realy nifty

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111 222 octave

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222 333 5th

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that would be the 5th above the octave

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the tonic is 111

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octave 222

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5th 333

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octave of the 5th 666

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so I would imagine that at sometime in the history of this development of the measurmement of the circle

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666 was very much a candidate

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111 to 666 gives a double octave plus a 5th

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666 was probably even used unless it was too sacred

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111 222 333 666 is the very foundation of music

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those relatios

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relations

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what is missing is the other intervals of the scale

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those intervals can be provided by one number 360

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had all of the intervals been derivable from 666

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we would probably have a 666 degree circle

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it can divide the circle in half

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in 3rds

#

it will not divide into quarters

#

so it has some shortcomings

#

so does 360

#

360 is not a triangular number

#

they would have wanted a number that was perfect, triangular, square, and produced all of the basic harmonic intervals

#

Good question for the curious"

#

Does such a number exist?

#

Try that on AI

#

use this command

#

***find a number that is perfect, triangular, square, and produces all of the basic harmonic intervals and search the highest numbers until you find one

#

another interesting divisor of 666 is 37

#

a prime

#

this is important also because only a prime can generate a never before detected interval

#

a new interval that was not heard previously

#

this one is also in hearing range

#

if we construct intervals using all of the divisors of 666 we are going to get some strange never before heard intervals

#

no one has ever heard tham

#

now this is the gateway into the music of the spheres or the harmony of the universe

#

and there are great ideas and philosphical mysteries to be revealed in this whole number mathematics of harmonics

#

meaning that it's a whole number sytsem of harmonics

#

system

#

so 360 is only an initiation into this mathematics

#

666 is the ultimate

#

666 is calling

#

operator get me klondike 666

#

666 is your magic number

#

666 iwill make your day

#

actually my proof that similar figures are multiples was unecessary since this is proven in Euclid

#

also my proof contained a superfluous step

#

there was no need to position the tqo squares on teh same center

#

so I could make the proof briefer

#

they are proportional whatever their positions

#

and geometrical proportionals are multiples

#

so triangular numbers are proportional

#

I am hearing no objections to this

#

3 6 10 15 21 28 36 is a series of triangular multiples

#

another observation:

#

we can deduce that triangular numbers have a direct relation to squares because each figure resolves into right triangles

#

well

#

not true

#

almost

#

they have a relation to rectangles

#

that relation holds I think

#

No

#

I think I was right the first time

#

these numbers must resolve to squares someway

#

maybe not

#

what do you think?

#

no it won't

#

the side is incommensuable with the bisecting line

#

AI" "This discovery of incommensurability with the equilateral triangle is attributed to ancient Greek mathematicians like Pythagoras and his followers. It challenged the earlier belief that all lengths could be expressed in whole numbers or ratios of whole numbers.

#

Here's a sketch of the proof (be aware that this is a more advanced concept):

#

Consider an equilateral triangle ABC with side length a.
Draw the altitude from vertex C to the side AB, intersecting AB at point D.
Let's assume for the sake of contradiction that a and the altitude CD (which we can denote as h) are commensurable. This means there exist positive integers m and n such that a = mh and CD = nh.
Now, focus on right triangle ADC. By the Pythagorean theorem, we know that AC^2 = AD^2 + CD^2 (where AC is the side of the equilateral triangle). Substitute the values we derived from assuming commensurability: a^2 = (m^2 * h^2) + (n^2 * h^2).
This simplifies to a^2 = h^2 (m^2 + n^2), which means a^2 is a multiple of h^2. Since a^2 is the square of the side length, it should also be an integer multiple of itself (a is the side length).
However, analyzing (m^2 + n^2), we can see that it will always be odd. The square of any even number is even, and the sum of two even numbers is even. The square of any odd number is odd, and the sum of an odd and even number is odd. Therefore, (m^2 + n^2) can never be even, which contradicts our conclusion that a^2 is a multiple of itself.
This contradiction proves our initial assumption that a and h are commensurable must be false. The side and the altitude of an equilateral triangle are incommensurable.

#

right triangle : square : : equilateral triangle : : rhombus

fallow mesa
#

so what is a "multiple" anyway?

#

wow i really like that 666 number

fallow mesa
#

anyway, if the sides of similar figures are proportional are not their areas proportional also?

#

yes

#

then they are multiples

#

that is hard rto wrap your mind around

#

Here is the breakdown of divisors for 300, 360 and 666

#

AI: Divisors of 300: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 25, 30, 50, 60, 75, 100, 150, 300
AI: Divisors of 360: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360
AI: Divisors of 666: 1, 2, 3, 6, 9, 18, 37, 74, 111, 222, 333, 666

#

Let's take a close look at each series

#

the divisors of 300 generate superperticulars up to 6, the 6th term in the series

#

let's look at that section of the whole series 1, 2, 3, 4, 5, 6

#

every one of these is a superparticular

#

2/1 is an octave this interval involves a prime and a prime always introduces an interval that was not previously heard, known or detected

#

so in this section of the series we can pickout other doubles like 2/4 3/6

#

and though they are higher and might sound different because of pitch

#

we have already heard the octave in 2/1

#

the next superparticular is 3/2 this is the 5th and this is the first time the 5th sounds in the series

#

notice that since this is whole number math we cannot insert any other interval such as a 5th between 2/1

#

that whole octave is occupied and closed

#

though in modern math we know that we could insert an infinite number of octaves between 2 and 1

#

1.5 for instance

#

wait is that correct?

#

no t's not

#

we could put an octave under 1 with 0.5

#

this could not be done by the ancients because their math did not allow it

#

since we can not divide any number by a number larger than it

#

a corollary to this rule is that 1 is indivisible

#

if you are also following the 1 + 1 thrfead take note of this

#

we may need that

#

1 = 1 thread

#

OK so we are examining this series:

#

Divisors of 300: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 25, 30, 50, 60, 75, 100, 150, 300

#

and particularly we are examining a section of the series:

#

1, 2, 3, 4, 5, 6,

#

my point now is that every prime introduces a new kind of interval which has never before been heard in the series

#

note how few primes there are in this series

#

there are a few

#

absolute and relative primes

#

Now in 1, 2, 3, 4, 5, 6, we have the series 1, 2, 4 which forms two octaves

#

and if we move our tonic to 3 then 6 forms an octave with it

#

but all octaves reduce to 2/1

#

so we are not hearing new intervals there

#

so whateverv number you could choose

#

1001 say

#

2002 is it's octave

#

all numbers have octaves

#

all whole numbers

#

all rational numbers

#

even in modern math 9999,5 has an octave

#

it's the double of that number

#

1, 2, 3, 4, 5, 6,

#

there is the next superparticular relation to consider: 3/2

#

because a prime has entered for the first time this will form an interval that we have not heard yet in the series

#

this is the 5th

#

but note this is not the 5th above the tonic but the 5th above the octave

#

we have tonic, octave, 5th

#

no 5th and no other inteval can be positioned within the octave 2/1

#

how many 5th are in this series?

#

6/4 is a 5th because it reduces to 3/2

#

we have heard this relation before

#

it's just in a higher register

#

the next superparticular is 4/3

#

if a prime has entered or the two numbers are prime to each other it's a new never before heard interval

#

this is the 4th

#

because in any superparticular the numbers are consecutive and therefore relatively prime to each other

#

some of the numbers being introduced are absolutely prime

#

other are relatively prime because they are consecutives

#

5/4 and 6/5 are such

#

now lets's go back to the whole series

#

1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 25, 30, 50, 60, 75, 100, 150, 300

#

there is a disjunct interval at 10/6

#

ths reduces to 5/3 which is disjunct and superpartient

#

it may confuse your eye that the numerator is larger than the denominator but that's how they did this

#

no fractions are allowed

#

I would much rather write 3/5 than 5/3 because it corresponds to the layout of the series

#

you can invert these when you read them if it helps

#

a disjunct interval leaps

#

after this disjunction something interesting happens in the next part of the series

#

let's see what it is

#

12/10 reduces to 6/5 this is not a new interval

#

we just heard it before

#

we heard it just before the disjunction

#

so it looks like the intrvals are reversing

#

this is an inversion

#

the next is 15/12 which reduces to 5/4

#

we heard that just before 6/5 earlier in the series

#

what is this called ?

#

inversion?

#

next is 20/15 which reduces to 4/3

#

the series is inverted

#

none of these in the upper part of the series is prime or relatively prime

#

or introduces a new interval

#

25/20 reduces to 5/4 now it is going in the opposite direction

#

we have heard this one several times now

#

30/25 is 6/5

#

50/30 reduces to 5/3 the disjunct interval

#

1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 25, 30, 50, 60, 75, 100, 150, 300

#

60/50 is 6/5

#

75/60 = 15/12 = 5/4 it's going back again

#

100/75 = 4/3

#

150/100 = 3/2

#

300/150 = 2/1

#

we are back where we started

#

that was an interesting excursion

#

we could almost say that it came around full circle

#

except those bumpy disjunct intervals

#

but those too are harmonic intervals in their own right

#

5/3 is absolutely and relatively prime

#

the two numbers are prime

#

so that was a truly interesting trip to take

#

we'll take a look at 360 and 666 soon

#

to preview the series I am or will be referring to just search for divisors in this channel

#

or interval

#

whta does that series of relations look like if we lay it out in one string?

#

2/1, 3/2, 4/3, 5/4, 6/5, 5/3, 6/5, 5/4, 4/3, 5/3, 5/4, 6/5, 5/4, 4/3, 3/2, 2/1

#

the entire series is composed of 5 superparticulars and 1 superpartient relation

#

there are 6 kinds of relations in this series

#

among others

#

we could take other pairs of numbers also

#

I do 't think any other relations will generate tones or intevals that we have not heard before

#

for instance 300/4

#

these are not prime nor relatively prime

#

300/4 is 75 and that is a tone that we have heard before

#

so there seems to be only 6 relations here

#

except I haven't done a very thorough look

#

what interests me also nis the curve that this series of relations outlines

#

as we progressed through this series the intervals first became narrower

#

the interval narrowed until that disjunct leap

#

that was 2/1, 3/2, 4/3, 5/4, 6/5, 5/3 including the disjunction

#

beginning from after this first disjunction - the intervals widened until the second disjunction 5/3, 6/5, 5/4, 4/3, 5/3,

#

after the second disjunction they continued their opening up 5/3, 5/4, 6/5, 5/4, 4/3, 3/2, 2/1 until we arrived at the 8ve relation again

#

that is inversion

#

the intervals got smaller and smaller and then larger and larger

#

1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 25, 30, 50, 60, 75, 100, 150, 300

#

you can also pick out other octaves in the series and 5th etc

#

they are simply in higher registers

#

for instance 25, 50, 100

#

I don't vthink there s a new interval among these though

#

AI: Divisors of 360: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360

#

AI: Divisors of 666: 1, 2, 3, 6, 9, 18, 37, 74, 111, 222, 333, 666

#

these are the next series up for inspection

empty apex
#

this guy from 1st grade at age 99 is straight out of heaven, lmoa

fallow mesa
#

@X suggested: "144 would be easy it divides into 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, and 144"

#

@X also suggested 840

fallow mesa
#

why can't a unit circle have a circumference of 1?

#

why can't pi = 1? It's a convention. ASll you have to do is designate that pi=1

#

why don't you designate that pi=1/2? then proceed to it's mathematical results

#

just draw a line, call it pi and make pi=1

#

Imean if you want to visualize it

#

you might use any shap

#

call pi a square number and see what happens

eternal flower
fallow mesa
#

what do these numbers have to do with the circumference of a circle? Is the 10 in your formula a triangle? how can you measure the circle withthat formula?

#

we agree that math is a language

empty apex
#

math is something more than a language

fallow mesa
#

right

#

yes

#

ok

#

it's base 60

#

you posted this question?

#

what do you know about Greek culture? do you know the greek language, literature, history, math, sciences?

#

there is a lot of math that is still lost to us westerneres because it still has not been translated. I think they are still working on getting all of Archimedes together and translated

#

you mean, that we cannot change the 360 degree system because it is so fixed?

#

the perfection of any of the numbers I would like to consider is how well they perform in both measurements of space and time

#

because the measuement of any harmonic interval is a measurement of time

dense grove
fallow mesa
#

this is correct since pi is a convention. we can drive on either the left or right hand side of the road as long as we put the signs on the correct side

dense grove
dense grove
grim plaza
#

I personally use ฯ€=1260

fallow mesa
#

have you checked this observation because it may be worth looking into. I'm going to dive deeper into 360 next

grim plaza
#

So yea it's not fixed

fallow mesa
#

dive into the harmonics that can be produced

dense grove
#

how did u get that?

grim plaza
grim plaza
fallow mesa
#

how did you introduce 37 into the discussion? 37 is going to be prominent in our examination of 666, I think

dense grove
#

it it was

#

this

#

$7\pi\ = 1260$

woven reefBOT
#

ExplorerAC | #SendWolfToZoo

877077191135985705.png
dense grove
#

then I could agree

grim plaza
#

ฯ€=360

dense grove
#

2pi is 360 rad

grim plaza
#

This guy said

fallow mesa
#

1225 is an interesting number. that's a square. I don't think any number ending with 5 can serve for any deisions other than 5

#

that will not work for measuring space and certainly can't get us anywhere in producing a rational set of harmonic intervals

#

why are you choosing these kinds of numbers? what are you looking for?

grim plaza
#

But yea I usually chance it according to the question cause ฯ€=22/7 and if there is angle that we need to calculate and we have pi in the equation 7 and 7 cancle out very well

eternal flower
fallow mesa
#

I'm reading through this thread this morning and see that I made some few mistakes but too often before catching myself.

#

CORRECTION 36 is not perfect. 360 is not perfect and not a triangle

#

36 IS a square and a triangle

#

I ma reading around this post now

#

this is correct

#

I also said that there was no decimal 10 in sexagesimal. this is because I was under the wrong impression that sexagesimal is base . someone later corrected this to base 60

#

still, the question remains - in the sexagesimal system, 10 is not a milestone like 10 20 30.... 100 1000 10,000

#

only multiples or submultiples of 60 would serve as milestones

#

then there are triangles and squares

#

here is where I was corrected and directed to base 60

#

I still don't understand this

#

don't understand this either

eternal flower
#

x_6 means x is written in base 6 so for example 21_6 is 2*6+1 = 13

fallow mesa
#

at that time I still did not understand that the sexagesimal system had the decimal 10

#

here cute rizzly caught the fact that 360 is not a triangle. good job

eternal flower
#

also if you could turn off ping while reminiscing that would be spectacular i dont really want to get pinged 20 times cuz of your trip down memory lane

fallow mesa
#

Actually, I started making these grosser errors when I began consulting with AI for quick easy answers

#

other errors like thinking that we were talking about base 6 are my own

#

BTW take note of how AI corrected itself back then "You are absolutely right! I apologize for the mistake in my previous responses about 360 being a triangular number.
Yes, 360 is indeed a triangular number."

#

WOW! I'm having fun AGAIN just reading through this thread

#

DOUBLE FUNS !!!!

#

is that high or low? good lucky

#

what is your reasoning behind such an analogy?

eternal flower
fallow mesa
#

same here

#

same old same old

#

same old same old

#

same old same old

eternal flower
fallow mesa
#

good question

#

probably because if you are going to arrange objects into geometrical shapes the most logical place to start is with triangles

eternal flower
fallow mesa
#

so the discovered and explored the triangular numbers first

dense grove
fading nebula
#

lmao

rancid sage
rancid sage
#

Prime factorization: 360=2^3*3^2*5

#

Maybe they wanted descending primepowers?

obtuse pond
#

Holy man

#

Man thought a lot

scenic onyx
carmine shard
#

The circle is a fundamental geometric shape that has captivated mathematicians, scientists, and philosophers throughout history. One of the most intriguing aspects of the circle is the fact that it is commonly divided into 360 degrees. This division of the circle into 360 distinct units has practical applications and deep historical roots that reveal insights into the development of mathematical and scientific thought.

#

The origin of the 360-degree circle can be traced back to the ancient Mesopotamian and Babylonian civilizations. These early cultures developed sophisticated systems of mathematics and astronomy, and they were particularly interested in tracking the movements of celestial bodies. The Babylonians, for example, observed that the sun appeared to travel through the sky at a relatively constant rate, completing a full circuit around the earth once every 360 days. This observation led them to divide the circular path of the sun into 360 equal segments, each representing a single day of the year.

#

The 360-degree circle also has connections to the ancient Egyptian calendar, which was based on a year of 365 days. The Egyptians divided the circular path of the sun into 360 equal segments, with an additional 5 days added at the end of the year to account for the extra quarter-day in the solar year. This 360-day calendar system, combined with the Babylonian observation of the sun's annual circuit, likely contributed to the widespread adoption of the 360-degree circle in the ancient world.

#

Beyond its astronomical origins, the 360-degree circle also has practical applications in various fields of mathematics and engineering. The divisibility of 360 into numerous whole-number factors, such as 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, and 180, makes it a versatile and convenient unit of measurement. This divisibility allows for the easy calculation of angles, the construction of regular polygons, and the division of circles into equal parts, all of which are essential in fields such as architecture, navigation, and surveying.

#

The 360-degree circle has also influenced the development of other mathematical and scientific concepts. For example, the concept of the degree as a unit of measurement has been extended to other circular and angular measurements, such as the 360-degree rotation of a full circle, the 180-degree angle of a straight line, and the 90-degree angle of a right angle. Additionally, the 360-degree circle has been used as a basis for the development of trigonometric functions, which are fundamental to various branches of mathematics and science.

#

The enduring presence of the 360-degree circle in modern mathematics and science is a testament to its practical utility and its deep historical roots. While alternative systems of circular measurement, such as the 400-degree "grade" system, have been proposed, the 360-degree circle remains the dominant and most widely used system, reflecting its versatility and the ingenuity of the ancient civilizations that first developed it.

#

In conclusion, the 360-degree circle is a remarkable mathematical and scientific construct that has stood the test of time. Its origins in ancient astronomy, its practical applications in various fields, and its influence on the development of other mathematical and scientific concepts all contribute to its continued relevance and importance in the modern world. The story of the 360-degree circle is a testament to the power of human observation, ingenuity, and the ongoing quest to understand the world around us.

fallow mesa
#

nice reminder though i don't eccept that as a complete explanation

#

like someone else suggested before

#

AI is very unreliable

fallow mesa
#

why not use 5040? since it's divisors 1, 2, 3, 4, 5, 6, 7, 8

#

No it's divisors are multiply 1, 2, 3, 4, 5, 6, 7

#

you don't get 8 & 9

#

well, you get 8

#

hey you get 9 too?

#

well then that's it

#

we can do astronomy and harmonics with 5040

#

it divides by 360 also

#

360*14=5040

#

we can get all the superparticulars up to 10

#

this number don't poop out til 11 and 13

#

5040 can handle 12 also

#

it accepts 14 as a divisor

#

15*336

#

16*316

#

17 is prime

#

18*280

#

19 is prime

#

20*252

#

this number is better than 360

#

especially for doing harmonics

#

maybe 360 got established with all of it's shortcoming before they had symbols for numbers in the thousands

#

I mean math has to evolve

#

so there was a first time that someone counted to 6

#

to 10

#

to 100

#

to 1000

#

maybe the system got started before they had counted up to maore that 1000?

#

100 must have been an incredibly high nember once upon a time

#

another thought

#

we are talking about it in terms of dividors

#

maybe that's wrong

#

you have to think of high number to get divisors

#

spose any number for these observation was the slow evolution of the system by way of multiplication?

#

spose they started out by divideing the sky into two parts

#

then they tried another system with three parts

#

3 euilateral tringles in the circle

#

equilateral

#

triangles

#

they would multiply up with every stage in the develpoment of the 360 degree system

#

so there were primitive stages in the development

#

2*3=6

#

now they could study the sky in six parts

#

and I think they might have been trying to find evidence for religious and other supersticious ideas

#

numerology

#

so they liked a number like 3 because it's the first triangle

#

and they were learning early hgeometry by placing objects in symetrical formations

#

which is why i thinknthe triangle was of first importance for them

#

then they like 6 because it is tri and perfect

#

and they had this thing about perfect numbers

#

the idea of perfect numbers sparks of numerology

#

and they were more preoccuied with addition than multiplication

#

in their primitive math

#

addition comes before multiplication developemtally speaking

#

historically addition comes first

#

and triangular numbers are a sort of primitive multiplication

#

by way of addition

#

we get these numbers by addition

#

so then they put 3 and 6 side by side and get 36

#

and they just like the aestetics of this number

#

two triangles sort of spacilly "added" together

#

they are related anyway

#

and 36 is a triangle and a square

#

so this system developed through multiplication

#

not by way of division

#

they might have been terrified of bigger numbers

#

like they had a tribal taboo

#

"no numbers bigger than yer head"

#

they might have had to sort of tip toe up to a number that was higher than 100

#

they may have needed a special ceremony navigated by a witch doctor

#

who cleansed the area before an after using such numbers

#

math : numerology : : astronomy : astrology

#

5040 may have been a number too high for any one to imagine

#

sure

#

tere had to be a time when 5040 did not exist

#

just like you have to first invent 100 before you can go on and invent 1000

fallow mesa
# grim plaza I personally use ฯ€=1260

cool kid, you have the right attitude about math. Math is nothing but conventions and we are free to assign values to any symbols we like. We can also introduce new ideas and give symbilic notation to them.That is how pi came into being

rancid sage
#

Both 60 and 420 are nicer than 360 in that they have divisors 1,2,3,4,5,6 resp. 1,2,3,4,5,6,7

fallow mesa
#

for example: what difference does it make whether a number id "odd" or "even"

#

what would happen if we called all of the even numbers "odd" and the oddd numbers "even"?

#

nothing

#

we would get the same results

fallow mesa
#

we need 8 and 9 to generate a semitone

rancid sage
#

If you also want 8, there is 840, and if you want 9 too there's 2520

#

Wait

fallow mesa
#

let me check

#

yep

#

9 works

#

ok that is good

rancid sage
#

What if the system was created in year 360 according to their timekeeping?

fallow mesa
#

8 & 9 check out

#

I think tat we are so accustom to seeing 360 in association withb geometry and astronomy that we are overlooking the importance of harmony in their sciences

#

harmonics

#

harmonics had to intrigue them immensely

#

it's like looking for the ether or tracking a particle

#

harmonics was state of the art math and physics

#

that was where you got your phd

#

so I don't buy any explanation that neglects harmonics

#

and the unified theory

#

well, I'll admit

rancid sage
#

Okay

#

So maybe they removed some divisors on purpose

fallow mesa
#

I just multiplied 1, 2, 3, 4, 5, 6, 7,

fallow mesa
#

we need every superparticular

#

there is no reason to omit 7

rancid sage
#

What if it was an accident

fallow mesa
#

1234567*8=40320

#

that;sout of hearning range

#

x2x3x4x5x6x7x8=40320

#

1x2x3x4x5x6x7x8

#

5040 divides by 8 and 9

#

your number does also

#

and it include 7

#

what was your number?

#

i want to work with that one

#

2520?

rancid sage
#

Yes

fallow mesa
#

that's 1/2 of 5040

rancid sage
#

Just the lcm of 1,2,3,...,9

fallow mesa
#

5040/2520=octave

#

i thin k i might have already tried this one

#

2520

#

you get one problem somewhere

#

it doesn't generate all of the scale with whole numbers

#

try it out

#

5040 does that

#

kind of wieldy for astronomy though

rancid sage
fallow mesa
#

idk

#

with min and sec astronomy has some clumsy math

#

plus the base 60

#

it's not easy t calculate with

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degrees minutes seconds

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that's like trying to divide and muliply hours inutes and seconds on the clock

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give me UNIX or the SWATCH BEAT

rancid sage
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360 in base 10 is 260 in base 12

fallow mesa
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I can use those

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try to add and subtract the time stamps in this conversation

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i could do it if it were unix

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in a snap

fallow mesa
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harmonics

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the system has to be friendly for harmonic calculations

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360 is unsuitable

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it's ok

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but it's very limited

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there are only 6 unique intervals among all of it's divisors

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it has 24 divisors

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but only 6 unique harmonic intervals

rancid sage
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What are harmonic intervals?

fallow mesa
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the suoerparticulars first

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the relation between any two consecutive numbers

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1/2 2/3 3/4

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those are all foundational to musical harmony

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octave 5th and 4th

rancid sage
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And why do you think they found those important in their number system?

fallow mesa
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because they were attempting a unified theory the universe that they called the harmony of the spheres

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they wanted to harmonize all of their science just as we do today

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they believed that the planets and stars moved and disturbed the "air" around them and produced sound

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they wanted to understand the whole universe in harmonic proportion

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that was what they were looking at the heaven for

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however, 360 might have been very well established before the study of harmonics was even found

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O'm not committed to anything

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I just think it's an interesting possibility

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anyway you should try working out the intervals sometime with a calculator

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there's a lot of nifty things to discover there

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lots of patterns

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but it's whole number math

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whole number math is impresssive

fallow mesa
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like the Incas build their temples using a totally foriegn measuements ystem and methods

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they were thinking differently

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my impression is that perfect numbers were exalted beyod reason

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we should think them ridiculous for the elevation of so-called perfect numbers

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they had obviously come to apoint of exhausting their arithmetic

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the triangular number gtoo

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it's very likely that they wer learning math by playing with shapes

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just think

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they could have founded and handed down to us a completly diffecrent math than we now know

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for example they could have said that all figures with equal sides were equal to each other

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a triangle is equal to a square because they both have equal sides

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the rtiangle has 3 equal sides and the square has 4 equal sides

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therefore ethey are equal figures

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they could have said that 2 was odd and 3 was even

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they could have establish all kinds of strange ideas as conventional, customary and logical

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and we would have inherited those ideas and anyne espousing modern math would be thought a lunatic

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I think 360 was chosen for silly, nonsensical, religious and supersticious "reasons"