Old school math

msasa79

Jedi Council Member
The C's have noted that knowledge if not shared is STS, which is rather straightforward to understand, but if nobody is asking, then it seems like imposing.
Having that in mind, the content of the post corresponding to the title of the thread was placed in the spoiler box.

The C's saying that numbers 1-2-3 is all we need to construct other numbers, and arrangement of C's crystals on the table this evening, brought an idea to mind how that could be done. Well, there was also question on my mind what does the C's statement that counting is arbitrary really mean, because that seemed totally counterintuitive and ilogical. And it still does, so I'm inclined to think that was food for thought, if I remember correctly that C's statement.

Number 1 can be written as a dot, . , and it seems that in this type of notation it is not needed anymore for construction of other numbers, which makes me wonder also about that C's statement. In other words, counting does not seem to be arbitrary and number 1 is practically not needed except to know that after number 2 comes number 3 and after number 3 comes number 4, and so on, which basically amounts to counting. Well, except if . is not used for truncating other, higher valued numbers which very quickly in this notation start to appear like a long string of | and =, which looks rather clumsy to work with.

Number 2 is represented as a horizontal line, - , and number 3 as a vertical line, | . Number 4 then simply becomes two horizontal lines, one next to the other, --, or one above or beneath the other, which then gives additional symbol, = , making number 4 almost on equal footing to number 2 and number 3. In that manner, we have four, i.e. three symbols, leaving . aside, to work with:
. 1
- 2
| 3
-- or = 4.

Now, if we take number 4 as -- and convert one - into |, we get number 5 as -|, i.e. 2 and 3 (using notation where lower valued numbers are on the left to higher valued ones), and that's an indication when new symbol appears, i.e. we work with the base 4. Number 6 is then just ||, but it can also be written as -=, and to get number 7, we either convert one | into = or - into | to get |=. Number 8 comes to be when that | becomes =, i.e. ==. Now, that can also be written as -||, which makes ground to work with three symbols and continue 'counting' further. Convert - into | and get number 9 as |||, then | into = to get number 10 as ||=, number 11 as |== and number 12 as ===.

And here we get a 'rule' how and when to add new symbols for higher valued numbers, as number 12 can also be written as four |, |||| , meaning that from three symbols, ===, we switch to four symbols, ||||, and continue converting | into = to get the next number in order. When all | are converted into =, the same switch is applied, i.e. === -> ||||, and the conversion of | into = starts again to go higher in order.

1 .
2 -
3 |
4 = (or --, i.e. transition from one symbol to two)
5 -|
6 || (or -=)
7 |=
8 == (or -||, i.e. transition from two symbols to three)
9 |||
10 ||=
11 |== (or -|||, but we don't need four symbols just yet)
12 === -> |||| (transition from three symbols to four, which makes the basis for further transitions)
13 |||=
14 ||==
15 |=== (which can be written as |||||, but we don't need five symbols yet)
16 ==== -> ||||= (transition to five symbols)
17 |||==
18 ||=== (which can be written as ||||||, but we don't need six symbols yet)
19 |====
20 ===== -> ||||== (transition to six symbols)
21 |||=== (which can be written as |||||||, but we don't need seven symbols yet)
22 ||====
23 |=====
24 ====== -> ||||=== (transition to seven symbols)
and so on we go ...


An interesting thing about this notation is that it makes it rather easy to identify prime numbers, as the numbers which can only be divided by themselves and 1: they are those numbers/symbols which can't be 'decomposed' into identical groups of symbols used for lower valued numbers. In other words, for example number 22 as ||==== can be split into two numbers 11, |== and |==, which then can't be split into the identical groups of symbols anymore, similarly to number 23, |=====, or number 17 |||==. The same goes for number 10, ||=, which can be split into -| and -|, i.e. two numbers 5, which then give five symbol notation for that same number 10, -----, i.e. five numbers 2.

That's all for now, and at least until that dilemma outside of the spoiler box is elucidated. There are other things that I'd like to share, but as nobody is asking, that I'm aware of, I've restrained myself from doing so.
 
Hey Mytia, is this your own system? I've played around with inventing alternative number systems for worldbuilding and so on before, making base 4 or 16, etc. I found the lower the base is, the harder it is to write out longer numbers. It seems like some way of writing this is meant to make simpler math problems and so on easier? I see there's multiple ways of writing out a given number, and the way to write each number can be expressed as a 4 dimensional coordinate (x,y,z,w) where x, y, z, and w represent the number of ., -, |, and = in each number. In a sense our current math language already has similar "synonyms," like for example 4 can be written as 2+2. If everyone was automatically grouped into units of 1, 2, 3, and 4, it would probably make operations involving multiplication or division of those four numbers easier, but it also might make other types of operations (e.g. with very large numbers) more cumbersome. The social and cultural forces shaping numbers and their use probably plays a large role in this as well. OSIT.
 
Hey Mytia, is this your own system? I've played around with inventing alternative number systems for worldbuilding and so on before, making base 4 or 16, etc. I found the lower the base is, the harder it is to write out longer numbers. It seems like some way of writing this is meant to make simpler math problems and so on easier? I see there's multiple ways of writing out a given number, and the way to write each number can be expressed as a 4 dimensional coordinate (x,y,z,w) where x, y, z, and w represent the number of ., -, |, and = in each number. In a sense our current math language already has similar "synonyms," like for example 4 can be written as 2+2. If everyone was automatically grouped into units of 1, 2, 3, and 4, it would probably make operations involving multiplication or division of those four numbers easier, but it also might make other types of operations (e.g. with very large numbers) more cumbersome. The social and cultural forces shaping numbers and their use probably plays a large role in this as well. OSIT.

Although it was written down last night, I'm pretty confident it's not my invention, at least not in a conventional way.
It 'feels' like remembering, maybe past life stuff, and it also 'feels' like coming from a long time ago, hence the title "old school".

Truncation can be used starting at number 16, ==== -> ||||= , where we see that last = stays there unchanged as we go on with counting, so we can replace it with a dot, . , and continue working just on that left part, ||||. Then number 20 would be written as ||||.-, meaning |||| and == or number 12 on the left and number 8 on the right. Number 24 would then be ||||.|, number 28 ||||.= and number 32 ||||.-|, standing for number 12 on the left side and number 20 on the right side. In that way we would basically be working on both sides simultaneously, adding numbers on the left in 'ordinary' way and on the right for truncations, i.e. multiples of number 4. Highest value obtained in this way, i.e. ====.==== , would be number 80.

Another possibility would be to use dot instead of number 16, i.e. number 16 would still be ====, but also . , which then makes number 17 as two dots, .. , i.e. number 1 and number 16. Number 18 would be -. , number 19 |. and number 32 as ====. would also be written as .- , i.e. two times number 16. This option seems more economical; we would be starting anew on the left with each ==== reached there, and add multiples of number 16 on the right, which would enlarge our set to number 272 as ====.==== . On the other hand, if we truncate with number 80, highest value when truncation with number 4 is used, so to prolong our set from there, we can reach number 1296 written as ====.==== or the highest value expressible.

With these options in mind, the C's statement about 1-2-3 being all we need, makes more sense.

Regarding the applications, well, as seen from what's written, I'm currently re-learning or re-remembering the very basics, but I'm also confident that the other things would follow, in their due time.

This was also inspired by having prime numbers in mind, and what's said in the opening post is still valid when we use a truncation method, i.e. if a number can't be split into identical lower valued groups, then it's a prime number. Truncation with number 4, i.e. when we use a dot for =, seems rather appropriate to identify so called Pythagorean primes, which are of form (4×n+1), and represent prime numbers which can be written as a sum of two squares (wiki link). Those prime numbers seem to play important role in geometry, and identifying them and relations among them was very productive while working on those Tables presented in "Hidden Secrets of all existence in Prime Numbers?" thread.
 
Although it was written down last night, I'm pretty confident it's not my invention, at least not in a conventional way.
It 'feels' like remembering, maybe past life stuff, and it also 'feels' like coming from a long time ago, hence the title "old school".

What are you hoping to get out of these explorations?
 
What are you hoping to get out of these explorations?

A homage to a great, remarkable man, veritable giant who's legacy illuminates the path for all of us who choose to follow his footsteps.

Pierre_Lescaudron.jpg


:love:
 
Pierre I met and got to know was rather skilled and creative at hands-on craftsmanship, so in that spirit I embarked on bringing the prime cube discussed in the series of posts (for example this one) in the other Forum thread into the hard core 3d reality.

It quickly became evident that the direct, seemingly simplest approach in constructing the 12×12×12 prime cube, that of assembling it from all the small cubes or cubical blocks or stones, was the least efficient one, as there would be more than 2000 of these stones that go into the final product. The next line of actual construction, that of using pillars or columns attached to the basal plane, which initial result's depicted in the image below, also turned out to be rather inefficient and both time and resource consuming, as there would still be more than 200 of these columns or pillars on just one side of the basal plane.

IMG_20240917_210847.jpg

Living in a multi-store building in a urban setting without proper place for a workshop or the adequate machinery to carve real 'hard' wooden material, for the time being the choice of actual assemblage settled to using sort of planes made of cork wood. I'm open though to any other viable option that might be out there and that haven't crossed my mind so far. To make the process of final carving the cork wood using scalpel easier, first a dummy versions of the planes have been made out of cardboard (image below shows the dummies, not all of them in their final form, of the basal plane and first four planes above/below it). And that's where things got sort of sidetracked into the mathematics, geometry, prime numbers, Pythagorean triples and quadruples, Fermat's and Legendre's theorems, distances and/or metrics of hyperdimensional spaces, and related stuff.

IMG_20240922_204058.jpg

In practical terms, a square spanned by diagonals of 24 units (no matter which ones, but in my case it's 24cm), representing the 'outer' basal plane of 12×12×12 prime cube (prior to cutting or carving out those notches/folds to get squarish or cubical outer ring of prime cube's corner stones for example), effectively is composed of 4 isosceles right-angle triangles with their sides equal a=b=24/2=12, that is the dimension or side of our 12×12×12 prime cube. The hypotenuse of such a triangle, which is effectively the side of the dummy 'outer' basal plane prior to cutting out those notches, is then c2 = 122 + 122 = 288 which is very close to 172 = 289, and in fact when working in metric system, as sqrt(288) = 16.97 that is 169.7mm, is basically only cca half of a thin line width, which is 0.5mm, difference from actual 17cm. So for all the practical purposes, the dummy basal 'outer' plane to start with is a square with its side of 17. Which also means that 17 is a spatial diagonal of a cuboid or a 'hypotenuse' in Pythagorean quadruple of (1, 12, 12, 17) as 172 = 12 + 122 + 122.

On top of that, when calculating exactly how many stones there are in the prime cube, using their numbers in the planes, another interesting thing popped up. Starting from the top or the bottom, first plane is just 1 stone, in the case of the 'imperfect' prime cube that of 9737333 at the top or the bottom of the central pillar/column, while in the case of the 'perfect' symmetric prime cube where there's an additional step further up or down, that of the 9737333rd prime number in order, that is 174 440 041 (it can be seen in the 12×12 table in this post). Next plane, second from the top or the bottom, gets additional 4 stones (5 in total), sort of in a manner to cover free sides of that central stone in the first plane. Next plane, third from the top or the bottom, gets additional 8 stones compared to the second one (13 in total), covering all new free sides of the stones in it. The forth plane from the top or the bottom gets 12 more stones (25 in total) compared to the previous plane, again sort of covering all the free sides of the stones in the plane above or below it.

Since that manner of covering free sides of the stones continues with each successive plane, that is each new plane gets 4 additional stones to the covering stones of the plane before it, we can simply write that nth plane from the top or the bottom consists of an = an-1 + 4×(n-1) stones, with the first element of the sequence or series being a1=1.
The 12th plane or the basal one in our case, a12, will then have 4×(12-1) = 44 additional stones compared to the first plane above or below it, or a11 in the notation used here, which is exactly the number of corner stones with number 31 on them in that outermost ring of the 'imperfect' prime cube, or number 37 which is 12th prime number in the case of the 'perfect' symmetric prime cube.

When we put all this together to get a general term of the sequence or series, it turns out that it can be written as an = 1 + 4×Sumi=0n-1 (i), where the Sumi=0n-1 (i) simply stands for an addition of consecutive natural numbers, starting from 0 up to (n-1). So we get a1 = 1 for the first element of the sequence, as that 1 stone at the top or the bottom of the prime cube, and a2 = 5, a3 = 13, a4 = 25, a5 = 41, a6 = 61, a7 = 85, a8 = 113, a9 = 145, a10 = 181, a11 = 221 and a12 = 265 stones for the basal plane. By simple summation it turns out that there would be 2047 stones in total in the prime cube, that is 11 planes above and 11 planes below the 12th basal plane in the middle, and that there would be a11 = 221 pillars or columns attached on both sides of the basal plane if that would be our choice of constructing the prime cube.

The interesting and "funny" thing about that general term expression, an = 1 + 4×Sumi=0n-1 (i), is that that's exactly the form of Pythagorean primes, (4×n+1), mentioned in previous post in this thread, and which can be expressed as the sums of two squares. Of course, not all numbers of the stones in our prime cube's planes are Pythagorean primes nor all the numbers of the form (4×n+1) are prime numbers to begin with, like (4×2+1) = 9 or (4×5+1) = 21 or (4×6+1) = 25 for example, but in the case of our 12×12×12 prime cube, except the singular stones at the top and the bottom, all numbers of the stones in its planes are either Pythagorean primes or the product of the two of them.

Since the 7th prime number (list), our number 17 from several paragraphs in the post above, is the 3rd Pythagorean prime (list), meaning that it equals to a sum of two squares itself, 17 = 12 + 42, (17 is also a quartan prime, that is 17 = 14 + 24, and in addition it can be expressed as a sum of three squares, 17= 22 + 22 + 32, and also as a sum of three cubes, 17= 13 + 23 + 23, but let's perhaps leave that for later), and also the hypotenuse in a primitive Pythagorean triple, that is part of a Pythagorean triple (8, 15, 17) meaning 172 = 82 + 152, it was only natural to check potential relationship between the Pythagorean prime numbers and primitive Pythagorean triples.

And lo and behold, it turns out that hypotenuses in primitive Pythagorean triples are all either Pythagorean primes or the product of the Pythagorean primes. Well, contrary to my initial excitement, that's not exactly some new news, as it's written there on the Wikipedia page that for a Pythagorean triple (a, b, c):
  • The hypotenuse c (which is always odd) is the sum of two squares. This requires all of its prime factors to be primes of the form 4n + 1.[16] Therefore, c is of the form 4n + 1.

What is sort of a new, that is at least not stated on the wiki page, is that, specifically, if the hypotenuse is a product of two different Pythagorean primes, for example 65=5×13 or 85=5×17 or 221=13×17, then there would be possible to construct more than just one Pythagorean triple with it (two for a product of two different Pythagorean primes, for numbers up to 300), like (16, 63, 65) and (33, 56, 65) or (13, 84, 85) and (36, 77, 85) or (21, 220, 221) and (140, 171, 221), while when the hypotenuse is a Pythagorean prime squared or cubed or to the forth power and so on, that is a product of just one Pythagorean prime, there would be only one triple with it, for example (7, 24, 25=5×5) or (44, 117, 125=5×5×5) or (119, 120, 169=13×13) or (161, 240, 289=17×17).

When checking higher valued products of Pythagorean primes for a hypotenuse in the primitive Pythagorean triple with the online calculator (section 7.1 on this page), it turned out that there seems to be a progression for possible primitive triples of 2n-1, where n is the number of different Pythagorean primes used in the product. Meaning that for example if product used just one Pythagorean prime, n=1, there would be just one possible triple with it, that is 21-1=20=1, no matter how many times that prime was multiplied with itself. If there were two different Pythagorean primes, n=2, in the product, then there would be two possible Pythagorean triples with that hypotenuse, 22-1=21=2, again no matter how many times we multiply those two primes. If the product was made of three different Pythagorean primes, n=3, then there would be four possible triples with that hypotenuse, 23-1=22=4, if there were four different primes multiplying with each other, there would be eight possible triples, 24-1=23=8, no matter how large the product was, 16 for a product of 5 different Pythagorean primes, 32 for 6 of them, and so on.

As this post already turned out to be longer than initially excepted, let's leave other possible/potential relations and relationships and implications among/with those numbers for some other time. If there would be interest for that on the Forum of course.
 
Well, upon reflection after playing with the Pythagorean spiral (see attached image) and nudged a bit by some things expressed in the latest post on Ark's blog, have realized, to my great embarrassment, that what had called "(hairy) prime cube" in many of my posts on the Forum at the time, is in fact no cube at all, but its dual geometric solid, that is octahedron or sort of a 'hairy dipyramid' as in eight-triangle-faced geometric body/solid.

So, apologies for basically spreading disinformation in all those previous posts of mine.

IMG_20250410_164100.jpg
 
Back
Top Bottom