Conmath base-10 permutation

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dwnielsen
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Conmath base-10 permutation

Post by dwnielsen »

Okay, this was something that came to mind last night, and, since there's pretty much never a serene, focused moment around here, it's just 1/2-baked. Still, I'll try to piece it together as I go along.

A region of a certain universe is inhabited by a people called the Kiubrutes. Understanding quantities is a bit different for them. Our 3-dimensional volumes appear to the Kiubrutes as cube-root lengths. So, eg, 8 cubic meters are observed as a length of 2 meters. And whereas we are used to making a mark with chalk and counting one, then making another and counting two, the Kiubrutes make a mark and count one, then must try 7 more times before the second appears.

But this is not really the true story of Kiubrute perception, just an illustrative lie. You see, 10 is a special number in Kiubrutanica, and the patterns I have described only fall within a range of 10. The Kiubrute decimal digits appear the following way to us, as the final digit of the cube:

Code: Select all

x      x^3 mod 10
-----------------
0      0
1      1
2      8
3      7
4      4
5      5
6      6
7      3
8      2
9      9
Nicely, we still make out unique values for all the possible digit values, only 2, 3, 7, 8 are permuted from what we normally observe. (Notice this one-to-one mapping would not have occurred were we using a power of 2, 4, 5, or 6, but the same pattern as above does occur if we take the 7th power. Notice also that the mapping is equal to its own inverse, so, eg, 3->7 and 7->3, etc, meaning the Kiubrutes observe our perceptions as weird in exactly the same way!)

Since 10^3 = 0 (mod 10), and therefore
11^3=1 (mod 10)
12^3=8 (mod 10)
..
we see that the decimal digits otherwise work just like ours.

Would anyone care to describe Kiubrutanica a little further?
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Re: Conmath base-10 permutation

Post by dwnielsen »

Sorry, I forgot there was a general conmath thread on this board. If this should have gone there, mea culpa.

Anyway, thought I'd mention one tiny point: since this system uses an odd power, negatives also work out okay.
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Re: Conmath base-10 permutation

Post by dwnielsen »

This is approximately how our unit circle centered at (0,0) (if there is such a thing in the real world) appears to the Kiubrutes (ETA: fixed image).
Image
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Re: Conmath base-10 permutation

Post by Yačay256 »

This is very clever: Changing integers instead of units for exponential functions, quite creative.
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Re: Conmath base-10 permutation

Post by dwnielsen »

Thanks, Yačay, it just so happens that 60 (and 59), 30 (and 29), 20 (and 19), and 10 are special numbers in my conproject, so a base-10 approach seemed natural. I found it interesting that the cube power mod 10 gives all the numbers 0 to 9.

Here is an animation of the unit circle moving up and down a little from (0,0) (though not very smoothly :roll:). Still, kinda looks like some faces in there. :-)
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Re: Conmath base-10 permutation

Post by zelos »

im not sure exacly what your idea here is
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Re: Conmath base-10 permutation

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It's not entirely clear to me either, but I'll be posting more about it here as more related information is discovered.
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Re: Conmath base-10 permutation

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I mentioned that 10, 20, and 30 are special numbers in the conproject I'm pursuing. The following experiment seemed to fall in line with former approaches. Here's what was done.

A strip of paper was cut, and the numbers 0 to 4 were written along the left edge from top to bottom with approximately uniform spacing. On the back side 5 to 9 were written (also on the left). By the right edge, a 0 was placed next to the 0 on the left. The reverse count (mod 10) was then written along the right edge front and back, so 0,9,8,7,6,5,4,3,2,1. A special mark was placed next to these numbers to indicate that they belong to the 2nd set; for that reason, I'll call them 0*,9*,8*,.. . Between the left and right, another count was then written, this one identical to the count on the left (although, due to symmetry, it could be identical to either that on the left or on the right), except given a different identifying mark, let's say "**".

A Möbius strip was constructed by joining the top and bottom edges of the strip after half-twisting one of them. Then a cut was made beginning between two sets of numbers and continuing until the cut met itself and produced the object shown below.

Image

What I like about this form is that it contains sets of 10, 20, and collectively 30 numbers. It contains a monadic piece that looks like our written 0 (with a twist, or at least a half-twist :-) ), and a symmetrical dyadic piece that resembles the infinity sign with a half-twist in each lobe (had we only used 2 sets of numbers and cut between them, this is the only piece that would have resulted). The piece containing 20 numbers has 10 on one side, and 10 on the other (in the same positions but on opposite sides of the paper). The 20 consonants in the related conlang has 10 mouthy-throaty pairs (20 consonants total); additionally there are 10 vowels that sort of form their own set.

Pinching the symmetrical 2-lobed piece at a number so that the written 0 and 0* are as far from the center as possible means that under my thumb in the picture are the numbers 8 and 2*, and on the side with my index finger are 3 and 7*. Of course, due to symmetry, {2,8*} and {7,3*} also work this way. You may recall that these define the (n^3 mod 10) transform mentioned in my last post.

I also mentioned that n^3=n^7 (mod 10). I'm no mathematician, so, without proof, I'll just note that n^3=n^(3+4m) (mod 10) for m,n>=0. Notice (3+4m)=3,7,11,15,19,.. . To get even more numerological for just a moment (let the eye-rolling commence), 3 and 11 are the number of spatial dimensions and string-theory dimensionality respectively, and 3 and 7 are Western holy numbers as well as a pair in the transform. 15 is half of the collective 30 numbers. 19 is important in this conproject, but for other reasons.

Perhaps some one or more of these associations and forms could inform inferences made about the weird and informal system mentioned in the OP. Oh well, I'm noting them anyhow.
Last edited by dwnielsen on 08 Feb 2011 01:38, edited 2 times in total.
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Re: Conmath base-10 permutation

Post by eldin raigmore »

(1) This looks very interesting; I look at it every time I log on to the CBB.
(2) After several such looks, I still have no f***ing idea what you're talking about.
(3) I guess I'll keep trying.
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Re: Conmath base-10 permutation

Post by Micamo »

As far as I can tell by the "unit circle" thing, their visual field is distorted by applying the complex map z->z^3. The first question to ask is why the heck this would evolve in the first place!
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Re: Conmath base-10 permutation

Post by dwnielsen »

@eldin: Thank you for letting me know how this is perceived. This thread is kind of a public journal in which I've tried to make things clear enough so that, were I to look at this 5 years from now, perhaps I could piece together what the hell I was talking about. I'm assuming a little familiarity with some math concepts. Not all terminology is always being used correctly, and sometimes it takes me a little while to get clear myself. I am also trying to state the motivation for why I was doing these things in the first place. Since initial motivating notions often are not always specifiably logical, this may also contribute to some confusion.

If there are some particular terms I can clarify, please let me know. This may be like a TV show in which some episodes may make more sense out of context than others. Regarding ideas, I've had a lot of loose-hanging threads over the years, so I may pull them in from time to time. Some of them may be based on rather in-depth papers I've read and studied in the past. If so, I'll try to reference them. Also, some loose threads may just be cut after some time, if they don't work themselves in.

@Micamo: I can see how you might think that, but there's no complex map z->z^3. Instead, it's actually just a transformation of the individual digits in a decimal number, where 2->8; 3->7; 7->3; 8->2.
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Re: Conmath base-10 permutation

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As Micamo queried, why would such a system have evolved in the first place? I don't know that it evolved necessarily in the biological sense. The Kiubrutes may simply be us, but as we view each other through a special lens. (OT: Did anyone see that episode of Futurama in which the characters visit another "dimension" in which flipped coins land on the opposite side from ours? :-) ) I tend to think that things are "allowed" by an imagined universe as long as they are logically consistent. These allowances may be greater than we currently understand. It is interesting sometimes to explore these limits, especially if interesting symmetries and properties are discovered along the way.

However, there may be some constraints here. I was looking on a math forum the other day for a little clarity on the implications of this system, and the moderator over there (a very insightful person by the screenname of CRGreathouse, and oddly enough a member of ZBB) helped me a bit. Here are a couple of significant points he made.

____________

CRG: ..It's easy to show that 1.999... = 2 (where the "..." means an infinite number of decimal places and we're implicitly working in the real number system). They would write this as 1.999... = 8, so now a number like 4 is both less and greater than 8.

ME: Ah, how true! Might it be possible analytically to define 1.999..=8 (and do the same with other otherwise-inconsistent values) after the transform (in Kiubrutanica) and =2 before it?

CRG: Sure -- we already have one discontinuity with reality, and this seems the lesser of the two.

..

CRG: ..if there was contact trade would depend heavily on how this worked. If I could buy "3 pounds" (7 lbs) of gold and then sell 2 lbs ("8 pounds") of it back for more than the price I paid, I'd just keep doing that (and changing units appropriately) until I had essentially all of their gold (and they were happier for it!).

ME: In that case the quantity of gold is not conserved. Us: 7-2=5. Kiubrutes: -3+8=5 -> 5 (to us).
(To create conservation, one of the above results should be -5 instead.)

Since in general it seems there is not conservation of "information", then you're right, all sorts of "leaks" and "growths" occur in exchanges. But then perhaps there is the possibility of observation without communication, ie, only one side can observe the other. Another alternative might be that neither realm can observe any part of the other realm that has observed its own realm, meaning that as the realms continue observations on each other, their view of each other diminishes, unless the observations possible continue to increase. Or maybe there is another "correction" mechanism. :-s
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Re: Conmath base-10 permutation

Post by Micamo »

dwnielsen wrote:CRG: ..It's easy to show that 1.999... = 2 (where the "..." means an infinite number of decimal places and we're implicitly working in the real number system). They would write this as 1.999... = 8, so now a number like 4 is both less and greater than 8.
You can't do that. Just switching the referents of the label doesn't change anything, but here you're explicitly taking the axiom that the underlying referents are the same, which leads to a contradiction.
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Re: Conmath base-10 permutation

Post by dwnielsen »

I think what is required is that the observing side be distinguished from the observed side.

Quantities in the observed realm (if assuming we are the ones performing the observation, then this would be Kiubrutanica) will be marked with an asterisk (*).

It may seem inelegant to do this, but there are some reinforcing notions that support doing this.

(1) Communication does not appear to be possible between realms, instead, only 1-way observation. Therefore the observing and observed systems already seem to require some distinguishing.

(2) The following post will give a minor "practical justification" for doing this, I think.
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Re: Conmath base-10 permutation

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In the previous post, I said I was going to give a "practical justification" for making analytical distinctions on the "input" and "output" sides of the transform, but that may not happen in this post. Hopefully the next few posts will implicitly make that more justified, however.

Anyway, for some reason it occured out of the blue the other night that the sum totals of electrons as atomic subshells are filled (according to the Aufbau principle: 1s,2s,2p,3s,..), contain the numbers 10, 20, and 30 - viz they are 2,4,10,12,18,20,30,.. . (The subshells themselves contain 2,2,6,2,6,2,10,.. electrons.) This could be handy, since these initial 30 elements are the most essential to supporting complex life on Earth (in fact, element 30 is zinc, and then immediately after it follow elements that are much less biologically desirable for humans in anything more than very trace amounts). Even more strangely, the arrangement of the 20 consonants in the related conalphabet seems to reflect the electron filling sequence somewhat (but I hope to get to that later).

(Also possibly worth mentioning, the subshell capacities are 2,6,10,14,18,22,26. If we added 1 to each of these, we would get 3+4*m, and remember that n^3=n^(3+4*m) (mod 10).)

First, let's look back at the Möbius object comprised of two linked loops of paper (one large and one small) with numbers written on them. Recall that the larger loop had 2 sides (with 10 numbers on each side), and the smaller was a true Möbius strip with only one side (with 10 numbers). If we consider a "pair" of numbers to be made up of those numbers that are on the same position of the large loop strip, but written on opposite sides of the paper, then we can list all the numbers as the following:

Code: Select all

LARGE
0 5*
1 4*
2 3*<-
3 2*<-
4 1*
5 0*
6 9*
7 8*<-
8 7*<-
9 6*

SMALL
0**
1**
2**
3**
4**
5**
6**
7**
8**
9**

(<-The arrows indicate the {2,3,7,8}-positions, which are used by the transform mentioned in the previous posts.)

(* **Asterisks are used to keep track of the sides of the paper.) 
Let the 20 numbers on the large loop represent the first 20 chemical elements, viz H to Ca. Let the 10 elements on the small loop represent the next elements, viz the transition metals Sc to Zn.

In the above list, the 10 pairs each sum to 5 (mod 10). In chemical reactions, however, we often would like to sum the number of valence electrons to 2 or 8 (eg Al (3) + P (5) = AlP, aluminum phosphide). This is what I mean when I say valence, although the terminology may be technically imprecise. Let's set up these pairs as though each were a chemical bond producing a neutral compound:

Code: Select all

LARGE
Valence    Corresponding elements
1 1*       H  K*  (potassium hydride)
2 8*       He Ar* (noble helium,noble argon)
1 7*       Li Cl* (lithium chloride)
2 6*       Be S*  (beryllium sulfide)
3 5*       B  P*  (boron phosphide)
4 4*       C  Si* (carborundum)
5 3*       N  Al* (aluminum nitride)
6 2*       O  Mg* (magnesium oxide)
7 1*       F  Na* (sodium fluoride)
8 2*       Ne Ca* (noble neon,calcium ion)
You might notice that in each column of numbers above, we have a repetition of (1,2), viz the common sequence to both columns is 1,2,1,2,3,4,5,6,7,8. Let's redefine this as 9,0,1,2,3,4,5,6,7,8, calling these numbers val instead of valence:

Code: Select all

LARGE
Val        Corresponding elements
0 8*       He Ar* (noble helium,noble argon)
1 7*       Li Cl* (lithium chloride)
2 6*       Be S*  (beryllium sulfide)
3 5*       B  P*  (boron phosphide)
4 4*       C  Si* (carborundum)
5 3*       N  Al* (aluminum nitride)
6 2*       O  Mg* (magnesia)
7 1*       F  Na* (sodium fluoride)
8 0*       Ne Ca* (noble neon,calcium ion)
9 9*       H  K*  (potassium hydride)
So valence=val (mod 8) except for the cases of

Val Element
0...He
0...Ca

Helium is easily explained, because it lies in the first period of the element table, so is rightly mod 2, not mod 8.

Ca is not quite as simple. Like He, Ca also has valence 2. Since Ne is a noble gas, Ne and Ca cannot react anyhow, so it seems possibly acceptable to assign Ca a different value. But that's not a good plan, because Ca must be able to react with other elements; for this reason and to create symmetry of the neutrals (He, Ar, Ne), Ca was replaced with CaH2, calcium hydride (or hydrolith). Ca has an oxidation number of +2, and H has an oxidation number of -1 in this molecule, so CaH2 is neutral overall. Also, CaH2 reacts strongly with water to produce hydrogen. In this way, it sort of brings us full circle back to element H.

Code: Select all

LARGE
Val        Corresponding elements
0 8*       He Ar*   (noble helium,noble argon)
1 7*       Li Cl*   (lithium chloride)
2 6*       Be S*    (beryllium sulfide)
3 5*       B  P*    (boron phosphide)
4 4*       C  Si*   (carborundum)
5 3*       N  Al*   (aluminum nitride)
6 2*       O  Mg*   (magnesia)
7 1*       F  Na*   (sodium fluoride)
8 0*       Ne CaH2* (noble neon,hydrolith)
9 9*       H  K*    (potassium hydride)
The subshells are associated with val as the following (ellipses (..) indicate that the left edge is connected to the right edge):

Code: Select all

Filling order: 1s,2s,2p,3s,3p,4s 
        |  |           |  |
        V  V           V  V
  0  1  2  3  4  5  6  7  8  9
..__ _____ _________________ __..
 1s    2s          2p

  |  |           |  |
  V  V           V  V
  8* 7* 6* 5* 4* 3* 2* 1* 0* 9*
  _________________ _____ _____
         3p          3s    4s
Notice that the {2,3,7,8}-positions (those used in the transform) appear symmetrical, one set in shell 2, and the other in shell 3.

Let's try to add some more symmetry. Consider what happens if we "cut in half" the p subshell, defining subshell pa to contain the first 3 electrons in the p-subshell, and the other "half" of the p-subshell will be called subshell pb. So now pa and pb contain 3 electrons each, and the s-subshells still contain 2 each. If we add together an s-subshell with either pa or pb, we contain a total 5 electrons, viz we get the following:

Code: Select all

Filling order: 1s,2s,2pa,2pb,3s,3pa,3pb,4s 
        |  |           |  |
        V  V           V  V
  0  1  2  3  4  5  6  7  8  9
..__ ______________ ___________..
         2s2pa         1s2pb
    |     |        |     |
  | ||    |      | ||    |
  V |V    |      V |V    |
  8* 7* 6* 5* 4* 3* 2* 1* 0* 9*
..________ ______________ _____..
 4s3pb           3s3pa
In the above figure (sorry if it looks a bit busy), I find it interesting that each break in the line segments representing our newly defined "subshells" split the {2,3} and {7,8}-positions in the other sequence. I've tried to mark where these splits are by drawing lines in the figure.

What rationale might we have for defining these new "subshells"? I don't exactly know.
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Re: Conmath base-10 permutation

Post by Micamo »

I'm officially convinced you have no idea what this is either.
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Re: Conmath base-10 permutation

Post by dwnielsen »

Micamo, it's not all clear, but
(1) It could still be useful, even if it were just for book-keeping or memorization purposes, and it could still yield insight
(2) This could be considered an alternate interpretation of things, even if it wound up being, say, a more primitive model developed by a conculture
(3) I'm enjoying it

I'll try to state explicitly something that made all this seem interesting:

Take, for example, 64 (or 4^3).
64=4 (mod 10), so consider 64 to be like the number 4 for our purposes.

64^3 = 4 (mod 10)
64^(1/3) = 4 (mod 10)

So we see that if we cube or cube-root our number, we get the same value. In this way, it might be possible to consider dimensionality to be repetitive like a ring, not a simple progression. I'm not saying that's necessarily the best interpretation, but it seems a possibility.
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