Friday, June 28, 2013

Delta-characteristic of a graph

Suppose you have a graph like the following:
I am defining the Delta-characteristic to be, for each vertex set up a distance-like formula (hence the delta in the name) relating the vertex to the vertices it is connected to. For example C is connected to B,D, and J so one formula is:

I found that Maple solves a system of this type of equations readily but won't do anything with the system of actual distance formula equations, and I think this one is enough to accomplish what I need from it. 

Now since there will be one equation for every vertex the system can be solved:
and it turns out the solution is:
Since the labeling of the graph was arbitrary, it was only the way the vertices were connected that mattered, this result can form a list(ordered by magnitude and rounded to the digit necessary to distinguish the numbers):

Isometric graphs will form the same set, because the equations don't take into account the drawing of the graph, and the reordering of the set numerically will end up the same no matter the original labeling. 
I don't know whether two non-isometric graphs can have the same Delta-characteristic but it seems unlikely.

Another interesting aspect of the Delta-characteristic is consider this graph instead:

A vertex L has been added:
the Delta-characteristic works out to:

See the only difference is one more number in the list, and it is a repeat of 3.353, this is because vertex A and vertex L are indistinguishable in the graph apart from labeling, they are connected to the same vertices. 

Maybe I should explain some more, the way I see it the purpose of the square roots is to prevent mixing of the variables, like if I just used the equation A=B+C and B=A+D+E for example when the equations are solved the distinction between variables that occurred on the left side vs the right side and when equations are say, added the distinction between which vertices each variable are connected to are also lost. So the square root is sort of forming logical divisions between how the variables occur in the equations rather than for their numerical significance. 

Here is an tabulation of the Delta-characteristic of all the simple graphs with 5 vertices:
G1:  1     1     1.259 1.259 1.587
G2:  1     1     2     2     2
G3:  1.341 1.341 1.799 1.799 1.897
G4:  1.375 1.484 1.484 1.892 2.205
G5:  1.587 1.587 1.587 1.587 2.519
G6:  1.587 1.587 2.245 2.52  2.52
G7:  1.673 1.673 2.246 2.246 2.799
G8:  1,406 1.979 2.161 2.161 2.510
G9:  1.415 2.005 2,259 2.505 2.601 
G10: 2     2     2     2     2
G11: 1,640 2.368 2.692 2.804 2.804 
G12: 1.751 2.425 2.425 2.814 3.068
G13: 2.213 2.213 2.318 2.686 2.686
G14: 2.289 2.289 2.289 2.620 2.620
G15: 2.314 2.314 2.314 2.314 3.042
G16: 1.823 3.079 3.079 3.079 3.326
G17: 2.579 2.579 2.579 3.326 3.326
G18: 2.400 2.881 2.881 2.952 2.952
G19: 2.503 2.503 2.961 2.961 3.306
G20: 2.667 3.213 3.213 3.556 3.556
G21: 3.130 3.130 3.130 3.130 3.538
G22: 3.368 3.368 3.781 3.781 3.781 
G23: 4     4     4     4     4

As you can see, no two non-isomorphic graphs gave the same Delta-Characteristic. In fact only G1 and G2 have any numbers in common... 

Saturday, June 8, 2013

Golden frame

Suppose you bought a painting that the painter did in the Golden ratio of width to height. And you want to find a frame that does it justice. I suggest this:
The painting is the inner square and the frame is the outer. There are some nice things about this choice of frame here's the math:
Little w and h are given by the painting being in the golden ratio, but here the frame is in golden ratio by area, the big width times the big height is exactly the golden ratio times the area of the small frame(w*h*1.618=WH). And nicely that makes the big width almost precisely double the small height. There's sort of an optical illusion going on that the base looks almost 2.5 times the small height but if you measure it is really close to double.
TLDR; The two boxes are in the golden ratio by their lengths to widths and also comparing their areas to each other. And that makes it so the bottom of the big rectangle is almost exactly double the side of the small rectangle.

a way to get close to big square roots in your head

First take your number, say 25347 and round that a bit to
note that the 10^4 is an even power of 10 so use this formula:
so .8 + (1/3)2.5 = 1.63 and subtract .0625 to get 1.57
for every even power of 10 above 0 move a decimal place to the right, so since 25347 was *10^4 move 2 decimal places over. That gives you 157 which is close to the square root of 25347 (it is actually 159.2)

For odd powers such as 253478
use this formula:
2.4+1.1*x - (1/30)*x^2
so start the same round the number to
2.5 *10^5
2.4 + 1.1*2.5 = 2.4 + 2.75 = 5.15 and then do 1/3 of 2.5 .83 and shift to .083, 5.15-.083 = 5.067
then shift the decimal to the right for every odd power of 10 over 1, 506 the answer is actually around 503.

So with a little practice one can get pretty good at getting close to square roots in one's head.

Why it works:
Here's a graph of the function I listed vs. the square root from 1 to 10
The shape of the graph repeats for 100-1000 with the y axis multiplied by 10
Every even power of 10 such as 4.5*10^4 looks just like 4.5 but with the axis multiplied by a 100. 

The other function works the same way. 

Friday, June 7, 2013

Modulus algorithm

Suppose you want to see if 897354 divides 54
the leading eight of 897354 has 5 digits to the right of it, so looking at the fraction 8/54
.14814814 we can just look at the fractional part when this decimal is shifted to the right 5 places
14814.814 and just the part to the right of the decimal is .814 (rounded to thousandths)
The 9 in 897354 has 4 digits to the right of it so 9/54 is:
.166666 and shifting the decimal 4 digits to the right and dropping the whole number part is .666
so so far we have .814, .666 and doing the same thing for the rest is .63, .555, .925, .074
A fast way to do find these numbers is to calculate 1/54 and then multiply by 8,9,7,3,5,4
We can add these 5 numbers and get: 3.664 the part to the right of the decimal is .664 which is within a couple thousands of the remainder when 897354 dividing 54 = .666 so 897354 does not divide 54 (it has a remainder of 2/3rds.

The nice thing about this algorithm is it takes a process that is usually serial in nature, division you usually do long division from left to right and makes it parallel, if you had 10,000 digits of a number you were dividing you could send to 10,000 computers for example if the 573rd digit was 9,
573, 9, 4714
which would tell it return the 573rd place of the decimal 9/4714:see section B,   and it would return something and then one thread could add up all the returned numbers to find out whether the 10,000 digit number divides 4714.

-B- to be completed

Monday, June 3, 2013

Universal alphabet

Here is a sample of how it looks written...
This says: "My name is Ben."

It looks cryptic but it is really simple, every individual sound is a group of four symbols in a square such as in the above:
This is the B sound in "Ben". The top left symbol in this square is one of x,1,2,3,4,5 and indicates how open the lips of the mouth are, x for closed and 1,2,3,4,5 for progressively more open. If the lips are rounded to form the sound there is a circle around this symbol. As you round and close the lips to form the b sound this is an x with a circle around it.
The bottom left symbol is a letter one of a,b,c,d,e,f,g and indicates where your tongue is in your mouth as the sound is formed. a is top of the mouth against the teeth, b is middle of the roof of the mouth, c is towards the back and roof of the mouth, d is forward near the teeth but halfway between the bottom and top of the mouth, e is back middle of the mouth, and f is bottom and forward against the teeth and g is bottom of the mouth back away from the teeth.
The top right symbol indicates the tone of voice as in music, a is the neutral note and b,c,d,e up to a gain are progressively higher tones of voice.
The bottom right symbol indicates how open the jaw is x for closed and is circled if the throat is closed as in "guh". , 1,2,3,4,5 for more and more open.
A word like "Ben" has a beginning, middle, and end sound hence three groups of four symbols make up the word. If the word is multiple syllables a comma is placed between syllables. Also the stressed syllable of the word has an apostrophe following it.

And that is all there is to it, with this way to describe mechanically the way the lips, tongue, jaw, and voice box are acting every dialect of every language can be written phonetically.