COVER math

I had this idea for something I call COVER math, which is an acronym that stands for constants, operands, variables, expressions, replacements.
For constants you have a set perhaps infinite of values that I will write as lower case letters {a,b,c,...}
For variables you have a set perhaps infinite of symbols that can represent any value in the above set, written with Greek letters {alpha, beta, gamma, ...}
For operands you have a set perhaps infinite of one to one and onto relations from the set of constants to the set of constants represented by upper case letters {A,B,C,...}
An expression is any combination of operands working on variables or constants with parenthesis suggesting the order of operation e.g.

would be the constant a and b under the operand A, the result of which is under operand B with variable alpha. 

A replacement is a list of rules for rewriting an expression for example:

Example:

Ordinary algebra is a COVER math, using the real numbers as constants, variables having their usual meaning, operands being the usual plus, minus, multiplication, and division, and operations such as distributivity of multiplication written as replacements. 

A question is every type of math and logic a COVER math? 

2. A simple but much different example could be constants {a,b,c,d,e}, operands {A,B,C}, variable {Alpha}, and no replacement rules. to define the operands we can use a set of tables from {a,b,c,d,e}X{a,b,c,d,e} such as the following...

An example expression:

simplifies to:

and can be graphed as:

with a domain over alpha and range in {a,b,c,d,e}

art math music and invention part 2

A video I made showing stuff I've been working on the last couple months set to some music I made in Garage Band today...

Guass's circle problem revisited for about the tenth time

Gauss's circle problem is asking how does the number of lattice points inside or on a circle centered at the origin for a certain radius relate to pi*r^2, namely to find a bound in terms of r on how much must be added to pi*r^2.. Here for example is a radius 11 circle on a lattice...

My thought was to draw two polygons, they both have 2 sides along the radius of the circle ,the black  and green polygon starts 2 grid points below the blue at the top of the circle, moves only right or down (starting with right from 1 point below the top of the circle) and visits the nearest points inside the circle, and the blue polygon does the same but only visits points outside the circle starting from 1 unit outside the circle. ...

The black section will have a perimeter of 2*(radius-1) grid points and the blue will have 2*(radius+1) gridpoints. Remember for example the black border has to have the same perimeter as the green because they have to move the same amount moving only down and right no matter the order they do so in. From here I will consider the full polygons around and inside the circle which would be 4 times as large as each of these quarter arcs and then subtract 4 for the 4 overlapping points. . 

http://en.wikipedia.org/wiki/Pick's_theorem

Pick's theorem says the enclosed area of a polygon: A is related to the interior grid points i, and the number of points around the border b in the above way. I will use two different versions of this formula one for the blue polygon around the circle (variables sub b) and one for the black polygon (variables sub l ) and average the two formulas together to get a third. 

To make some substitutions to the bottom formula, I will suppose that the average value of the area of the (A[b]+A[l])/2 is (pi*r^2) because the outer is seemingly just as much outside the area of the circle as the inner one is inside.And I'll use 8*(radius+1)-4 for the b[b] and 8*(radius-1)-4 for b[l], and get:

Now, in our example r = 11. But see how i[b] should be all the interior points for r=11 and i[l] would be the interior points for a circle 2 units smaller in radius or r=9. So we might figure the average of i[b] and i[l] should be something close to the interior of a circle with r=10.  So substituting G[r] for the average interior and r+1 for r we get:

This figure is always too high because I took G[r] to be halfway between i[b] and i[l] but it will be actually closer to i[l] because i[b] and i[l] grow quadratically with respect to the radius of the circle.  

The question was how much more than pi*r^2 is this expression so subtract pi*r^2 from the above to get:

So it's roughly 1/4 the size of the upper bound Gauss found as r gets larger and dominates the expression.  

Beyond the part of the wikipedia article discussing Gauss's upper bound I don't understand the modern work that's been done on the subject so I don't know if this is the best that's been found or not. 

Low Kolmorogov complexity but never repeating series?

I'm looking at the series generated by starting with n=1 and make r equal to the remainder of the sum of the digits of n when divided by 5, plus 1. Finally make n = n+r.  So n goes:
1
then the sum of the digits of 1 is 1, and the remainder when divided by 5 is 1, plus 1 is 2. n  becomes n+2 = 3.

3
7
10
12
the sum of the digits of 12 is 3 so the remainder of when divided by 5 is 3 plus 1 is 4, n+4 = 16.
16
19
20
23
24
...
[1, 3, 7, 10, 12, 16, 19, 20, 23, 24, 26, 30, 34, 37, 38, 40, 45, 50, 51, 53, 57, 60, 62, 66, 69, 70, 73, 74, 76, 80, 84, 87, 88, 90, 95, 100, 102, 106, 109, 110, 113, 114, 116, 120, 124, 127, 128, 130, 135, 140, 141, 143, 147, 150, 152, 156, 159, 160, 163, 164, 166, 170, 174, 177, 178, 180, 185, 190, 191, 193, 197, 200, 203, 204, 206, 210, 214, 217, 218, 220, 225, 230, 231, 233, 237, 240, 242, 246, 249, 250, 253, 254, 256, 260, 264, 267, 268, 270, 275, 280, 281]
Just looking at r each time gives:
[2, 4, 3, 2, 4, 3, 1, 3, 1, 2, 4, 4, 3, 1, 2, 5, 5, "1", 2, 4, 3, 2, 4, 3, 1, 3, 1, 2, 4, 4, 3, 1, 2, 5, 5, "2", 4, 3, 1, 3, 1, 2, 4, 4, 3, 1, 2, 5, 5, 1, (2, 4, 3, 2, 4, 3, 1, 3, 1, 2, 4, 4, 3, 1, 2, 5, 5, 1, 2, 4, 3, 3), 1, 2, 4, 4, 3, 1, 2, 5, 5, 1, 2, 4, 3, 2, 4, 3, 1, 3, 1, 2, 4, 4, 3, 1, 2, 5, 5, 1]
See the first 18 numbers in r almost repeat but the 18th number is "1" and the 36th number is "2", then in parenthesis is the first 22 numbers but the last number is different.

These runs of repeating numbers get longer and longer but the series as a whole doesn't repeat. For example, looking at the first 51,000 numbers the first 1,000 repeat starting at:
repeats

1800,5040,8280,11520,13320,14760,16560,18000,19800,23040,26280,29520,31320,32760,34560,37440,40680,43920,45720,47160,48960,50400
There doesn't seem to be a consistent long range pattern of these numbers repeating. But there are many interesting patters, such as if we look at when the first 10,000 numbers repeat we find:
18000, 50400, 82800,... basically 10x as spread out as the repetition of 1000 numbers, so there is a sort of fractal arrangement of runs of numbers.

Of the first 1,000,000 numbers in the series, the distribution of 1's, 2's,3's, 4's, and 5's are:
[222,223, 222,222, 222,223, 222, 222, 111,110]
So there are almost exactly as many of every number but 5, which there are almost exactly half as many.

I say this series has low kolmogorov complexity because series with lots of runs of the same numbers can be written shortly by saying when these long runs occur and writing it once, and it works on every scale on this series, but I think it never does repeat exactly which makes it interesting as well.

Here's a picture of the first million terms of r, the first 1000 on the bottom row, the next 1000 on the next row up from left to right, with 1 as black, 2 as red, 3 as green, 4 as blue and 5 as white.

Zooming in somewhere near the middle:

and zooming about 8 times further in somewhere near the middle:

Maps can be rectangularized maintaining borders

You can take any map, say this one of the southeast U.S....

And do what I call rectangularizing it maintaining borders. So the first step is to draw every border including the ones with the blank area of the map as straight lines. A peninsula such as Florida is made into a triangle. 

The distinct regions, in this case states can be labeled or numbered and removed from the original map. 

Now imagine you draw an infinite number of lines across this map from north to south, and group together any infinite collection of lines that cross through the same regions in the same order. For example one group will be:

Every line drawn between these two north south lines (they're supposed to be parallel but I was drawing by hand) passes through region 11, then 7, then 3, then 1. So doing this all the way across the map...

These are supposed to be parallel but I think you get the idea...

And now the next step is to do the same thing but with horizontal lines...

In this case you end with a 5x12 grid where each cell contains a part of 1 region or none at all or it is split by a diagonal line in which case you can draw extra lines and give the left half to one region and the right half to the other region. So This can be translated to a grid...

This is the rectangularized version. You can see that for instance from a part of the 8 region you can go north to get to 11 from another part you can only get to 10 from going north, etc... And I think every map can have this done to it. I'm not sure if it could lead to an easier proof of the four colour theorem or not...

driveby

I took a video of driving past these five houses, an individual frame of the video looked like:

And I took the single middle column from each frame and put them together into one image to get:

So the individual column on the far right was actually taken a minute later after driving down the street than the one on the far left.

Bearing/Distance curve

Suppose a curve C has a certain starting point in a chosen coordinate system. For example -3, 2 in Cartesian coordinates.

My idea was that you could have a Bearing/Distance plot that for a curve without discontinuities takes the form of another smooth curve Bd

The distance is an amount d that is the arclength of the curve C, and the bearing varies mod 2 between 0 and 2 and indicates the direction the tangent line to the curve C is pointing when the arclength equals d.
The bearing direction is mapped as follows.

 But any direction within the range -inf..inf is valid the direction is mod 2, but a value of 6 might indicate that you've turned a full circle 3 times. So Bd generates a curve C as shown.

See the red curve C starts heading in direction 0 (also direction 2) but rapidly turns it's bearing until it is facing in the 1 direction, stays heading in that direction for about 3 arclength and then gradually turns until it is not quite facing in the 0 or 2 direction.

I think that one nice thing about this setup is that because straight lines in the Bd plot generate circles of varying diameter, the curvature of the C plot at a certain point is the slope of the Bd plot at the corresponding point, with decreasing curvature in the C plot corresponding to a flatter line in the Bd plot.