Following on from a previous post (http://tglad.blogspot.com/2010/07/that-crazy-formula.html) it is an interesting exersize to push the number of operators back another notch. Here's a somewhat speculative idea.
Take the ultra-operators of the previous post and increase their position by one, so operator 1 is equality, 2 is addition, 3 multiplication etc. Then make the 0th operator the comma operator.
The previous post talked about having a number-of-values (rather than a set of), this concept is defined by the use of ',' to allow more than one. So the value of a,b is simply the two values a and b, i.e. well it is just a,b i.e. it doesn't really operate on the two value other than to allow them to exist side by side.
This could extend the previous crazy formula to:
e (4) i (3) pi (2) 1 (1) 0 (0) .
i.e. e^i*pi + 1 = 0,.
where . is the empty element, or void, or the contents of the empty set.
It leads to a new creation story for numbers (going from right to left in above equation)...
In the beginning there was nothing (void)
For there ever to be more than this, the concept of a plurality (',' operator) is added
The first value can now exist alongside void, and since there is no arithmetic yet it is and can only be nothing/zero
Being a proper value it equates to something, hence the '=' operator can exist
The existence of equality allows for inequality (something that doesn't equal 0), we name that something 1, the unit.
What is to 1 as 1 is to 0? this recursive question leads us to define the set of integers and consequently the addition operator. (and inversely the subtraction operator as seen by seeing the above equation as -1 + 1 = 0)
Repeated addition on the integers leads to a definition for the '*' operator (and inversely divide) which leads to the quotient numbers.
This somehow leads to a new type of number not represented so far, pi the architypal transcendental.
Repeated multiplication leads to exponentiation operator
This (and its inverse, logarithm) require the new number e
Exponentiation leads to the question x^2 = -1 what is x? which leads to the development of the imaginary number i.
Writing this made me think, most operators come in pairs, so what are the inverses of ultra operators 0 and 1?
I would guess that the inverse of equals is 'doesn't-equal' (!=), but what does it evaluate to? Well wrap the values inside sets and '=' becomes the intersection operator I think.. so '!=' would probably be the equivalent of union minus intersection, or a xor b.
Harder it the ',' operator, what is its inverse? (call it ';') what is a;b?
My guess at the correct answer is that it is its own inverse (or equally it has no inverse). What do you think?
Monday, September 6, 2010
Thursday, September 2, 2010
a 3d mandelbrot
This is an idea for a mandelbrot set taken to the next dimension.
The boundary of the mandelbrot set can be thought of as a set of bifurcation points, each bifurcation point represents a stable orbit, the stable orbit takes up a small space so variations of it allow an expanded attraction basin, these are seen as the disks that sit off the main cardoid. The stable orbits can be equally considered to be polygons.
From the first disk clockwise towards the real axis the largest disks come off at the main bifurcation points that represent orbits that are: a line, a triangle, a quadrilateral, a pentagon, hexagon etc. They aren't regular polygons. Other smaller disks appear at stars and more complex polygons.
A 3d equivalent of the boundary would be a set of points that represent the stable polyhedrons that exist. Thus you may expect the bulbs to appear at a plane, tetrahedron, cube, octahedron etc.
In order for an iteration to trace out a plane it needs to operate with a full orientation and the iteration needs to branch, in fact produce two children each iteration to represent the 2 dimensions of the surface of a polyhedron. Inside would be if all branches stay bounded.
An equivalent of the mandelbrot would also have universal properties, one necessary condition seems to be the doubling of the angle.
What would such a set look like? well you would expect offshoot bulbs at distorted versions of all the regular polyhedrons and stellated polyhedrons. It seems like the number of bulbs will be far fewer as stable polyhedrons are rarer than polygons. It also seems like the large bulbs will be finite in size, so the surface will be quite different to the mandelbrot set.
Here is an attempt at the formula, which at least gives an indication of how such a mandelbrot might be constructed:
Julia set:
The julia branching iteration requires a rotation for each branch, which is 2 quaternions, so the map is 8D.
branch1: q1 = q1*q1 + c; q2 = q2*q1 + c;
branch2: q1 = q1*q2 + c; q2 = q2*q2 + c;
The idea is that each iteration the orientation (representing an polyhedron face) is rotated in two different directions, deformed versions of all platonic solids should be possible from this if c=0. If not then this formula is wrong.
Mandelbrot set:
the 4d set of points c for which the corresponding julia set is connected
A problem with this is that the set is intractable, the computation doubles each iteration.
Another construction may be mathematically simpler and in this way may be more likely to be the correct extension...
Instead of quaternion pairs you operate on octonions O, so the julia set needs to produce two versions of O^2 (for branch1 and branch2), the first uses just the octonion multiply. The second is the octonion multiply but with the first 4 and last 4 components swapped. This recognises the fact that the octonion is two quaternions that have equal precedence.
The boundary of the mandelbrot set can be thought of as a set of bifurcation points, each bifurcation point represents a stable orbit, the stable orbit takes up a small space so variations of it allow an expanded attraction basin, these are seen as the disks that sit off the main cardoid. The stable orbits can be equally considered to be polygons.
From the first disk clockwise towards the real axis the largest disks come off at the main bifurcation points that represent orbits that are: a line, a triangle, a quadrilateral, a pentagon, hexagon etc. They aren't regular polygons. Other smaller disks appear at stars and more complex polygons.
A 3d equivalent of the boundary would be a set of points that represent the stable polyhedrons that exist. Thus you may expect the bulbs to appear at a plane, tetrahedron, cube, octahedron etc.
In order for an iteration to trace out a plane it needs to operate with a full orientation and the iteration needs to branch, in fact produce two children each iteration to represent the 2 dimensions of the surface of a polyhedron. Inside would be if all branches stay bounded.
An equivalent of the mandelbrot would also have universal properties, one necessary condition seems to be the doubling of the angle.
What would such a set look like? well you would expect offshoot bulbs at distorted versions of all the regular polyhedrons and stellated polyhedrons. It seems like the number of bulbs will be far fewer as stable polyhedrons are rarer than polygons. It also seems like the large bulbs will be finite in size, so the surface will be quite different to the mandelbrot set.
Here is an attempt at the formula, which at least gives an indication of how such a mandelbrot might be constructed:
Julia set:
The julia branching iteration requires a rotation for each branch, which is 2 quaternions, so the map is 8D.
branch1: q1 = q1*q1 + c; q2 = q2*q1 + c;
branch2: q1 = q1*q2 + c; q2 = q2*q2 + c;
The idea is that each iteration the orientation (representing an polyhedron face) is rotated in two different directions, deformed versions of all platonic solids should be possible from this if c=0. If not then this formula is wrong.
Mandelbrot set:
the 4d set of points c for which the corresponding julia set is connected
A problem with this is that the set is intractable, the computation doubles each iteration.
Another construction may be mathematically simpler and in this way may be more likely to be the correct extension...
Instead of quaternion pairs you operate on octonions O, so the julia set needs to produce two versions of O^2 (for branch1 and branch2), the first uses just the octonion multiply. The second is the octonion multiply but with the first 4 and last 4 components swapped. This recognises the fact that the octonion is two quaternions that have equal precedence.
Measuring those awkward things
Imagine what the world would be like if only knew about whole numbers... no fractions or decimals. It would be very hard to live the life we do now without using continuous 'real' numbers, I think you would agree. Imagine the difference it makes to every part of study and work to be able to use these in-between, fractional numbers.
Now imagine yourself in the future looking back at today, a time when we only ever measure things using a whole number of dimensions. We can only ever measure the length of a river by pretending it is measured in metres^1, we can only count the branches on a tree by choosing a minimum size to be considered a branch.
In the real world most things do not have an integer number of dimensions, and pretending they do leads to problems in measuring them, an example of this is if you look up the length of Britain's coastline, you should find the quoted figures vary hugely.
Most things in the real world such as clouds, trees, land, water surface have a fractional number of dimensions.
OK, it seems reasonable to argue that nothing in reality is a fractal because nothing keeps its detail to infinite depth (to its atomic structure and beyond). However, the same argument can be used for whole numbers of dimensions, nothing in nature is made of absolutely straight lines or perfectly smooth curves either. The fact is that in both cases we approximate the real world within the range that is important to our measurements.
How do fractal measurements work? well the theory seems to already be known, but not apparently well used. You can use the Hausdorff measure to find the 'size' of a fractal structure, so if a coastline has fractal dimension 1.2, then you can measure its size in metres^1.2, e.g. Britain may be 200,000m^1.2.
So what would the world be like if we all started using fractional dimensions? Here are some areas with made up examples:
Geography:
measure the area of mountainside in metres^2.3
count the boulders on a scree slope in metres^0.5
measure the size of a river system in metres^1.4
Meterology:
measure the cloud cover in humidity^1.3
measure the wind levels in (metres per second)^1.2
Town planning:
measure the number of roads in metres^1.4
Astronomy:
count the moon's craters in metres^0.5
count the number of asteroids in the asteroid belt in kg^0.6
Biology:
measure lung size in metres^2.5
measure the number of neurons in the brain or the size of the cortex (both fractal)
count the number of animals on the planet in kg^0.8
measure the size of a forest by more than just the number of trees, include all smaller vegetation.
Chemistry:
measure the speed of a chemical reaction in seconds^0.5
Maths:
compare the size of various algebraic fractals
Economics:
stockmarkets fluctuate in a fractal distribution, measuring the size of different distributions is helpful
Business:
company pay follows a power law, so is fractal, so can be measured.
In fact, the point of this long post is really to say that pretty much everything follows a power law of some kind, so the only sensible way to measure it is a with a fractal measure. Something that as a whole we really don't do at the moment.
Next post I will go through the maths of measuring things with power laws.
Now imagine yourself in the future looking back at today, a time when we only ever measure things using a whole number of dimensions. We can only ever measure the length of a river by pretending it is measured in metres^1, we can only count the branches on a tree by choosing a minimum size to be considered a branch.
In the real world most things do not have an integer number of dimensions, and pretending they do leads to problems in measuring them, an example of this is if you look up the length of Britain's coastline, you should find the quoted figures vary hugely.
Most things in the real world such as clouds, trees, land, water surface have a fractional number of dimensions.
OK, it seems reasonable to argue that nothing in reality is a fractal because nothing keeps its detail to infinite depth (to its atomic structure and beyond). However, the same argument can be used for whole numbers of dimensions, nothing in nature is made of absolutely straight lines or perfectly smooth curves either. The fact is that in both cases we approximate the real world within the range that is important to our measurements.
How do fractal measurements work? well the theory seems to already be known, but not apparently well used. You can use the Hausdorff measure to find the 'size' of a fractal structure, so if a coastline has fractal dimension 1.2, then you can measure its size in metres^1.2, e.g. Britain may be 200,000m^1.2.
So what would the world be like if we all started using fractional dimensions? Here are some areas with made up examples:
Geography:
measure the area of mountainside in metres^2.3
count the boulders on a scree slope in metres^0.5
measure the size of a river system in metres^1.4
Meterology:
measure the cloud cover in humidity^1.3
measure the wind levels in (metres per second)^1.2
Town planning:
measure the number of roads in metres^1.4
Astronomy:
count the moon's craters in metres^0.5
count the number of asteroids in the asteroid belt in kg^0.6
Biology:
measure lung size in metres^2.5
measure the number of neurons in the brain or the size of the cortex (both fractal)
count the number of animals on the planet in kg^0.8
measure the size of a forest by more than just the number of trees, include all smaller vegetation.
Chemistry:
measure the speed of a chemical reaction in seconds^0.5
Maths:
compare the size of various algebraic fractals
Economics:
stockmarkets fluctuate in a fractal distribution, measuring the size of different distributions is helpful
Business:
company pay follows a power law, so is fractal, so can be measured.
In fact, the point of this long post is really to say that pretty much everything follows a power law of some kind, so the only sensible way to measure it is a with a fractal measure. Something that as a whole we really don't do at the moment.
Next post I will go through the maths of measuring things with power laws.
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