Posts Tagged ‘B’

About Perspective

Thursday, February 2nd, 2012

Effective Ways To Improve Production Efficiency For Your Manufacturing Business

While all business owners should be taking measures to enhance efficiency, as this aspect of business growth is of extreme importance regardless of the industry, manufacturing businesses should take measures to boost production efficiency for several other reasons. These reasons range from improving workplace safety to saving on costs and everything in between. Manufacturing businesses that claim the lion’s share of the market are always searching for new ways to improve production efficiency with the intention of ultimately improving their bottom line.

So, to ensure your manufacturing business is constantly making notable strides towards improving production efficiency, we have rounded up five practical ways for you to achieve this. With that said, these five solutions break down the five fundamental focus points of manufacturing efficiency. Therefore, you can use these practices for years to come.

Waste in the manufacturing industry is a cost that should be avoided. And waste is also a broad term in this industry as it refers to employee hours, energy consumption, materials, and a few others. However, material waste is usually the biggest concern in the manufacturing industry. When searching for ways to reduce waste, you will need to utilize your conclusions from evaluating your production line, learn more detailed information from this top tier short cut fibers manufacturer.

When identifying waste, each of your production line processes will create some waste; identifying the processes responsible for making the most waste is the best way to reduce overall waste. However, you can also reduce waste by recycling or reusing waste instead of dumping it. You could also consider selling your waste to a business that can properly use it if you cannot eliminate the bulk of your waste.

The production line in your manufacturing business is the core of your business functions. When evaluating the production line, you should assess all the details as well, including the commonly overlooked details, such as loading dock bumper and other essential loading dock equipment your business needs. When it comes to sourcing and maintaining loading dock bumpers, you can consider Miner Corp or other leading industry service and equipment providers to ensure your production line has everything it needs to function as it should.

Throughput is another primary focal point to consider; you need this metric to measure the number of units being produced in a certain period on average. While quality equipment is crucial in ensuring your production line can run smoothly, evaluating your throughput will help you identify issues in your production line on occasions when the throughput is not entirely up to par. Capacity utilization is another element of your production line to evaluate, and this refers to calculating your factory’s total output capacity. This will allow you to determine your production line’s performance at all times.

While you are evaluating your production line, you will probably discover the most significant breakdowns in production. These are otherwise referred to as bottlenecks. Because having to shut down operations even for half a day can have a massive impact on your business’s reputation and overall profits, identifying sources of breakdowns and taking measures to prevent them from reoccurring is vital to all companies operating in the manufacturing industry.

About Number and Magnitude

Monday, January 9th, 2012

We have lost the relationship between Number and Form or Number and Magnitude as the Ancient Greeks called their Forms.

A few years ago a Revolution in Mathematics and Physics has started. This revolution is caused by Geometric Algebra.

In Geometric Algebra the Ancient Theories of Euclid and Pythagoras are reevaluated.

Numbers are Scalar (Quantum) Movements of Geometric Patterns and not Static Symbols of Abstractions that have nothing to do with our Reality.

Movements and not Forces are the Essence of Physics.

The basic rule Movement = Space/Time (v=s/t) shows that  Time and Space are two Reciprocal 3D-Spaces. Our Senses Experience Space and not Time.

The Simple Rule N/N=1/1=1 balances the Duals of Space and Time. One Unit Step in Space is always Compensated by One Unit Step in Time.

Geometric Algebra has a strange relationship with Pascals Triangle. This Triangle, also called the Binomial Expansion, contains all the Possible Combinations of two Independent Variables. Our Universe is a Combination of Combinations exploring Every Possibility.

The last and perhaps most important Discovery in Mathematics called Bott Periodicity shows itself in Pascals Triangle.

Bott Periodicity proves that we live in a Cyclic Fractal Universe, the Wheel of Fortune, that is Rotating around the Void, the Empty Set. The Empty Set contains Every Thing that is Impossible in our Universe.

This blog is not a Scientific Article. I have tried to connect the Old Sciences and the New Sciences in my own Way.

It contains many links to Scientific Articles and even Courses in Geometric Algebra.

So if you want to Dig Deeper Nothing will Stop You.

About the One and the Dirac Delta Function

Every Thing was created out of  No Thing, the Empty Set, ɸ, the Void, the Tao. The Empty Set contains 0 objects.

The Empty Set is not Empty. It contains Infinite (∞) Possibilities that are Impossible.

Every impossibility has a probability of 0 but the sum of all possibilities (1/∞=0) is always 1. In the beginning ∞/∞ =1  or ∞x0=1.

This relationship is represented by the Dirac Delta Function. It is used to simulate a Point Source of Energy (a Spike, an Explosion) in Physics.

The Delta is reprented by the Symbol Δ, a Triangle. The Delta is called Dalet in the Phoenican and Hebrew Alphabet. Daleth is the number 4 and means Door.

The original symbol of the Delta/Daleth contains two lines with a 90 Degree Angle. Two orthogonal lines create a Square or Plane.

The Dirac Delta Function is defined as a Square  with an Area of 1,  a Width of 1/n and a Height of n where n->∞.

The Dirac Delta Function is a Line with an Area of 1.

In the Beginning a Huge Explosion took place that created the Universe.

The Dirac Delta Function δ (x) has interesting properties: δ (x) = δ (-x), δ (x) = δ (1/x). It has two Symmetries related to the Negative Numbers and the Rational Numbers.

When we move from 2D to 1D, the Number Line, the Delta Function becomes the Set of the Numbers N/N =1.

The Tetraktys of Pythagoras

The Monad (1) of the Tetraktys of Pythagoras, the Top of the Triangle, was created by Dividing the One (1) by Itself without Diminishing itself. The Monad (1/1=1)  is part of  the 1D Delta Function.

Creation is an Expansion of the 1/1 into the N/N, adding 1/1 all the time,  until ∞/∞ is reached. At that moment every Impossibility has been realized.

File:Dirac function approximation.gif

The Dirac Delta Pulse

 

To move Back to the Void and restore the Eternal Balance of  the One,  Dividing (Compression) has to be compensated by Multiplication (Expansion).

At the End of Time N/M and M/N have to find Balance in the N/N,  move Back to  1/1, Unite in the 0 and become The Void (ɸ) again.

About the Strange Behavior of Numbers

The big problem of the Numbers is that they sometimes behave very differently from what we Expect them to do.

This Strange Behavior happens when we try to Reverse what we are doing.

It looks like the Expansion of the Universe of Numbers is Easy but the Contraction creates many Obstacles.

It all starts with the Natural Numbers (1,2,3,).

When we Reverse an Addition (Subtract) and move over the Line of the Void Negative Numbers appear. Together with the Natural Numbers they are called the Integers.

The same happens when we Reverse a Division and the Fractions (the Rational Numbers) (1/3, 7/9) suddenly pop up.

An Integer N is a Rational Number divided by 1 (N/1).

The Integers are the Multiples of 1, the Fractions are its Parts.

Numbers behave even stranger when we want to Reverse a Repeating Repeating Addition (Irrational Numbers) and want to calculate a Rational Power (2**1/2).

The Complex Numbers (or Imaginary Numbers), based on the Square Root of -1 called i, are a combination of the Negative Numbers and the Irrational Numbers.

Irrational Numbers ( the Pythagorean Theorem), Fractions (a Piece of the Cake) and Negative Numbers (a Debt) are part of our Reality but the Strange Number i represents something we cannot Imagine.

About the Duality and the Expansion of Space

In the beginning the only One who was in existence was the 1.

When the One divide itself again the number -1, the Complement of 1, came into existence.

1 and -1 are voided in the No Thing, the Empty Set, 0:  -1 + 1 = 0.

The Two, the Duality, both started to Expand in Two Opposite Directions (<– and +->) both meeting in the  ∞/∞. This expansion is what we call Space.

Space is a Combination of the Strings S(1,1,1,1,1,…) and -S = (-1,-,1,-,1,-1,…) where S+S=(0,0,0,0,0,0,…).

The Expansion pattern of Space is a Recursive Function S: S(N)=S(N-1)+1 in which + means concatenate (or add) the String “,1″.

An Addition X + Y is a concatenation of S(X) and S(Y). A Substraction X-Y is a concatenation of S(X) and -S(Y). In the last case all the corresponding combinations of 1 and -1 are voided. (1,1,1,1)-(1,1,1)=(0,0,0,1)=(1).

Multiplication XxY is Adding String S(Y) every time a “1″ of S(X ) is encountered: 111 x 11 = 11  11  11. Dividing X/Y is Subtracting S(X) every time a “1″ of S(Y) is encountered:.111  111  1/111=11 1/111. In the last example a Fraction 1/111 appears.

This Number System is called the Unary Number System.

About the Trinity and the Compression of Space called Time

The Strange Behavior of Numbers is caused by the Limitations of our Memory System. We are unable to remember long strings that contain the same Number.

To make things easy for us we Divide Space into small Parts so we were able to Re-Member (Re-Combine the Parts).

When we want to Re-member, Move Back in Time, we have to Compress Expanding Space.

Compressed Space is Time.

Time and Space have a Reciprocal Relationship called Movement (Velocity = Space/Time).

There are  many ways ( (1,1,1), (1,1,1),..) or ((1,1),(1,1))) to Compress a String in Repeating Sub-Patterns.

In the blog About the Trinity I showed that the most Efficient Way to group the One’s is to make use of a Fractal Pattern (a Self Reference) and Groups of Three Ones.

The Trinity applied to the Trinity ( A Fractal) is a Rotating Binary Tree. Binary Trees represent the Choices we make in Life.

The rotating Expanding Binary Trees generate the Platonic Solids (see linked video!) when the (number)-parts of the Binary Tree Connect.

The Ternairy Number System is represented by the Binary Tree

When we connect Three Ones (1,1,1) by Three Lines (1-1,1-1,1-1) a 2 Dimensional Triangle Δ is Created.

If we take the Δ as a new Unity we are able to rewrite the patterns of 1′s and -1′s into a much Shorter Pattern of Δ’s and 1′s: (1,1,1),(1,1,1),(1,1,1), 1,1 becomes Δ,Δ,Δ,1,1.

We can repeat this approach when there is still a Trinity left: Δ,Δ,Δ,1,1 becomes ΔxΔ,1,1.

This Number System is called the Ternary Number System.

About Ratio’s and Magnitudes

According to EuclidA Ratio is a sort of relation in respect of size between two magnitudes of the same kind“.

A Magnitude is a Size: a property by which it can be compared as Larger or Smaller than other objects of the Same Kind. A Line has a Length, a Plane has an Area (Length x Width), a Solid a Volume (Length xWitdth x Height).

For the Greeks, the Numbers (Arithmoi) were the Positive Integers. The objects of Geometry: Points, Lines, Planes , were referred to as “Magnitudes” (Forms). They were not numbers, and had no numbers attached.

Ratio, was a Relationship between Forms and a Proportion was a relationship between the Part and the Whole (the Monad) of a Form.

Newton turned the Greek conception of Number completely on its head: “By Number we understand, not so much a Multitude of Unities, as the abstracted Ratio of any Quantity, to another Quantity of the same Kind, which we take for Unity”.

We now think of a Ratio as a Number obtained from other numbers by Division. A Proportion, for us, is a statement of equality between two “Ratio‐Numbers”.

This was not the thought pattern of the ancient Greeks. When Euclid states that the ratio of A to B is the same as the ratio of C to D, the letters A, B, C and D do not refer to numbers at all, but to segments or polygonal regions or some such magnitudes.

The Ratio of two geometric structures  was determinated  by fitting the Unit Parts of the first geometric Stucture into the Other.

The Perfect Triangle of the Tetraktys contains 9 = 3x3 Triangels. A Triangle contains 3 Lines and 3 Points.

An Example:  The Tetraktys is a Triangle (A Monad) and contains 9 Triangles (a Monad). The 1x1x1-Triangle Δ, a Part of the Tetraktys,  is Proportional to the Whole of the Tetraktys (T) and has a Ratio T/Δ = 3= Δ -> T = Δ (3)  x Δ (3) = 9.

The Mathematics of Euclid is not a Mathematics of Numbers, but a Mathematics of Forms.

The symbols, relationships and manipulations have Physical or Geometric Objects as their referents.

You cannot work on this Mathematics without Knowing (and Seeing) the Objects that you are Working with.

About Hermann Grassman, David Hestenes and the Moving Line called Vector

Hermann Grasmann lived between 1809 and and 1877 in Stettin (Germany). Grassmann was a genius and invented Geometric Algebra a 100 years before it was invented.

In his time the most important mathematicians did not understand what he was talking about although many of them copied parts of his ideas and created their own restricted version. None of them saw the whole Grassmann was seeing.

When he was convinced nobody would believe him he became a linguist. He wrote books on German grammar, collected folk songs, and learned Sanskrit. His dictionary and his translation of the Rigveda were recognized among philologists.

Grassmann took over the heritage of Euclid and added, Motion, something Euclid was aware of but could not handle properly.

angle between vectors in 2 dimentions

A Displacement or Bivector

Grassmann became aware of the fact your hand is moving when you draw a 2D Geometric Structure. He called the Moving Lines, that connect the Points, Displacements (“Strecke”).

screw theory 2

A Displacement and a Rotation of a Vector

In our current terminology we would call the Displacements “Vectors”.

blades

Vector algebra is simpler, but specific to Euclidean 3-space, while Geometric Algebra works in all dimensions. In this case Vectors become Bi/Tri or Multi-Vectors (Blades).

The Trick of Grassmann was that he could transform every transformation on any geometrical structure into a very simple Algebra. Multi-Dimensional Geometric Structures could be Added, Multiplied and Divided.

The Greek Theory of Ratio and Proportion is now incorporated in the properties of Scalar and Vector multiplication.

add-bivectors

Combining (Adding) Bivectors creates a Trivector

About a 100 years later David Hestenes improved the Theory of Grassmann by incorporating the Imaginary Numbers. In this way he united many until now highly disconnected fields of Mathematics that were created by the many mathematicians who copied parts of Grassmanns Heritage.

About Complex Numbers, Octions, Quaternions, Clifford Algebra and Rotations in Infinite Space

Grassmann did not pay much attention to the Complex Numbers until he heard of a young mathematician called William Kingdon Clifford (1845-1879).

Complex numbers are ,just like the Rationals (a/b), 2D-Numbers. A Complex number Z = a  + ib where  i**2=-1. Complex Numbers can be represented in Polar Coordinates: Z = R (cos(x) + i sin(x)) where R = SQRT(a**2 + b**2).  R is the Radius, the Distance to the Center (0,0).

When you have defined a 2D-complex Number it is easy to define a 4-D-Complex Number called a Quaternion:  Z = a + ib + jc + kd or a 8-D Complex Number called an Octonion.

William Rowan Hamilton, the inventor of the Quaternions, had big problems to find an interpretation of all the combinations i, j and k until he realized that i**2 =j**2 = k**2 = ijk=-1.

What Hamilton did not realize at that time was that he just like Grassmann had invented Vector Algebra and Geometric Algebra.

Quaternions are rotations in 4D-space

This all changed when William Kingdon Clifford united everything in his new Algebra.  Clifford’s algebra is composed of elements which are Combinations of Grassman’s Multivectors.

The Clifford Algebra that represents 3D Euclidean Geometry has 8 = 2**3 components instead of 3: 1 number (Point), 3 vectors (Length), 3 bivectors (Area) and 1 trivector (Volume).

It turns out if you use combinations of these elements to describe your geometric objects you can do the same things you did before (you still have 3 vector components).

In addition, you can have additional data in those other components that let you find distances and intersections (and a lot of other useful information) using simple and (computationally) cheap numerical operations.

The most important Insight of William Kingdom Clifford was that the Complex Numbers are not Numbers all.

They are Rotations in higher Dimensional Spaces.

About Pascal’s Triangle and Mount Meru

The String 1,3,3,1 of Clifford’s 3D Geometry is related to the 4th Level of Pascal’s Triangle. Level N of Pascal’s Triangle represents N-1-Dimensional Geometries.

The Sum of every level N of the Triangle is 2**N. This Number expresses the Number of Directions of the Geometric Structure of a Space with Dimension N.

A Point has 0 Direction, while a Line has 2 Directions, relative to its Center point, a Plane has 4 Directions, relative to its Center Point, and a Cube has 8 directions, relative to its Center point.

Mount Meru (Pascal's Triangle in Vedic Mathematics)

Pascal’s Triangle is also called the Binomial Expansion. This Expansion shows all the Combinations of two letters A and B in the function (A+B)**N. Level 1 of the Triangle is (A+B)**0 = 1  and level 2 is A x A + 2 A x B + B x B -> 1,2,1.

The Binomial Expansion converges to the Bell-Shaped Normal Distribution when N-> ∞.

The Diagonals of Pascal’s Triangle contain the Geometric Number Systems (Triangular Numbers, Pyramid Numbers, Pentatonal Numbers, ..) and the Golden Spiral of the Fibonacci Numbers.

Pascal’s Triangle is a Repository of all the Possible Magnitudes and their Components.

The Normal Distribution shows that the first level of the Triangle (the Tetraktys) is much more probable than the last levels.

The first four Levels of the Triangle of Pascal contain the Tetraktys of Pythagoras.

 

The Tetraktys  is an Ancient Vedic Mathematical Structure called the  Sri Yantra, Meru Prastara or Mount Meru.

About Numbers, Operations and the Klein Bottle

The Complex Numbers are not “Numbers” (Scalars) at all.

They are “Operations” (Movements) that can be applied to Magnitudes (Geometries) and Magnitudes are Combinations of the Simple Building Blocks of the Tetraktys, Points and Lines.

The Tao of Ancient China was not for nothing represented by a Flow of Water. According to the Ancient Chinese Mathematicians Every Thing Moves.  In the Beginning there was only Movement.

In the Beginning only the One was Moved but when the Duality was created the Two moved around each other never getting into contact to Avoid the Void.

When we look at the Numbers we now can see that they are the result of the Movements of  the first Diagonal of Pascals Triangle,  the 1′s (Points) or better the Powers of  the One: 1 **N (where N is a Dimension).

Even in the most simple Number System, the Unary Number System, Concatenation is an Operation, An Algorithm.

The Mathematician John Conway recently invented a new Number System called the Surreal Numbers that contains Every Number you can Imagine.

The Surreal Numbers are created out of the Void (ɸ)  by a simple Algorithm (Conway calls an Algorithm a Game) that describes Movements (Choices of Direction: Up, Down, Left, Right, ..)  that help you to Navigate in the N-Dimensional Number Space.

The Ancient Chinese Mathematicians played the same Game with the Numbers.

Algorithms were already known for a very long time by the Ancient Vedic Mathematicians. They called them Yantra’s.

SriYantra

Sri Yantra

Geometry is concerned with the Static Forms of Lines and Points but there are many other more “Curved” forms that are the result of  Rotating Expansion and Compression. These forms are researched by the modern version of Geometry called Topology.

The most interesting 4D Topological Structure is the Klein Bottle.  The Klein Bottle is  a combination of two Moebius Rings. It represents a Structure that is Closed in Itself.

It can be constructed by gluing both pairs of opposite edges of a Rectangle together giving one pair a Half-Twist. The Klein Bottle is highly related to the Ancient Art of Alchemy.

The movement of the Duality around the Void can be represented by a Moebius Ring the Symbol of Infinity ∞.

Later in this Blog we will see why the Number 8 is a Rotation of ∞ and the symbol of Number 8 is a combination of the symbol of the number 3 and its mirror.

First we will have a look at the Reciprocal Relation between Space and Time.

The Klein Bottle, The Universe Closed in Itself, the Basic Structure behind Alchemy.

The Klein Bottle, The Universe Closed in Itself, the Basic Structure behind Alchemy.

About Dewey B. Larson, Velocity and Time

Dewey B. Larson (1898 – 1990) was an American Engineer who developed the Reciprocal System of Physical Theory (RST).

Larson believed that the failure to recognize that Motion is the most basic physical constituent of the universe has handicapped the progress of the traditional study of physics, which focuses on Forces.

The definition of Motion stems from the Equation of Velocity, v = ds/dt.

Instead of depending upon the change of the location of an object to define an arbitrary “quantum” of space per “quantum” of time, such as miles per hour, or meters per second, the RST assumes that the observed universal passage, or progression, of time is one aspect of a universal motion that necessarily must be accompanied by a universal “passage,” or progression, of space.

The Units of Time fill up the Units of Space. Space and Time are Duals.

Space is not-Time and Time is not-Space. Time is Non-Local, Cyclic and represented by the Rotating Imaginary Numbers. Space is Local, Linear and Represented by the Scalar Numbers. Space is the Vacuum and the Nothing and Time is the non-vacuum, the Every Thing, the Solids represented by the Cube of Space.

The Cube of Space is the structure behind the Tetraktys but also behind the Book of Genesis.

Our Reality contains two Reciprocal 3D-structures related to Space and Time. Space and Time are related by the Simple Formula N/N=1/1=1, the Formula of Diracs Delta Function.

We are able to perceive the Real 3D-Structure of Space. The 3D-Structure of Time is Imaginary. It is situated in the Imaginary Number Space of i.

LarsonsScalarCube.jpg

Larson’s Cube, the Geometric Representation of the Octonion.

Larson, a Self Thought Genius like Grassmann, developed Geometric Algebra without knowing anything about Geometric Algebra but he also invented String Theory long before String Theory was invented.  The Mathematics of Larson is also the Mathematics of the Tetraktys of Pythagoras without even knowing anything about it.

wom_image3.jpg

The Periodic System of Larson

Larson was able to Calculate all the important Physical Numbers without any problem and was also able to Calculate Chemical Structures and Reactions.

About the Bott Periodicity

The fourth line of Pascals Triangle and the Tetraktys contains 8 Directions in the Four Geometric Dimensions: 0, 1, 2, and 3.

Mathematicians are intrigued with this number 8, because they find it popping up unexpectedly in advanced mathematics.

In fact, expanding the Binomial Expansion to 8 dimensions just creates an inverse copy of these first Four Dimensions, and then the pattern just repeats itself with a half-twist and back from there, ad infinitum.

This is called Bott Periodicity discovered by the mathematician Raoul Bott (1923-2005).

The mathematician John Baez wrote an article in which he relates this 8-fold Periodicity to the Scalars (1), the Complex Numbers (2), the Quaternions (2×2), and the Octonions (2x2x2 = 2**3).

Bott Periodicity

The Universe of Numbers and Magnitudes  is Cyclic and Fractal.

Our own Reality, symbolized by the Tetraktys,  repeats itself in Higher Dimensions until Infinity.

The Tetrad, represents Completion, because it contains all its Previous Numbers, the 1, 2, 3, and itself, 4, in One Number, 10 = (The One) +  9 (= 3 (Trinity)x 3 (Trinity) = Tetraktys).

As you can see in the Picture above the Fractal Pattern of 8 contains two kinds of Trinities/Triangles, an Upside and a Downside (Rotated by 180 Degrees) Triangle. When you Rotate by 180 Degrees the 1 becomes -1 and 1 + -1 =0 is the Void.

The Square is a combination of two Triangels. It is represented by the Of Star of David, the Symbol of the Heart Chakra.

The Star of David, the Symbol of Human Center, the Heart Chakra.

The Multi Dimensional Rotations of the Octonions always Come Back to Square 1/1=1, the One and keep Rotating around the Center, the Nothing,   Until Infinity.

LINKS

About the Tetraktys (1)

About the Tetraktys (2)

About Triangular Numbers and Pascal’s Triangle

About the Empty Set

About the Relationship Between Geometry and Music

About the Trinity

About the Game of the Surreal Numbers

About Larson and the Unification of Mathematics

The Collected Works of Dewey B Larson

About Number and Magnitude

About Ratio and Proportion

About Ratio and Proportion by Euclid

A book of Augustus deMorgan about “The Connection between Number and Magnitude”

The text of the Fifth Book of Euclid

An Educational You Tube Channel called Insights in Mathematics

About the History of Geometric Algebra

About the Sri Yantra

About Geometric Algebra

Free Software to use Geometric Algebra

About Clifford Algebra

About Yantra’s

About Movement

About Topology

About the Digital Root Patterns

About the Heart Chakra

A Video that shows how the Platonic Solids are created out of the Trinity Numbers

All you want to know about Geometric Patterns

About the Law of KLeiber

Friday, October 29th, 2010

A power law relates one variable to another raised to a constant power. The general form takes y = xa, where y and x are variables, and a is a constant exponent.

A power law exhibits the property of scale invariance. When you multiply the Scale (x) with a factor b the function (y =  baxa) does not change its Shape.

In 1817 Goethe wrote his book ”Zur Morphologie“. This book was the start of a new science called Morphology, the Science of the Shapes.

In his book Goethe describes the so called Uhrplant, the Primal Plant, which is based on the shape of the Leaf. Goethe believed that every Plant was a Leaf within a Leaf within a Leaf.

At the time of Goethe the concept of the fractal was not known. It was developed in 1975 by Benois Mandelbrot (“The Fractal Geometry of Nature”).

A fractal is a self-similar structure. It’s shape repeats itself on every level of expansion.  Some fractals are scale-invariant.

Scale Invariance in the Leaf

The scale invariant fractal structure of the Leaf

About Kleiber’s Law

Metabolism is the process by which your body converts what you eat and drink into energy. During this complex biochemical process, calories in food and beverages are combined with oxygen to release the energy your body needs to function.

The oldest and best known Power Law is Kleiber’s law devised by the Swiss-American zoologist Max Kleiber in 1932.

Kleiber’s Law, MR = W3/4, describes the relationship of the metabolic rate (MR) to the biomass W, raised to an exponent.

Kleiber’s law means that a cat’s metabolic rate is not a hundred or 21.5 times greater than a mouse’s, but about 31.6 — 100 to the three-quarter power.

This relationship seems to hold across the animal kingdom  and it has since been extended all the way down to single-celled organisms, and possibly within the cells themselves to the internal structures called mitochondria, the cellular powerplants, that turn nutrients into energy. Mitochondria have many features in common with bacteria.

The law of Kleiber is  also applicable to super-organisms” like ant colonies, Cities and Eco-Systems.

Because of Economies of Scale, larger and more complex forms of organisms need less energy for each individual cell. They grow and reproduce more slowly and they live longer.

Kleiber's law

Kleiber's law

About Fractals and Kleiber’s Law

Kleiber’s law can be explained from a general model that describes how essentialmaterials are transported through Space-Filling Fractal Networks of Branching Tubes.

Goethe was right. The leaf is a one of the fundamental structures of Nature.

The factor 3/4 (3/3+1)  is a consequence of the fact that a Fractal Structure has to incorporate a not-Fractal Structure, 3-Dimensional Space.

One of the Bilateria: The Octopus

One of our fellow Bilaterians: The Octopus

About Bilateria

About 590 million years ago, the Central Nervous System (CNS), the Brain, appeared.

The organisms with a CNS (including the Humans), the Bilateria, are able to Act and React to a Possible Harmful Stimulus.

The CNS of the Bilateria is  a result of  an Increase in Competition between the Life Forms that came out of the Continuing Fusion of  the Cooperative Life Forms, the Bacteria. The first step in this proces was the Tube of the Sponge.

The fundamental Bilaterian Body Shape is a Tube running from Mouth to Anus, and a second Tube called the Notochord, with an especially large Sphere at the front, called the “Brain“.

The Bilateria have Five Body Spheres (1) the Brain; (2) the Spinal Cord; (3) the Heart and Lungs; (4) the Digestive Organs and Kidneys; (5) the Bladder and Reproductive organs.

Lung

About the Tiny Spheres of the Lungs

When we look at the Lungs, one of the Fractal Tube-like branching-structures of the human organism, we can see that the Branches end in Nodes called the alveoli (“little cavities“). The end-nodes of the branching system are tiny Spheres.

Alveoli

The Alveoli use another basic structure of Nature: The Sphere

In the tiny Spheres the Exchange takes place of Carbon Dioxide and Oxygen between the Lungs and the Blood-Vessels. Each human lung contains about 600 million alveoli.

Water diffuses from the alveoli cells into the alveoli so that they are constantly moist.  Oxygen dissolves in this water before diffusing through the cells into the blood.

The Oxygen-rich blood returns to the Heart via the pulmonary veins to be pumped back into circulation. The Carbon Dioxide is pumped out by the Lungs.

In the many million Small Spherical End-Nodes Two Circulatory Systems, the Cardiovascular System (Heart) and Pulmonary System (Lungs), are Connected.

The Big Structures of Nature are able to Scale because the Connection-Points of the Networks are Very Small.

They are reusable on Every Scale that is Bigger than the Scale of the Connection Points.

We will see that all the other Fractal Systems in our Body are based on the Same Principle.

Villi

Villi

About the Tube of the Digestive System

The Tube contains the Digestive System. It breaks-down larger food molecules into smaller ones that can be absorbed into the blood stream. This happens in the Small Intestine in the vili and microvilli.

The villi (“shaggy hair“)  are tiny, finger-like projections that are approximately 0.5-1mm in length. The microvilli are mechanosensors and have a lot in common with the flagellum (“the roter“) of a bacterium. Microvilli appear in many places in the body. They are also of importance on the cell surface of white blood cells, as they aid in the migration of white blood cells.

The Tube of our Digestive System is highly similar to the Tube of the Sponge, the first fusion of the Bacteria in a  more efficient metabolic structure.

The Output part of the Tube is called the Large Intestine (Colon). Its function is to absorb water from the remaining indigestible food matter, and then to pass Useless Waste Material from the body. The large intestine houses over 700 species of bacteria that perform a variety of functions.

Liver

The Liver Fractal

About the Chemical Factory of the Liver Cell.

The Liver is  a Fractal Branching System that detoxifies harmful substances absorbed via the Small Intestines.  It’s basic structure is the Liver Cell.

The Chemical reactions in the liver cells produces a lot of waste heat. This is carried round the body in the blood and warms less active regions.

The Liver regulates the amount of Blood Sugar, Lipids, Amino Acids.  The Liver is the Storage House of  Blood, Iron, Vitamine A, D, B12  and Clycogen, the Source of Energy of the Body.

The Liver produces Bile that is stored in the Gall Bladder. Bile is used to dissolve fat.

White Blood Cell

The White Blood Cell

About the Spleen and the Immune System

Another Fractal Branching System, the Lymphatic System ( “The Immune System“, Spleen),  maintains the health of the body by protecting it from invasions by harmful pathogens, such as bacteria, viruses, fungi, and parasites. These pathogens are the cause of many diseases, so it is necessary to detect and eliminate them rapidly.

The Lymphatic System is connected to the Blood System and produces the White Blood Cells (Lymphocites).  The Lymphocites are the Basic Unit of the Human Defense System. They look like Bacteria.

A lymphocite fits with a pathogen

A lymphocite fits with a pathogen

The surface of a lymphocyte is covered with a large number of identical receptors. Recognition and destruction occurs when the receptors of the lymphocyte fit like a key into the surface of the pathogen.

The Immune System is a Highly Adaptive System. It is able to generate new types of Lymphocites out of a Library of DNA-components.

Nephron

Nephron

About the Kidneys and the Bladder

The Kidneys serve the body as a natural filter of the blood, and remove wastes which are diverted to the urinary bladder. The Nephron is the basic structural and functional unit of the kidney. In humans, a normal kidney contains 800,000 to 1.5 million nephrons. Nephrons are wave-like structures. Its chief function is to regulate the concentration of water and soluble substances like sodium salts by filtering the blood.

About the Electro Magnetic System

The last  Fractal Branching System, the Meridians, is a Fast Electro-Magnetic Channel that connects to Organs to the Spine and the CNS.

The Ancient Chinese Scientists believed that this System pumps its Energy, the Chi Force, out of the Earth Magnetic Field.

The Ancient Egyptian Scientists believed the same thing. They named  the Four Cavities the Four Suns of Horus and the Spine, the Djed Pillar (Dj means “Snake”).

The Five Organs of Acupuncture

The Five Systems of Acupuncture

About the number Five

The number Five (4 +1 ) plays an important role in Ancient Chinese Medicine.

The Central Fractal System, the One (1), the Fire System, the Blood System, with its Center the Heart,  contains Four (4) Links to the Other Systems.

The Four Systems are combinations of Two Forces, Expansion and Compression.

The Two Forces came out of One Force (“The Void“).

Two Systems contain the same combination (Expansion x Expansion (Wood, Gall Bladder, Liver, Three Heater), Compression x Compression (Metal, Lungs)).

Two Systems combine Expansion and Compression in a different order (Expansion x Compression, Compression x Expansion). They create a Wave (Water, Bladder, Kidney) or Spiral -like (Earth, Immune System) structures.

The Five Spheres of the Body are incorporated in one Super Protective Sphere, the Coelum, the Skin.

In the Human Embryo the First step of Division of the Cells is between the Coelum,the Multi-Cellar Body, and the Brain, the CNS. In the first step the One was divided into the Two, the Actor and the Monitor.

The CNS  “mirrors” the activities of  the Body and acts as a “predictive simulator”.

The Shape of the CNS is a mirror (Up-Side-Down) of the Shape of the Multic-Cellular Body.

The Bilateria are a Fuse of two Organisms in which one organism changed into the Body and the other changed into the Brain.

These two structures are always competing in the Human Being.

The Brain is Looking Up at the Sky. The Body is Looking Down at Mother Earth, it’s Creator.

The Brain and the Body are United in  the Heart, the Balancer of Body and Soul and the Keeper of the Rhythm (The Pericardium).

Sponge

Sponge, The Tube

About Bacteria

The Body Shape of the Tube was inherited from the first Multi-Cellular Organism, the Sponge. The Sponge is a  Static Cluster of Amoeba, free moving bacteria, that are propelled by their flagellum (a Roter).

Bacterium

The Basic Building Block of the Body, the Bacterium

Bacteria are small chemical factories that are able to share and combine their production processes by exchanging DNA.

With their flagellum the Bacteria Explore the environment to find the chemical food they need. When they have found food the Circulation of the Flagellum moves into the Opposite Direction.

The bacterium integrates incoming chemical signals during a few-second period of its travels, and adjusts its direction of advance accordingly.

The integration is achieved through temporary chemical modifications to molecules located in the bounding membrane, which transfer nutrients to the cytoplasm, and also through changes to certain other molecules in the interior of the cell. Such integration is essentially a short-term memory mechanism.

Bacteria exchange their DNA

Bacteria exchange their DNA

At a certain moment the Bacteria got together because the Sponge is a much more Efficient Metabolic Structure than a group of Free Moving Bacteria.

Strangely enough the DNA of the Sponge already contains the complete Bleuprint of the Humans.

Scientists now believe that most species on earth are a result of a loss of DNA from the original Bleuprint that was created when the Bacteria finally fused into one Organism.

About the Nano-Level

Recently scientists have found Electric Fields as strong as 15 million volts per meter in the Nano-Parts of the Cells. These fields are as strong as those produced in lightning bolts.

It’s not clear what causes these strong fields or what they might mean but they could account for a until now unknown (by Science) or well-known (by the Old Scientists, Chi) Source of  Bodily Energy.

The nano-parts of the Cell could be the very very very small universal building blocks of Nature.

What has Happened?

In the beginning the Chemical Soup generated Self-Reproducing Chemical Factories.

These Factories combined into bigger Factories by exchanging and combining their factory-designs.

Every time when the Factories fused they became more Efficient, Bigger, Stronger, Older and therefore more Competitive.

When the Factories became more competitive they had to Protect themselves against the other Factories.

To Protect themselves the Multi-Cellular Systems started to Sense in Many Directions.

In the first step a Nerve-Net was created.

In the second step one part of the Organism turned Upside-Down and became a Specialized Predictive System (the CNS).

This part fused with the Bodily Part creating one Organism that contained Two Organisms.

The Bilateria, the Organisms with Two Complementary Parts, Body and Mind, were born.

Some of the Bilateria called Humans fused in Social Structures and the Social Structures fused and fused and became more efficient, bigger, stronger, older and therefore more competitive.

In due time they will Rule the Earth and will start to create bigger and bigger systems until they will rule the complete Universe.

To make this possible the Humans need every other part of Nature to Sustain their Growth. This will certainly result in a huge Collapse of their own Eco-System.

The main reason this is happening is that the Humans forgot to Copy the Designs of Mother Nature.

The Brain (Thinking and Sensing, Left Brain), Looking Up at the Sky,  believed it could do a much Better Job than it’s counterpart the Multi-Cellular Body (Emotions and Imagination, Right Brain), Looking Down at the Earth, the Source of All Creation.

Kleiber’s Law shows that we are now using 122 times more Energy per Person than we really would need if we would Scale in the right way.

What can we learn from Kleiber’s Law?

All organisms including the humans depend for their maintenance and reproduction on the close integration of numerous subunits.

These components need to be serviced in a relatively `democratic’ and efficient fashionto supply energy, remove waste  and regulate activity.

Natural Selection solved this problem byevolving hierarchical fractal-like branching networks, whichdistribute Energy, Information and Materials between Big Reservoirs (“Lungs”) and Small Connection Points (“Alveoli”) of other Circulatory Systems (“Blood”).

This Fractal Bottum-Up approach is highly effective and efficient.

The not-fractal, Top-Down,  machinery designed by the Humans is Scaled with a factor 1 so we can improve a lot by copying Mother Nature.

LINKS

About Goethe and Morphology

About the Heart Chakra or Why the Heart connects the Brain and the Body

How to use the Heart to find Balance

About the Heart and Ethics

Why Humans look a lot like Bacteria

About Kleibers Law and the Rain Forest

About the Five Worldviews

A movie about Fractal Structures in Nature

About the Fusion of Water in the Cell

About the Number Five in Chinese Medicine

Why Humans are part of Super Organisms

How the Humans evolved out of the Bacteria

About the Left and the Right Brain

About the Void

How the Heart synchronizes the Body

About the Fast Transmisson Channel in the Body

About Kleiber’s Law

A general model to explain Kleiber’s Law

About Kleiber’s Law and the Growth of Cities

About the Immune System

About Bacteria

About the Law of Kleiber and Ant Colonies

The terrestrial evolution of metabolism and life

A movie about the origin of vertebrates

About Morphology or How Alan Turing Made the Dream of Goethe Come True

Tuesday, November 17th, 2009

The Ancient Greeks believed that the images of waking life and dreams came from the same source, Morpheus (Μορφέας, Μορφεύς), “He who Shapes“.

The Science of the Shapes, Morphology, was created and named by Goethe in his botanical writings (“Zur Morphologie“, 1817).

Goethe used comparative anatomical methods, to discover a primal plant form that would contain all the others-the Urpflanze. Goethe being a Romantic Idealist hoped that Morphology would Unify Science and Art.

The Uhrplant shows itself also in the Lungs and Riversystems

The Uhrplant shows itself also in the Lungs and Riversystems

“The Primal Plant is going to be the strangest creature in the world, which Nature herself shall envy me. With this model and the key to it, it will be possible to go on forever inventing plants and know that their existence is logical”. Nature always plays, and from which she produces her great variety. Had I the time in this brief span of life I am confident I could extend it to all the realms of Nature – the whole realm“.

Goethe (wikipedia)

Goethe (wikipedia)

Hundred years later in the 1920s Goethe’s dream came true. Morphology moved outside Biology to other parts of Science due to the works of D’Arcy Thompson’s On Growth and Form, Oswald Spengler Morphology of History, Carol O. Sauer Morphology of Landscape, Vladimir Propp, Morphology of the Folktale and Alfred North Whitehead Process and Reality.

Goethe observed nature and reflected on similar structures. He believed that there was something behind this similarity, an archetypal plant.

According to Goethe the archetypal plant was the leaf (“While walking in the Public Gardens of Palermo it came to me in a flash that in the organ of the plant which we are accustomed to call the leaf lies the true Proteus who can hide or reveal himself in all vegetal forms. From first to last the plant is nothing but leaf“).

At this moment scientists know the reason why the leaf is the most important structure of the plant. It is a solar collector full of photosynthetic cells.

The energy of the sun provides the energy to transform water from the roots gathered by the leafs and carbon dioxide out of the air also gathered by the leafs, into sugar and oxygen. Plants are structures with many leaves. These leafs shield other leafs from collecting sunlight and water.

To solve this problem a plant has to optimize its structure to collect enough Sunlight and Water. The process of Optimization is not a Central Coordinated action. Every leaf tries to find the best place in the Sun on its own. This place determinates the growth of the next level of branches and leafs.

Goethe observed a pattern and deduced a structure, the leaf, the Uhrplanze. What Goethe really observed was not a Static Uhrplant but the Dynamic Process of the Branching of all kinds of leaves in all kinds of plants (Morpho-Genesis).

The leafs of the plants are not the main target of the morphogenesis of the plant. The visible External and the invisible Internal Forms or Organs are one of the many solutions of an equation with many variables and constraints. The optimal solution is reached by experimenting (“Nature always plays”).

Many solutions fail but some survive (Evolution of the Fittest). When a solution survives it is used as a Foundation to find new rules for more specific problems (Specialization). When the environment, the context, changes old rules have to be replaced by new rules (a Paradigm Shift).

The Fractal Geometry of Nature

The Fractal Geometry of Nature

New mathematical paradigms in the field of the Machines and Languages (Alan Turing, The Chemical Basis of Morphogenesis) and the Self-Referencial Geometry of Nature (Benoît Mandelbrot, The Fractal Geometry of Nature) have stimulated further investigation in the Field of Morphology.

In 1931, in a monograph entitled On Formally Undecidable Propositions of Principia Mathematica and Related Systems Gödel proved that it is impossible to define a theory that is both Self-Consistent and Complete. The paper of Gödel destroyed the ambitions of the Mathematicians at that time to define one theory that explains everything.

In 1936 Alan Turing produced a paper entitled On Computable Numbers. In this paper Alan Turing defined a Universal Machine now called a Turing Machine. A Turing machine contains an infinite tape that can move backwards and forwards and a reading/writing device that changes the tape. The Turing Machine represents every Theory we can Imagine.

Turing proved that the kinds of questions the machine can not solve are about its own Performance. The machine is Unable to Reflect about Itself. It needs another independent machine, an Observer or Monitor to do this.

It can be proved that Turing proved the so called Incompleteness Theorem and the Undecidability Theorem of Gödel in a very simple way.

eniac

The Eniac

In 1943 Turing helped to Crack the Codes of the Germans in the Second World War. At that time the first computers were build (Eniac, Collossus).

It was very difficult to Program a Computer. This problem was solved when Noam Chomsky defined the Theory of Formal Grammars in 1955 (The Logical Structure of Linguistic Theory).

When you want to define a Language you need two things, an Alphabet of symbols and Rules. The symbols are the End-Nodes (Terminals) of the Network of Possibilities that is produced when the Rules (Non-Terminals) are Applied. The Alphabet and the (Production- or Rewriting) rules are called a Formal Grammar.

If the Alphabet contains an “a” and a “p” the rules S→AAP, A→”a” and P→”p” produce the result “aap”. Of course this system can be replaced by the simple rule S→”aap”. The output becomes an infinite string when one of the rules contains a Self-Reference. The rules A→a and S→AS produce an Infinity String of “a’-s (“aaaaaaaaaaaaaaaaaa….”).

The system becomes more complicated when we put terminals and rules (non-terminals) on the Left Side. The System S→aBSc, S→abc, Ba→aB and Bb→bb produces strings like, “abc”, “aabbcc” and “aaabbbccc”. In fact it produces all the strings a**n/b**n/c**n with n>0.

The inventor of the theory of Formal Grammar, Chomsky, defined a Hierarchy of Languages. The most complex languages in his hierarchy are called Context-Dependent and Unrestricted. They represent complex networks of nodes.

A language where the left-hand side of each production rule consists of only a single nonterminal symbol is called a Context Free language. Context Free Languages are used to define Computer Languages. Context Free Languages are defined by a hierarchical structure of nodes. Human Languages are dependent on the context of the words that are spoken.

It is therefore impossible to describe a Human Language, Organisms, Organisations and Life Itself with a Context Free Computer Language.

Context Free Systems with very simple rule-systems produce natural and mathematical structures. The System A → AB, B → A models the Growth of Algae and the Fibonacci Numbers.

A Recognizer or Parser determinates if the output of a formal grammar is produced by the grammar. Parsers are used to check and translate a Program written in a Formal (Context Free) Language to the level of the Operating System of the Computer.

grammarRegular and Context Free Grammars are easily recognized because the process of parsing is linear (causal, step by step). The stucture of the language is a hierarchy.

The recognizer (now called a Push-Down Machine) needs a small memory to keep the books.

Context Dependent (L-systems) and Unrestricted Grammars are difficult to recognize or are not recognizable in practice because the parser needs a huge sometimes Infinite Memory or Infinite Time to complete its task.

To find the Context the Recognizer has to jump backwards and forwards through the infinite string to detect the pattern.

If the network loops the recognizer will Never Stop (“The Halting Problem“).

Turing proved that the Halting Problem is Undecidable. We will Never Know for Sure if an Unrestricted Grammar contains Loops.

The Rules and the Output of Unrestricted Grammars Change and never stop Changing. Our Reality is certainly Context Dependent and perhaps Unrestricted.

Parsing or Recognizing looks like (is similar with) the process of Scientific Discovery. A theory, a Grammar of a Context-Free Systems (“aaaaaaaaaaa…”) is recognizable (testable) in Finite Time with a Finite Memory. Theories that are Context Dependent or Unrestricted cannot be proved although the Output of the Theory generates Our Observation of Nature. In this case we have to trust Practice and not Theory.

cellular automata

A 3D Cellular Automaton

In 2002 the Mathematician Stephen Wolfram wrote the book A New Kind of Science.

In this book he tells about his long term Experiments with his own Mathematical Program Mathematica. Wolfram defined a System to Generate and Experiment with Cellular Automata.

Wolfram believes that the Science of the Future will be based on Trial and Error using Theory Generators (Genetic Algorithms). The big problem with Genetic Algorithms is that they generate patterns we are unable to understand. We cannot  find Metaphors and Words to describe the Patterns in our Language System.

This problem was adressed by the famous Mathematician Leibniz who called this the Principle of Sufficient Reason.

Leibniz believed that our Universe was based on Simple Understandable Rules that are capable of generating Highly Complex Systems.

It is now very clear that the Self-Referencial Structures, the Fractals, of Mandelbrot are the solution of this problem.

The Scientific Quest at this moment is to find the most simple Fractal Structure that is capable of explaining the Complexity of our Universe. It looks like this fractal has a lot to do with the Number 3.

It is sometimes impossible to define a structured process to recognize (to prove) a Grammar. Therefore it is impossible to detect the rules of Mother Nature by a Structured process. The rules of Mother Nature are detected by Chance just like Goethe discovered the Uhrplanze. Science looks a lot like (is similar with) Mother Nature Herself.

When a Grammar is detected it is possible to use this grammar as a Foundation to find new solutions for more specific problems (Specialization, Add More Rules) or when the system is not able to respond to its environment it has to Change the Rules (a Paradigm Shift). All the time the result of the System has to be compared with Mother Nature herself (Recognizing, Testing, Verification).

Turing proved that if Nature is equivalent to a Turing machine we, as parts of this machine, can not generate a complete description of its functioning.

In other words, a Turing machine, A Scientific Theory, can be a very useful tool to help humans design another, improved Turing Machine, A new Theory, but it is not capable of doing so on its own – A Scientific Theory, A System, can not answer Questions about Itself.

The solution to this problem is to Cooperate. Two or more (Human) Machines, A Group, are able to Reflect on the Other. When the new solution is found the members of the Group have to Adopt to the new solution to move on to a New Level of Understanding and drop their own Egoistic Theory.

Each of the individuals has to alter its Own Self and Adapt it to that of the Group. It is proved that Bacteria use this Strategy and are therefore unbeatable by our tactics to destroy them.

Turing proved that Intelligence requires Learning, which in turn requires the Human Machine to have sufficient Flexibility, including Self Alteration capabilities. It is further implied that the (Human) Machine should have the Freedom to make Mistakes.

Perfect Human Machines will never Detect the Patterns of Nature because they get Stuck in their Own Theory of Life.

The Patterns of Turing

The Patterns of Turing

The Only ONE who is able to Reflect on the Morphogenesis of Mother Nature is the Creator of the Creator of Mother Nature, The Void.

Gregory Chaitin used the theory of Chomsky and proved that we will never be able to understand  The Void.

The Void is beyond our Limits of Reason. Therefore the first step in Creation will always be  a Mystery.

At the end of his life (he commited suicide) Alan Turing started to investigate Morphology.

As you can see the Patterns of Alan Turing are created by combining many Triangels. The Triangel is called the Trinity in Ancient Sciences.

According to the Tao Tse King, “The Tao produced One; One produced Two; Two produced Three; Three produced All things”, which means that the Trinity is the Basic Fractal Pattern of the Universe.

In modern Science this pattern is called the Bronze Mean.

It generates so called Quasi Crystals and the Famous Penrose Tilings.

The Bronze Mean is represented by the Ancient Structure of the Sri Yantra (“Devine Machine”).

Goethe was not the real discoverer of Morphology. The knowledge was already there 8000 years ago.

LINKS

About the Observer and Second Order Cybernetics

A PDF About the Morphology of Music.

The origins of life and context-dependent languages

A Website About the Morphology of Botanic Systems

A Website About the Morphology of Architectural Systems

A Plant Simulator using Morphology

About Intelligent Design

The Mathematical Proof of Gödel of the Existence of God

About Bacteria 

About the Bronze Mean

About the Trinity