About the Geometry of Change Version 2

This is Version 2. In this version I have finished the Educational part of this blog (“Chapter 1) and I have started to write in Chapter 6 about the use of the Geometry of Change. In this part I advice you to spend some time on a video of George Lakoff one of the Leading Scientist in Embodied Cognition . This part of Cognition Theory is based on contradiction (“Bisocation“).

This is a Part of long series about the Philosophy of Science in which you see the Perspectives of Paths of Change come back united with an idea about the How to do Science being a Process.

0 Introduction

About Contradiction:

The first two video’s are about the Foundation of Mathematics which is based on Contradiction . The third video is about the Human Relationship theory of Alan Fiske.This theory is a Bridge between Geometry and Change Management. If you want to start with this subject go to Chapter 6. This blog starts at Chapter 1.

Gregg Zuckerman About the Mathematics of Consciousness based on Non-russell Sets. Consciousness is based on Contradiction.
Alan Fiske has developed a widely accepted and used theory about how people coordinate and cooperate. This theory is compatible with Paths of Change. The theory is also compatible with the results (see chapter 2) of Measurement theory where the types of measurement are called Scales. Scales can be related to Geometries. In the blog I start with explaining geometries and move in Chapter 6 to the field of change Management where I Apply the Geometries.

In Version 1 I have finished my educational objective and will start with a new Chapter 6 in which I Apply the Geometry of Change. This blog starts at Chapter 1.

The reason behind this blog is explained in the last chapter of my blog About the Ecology of the State.

In the blog about the Ecology of the State I reused a blog about Anti-Fragility, a new name for Resilience. Resilience is a major variable in the theory of Panarchy which is a theory about Ecologies.

I combined Panarchy with Paths of Change also a Theory about change and the theory of Alan Fiske about Human relationships. This combination gave me the insight that there is a shared Geometry behind many of the theories tabout Change.

This blog gives you an opportunity to combine the theories yourself. I also will show you that this geometry was known for a very long time ago mainly starting in India (See “Vedic Mathematics“). In the interpretation of the Theory of Alan Fiske the Scales of Measurement play an important role. I have created a special Chapter 2. for this subject.

In this blog I want to explore the Geometry of Change. To make it possible that others are also able to think about Geometries this blog contains many Educational Videos’.

1 About Geometry

As you can see in this Chapter Geometry is very old. It was the Foundation of Art, Architecture , the State and Religion. People have been fascinated by Patterns like the visible patterns Up in the Sky of the Sun, Moon Planets and the Zodiac but also the patterns in numbers. Out of Geometry many sciences have grown that were grouped in schools sometimes called a University. Behind he patterns music was found and music was also created using patterns. The most important Music was the Music of the Spheres . This chapter is Big. It contains many Video’s that contain simulations. The main reason is that you have to See (“insight”) the Theory to understand the Practice.

1.1 How Physics, Movement, Symbols and Sound are supported by Geometry

Richard Feynman was able to recreate every theory at will and was therefore able to explain what the theory was about. His lectures are famous. This is a recontruction with a Simulation of his Last Lecture. It is a lecture about Geometry. Below you see a comparable lecture about a comparable genius Riemann who lived a short life but was able just as Feynman to see (Imagine) a lot. This shows the Power of Mathematics. It is not about Reasoning but about Insight. That’s why I have selected many video’s and will find more. Enjoy.
Why the Prime Numbers lie on a line but nobody knows why. A nice present of Quanta Magazine. about the Hypothesis of Riemann. Riemann comes back later when he solves another problem, the problem why geometries are different. He proved that this has to with the Curvature of space. Something you see below when you see the solution of Einstein to the problem of Gravity solved before that time by Newton who stole it from Leibniz, one of the last Magicians before the Golden Dawn (see video below about Mystic Geometry.) Einstein got the solution for his problem of his friend the Mathematician Minkowski.
The curved space of Einstein.
The role of Higher Dimensions in Physics and Mathematics. It started with Albert Einstein. He used a new 4D curved geometry to explain gravity. Kaluza found that Gravity and Electro-magnetism could be united by adding an extra 5th Dimension. This solution was improved by Oscar Klein by introducing Imaginary (2D) numbers . This solution transformed much later into 10 dimensional (vibrating) String /theory to integrate quantum mechanics, . As you can see in the next video the extra dimensions are put into a Very Small Closed in itself (“circular”) area in Space.
About Kaluza Klein Theory.
Part of a long series of lectures and simulations about symmetric shapes that played an important role in ancient physics and mathematics which was at that time related to the Order of phytogoras.
Transforming a flat structure into a 3D projection of a higher order structure made of Pentacles.

The Physicist Sean Carrol About Geometry and Topology in particular Riemannian non-Euclidean geometry, and topological invariants called “Homotopy groups.

The geometry of our space has zero curvature. In spaces with positive or negative curvature other rules are valid. George Bernhard Riemann developed a new geometrical concept by imagining a 2-dimensional bookworm living on a 3D crumpled sheet of paper.

The surface of the paper would seem to be flat and undistorted. Because the worm can only perceive two dimensions it would not be able to detect that his own body crumples with the paper in the higher dimension. The “force” that would keep the bookworm from moving in a straight line was not a mysterious “action at a distance”, but a result of the unseen warping from the third dimension.

You can imagine a similar experience by walking a straight line on a rotating platform. To you the shortest connection is a straight line while you move with a curve when observed by someone outside the platform. The geometry changes the idea that the shortest distance between two points is a straight line.

It even changes the idea that we return on the same point when we move in a loop (“circle”). This happens when our space contains one or more holes (“singularities”, “dividing by zero”).

With this example you can learn that it makes a lot of Difference when you are In (Intrinsic) or Outside (Extrinsic) the Context.

Geometry and the Riemann Space
Space Harmony: The work of Rudolf Laban about Movements. The same geometry can be used to make letters, sounds and meaningful symbols.
Kim Veltman about the history and geometry of Letters.

 

System Theory related to Buddism. A speedy introduction of system thinking leading to chapter six..2.
How to transform an infinite proces into a finite proces
A very rich source about Mystic Geometry. About India, Soemer, Egypt, Magic Kabbalah, the Rosicrucians , Blavatsky and of course the Golden Dawn.

2 About the Theory of Measurement

Measurements can be made through the human Senses or through the use of a measurement Instruments.

Measurement is the assignment of Numerals to a system according to a set of rules.  These rules are called a Theory.

Our senses or the instruments that are used to measure the system also represent a theory. This theory has a sometimes unknown impact on our measurement.

To avoid the Bias of the Theory behind our Measurement Systems a theory has to be a Structure Preserving mapping, a homomorphism. The Structure of the system has to stay the same when the rules are applied.

If the theory does not preserve structure the theory introduces its own theory and we are not measuring the original system.

Not only the primary theory has to be a homomorphism. The complete chain of theories we use has to preserve the structure (The Uniqueness Axioma) of the original system.

To avoid Bias a System of Measurement Scales is defined.

A video about Scales.
The Hierarchy of Scales
The Hierarchy of Measurement Scales

a Nominal Scale defines Sets. Sets contain the Same elements. When we want to preserve structure  we have to satisfy the constraint If s1 = s2 ⇔ f(s1) = f(s2) (“the Same stays the Same”). This is called the Symmetric group, the group of all Permutations of the objects in the Set.

The Ordinal Scale is part of the nominal scale. The relational structure preserved during the measurement process preserves equality and order. The set of admissible transformations are relations that satisfy the constraint If s1 < or >s2 ⇔ f(s1) < or >f(s2). This is called the Order-Preserving Group and is the group of monotonic increasing functions.

Monotonic Functions Increasing and Decreasing
Monotonic Increasing and Decreasing functions,

The Interval Scale has to satisfy the constraint  If s1−s2 = s3−s4 ⇔ f(s1)−f(s2)=f(s3)−f(s4).This is called the General Linear Group (or Affine Group).  Differences can only be preserved when the transformation is a Linear transformation Ax + B where A is a Rotation and B is Translation (Displacement).

The Ratio Scale has to satisfy the constraint If s1/s2 = s3/s4 ⇔ f(s1)/f(s2) = f(s3)/f(s4). This is called the Linear Group.The only transformation that satisfies this constraint is the functions f(x) = Ax.

Mathematician Prof. Norman Wildburger About the History of Mathematics focusing on geometry. . Norman has made thousands of video’s explaining everything you don’t know but want to know. THis video is about History.
An introduction to Group Theory. Group theory is about “being the Same“. This is a visualization.

Geometry, the Art of Transformation Size, Shape and Symmetry.


Figures shown in the same color are similar. A similar structure can be transformed by Rotation, Translation (Displacement) and Scaling (Multiplication), “blow up/down”).

Groups can be compared to numbers. Just like the primes the Simple Groups can be used to create other groups. The Biggest group is called the Monster.
The Geometry of the Space we live on, the Sphere, where Parallel Lines Cross at the Poles. This video contains nice visualizations.
Affine Geometry, The Geometry of Linear Transformations.

 

Projective Geometry is about Perspective and the projective plane, the space of one-dimensional subspaces of a three dimensional vector space (lines through the origin). This view leads to Hyperbolic and Eliptic Geometry finally framed in the language of Geometric Algebra . Affine geometry is the “world” of Linear Algebra and Vector Spaces leading to Euclidean and Einstein’s (Minkovski) ,Relativistic geometry.

With Geometric Algebra you can Calculate with Shapes. Geometric algebra is a very convenient representational and computational system for geometry. It is going to be the way computer science deals with geometrical issues. It contains, in a fully integrated manner, linear algebra, vector calculus, differential geometry, complex numbers and quaternions as real geometric entities, and lots more.

Geometric Algebra brings us back to the Tetractys of Pythagoras which shows the Fourth triangular number which is part of the Triangle of Pascal.also called the Yang Hui in China and Mount Meru in Sanskrit. The Triangle of Pascal contains many Fractal structures that were used a very long time ago in Music (Rythms), Art & Architecture (Proportions) and Numerology (“The Art of interpreting Patterns in Numbers).

Triangular Numbers.
The Tetractys (1+2+3+4=10 = 1 MOD 9) in a Mystical Vision of Jakob Böhme. In the center of the Heart you can see the Hebrew letter/numbers that are a copy of the comparable Sanskrit symbols. See the video of Kim Veltman above.

3 Mount Meru

the roots of all the geometries can be found in Ancient Indian Mathematics.

The five central towers of Angkor Wat symbolize the peaks of Mount Meru; Angkor, Cambodia.
Mount Meru Prastaara. When you Want to know more about the history of Sanskrit (“krit” = the Practice) and “sans = Speaking and the influence on many cultures (~f.i. Hebrew) of the culture of ancient India look at the Video of Kim Veltman above.

4 This Blog is a Fusion of many Blogs

A long time ago I wrote a blog about Projective Geometry and Geometric ALgebra that I will reuse in this blog.

The blog about the Ecology of the State is a spin-off of my blog about Anti-Fragility in which I combined Paths of Change (PoC), Panarchy and the theory of Alan Fiske about Human Relationships (see below).

 

The Colors of PoC (Red, Blue,, Yellow and Green) come back in every slide.

The Paths of Panarchy resemble a moebius Ring (8) that makes it possible to cross colliding lines. The three variables that determine Panarchy. Connection, Potential and Resilience (now called Anti-Fragile , Revolution in Panarchy , Agency in PoC and Yang in Taoism) are put into one Map. Observe the Heart in de Center.Also observe that the Connecting Lines in this Map are Directed Arrows (“Vectors“) and that the connected lines Move with the Clock..
Two connected Ecologies. They are connected by the Revolt-and-Remember-Jump.
Three connected Ecologies that could be a Human , a Social Network and a City.
The 3D-version of the model of Panarchy with the four Colors of PoC that correspond to the four Relational Models.
The model of Fiske related to PoC by its colors.
Theory of ALan Fiske explained by Steven Pinker.
The relationship between Alan Fiskes theory about Relationships and Measurement Scales. In the chapter about measurement you see that the Scales are related to Geometries. This implies that the Relationships are also related to Geometries.

It all comes back to Breathing

Realize that the 3D version of Panarchy is also a 3D version of the Movement of the Relational Models of Alan Fiske and their related Geometries in Cognitive Space and the movement of many Concepts that share the same Worldviews of Path of Change.

Also Realize that Panarchy is a result of a theory (Cultural Theory) that is based on the same Worldviews of PoC and of Alan Fiske that can be explained by a 2×2 Matrix containing the variables Agency and Communion that are the same as Yin and Yang.

Agency and Communion also determine the Big Five the most used Personality Classification (and many others)

Yin and Yang are also represented by 0 and 1 and therefore the Triangle of Pascal represents the many combinations of possible personalities on earth but is also a result of a completely different System (Geometries) to calculate Combinations. Think about (X>Y)**n (Affine) or (X-Y)**n (Euclidian) as an example (X>Y)**n is a Program (“Protocol”) to n times create all the possible combinations of X>Y.

It is therefore also not strange to call the total process of the movement of Expansion (Yin) and Compression (Yang) in space the Cosmic Breath named the Svara in Sanskrit. Svara स्वर (svara and svāra): Voice, musical note, sound; note of the musical scale, tone in recitation, accent in grammar, symbolic expression.

Hierarchy

The Ished Tree | Egyptian gods, Egyptian art, Ancient egyptian
Ished Tree in Egypt. Thot with the moon on his head (“the inventor) and Seshat with the wheel of the Seven Stars (Pleiades) and the symbol of the Bull, the daughter of Ma’at the Goddess of
Balance aquire the Magic leafs of the tree.The names of the ascended ones are written on its leaves.
The Benu Bird (Phoenix, the Fire Bird) lives on the top of the ished tree. in Heliopolis.
Yggdrasill, Old Norse Mimameidr, in Norse mythology, the world tree, a giant ash supporting the universe. One of its roots extended into Niflheim, the Underworld; another into Jötunheim, land of the Giants; and the third into Asgard, home of the Gods. Asgard was according to Kim Veltman situated in Tomsk Russia.

Balance

The Egyptian Goddess Ma’at was later called Sophia (Wisdom). Ma’at is the keeper of the Underworld. She is also the keeper of the Harmony in the universe. Ma’at weights the Heart (“the center”) with her feather which carries the symbol of the nothing (“the empty set”). When the Heart weights light as a feather the human was balanced in the Heart Chakra meaning that he lived a “Good Life” and was allowed to enter Heaven. Observe that the Heart is the Controller of the Upper (Head) and the Lower (Body) Cycle of Blood (“=Life“). Observe that the human is represented by an Ape with the moon of Thot on his head.
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5. About Perspectives and Projective Geometry

Projective Geometry PG)is a result of Perspective Drawing

IMAGE: Leonardo's perspectograph
The Perspectograph of Leonardo Da Vinci

As you can see in the picture above PG makes use of drawing Lines that connect the Point of the Eye (the Sensor) with the \Planes and Points of the Object of the Drawing. Observe that the object is a Sphere. and that the two lines connect with the sphere at the Horizon which is a line we often confuse with Infinity(“the point we will never reach”).. Also observe that the rear-end of the Sphere is invisible.

The pole of the Sphere can be connected to Parallel Lines that Never Cross. The lines connected to the eye of the artist and the Pole always have an Angle>0.

PG Connects many Points(“people”) by Lines (“Connectors“) and is there fore highly comparable with Communal Sharing of Fiske and Social of PoC.

What Perspectives have in Common.

What is a Point of View?

Two projections of the same object.
Alberti was the first to ask what two pictures have in common if two drawing screens (Planes) are interposed between the viewer and the object, and the object is projected onto both resulting in two different pictures of the same scene. Leone Battista Alberti (1404-72) was the writer of the irst treatise, about Perspective  Della pittura (1435). The solution to this problem was found in 1803 by  Lazare Carnot
In his book “Géométrie de Position” he proved that the so called Cross-Ratio is always preserved in a Projection with One Point of View.
The Cross-Ratio is the Ratio of the Ratio of the Four Points A,B,C,D lying on a Line that intersects the Four Lines defined by a,b,c,d that orginate at the Vanishing Point O. Carnot proved that (AB)(CD)/(AC)(BD) = (A′B′)(C′D′)/(A′C′)(B′D′).
How to unite all the geometries using a General Way of describing Lines and Points in higher dimensions. In this way the Horizon becomes a circle.
Cross-ratios are invariant under Möebius transformations. THis is a visualization.A Moebius transformation is a combination of many transformations, a reflection, a rotation, an inversion a translation and a homothety, a homogeneous dilation .
Geometries are types of Transformations (Rotation, Translation, scaling, projection and )and Things that Stay the same -> Invariants (Length, Angle, parallel lines, incidence (“crossing lines”, points) and the Cross-Ratio ( a complicated way to relateLlines or Angles that have the Same Point of View) ).

about Meta Geometries

Moving one step Higher in Abstraction.

About 4D Space and Quaternions : Quaternions are the Imaginary Numbers (x, i,j,k) of 4 D Space. The Imaginary Numbers are Rotations in 3D space (x,i) . This video gives you a way to train your 4D-view by starting in Flatland being an imaginary world in 2D. A Line is a projection of a Circle that starts at Infinity which is a point at the North Pole of the Sphere.

In 1872 Felix Klein, published a new Mathematical Research Program called the Erlangen Program under the title Vergleichende Betrachtungen über neuere geometrische Forschungen.

In this program Projective Geometry was emphasized as the Unifying Frame for all other Geometries.

Although lines in the Projective Plane meet in one point of Infinity Klein argued that there could be two points of Infinity if the Projective Plane was a Surface Closed in Itself.

When we look at the Origin of Projective Geometry, the Artist painting A Sphere, Earth, on a Flat Surface, it is not difficult to realize that this Closed Surface is a Sphere. This sphere is projected on a plane that contains the imaginary numbers that are solutions of the cross-ratio when the 4 points 𝐴𝐵𝐶𝐷 are not on a circle.

COnformal mappings are mappings that preserve angles. This video shows why the cross ratio is complex when the points are not on a circle.
About Kleins erlangen Program

6 Using The Geometry of Change

6.1 Refresh and Surprise

First I present a few Slides to refresh your knowledge about the Background of the Blog or to surprise you.

This image has an empty alt attribute; its file name is Slide42.jpg
The Paths of Panarchy resemble a moebius Ring (8) that makes it possible to cross colliding lines. The three variables that determine Panarchy. Connection, Potential and Resilience (now called Anti-Fragile , Revolution in Panarchy , Agency in PoC and Yang in Taoism) are put into one Map. Observe the Heart in de Center.Also observe that the Connecting Lines in this Map are Directed Arrows (“Vectors“) and that the connected lines Move with the Clock.
This is a >4 times Expansion of Paths of Change with the Human in the Middle and the Four types of humans that surround him/her. Here you see four types of Social Networks with a Production and Sales Cycle moving With- and Against the clock.The four Communities correspond with the four Relationship types of Alan Fiske.Remember that the four Relationships are related to four types of Geometries. This was proven by relating the Relationships to Scales of Measurement. Measurement theory (Chapter 2) is based on the idea that measurement has to be based on Structure-preserving maps from the outside (what we think we see with our senses) to the inside (what we see with our imagination which is related to an inner sensor (“the third eye”) which is the Pineal Gland.

Keep your Third Eye Open - Foods That Activate And Decalcify Your Pineal Gland of the Endocrine System — MOT MAG

Observe that explain means making something plane (flat) which is a projection from Knowledge (Affine Geometry) to the (flat) Image of the Imagination (Similarity Geometry). Observe also that Design (in Dutch Ont-Werp) is related to Throwing (projection) something at the right moment (ont). What target (Goal (Plan) ) do you want to hit? Realize that the energy for the projection comes out of the fusion of Complementary (Different) concepts. What is different? Not-the same. Not is a Negation.
Professor of Cognitive and Computational Neuroscience Anil Seth looks at the Neuroscience of Consciousness and how our biology gives rise to the unique experience of being you.THis presentation is mostly about the so called Be Bayesian Brain. In this theory the brain is a prediction machine but produces a “best guess” combining internal “memory” / “imagination” and external data produced by the senses. Damasio: ” Living systems are not static, but they act in the world in order to minimize surprise and persist in a future shaped by uncertainty” but humans Need Surprise to find life exiting.
“Organs are not machines” : The defining property of a living system is that it has organisational closure: every efficient cause that is necessary for its continued existence must be in turn generated from within the system. We look at organisational theories of the organism from Aristotle, via Kant, to Robert Rosen, who provided a basic relational diagram for a system with closure to efficient causation. 
Mathematical Foundations of Anticipatory Systems | SpringerLink
Robert Rosen: Anticipatory System.

ROBERT ROSEN AND GEORGE LAKOFF: THE ROLE OF CAUSALITY IN COMPLEX SYSTEMS – PowerPoint PPT Presentation

The presentation behind the link above explains the theory of Robert Rosen in combination of the theory of George Lakoff. For the last one move below. For the first one explore this blog about Meta-Engineering. As you will see is that the presentation tells a comparable story as Chapter 2 about measurement in the way that it explains that a model is a mapping from something we experience outside with our senses (A natural system) or with our instruments and we have to make a structure preserving mapping to avoid that we introduce a bias with our measurement instruments.

Body and Mind are connected by the Heart and are centers of cycles where the four views/types move around. The four views with one shared view create the “magic” number seven (3+1+3) . Observe that the cycles move with and against the clock creating the well known Moebius Ring (8). This model maps on the Indian Model of the Chakra’s and the Chinese Median Points. Service above Self is the motto of the Rotary where ROTAR is a permutation of TAROT.
Paths of Change mapped on the minor Arcana of the Tarot.
The path of the Hero (Fool) connects 2 triangular paths
The history of the Tarot. Rob Doctors van Leeuwen was visited by an Angel who asked him to do research about the pattern of the Tarot. He (and his brother Onno) detected two new cards called Truth (Blue) and Intuition (Yellow)
Paths of Change used in a model about the Production Cycle.
A picture out of my Blog about a new Ecology of the State that unites Panarchy, ALan Fiske and Paths of Change. It is an application of the Geometries. The most important Worldview is Yellow and was called Mythic by Will McWhinney in his book Paths of Change.
Mythic relates to Intuitive in the Nyers Briggs Model: “Intuitive people live in the future and are immersed in the world of Possibilities. They process information through patterns and impressions. Intuitive people value inspiration and imagination. They gather knowledge by reading between the lines. Their abstract nature attracts them toward deep ideas and concepts”

The first thing you have to realize is that you have been looking at pictures that describe the same subject with different names. All of them are about Change and Change looks like Breathing <-.-> (Contract/Expand). Below you see the Expansion Pattern of “nothing”/. This pattern is controlled by the Trinity, the number 3 and the Triangel. This pattern is called the Bronze Mean. It is a specialization of the Golden mean. The bronze mean contains a very important number 43.

The center is the Empty set (Yellow), the Nothing, the Foundation of creativity. Desire (Female) and Control (Male) create a child (Communion) Left you see Adam Kadmon the first Human and the Tree of Knowledge.
By playing with the Arrows (->) Expansion (Red) and Compression (Blue) are reproduced. The other 2 combinations create The Emotions (Green) and Creativity (Yellow).

The first picture of this paragraph contains a Cycle connected to another Cycle where both cycles move in opposite direction showing the well known Moebius Ring (8). The model contains a center that contains the model of the whole. This is called a fractal. When you compare this model with other models look at the colors. I have used the same colors of PoC all the time.

6.2 About Affine Geometry, Maps, Metaphors and Mappings

In the model of the Geometry of Change you see that Blue, named Unity in PoC and Knowledge in other models is called an Affine Geometry and is connected the concept Rank and the Constraint If s1 < or >s2 ⇔ f(s1) < or >f(s2) which is called the Order-Preserving Group. Knowledge is an Order but what order?

The solution to this puzzle can be found at the beginning of this blog where you read the sentence “The role of Higher Dimensions in Physics and Mathematics”. The concept Dimension is used to define a Ranking of Spaces.

Basic spatial dimension
Higher Dimensions are represented as Projections on a Plane. This 2D plane could be called a Scape (taken out of out of Land-scape).
George Lackoff researched the brain and detected that the Brain is based on the Body (“Embodiment”). When we are young we learn that it is difficult to learn to find balance to stand right up and to move. Thats why Up (Higher) is better dan Lower (Down). Thats why Heaven is Up and Hell is Down. Thats why a Tree is a Symbol of Knowledge. George also discovered that we always make representations of what we think. When we try to represent an abstract concept we use a prototype (one out of the set). He also found out that Our brain uses Frames to interact with the outside world: About Framing. This is also the subject of Goffman.

6.3 About Euclidian Geometry, Distance and the Horizon

beach birds calm clouds