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“).
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.
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
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.
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 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.
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.
Figures shown in the same color are similar. A similar structure can be transformed by Rotation, Translation (Displacement) and Scaling (Multiplication), “blow up/down”).
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 Pythagoraswhich 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).
The Colors of PoC (Red, Blue,, Yellow and Green) come back in every slide.
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 variablesAgency 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 possiblecombinations 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.
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.
PGConnects many Points(“people”) by Lines (“Connectors“) and is there fore highly comparable with Communal Sharing of Fiske and Social of PoC.
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.
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.
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.
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 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.
6.3 About Euclidian Geometry, Distance and the Horizon