Cognitary Stratus


trivergent, divergent, univergent, convergent.

“History does not repeat itself, but sometimes it rhymes.” –Mark Twain

Cognitary, Inc.


to Found

foresought and fidel,

forethought and factual,

familiar and friendly,

fair and full

to Fiat

seer and leader,

feeler and finder,

giver and taker,

seller and buyer


to Future

principle and power,

understanding and knowing,

safety and health,

prosperity and wealth

to Flow

vessel and berth,

heaven and earth,

table and hearth,

market and dearth


to Function

designing and engineering,

plotting and navigating,

crafting and smithing,

profiting and possessing

to Form

goal and person,

event and location,

service and product,

price and metric


to Fashion

control and command,

climate and terrain,

training and discipline,

currency and commodity

to Foot

sanctity and dignity,

certainty and verity,

testity and pacity,

quality and quantity

The above outline is the evolving strategic framework of my company Cognitary, Inc.  I am working to build a community of generalists to tackle client problems across the disciplines.



Universe: Czerepak Framework R0.2

I have been thinking about the terms “convergent” and “divergent” in Tim Brown of IDEO’s Design Thinking and realized that they were products of planar (2 dimensional) thinking.  This has lead me to alter my definitions of what convergent and divergent are and to also redefine “vergent” and add “trivergent”.  I also realized that the convergence point is at the center of the ellipsoid and each verge (radius) point is separate and distinct.


Converge, diverge, verge and triverge all come from the same Latin root “verto”, to turn out.  All of the polyhedron vertexes are representations of the intersections of radii with the surface of not a sphere, but an ellipsoid.  Therefore each vertex is a unique dimension or radius.  However, there is one thing that is still not recognized.


Roll, Pitch and Yaw ellipses alone are an incorrect representation of orientation in space because they fail to include orientation relative to the observer.  Roll, Pitch and Yaw are flat earth concepts.  You cannot represent an ellipsoid with three radii.  The minimum radial representation of an ellipsoid requires four points on the surface of the ellipsoid.  The tetrahedroid is the minimal representation of the inscription of an ellipsoid.


The above three ellipse object and four ellipse object are both ellipsoids, the only difference between them and the three ellipse ellipsoid above them is the perspective–they have been rotated in space.  Using the four dimensional representation gives us the table below:


While I was illustrating the above table it became apparent to me that it accurately reflected  John Boyd’s OODA Loop.  It also became apparent to me that the OODA Loop could be conceputally simplified to:

  1. OBSERVE: Range
  2. ORIENT: Direction
  3. DECIDE:  Elevation
  4. ACT: Fire

The OODA Loop or the Czerepak Framework cycle can be graphed as a simple two dimensional sine vertice:


Now, that’s all sure and fine and it provides a way of thinking with a minimum number of variables.  However, if we think about John Boyd as a military combatant it is not the right set of variables.  The model has to cater to the following needs:

  1. OBSERVE: Who and Why
  2. ORIENT: Where and When
  3. DECIDE: What and How
  4. ACT: How Many and How Much

Suddenly, it becomes obvious that in a system involving living organisms there is added complexity and layers of consciousness.  The following table is my first attempt to illustrate this:


The yellow row and column headers are what is of importance.  The naming of the white cells will have to come later.  Obviously, there are considerable changes in the order of the columns and rows, but I believe John Boyd is closer to the truth about the process than anyone else.  Therefore I am redefining everyone else’s concept to fit his.  What is important about the table is that in the columns each icon represents a set of ellipses that one ellipse at a time intersects with the ellipses above it to converge on a subset that is the target.


Colonel Boyd’s model was simple.  Deviations from it are based more upon misunderstanding than anything else.  This is the fundamental System Development Lifecycle (SDLC):

  2. ORIENT = INSERT = ANALYSIS = CLIMATE and TERRAIN = TACTIC = DIVERGENT THINKING = When and Where are the exceptions?
  3. DECIDE = UPDATE = DESIGN = FUNCTION and FORM = OPERATION = VERGENT THINKING = How and What are the exceptions?
  4. ACT = DELETE = DEVELOP = QUALITY and QUANTITY = GOAL = CONVERGENT THINKING = How Much and How Many are the exceptions?

“What are we deleting?” You may ask.  We are deleting exceptions that existed in the previous system whatever that system may have been.  We are never dealing with a non-existent system.  We are SELECTing a set of exceptions the current system does not handle.  We are INSERTing those exceptions into the current system.  We are UPDATEing the system to handle those exceptions.  We are DELETEing those exceptions from the system.  I still have to work to reconsider the names for each of the cells, but I am converging on that.  The differences between methodologies are really ones of scale and nothing else.  It’s how many exceptions do you intend to address at a time.

This effort is requiring a lot of work and rework because I have never dealt with eight interrogatives before, however the fit is conceptually the best I have ever had.

Universe: Interrogative Spaces


In my previous post I gave thought to Tim Brown of IDEO’s “design thinking”, Clayton Christensen’s “Innovator’s Dilemma”, Malcolm Gladwell’s “Tipping Point”, and Buckminster Fuller’s “Synergetics” concepts.  What emerged was the above Czerepak Framework.  My claim is this framework is fundamental to designing a system.

The thing that the above table shows is interaction within what I am now going to call the “Interrogative Spaces”: HowSpace, WhatSpace, WhySpace, WhoSpace, WhenSpace, WhereSpace, HowMuchSpace, HowManySpace.  Each ellipse I call a “vortice”.  The Interrogative Spaces are composed of one or more vortices.  The Framework above shows how Spaces are composed within the Interrogatives,  but what about interactions between the Interrogative Spaces?   A good example is speed or velocity.  Speed is the intersection of WhenSpace and WhereSpace:

v = r / t

Where v is velocity, r is radius and t is time.

If you are increasing Speed, which is acceleration, you have one dimension of WhereSpace and two dimensions of WhenSpace:

a = r / t’ * t”

Where a is acceleration, r is radius, t’ is the first clock and t” is the second clock.  You cannot measure acceleration with one clock. This uniqueness of every vortice applies to all the Interrogative Spaces and all inter-relationships between all of the Spaces.  .

Another way to look at the Interrogative Spaces is as sets and subsets.  The first row are the complete Space vortice sets.  The second row are the first Space vortice subsets.  The third row is the intersect between the row two and row three Space vortice subsets. And the fourth row are the intersects between the row two and row three and row four Space vortice subsets.

I do not believe that anything is constant.  Not the speed of light, not gravity, not cosmology.  Every intersection of dimensions creates a vortex in Universe and every one is unique.  We are simply unable to measure and manage the uniqueness of everything, therefore we make generalizations which create models that can always be falsified.

The Green Knight’s Quest

Universe: Hexahedron Theory

Hexahedron Schema:

  1. 4 Axes are Dimension Particle Sets
  2. 8 Vertexes are Space Particle Sets
  3. 12 Edges are Force Particle Sets

Additional Schema Components:

  1. 4 Axial Plane Sets
  2. 6 Edge Plane Sets
  3. 16 Axial Plane Triangulation Sets
  4. 24 Edge Plane Triangulation Sets

Look at the vertexes of the hexahedron as entities.

Entities are Sequence->Value->Type

Look at the edges and axes of the hexahedron as associations.


are: SourceEntity->VerbEntity->TargetEntity

or: SourceAssociation->VerbEntity->TargetEntity

The instances for the entities and associations are the sets we are working with.

The key is the universe is composed of particles of a broad variety.  But every particle is simply an association in the form of a set.  The lowest order particles are event and point.  They are one dimensional particles.  All subsequent higher dimension particles can be reduced to a subset of these particles.

I have revised my theory to include the observer in the system.  I am of the opinion that the observer is not unary but binary having two hemispheres to the brain.  Position and Velocity are composed of sets not points and are observed separately by the ordinal and cardinal hemispheres of the observer.  Consequently, the universe is not probalistic, but wholly deterministic.

Where – When : Space – Time

Sequa is an ordinal point set while frequa is a cardinal event set.

What – How : Mass – Light

Quala is an ordinal sequency set while Quanta is a cardinal frequency set..

Why – How Much :  Gravity – Energy

Grava is an ordinal quality set while Erga is a cardinal quantity set.

Who – Whom : Ordinality – Cardinality

Orda is an ordinal gravity set while Erga is a cardinal energy set.

I think there are even higher order entities and associations, but I have still to work them out.

Systema: Geodesates, Nodes and Links

“To every action there is an equal and opposite reaction.” — Isaac Newton

A predominant issue arising from my work is the discovery of the difference between a node and a link.  A node type represents a state type while a link type represents a transaction between state types.  However I am finding there are a limited number of node types (self-ordered states) and link types (self-ordered state actions).

In the diagram below, each polyhedron is a first frequency geodesate and has a unique polytrope/polytype combination.  A polytrope is the number of edges per polyhedron vertex.  A polytype is the number of polyhedron vertexes.  This is not the final version.  I am still working to purify my geodesate concept.

What I am revealing here is that each of the seven Node Types on the Left has only one Link Type on the right.  In the same way that an association is composed of a source node type and target node type, an association is composed of a source link type and target link type.

Here is an example of a homogenous Entity to Entity association:

Here is an example of a hetergeneous Entity to Positity association:

Having considered this it is now possible to conclude that there are a unique set of nodes each with a unique link which can be used to build homogeneous or heterogeneous associations.  In otherwords, each node type can perform only one action type.  It is the reaction type of the target node type that makes the action reaction combination unique in the system.

Let’s look at some examples of node type and link type associations:

  1. To identify a positity, positifies an identity.
  2. To objectify a projectity, projectifies an objectity.
  3. To chronify a chronity, chronifies a chronity.
  4. To projectify a quantity, quantifies the projectity.
  5. To qualify an identity, identifies a quality.

Fourty-nine possible type combinations exist.  I think there are even more types which I will explore with Archimedean Solids and higher frequency Geodesates in later posts.

Systema: Orientational, Decisional, Actual

Why – one of the six causalities
Who – one of the six identities within a causality
How – one of the six qualities within an identity
What – one of the six quantities within a quality
Where – one of the six spatialities within a quantity
When – one of the six temporalities within a spatiality

What I have just defined, finally, are the six dimensions of the interrogative hypercube. Six to the sixth power. That’s 46,656 unique combinations.

The temporal, spatial, quantal, qualal, idental and causal entities are each three dimensional, I hazard to guess, because each is a higher level abstraction of the same phenomena. Three dimensional space is an abstraction of three dimensional time; three dimensional quantity is a an abstraction of three dimensional space and so on.