Structural Geometric Topology II by Albert P. Carpenter

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This site is the second part of my first site entitled Structural Geometric Topology (geometrictopology1000.wordpress.com). This one is a continuation of the last and is constructed with the same idea that these models are of both molecular and mathematical interest.

Structural Geometric Topology is one of four mathematical disciplines that result from the systematic unification (integration) of geometry and topology. The others include the following: Structural Metageometry, Structural Topological Geometry and Structural Metatopology.

The minimum requirements for a fully functional mathematical disciplinary system are as follows:

  1. A clear definition of the discipline
  2. A clearly defined theory
  3. A clearly defined methodology
  4. A clearly defined taxonomic system
  5. A clearly defined ontology
  6. A clearly defined syntax

Disciplines to be unified: Geometry and Topology

Definitions: 

A. geometry: ontological definition-

B. topology: ontological definition-

C. structural geometric topology:

a. structural geometric topology: theoretical definition- the study of topological structures with geometric composition

b. structural geometric topology: theoretical and ontological definition-the study of topological structures including but not limited to knots, links, tori/handlebodies, Mobial structures, Kleinian structures, weaves, braids, cylinders, fulcrum, and annuli with geometric composition including but not limited to points, lines, polygons, and polyhedra.

Theory: Structuralism

Definition:

The unifying theory common to each is structuralism co-opted from theoretical chemistry and theoretical biology. So conceived, the relationship between each compound discipline is of components to structures.

Methodology: Integration

These objects are the product of a new methodology co-opted from cognitive science for use in cognitive mathematics. In the former field, it is known as conceptual integration, but it has been systematized using qualitative matrices and the process has been named “Systematic Conceptual Integration”.

Ontology: Objects of Study

A. Definition of Ontology: a catalog of objects

1. Ontology for geometry:

For geometry, the ontology of Euclidean geometry is used and is based on the classification of geometric objects according to their dimension. The ontology (as expressed by broadest categories) used for geometry includes: points, lines, polygons, and polyhedra.

2. Ontology for topology:

For topology, the ontology of contemporary topology is used and is derived from on-line sources. These are not classified based on dimension. The ontology used for topology includes: knots, links, tori, annuli, cylinders, tori/handlebodies, Mobius strips, braids, weaves, and Klein bottles, and fulcrum.

Taxonomy: Hierarchical and Nested

A. Classification: Hierarchical and Nested

B. Nomenclature: Binary Nomenclature (Hierarchical and Nested)

Ontogenic and Taxonomic Matrix

for

Structural Geometric Topology

(Topological Structures)

Operator: Integration Knots Links Tori/Handlebodies Mobial Structures
Points Pointal Knots Pointal Links Pointal Tori/Handlebodies Pointal Mobial structures
Lines Linear Knots Linear Links Linear Tori/Handlebodies Linear Mobial structures
Polygons Polygonal Knots Polygonal Links Polygonal Tori/Handlebodies Polygonal Mobial structures
Polyhedra Polyhedral Knots Polyhedral Links Polyhedral Tori/Handlebodies Polyhedral Mobial structures

(Table 2)

Note:

Geometric components are to be found in the furthest left hand rows of Table 1.

The ontology of topology could be extended to include braids, cylinders, fulcrum, weaves, Kleinian structures etc.

Syntax:  The spatial order, sequence, arrangement, orientation, and or location of the components of a structure.

A. Order:

B. Sequence:

C. Arrangement:

D. Orientation:

E. Location:

Chromology:

The use of color is intended to elucidate the substructures of which the objects are built. One color for each substructure is used to construct a single structure.

Data Sheets:

Ideally, for every structure in structural geometric topology, there would be an accompanying mathematical data sheet with its name, disciplinary identification, classification and properties etc. Unfortunately, that is beyond my resources at this time. However, included at this site is a sample data sheet that would be filled out for each object in the creation of a database. In addition to the mathematical data sheet, some day in the future there will also be a physical (molecular) data sheet for each object depicting its physical analog and physical properties.

I would like to acknowledge the pioneering work in Structural Polyhedral Topology of B.M. Stewart as published in his book; Adventures Among the Toroids, although he never recognized it as such.

Videography, Blogography and Bibliography:

  1. Albert P. Carpenter: Applied Structural Metamathematics (Video Presentations and Slide Shows) – https://youtube.com/channel/UCVfQupdDTOCdo87RnvCdeug
  2. An Introduction to Structural Geometric Topology: https://www.amazon.com/Introduction-Structural-Geometric-Systematic-Integration-ebook/dp/B01N1UFS6B
  3. An Introduction to Structural Metageometry: https://www.amazon.com/Introduction-Structural-Metageometry-Systematic-Integration-ebook/dp/B0721Q6GBJ/ref=sr_1_2?=digital-text&ie=UTF8&qid=1505725832&sr=1-2
  4. An Introduction to Structural Topological Geometry: https://www.amazon.com/Introduction-Structural-Topological-Geometry-Metamathematics-ebook/dp/B074MB3N/ref=sr_1_3?s=digital-text&ie=UTF8&qid=150725832&sr=1-3
  5. An Introduction to Structural Metatopology: https://www.amazon.com/Introduction-Structural-Metatopology-Systematic-Integration-ebook/dp/B074M9MDHF/ref=sr_1_4?s=digital-text&ie=UTF8&qid=1505725832&sr=1-4
  6. Structural Geometric Topology: https://geometrictopology1000.wordpress.com/
  7. Structural Metageometry: https://metageometry2.wordpress.com/
  8. Structural Topological Geometry: https://topologicalgeometry.wordpress.com/
  9. Structural Metatopology: https://metatopology2.wordpress.com/
  10. Porous Polyhedra: https://polyhedron100.wordpress.com/

 

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