Structural Geometric Topology by Albert P. Carpenter

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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.

Disciplines: Geometry and Topology


A. Geometry: (ontological definition)

B. Topology: (ontological definition)

C. Structural Geometric Topology: (structural ontological definition)

Theory: Structuralism


The unifying theory common to each is structuralism as found in theoretical chemistry and theoretical biology. So conceived, the relationship between each compound discipline is of components to structures. In Structural Geometric Topology for example, polyhedra, polygons, lines and points each function as components for topological structures such as knots, links, tori/handlebodies, Mobius strips, braids, weaves and Kleinian structures etc.

Methodology: Integration/Disintegration

These objects are the product of a new methodology for cognitive mathematics born of Cognitive Science. In the latter field, it is known as conceptual integration, but I have systematized it using qualitative matrices and named the process “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, monocylinders, polycylinders, handlebodies, Mobius strips, braids, weaves, and Klein bottles.

Taxonomy: Hierarchical and Nested

A. Classification: Hierarchical and Nested

B. Nomenclature: Binary Nomenclature (Hierarchical and Nested)


A. Order:

B. Sequence:

C. Arrangement:

D. Orientation:

E. Location:


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.


As regards the pointal models; there already exists an extensive study of these objects at: the Please go there to see them.

Videography, Blogography and Bibliography:

  1. Albert P. Carpenter: Applied Structural Metamathematics (Video Presentations and Slide Shows) –
  2. An Introduction to Structural Geometric Topology:
  3. An Introduction to Structural Metageometry:
  4. An Introduction to Structural Topological Geometry:
  5. An Introduction to Structural Metatopology:
  6. Structural Geometric Topology:
  7. Structural Metageometry:
  8. Structural Topological Geometry:
  9. Structural Metatopology:
  10. Porous Polyhedra:



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