Structural Geometric Topology by Albert P. Carpenter

Home » Journal II (for Structural Geometric Topology). Notes, Comments and Updates (19 Entries)

Journal II (for Structural Geometric Topology). Notes, Comments and Updates (19 Entries)


Lapse in posts due to change in meds…but ok now!

Still no Structural Metageometry! But soon I hope!….

Will post 2 new vids on structural metageometry Monday or Tuesday at Youtube.

Moving into other areas of structural geometric topology…

  1. Pointal topology
  2. Linear topology
  3. Polyhedral topology

Need to finish up with vids for structural metatopology…


New book, Structural Metageometry delayed. Editor on vacation. Working on the following:

  1. Monosurfaces
  2. Polygonal Topology: Hexagons and Squares; Triangles and Squares for new slide show presentation on Youtube.
  3.  Will finish up the video series for Structural Topological Geometry (videos 4 and 5) and if all goes well they will be posted on Monday.
  4. Sorry for the bad hair day on videos 1-3 for Structural Topological Geometry 🙂
  5. Need to work on Research for next seriesof videos in Structural Metatopology. Do not have as yet all the models needed to make the videos.

Registered (online) with the Redbud Topology Conference in April at U. of Arkansas, but have not heard back and am not sure if it is by invitation only or not?  Would like to go with some models and discuss them with people there if possible.


The first book of four, An Introduction to Structural Geometric Topology, is  published on (  The second book: An Introduction to Structural Metageometry is written and awaits editing, while the third (An Introduction to Structural Topological Geometry) is in a first rough draft form.

Currently working on models for Structural Metatopology. This is more challenging than the others as they are not immediately obvious to me, but I an determined to create them.

Have begun a study of one sided surfaces of which the Mobius strip is but one. Others include, Structural Hypercards, Structural Kleinian surfaces and what I’m calling ‘Structural Curlions’ which are surfaces part of which is cut and curled back onto itself.

Also am studying Bikelion, Trikelion and Tetrakelion etc. structures as well.

All for now. Happy New Year!


Finished the video series for Structural Geometric Topology at Youtube: (12 videos). Have an introductory video for Structural Metageometry up and the 2nd of 5 in post production.

I’m beginning to make models for Structural Topological Geometry for that video series/book including: Knotted Polyhedra, Linked Polyhedra, Toric/Handlebodied Polyhedra and Mobial Polyhedra to start. I know how to make models of these structures, but am yet to actually construct them. Soon!

I will not be exploring polygonal weaves at this time as planned, but will return to them later.

The book on Structural Geometric Topology is finished, but we are having difficulty uploading it in a format that’s Create Space will accept. Soon, I hope.


I have 11 videos (7 of me and 4 of just objects) up on Youtube now. All on or related to Structural Geometric Topology. I start tonight with my next video series of 5 videos on Structural Metageometry. They will follow the same pattern as the others.

Started working on Polygonal Weaves today. I will post a presentation on them in a week or two.

The book , An Introduction to Structural Geometric Topology is still in editing. Hope to have it on Amazon by Monday the the 12th or there abouts. Then we can start to edit the next book on Structural Metageometry.

Need to focus on making models for Metageometry now and stop making them for Geometric Topology.

I plan to have at least 30 videos posted when all  is said and done. It will take about 15-20 weeks to complete, if all goes as planned.

I also have a presentation prepared on the Structural Laws of Matter and Energy as applied to Chemistry that I will post on the 12th or 13th.


Getting close to completing final edits to book An Introduction to Structural Geometric Topology with new editor Tara Roys. First draft of An Introduction to Metageometry awaits editing. 5 videos now on Youtube, 2 more to go to finish out the series on Structural Geometric Topology (Linear Topology is in post production), then I’ll make the next series of videos on Metageometry. Also have a slide show presentation on 82 objects in Polygonal Topology.

Have made dozens of new Polygonal Topological structures that I will photograph when the fog lets up. Otherwise doing well, but not sleeping much (3-5 hours per night). Hypomanic and hence lots of energy that I am putting to good use.

There is a great need for the mathematical community to:

  1. Acknowledge these new fields of mathematics
  2. Establish journals and research groups to study these objects
  3. Join with Mathematical Chemistry/Chemists to work together to create molecular analogues of these structures
  4. Develop taxonomic systems and nomenclature for structures in Structural Geometric Topology etc.
  5. Develop software that will render Polygonal Topological Structures and Structural Geometric Topological structures, as well as, Structural Metageometric structures etc.

I use a pattern to structure the books I’m writing. They follow the structure of the disciplines themselves (Structural Geometric Topology etc.) and each is a kind of fractal book exhibiting self similarity at scale. Not only with in the introductory books, but in book series as well. So that with each book that delves deeper into each subject there is a zooming in of scale with added richness in detail.


Book (An Introduction to Structural Geometric Topology)  is still in editing mode. May be a week to 2 weeks till it is up on Amazon. 5th video is done and in post production: Structural Geometric Topology Part 3: Polygonal Topology. Will proceed with Book: An Introduction to Structural Metageometry and then 2-4 (of 100 possible books(one each for the binary categories of polygonal combinations)) books on Polygonal Topology: Homogeneous and Heterogeneous/ Periodic and Aperiodic  Polygonal Tori, Handlebodies, Knots, Links, Braids and Mobial Structures. These books will be much more detailed and will provide a sense of the richness of the field, while further refining the structure and organization of the disciplines involved.

Toying with the idea of building my own online database for these objects in Structural Geometric Topology (that I have built) modeled on those at wikipedia for polyhedra. Also, it will be a year or two before I can publish my work on synergy and dysergy in Chemistry.

A vision of the future for Structural Geometric Topology etc.:

  1. A peer reviewed journal and an online database for each discipline that results from the integration of Geometry and Topology to include: A. Metageometry
  2.                                                                                                          B. Geometric Topology
  3.                                                                                                          C. Topological Geometry
  4.                                                                                                          D. Metatopology
  5. Each of these will be organized by sub-disciplines based on the systematic integration of their respective ontologies. Thus, Geometric Topology will be organized by Polyhedral Topology, Polygonal Topology, Linear Topology and Pointal Topology. Each of these in turn is organized and structured by the further integration of the components used to construct the topological structures as for example homogeneous pentagons or heterogeneous pentagons and triangles etc.
  6.  An interdisciplinary committees will be formed to oversee the construction of the online databases, journals and also be responsible for the taxonomy, nomenclature etc. analogous to current chemical societies and communities.
  7. All this should be integrated with Molecular Metageometry, Molecular Geometric Topology etc.
  8. The establishment and integration of these two areas; materials science and mathematics should be modeled on existing efforts to study zeolites and their catalog online.


Very busy. Finished first draft of book; An Introduction to Structural Geometric Topology on the 11th of October. Laurance Friend is editing it. Hope to have final draft by  Monday/Tuesday the 24th and to publish on Amazon soon afterwards.  Second video up (The Systematic Integration of Geometry and Topology). Third video (An Introduction to Structural Geometric Topology Part 1). in post production to be uploaded on the week of 10/24. Fourth video (Structural Geometric Topology Part 2: Polyhedral Topology) scheduled for the 21st of October. First draft of An Introduction to Structural Metageometry is finished. Outline for An Introduction to Structural Metatopology and An Introduction to Structural Topological Geometry finished. Not making any more models for now. All books are but place holders until a committee can be formed to create database (one for each discipline) for objects and their mathematical properties similar to Online Zeolite Database which should be a model for the latter.


I’ve started making video presentations for the Systematic Integration of Geometry and Topology. I’ll begin with Geometric Topology and go from there. Started thinking about the Systematic Integration of Science and Mathematics. Ontological work is on hold (the making of models) as I work out the details of the above.


I have fully codified my architectonic vision for Geometry and Topology in terms of structure and composition.  It involves the integration of a 16×16 matrix in the formation of 256 Spatio-Dimensional categories. These then are interpreted in terms of points, lines, polygons, and polyhedra for Geometry and knots, links, tori/handlebodies and Mobius strips ect. for Topology!  We build objects not just compositionally i.e. atomically, but also build structures with other structures! In the same way that we build molecules with atoms, so to do we build larger molecules with smaller ones! Thank you Click Chemistry! The grandeur and beauty of this system is staggering!


Back to Geometric Topology, Topological Geometry and Metatopology:

  1. 3 Space Polyhedral Handlebodies (Geometric Topology) These are similar to Stewart Toroids but Polyhedral, Polygonal, Linear and Pointal.
  2. Anullated, & Sieved Braided Mobius strips etc. (Metatopology)
  3. Closed Weaves (as part of Geometric Topology)
  4. Braided; Lines, Polygons and Polyhedra (Topological Geometry) Need to study DNA polyhedra to see how to braid polyhedra.

Computer crashed! Need to get a new one and more Great Stella Software! Will do ASAP. When I get the $ Unit then will work on 2-4, but need computer for #1. JUnction Geometry and Syndimensional Geometry will have to wait. Will send out packages to universities when I get the cash to do it.


Been working on Junction and Link Geometry/Topology as it applies to Network Geometry, but then stopped. Need to figure out how to put graphics from Great Stella into power point . Output is low right now.  Goals for conference include; networking, exploring collaboration possibilities with an eye towards publishing and future speaking opportunities. Would like to advance this approach to Geometric Topology etc. via academic out reach. If nothing happens at conference, then I will explore other options including contacting local universities to show and talk about my work in Geometric Topology and art. Am especially interested in St. Louis University as they have a department with emphasis in Geometric Topology. Not going to load more animations because they slow down loading rates for my pages and take up too much space. Geometric Topology needs to establish/set up a database like the one for Zeolites online with over 5,0000,000 objects in it and then post it on the web. A committee needs to be set up to administer such a project. Hope to some day see that happen? Extended  my methodology (Systematic Conceptual Integration/analysis) to include Systematic Categorical Integration and Systematic Ontological Integration (with a spatial dimension added to it thus making it a cubic Matrix that nests each one within the other!).


Did not go to Colorado, but will try for the Unknot Conference in Ohio at Denison University. Made some studies of Interdimensional Mobius strips (those that occupy two planes and are thus not planar, but biplanar). About 10 of these and will post later at Geometric Topology II.  In a bit of a holding pattern. Not sure what to do next. Have made over 240 Geometric Topological objects for study. Will take some to the conference to talk about and for the presentation. I could go into and study the other 3 disciplines defined by the Integration of Geometry and Topology. We’ll see….


Uploaded first of the animations for Prismic Mobius structures ect. (11 animations in all-slow going!). Will post more later. Am working out variations of the Polygonal Trefoil, Tetrafoil, Pentafoil and Hexafoil Knots and Sieves with linear analogues up to Heptafoil and Octafoil varieties as well. Trying to get to the Workshop in Geometric Topology at Colorado College in Silver Springs, Colorado on the 8th-11th of June, 2016. May not be able to go. It may be too late to present. Even so would like to go anyway just in case. Will take photos of 30-40 new structures for Geometric Topology and post here as soon as the weather permits (been rainy lately).


Been working with Great Stella Software by Rob Webb to build geometric topological objects with Polyhedra. Having considerable success. Over 100 objects completed in 1 week. Some quite complex. Also taught myself how to make 2 Crown polyhedra using Stella and also working with B. M. Stewart’s Adventures Among the Toroids to create some nice polyhedral knots. (also found Dovesky’s work online very interesting).

One method I use to generate topological structures with geometric composition is to google search knots and then find those that can be “translated” into polygons, polyhedra points and lines.

Also looking at Molecular Geometry and Molecular Topology for ideas for pointal molecular models. I have in mind that Euclidean Geometry and Topology can be intertheoretically reduced to Molecular Geometry and Molecular Topology! Doing so would allow for the creation of novel Geometries based on molecular structures such as Biphenylene etc that suggest to me the possibility of  Syndimensional Geometry.

For those interested, I have another Geometry Journal at There are many math. and molecular models there as well.


Working on Cubic Knots, Links, Tori and Mobial Structures. Updated site to reflect Low Dimensional Geometric Topology instead of Two Dimensional (space) Geometric Topology. Will add Linear and Pointal models shortly in addition to polyhedral and polygonal models.

Note: Stewart Toroids are included here as examples of Polyhedral Tori and Handlebodies etc.


Working on presentation for Unknot conference at Denison University. If accepted?


Working on Discrete Closed Braids and Discrete Closed Links in 2 space.


I’m thinking now that what these objects represent are ones that belong to the discipline of Discrete Geometric Topology since they have discrete components (the polygons) and that each of the other disciplines: Topological Geometry, Metageometry and Metatopology are also defined here as being discrete. These are to be contrasted with shapes such as the torus which having no parts or components are thus continuous. So now have discrete and Continuous Geometric Topology, Continuous Topological Geometry, Continuous Metageometry and Continuous Metatopology.

The advantage of discrete over continuous Topology and Geometry is that discrete disciplines are more amenable to being physically instantiated as molecules.

Working on articles: Reductive Materialism and the Axioms of Euclid as well as Eliminative and Multiplicative Materialism in the Philosophy of Science.

Then will write: Synergy and Dysergy and the Subsumption of Arithmetic

And then finally when I have a computer and Great Stella software about Discrete Geometric Topology


Exhausted and fatigued. The stress of the this project is taking its toll. Need help with nomenclature and taxonomy for Geometric Topology. Want to base it on Nested Container cognition like that found in Biology’s taxonomic system and or Ecology’s system. Should be binary nomenclature with composition and structure as the defining characteristics.

Want to apply new method of Systematic Conceptual Integration to Science and Mathematics in the same way as I did for Geometry and Topology, but not clear as how to proceed. Wish I had help/someone to talk to. It would entail a table/ binary matrix with the following results:

  1. Metascience
  2. Scientific Mathematics
  3. Mathematical Science
  4. Metamathematics

But how would I define 1-4’s structure and composition?

  1. Metascience would be the application of the scientific method to science itself and would include: induction, empiricism and experimentation

2) Scientific Mathematics would be the application of the scientific method to Mathematics: i.e. induction, empiricism and experimentation.

3) Mathematical Science would be the application of mathematical methods to Science. (But what are mathematical methods in a general way?)

4) Metamathematics would be the application of mathematical methods to mathematics.

How, again, would I create a table for each with the same richness as that found in Neogeometric Topology?

Fixed the issue (of how to create a rich ontology) as can be found in the second part of the methodology page.

I fear these objects are so new and unlike previous conceptions of topological objects that they are being rejected by Google Image search. I must send examples to professors and a link to this page. I hope that will work, but what a loss if it does not.


Working today on the methodology and the glossary sections.

Need to provide names and figure numbers for all the objects on the web page.

Will have a second methodology section in a few days on interdisciplinary integration.

Will make a video lecture of the outlined methodology for Systematic Conceptual Integration tomorrow.

Will take some side view photos of Mobius strips to give a sense of their three dimensionality.


It took 7.5 hours (slow computer), but I got everything up. Looks good. Soon (Monday/Tuesday) I will make the video lecture for Cognitive Geometry and Topology.  Then, I will send out e-mails to various departments to see if anyone is interested in this body of work???


I plan to upload a large body of work tonight that represents my current research. It is just the tip of the iceberg of what is to come; a first step. I will explain how I did this and its significance in a later post.

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