In this section, I’ll update forth coming photos of polyhedra and journal entries. The journal entries are mostly my thoughts on Geometry, Chemistry, Nanoscience and Topology.
Please go to geometrictopology1000.wordpress.com for more journal entries and my most recent work on that topic. Geometric topology is to be defined as the study of topological structures such as knots, links, Tori/Handlebodies and Mobius strips/Klein structures with Geometric Composition such as polyhedra, polygons, lines and points. Conversely Topological Geometry is the study of Geometric structures with topological composition.
Outlined and posted my new methodology based on Systematic Conceptual Integration (SCI). Will do it at Deviantart as well.
There is also Systematic Conceptual Analysis that is possible too, but I will focus only on integration at this point. Putting concepts together as apposed to taking them apart. Both are needed, but for now the one is preferable to the other.
Have built the bulk of the web page: geometrictopology1000.wordpress.com with some very nice molecular and mathematical models of polygonal knots, links, tori and Mobius strips. It is one of 16 possible web pages for Geometric Topology! If anyone is interested please check it out.
I can now look at all that I have done in Topological Geometry (Porous Polyhedra) and Geometric Geometry (polygonal topology) and see where it fits in the taxonomy created by the SCI system for Geometry and Topology!
My thinking is accelerating at light speed. I will now make You tube videos about the topics listed below. Only they will have a transcript of the video instead of an actual database at the web pages. Although I continue to work feverishly in two dimensional geometric topology creating knots, links, Mobius strips and tori.
The true measure of a mathematical discipline is its ability to generate new shapes and in this Fractal Geometry was and is a tremendous success. But Topology and Geometry have stagnated. New kinds of Geometries and Topologies are needed and in order to do this new ways of thinking about them must be created.
“Ask not what Mathematics is, but rather what it could become.” APC
I can now see clearly the situation in Geometry and Topology. In two words, I would say it is ‘cognitive chaos’; a Gordian knot of thoughts. I must untangle it and rebuild these subjects from scratch.
I’m now solidly and purely working as a mathematician and not in the materials sciences. Not that they are not related, but that I am only interested in the math side of the equation right now. I’ll be working on 3 web pages:
- Geometric Topology: https://chaos477.wordpress.com/
- Topological Geometry: https://academic52.wordpress.com/
- Two Dimensional Geometric Topology: https://geometrictopology1000.wordpress.com/
Completed 22 Foil and Goldberg inspired polyhedra. Most are Foil polyhedra and include Icosahedral, Octahedral and Tetrahedral polyhedra, as well, as truncated versions of the same. Will post when weather permits.
Am also inspired to make porous (as in mesoporous) versions of Goldberg polyhedra.
I have set aside my investigations of Exploded polyhedra for now because I’m looking more closely at Foil polyhedra and Goldberg Geometry (and I mean Geometry not polyhedra). There are many other kinds of Goldberg polyhedra than those cited at wikipedia and I’m building models of them now. It is a vast area and will require much time and work, but in Porous Geometry, Goldberg can be reformulated into a full on Geometry and by Geometry I mean a complete ontology and what I mean by that is yet to be seen.
After an extended gestation period, I have new insights into Exploded polyhedra. The ones depicted thus far are linear explosions, however I now realize that they can be exploded in three dimensions; by length, width and height. Thus, each one can be done individually or in combinations of two and even all three dimensions at the same time. This affords eight combinatorial possibilities for dimensional explosion.
Explosion, or the insertion of one or more polygons between the polygons that make up any given polyhedra, is an operation that is performed on a polyhedra and can be applied to all the polyhedra I have created thus far including: Crown, Foil, Crinkled and Ring/Star polyhedra.
Explosion is an expansive operation there are also contractive operations as well, as for example, those that are applied to rings and result in the Crown polyhedra. Inversely the Crinkled polyhedra are another example of an expansive operation on a ring as apposed to a polygon.
P.S. When exploding a given polyhedra in terms of height the resulting polyhedra may be conceived of as a Hulled polyhedra or a Nested polyhedra.
It’s now becoming more clear what polyhedra I am working with. They would be characterized as Exploded Crown Polyhedra with the following characteristics: some though not all change identity when ‘exploded’ and all of them encapsulate crown polyhedra. Thus, in the same way that Crown Polyhedra encapsulate either, Archimedean, Platonic or Skew Polyhedra the Exploded Crowns encapsulate other Crowns. I’m also working with Dual or Catalan Exploded Crown Polyhedra that when exploded encapsulate the duals of Crown polyhedra. There is yet another class of polyhedra that I am working with that remain undefined and I suspect will continue to be so as they are rather odd, but also very interesting. Again, I will post photos when they are complete.
Been away for awhile. Been making art (painting and geometric sculptures) ,but have returned to making polyhedra (research) !: new kinds based on Crown Polyhedra, but for which I have neither a classification nor a name. I will post photos after my inventions are complete and the rain has stopped.
It seems that with these new kinds of Crowns that the possibilities are infinite and that my powers of inventiveness never cease to amaze me. It is as though I were not even consciously aware of what I am doing and only following the patterns they suggest/embody.
Not certain where these new crop of polyhedra will lead, but am excited to be adventuring again.
Uploading photos to front page today. Six new ones. Working on Belted Catalans now.
Completed Ringed Truncated Octahedron, three Face Truncated Polyhedra and the first of a new class of polyhedra: Strutted Polyhedra: i.e – Strutted Rhombic Triacontahedron. Will post photos ASAP. Will finish Strutted Multicrown Truncated Tetrahedron tomorrow.
The Strutted Polyhedra are the result of the substitution of regular polygons for vertices and polygonal lines for edges of various polyhedra. Only the Platonic polyhedra and the some though not all the Catalan polyhedra can be constructed in this way. At this point it appears that none of the Archimedean polyhedra can be made using this method.
Plan to work on the Ringed Polyhedra next and then more Strutted Polyhedra. Perhaps also some examples of Ringed Tessellated Polyhedra too.
Finished the following:
- Belted Multicrown Cube
- Belted Multicrown Octahedron
- Belted Multicrown Triangle
- Added to the Ring Star Multimobius Tessellation
Working on new:
- Multicrown Truncated Tetrahedron
- Face Trunacted Dodecahedron, Cube and Tetrahedron (with hexagons and triangles)
- Ringed Truncated Octahedron
- Ringed Great Rhombicuboctahedron
- Ringed Great Rhombicosidodecahedron
- and New Foil Snub Dodecahedron, Cube and Tetrahedron
- Ring, Star and Ring Star Tessellations (subunits of)
Also, modeled Foil Geometry on Ring Geometry so that there are Foil Polyhedra, Foil Crown Polyhedra, Foil Crinkled Polyhedraand Foil Ring Polyhedra. Not so sure about the Foil Crinkled Polyhedra though???
7/20/15 – Worked in wood for a while and made a small run of crown polyhedra for sale at Etsy. Focusing on Strutted Multicrown polyhedra (in paper) now. Will have photos soon. Defining 3 Dimensional Geometry by 3 dimensional objects. So a point is a polyhedron (i.e. a cube), then a line is a linear sequence of cubes and a plane is a square of cubes etc. same with 2 dimensional Geometry and so on. On the subject of Multicrown polyhedra: to date there are three ways to build with crown polyhedra:
A) with Tubes
B) with Crown tubes (ie antitubes)
c) with polygonal lines or struts
When I get a new computer and Great Stella by Rob Webb will be posting objects constructed from polyhedra. Also found new ideas in Chemistry:
1) MOFs-Metal-Organic frameworks
2) POMs- Polyoxometalates
Both are of interest….
Constructed a Trefoil Knot using only regular polygons and none touching as they knot each other.
For those not aware, I also post some, though not all, of my polyhedra at Deviantart: albertpcarpenter.deviantart.com
6/14/15 Sorry for the absence. I have been working on a different project for 3 weeks and did not have time for Geometry. In the last week, however, I’ve been working on tessellations: Zig-Zag, Crown and Mobius varieties, as well as, some Crown Foil polyhedra. Will post photos of them soon. The tessellations are most interesting as molecular structures. Will be publishing on Amazon: 1) A Beer with Bert-a screenplay inspired by My Dinner with Andre and about the integration of Philosophy, Science and Religion. Also to be published: the articles on 2) Occam’s sword and 3) Eliminative and Multiplicative Materialism. Will begin making gifts for donors to Kickstarter campaign after I get back from Baltimore.
5/11/15 Added a new page: The Structural Laws of Matter and Energy…with links to Occam’s sword and Eliminative and Multiplicative Materialism. Restated here as: 1) Occam’s Sword: https://occamsword.wordpress.com/ and 2) Eliminative and Multiplicative Materialism: https://eliminativeandmultiplicativematerialism.wordpress.com/ Both are articles I’ve written.
5/9/15 Added Twenty six photos three days ago, including: 1) Two to Carbon Nanostructures 2) Two to Archimedean and Platonic Zig Zag Tessellations 3) Two to Embedded Polyhedra 4) Five to Platonic Zig Zag Polygons 5) Five to Decomposed Crown Polyhedra 6) Several to Archimedean and Platonic Crinkled Polygons 7) One to Platonic Foil Polygons etc. Will add titles soon. Been thinking about The Principle of Substitution as it applies to regular Polygons and Ring/Stars. It can be done infinately thus rendering fractals. Been working on a second version of a Multicrown Dodecahedron/Dodecahedrane one quarter finished, but its boring and tedious. So it goes. Trudge away I must. At this rate I’ll be done in 15 more days.
4/30/15 Upon further reflection, Crown Geometry follows the same pattern as Tubular Geometry what that means is there can be Multicrown Benzene and Multicrown Propane etc. Instead of Hexagonal Crown Meta Tori etc. The only problem with this analogy is that bond lengths are fixed in Euclidean Chemistry, but variable in Tubular and Crown Chemistry. There is also this thought; each object in a given dimensional set with Euclidean and Crown Geometries being examples of base sets have fractal analogues in higher dimensional sets. For example, a point, line, plane and space in Euclidean Geometry are a base dimensional set, but the next set would start with the polyhedra from the first set as a point of the next set and forms the basis for the polyhedral line, polyhedral plane and polyhedral polyhedra of the second set and so on to infinity. In the case of Crown Geometry the Crown polyhedra become the points of the next dimensional set and the basis for crown polyhedral lines, planes and spaces and so on and so forth. These relationships can be expressed in tables of ever increasing complexity.
4/29/15 Been working with pentagons and pentagonal structures: Mostly regular pentagons in Zig Zag Geometry… Still too early to give any results, but it looks promising. Also, I have been studying Tubular Geometry or the Geometry of Nanotubes. One of my guiding principles in this pursuit is that of substitution, so that for any line (Bond) in Euclidean Geometry (Euclidean Chemistry) a nanotube can be substituted for the line in question. Thus, what would be Benzene in Euclidean Chemistry is Tubular Benzene in Tubular Nanogeometry. From this it can be stated that any Polycyclic Aromatic Hydrocarbon and (polyhex Fullerene) has a tubular analogue. Furthermore, instead of reinventing the nomenclature for such nanotube structures (Nanotori) one could and can simple add the adjective “Tubular” in front of the existing name for the analogous structure in Euclidean Chemistry. For example, what would be Napthalene, Biphenylene etc in Euclidean Chemistry would in Tubular Nanochemistry be Tubular Napthalene and Tubular Biphenylene etc. Or even Tubular Icosahedron, Tubular Octahedron and Tubular Tetrahedron (see pages so named). In addition to this, all the polyhedra/polyhedral molecules from Euclidean Geometry/Chemistry (i.e. the Platonic and Archimedean polyhedra) can be thought of as nanojunctions for larger nano tubular structures be they polyhedra or radial structures like those similar to the number line, 2 Dimensional graph and the 3 Dimensional Cartesian Co-ordinate system. These as tubular structures are important as endo- and exohedral structures as they apply to Tubular Nanopolyhedra etc. My thought on these topics has been influenced by the book, Periodic Nanostructures (2007 or 2008?) by ? Paid the entry fee for the Bridges Conference today. So, if all goes well I’ll be there in July/August. The internet as a repository for information and images on nanostructures is still in its infancy and is visually impoverished. Sad, but I have to rely on books for information about these structures and we all know that by the time a book gets published its 5 years out of date. I’m also frustrated by the trend at university libraries towards collecting e-books that are only available to students, faculty and staff at participating institutions. As an independent scholar, this leaves me out of the loop. Yet, I am convinced of the novelty of much of my work especially in regard to Crown Geometry and all the other Geometries I’m working on. I’ve probably already stated this, but another guiding principle in the construction of a Geometry is the idea that for every complete ontology there is a corresponding Geometry.
4/19/15 I now realize that Zig Zag Geometry is to be subsumed by Wave Geometry and that Wave Geometry is based on the negation of the second postulate of Euclid that the distance between two points is a straight line. Thinking of it in this way opens it up to other kinds of waves besides Zig Zag lines such as Saw Tooth and Sinusoidal waves ect. Also, in keeping witht the analogy with a number line, a flat or straight line is a zero degree wave line with others varying by degree. Chirality is a factor as well with left and right handed waves possible. These could be expressed as negative and positive waves.
4/17/15 New photos uploaded to: 1) Archimedean and Platonioc Crown Meta Polygonal Tori – 4 2) Archimedean and Platonic Polyhedra – 11 3) Platonic Crown Polygons – 8 4) Archimedean and Platonic Zig Zag Polygon Lines – 3 5) Archimedean and Platonic Reticular Polygon Lines – 2 6) Dodecahedral and Cunehedral Assemblies – 3 Polyforms have thus far been limited to Platonic varieties. They can be expanded to include Archimedean ones as well. These then become the basis for polyhedra. Deleted a lot of the PolyX pages. They were blank and I’ll add more when I start making actual models for them. Working on Platonic Zig Zag Polygons….then the Archimedeans…..then the tessellations. It’s a lot of work but it must be done. So, to recap, I am working on the following Geometries: 1) Crinkled, Crown and Ring Geometries 2) Zig Zag Geometry 3) Syndimensional Geometry 4) Polyvalent Geometry and the enumeration of Polyform Polyhedra…..(This one is HUGE!)
4/15/15 The Crown, Ring and Crinkled polyhedra (etc.) are all related by the operation of subtraction and addition on rings. Crowns are formed by subtracting one or more polygon from any given ring while crinkling involves adding them. This integrates three classes of polyhedra (etc.) Working on crown, ring and crinkled polygons next, then Zig Zag Geometry. Waiting for a sunny day to take photos of the Crown Meta Tori (Triangular, Square, Pentagonal and Hexagonal). Also, will take photos of 11 of the 13 Archimedeans in color.
4/14/15 Updating web page…Many new blank pages to reflect the followiing expansion of my taxonomy along these lines: 1) Polygonal lines 2) Polygons 3) Polyhedra 4) Tesselations 5) Tubes one each for the following: 1) Crinkled X 2) Crown X 3) Exploded X 4) Zig Zag X 5) Ring-Star X 6) Foil X 7) Regular and Semi-Regular X…..
4/13/15 Polyform Polyhedra: Not all polyforms make for polyhedra that are related to or based on the Archimedeans and Platonic polyhedra. Also, Polyforms have only been considered as Homogenous varieties and not heterogenous ones. For example, the homogenous ones are all of the same shape (regular polygon) while the heterogeneous ones would be composed of two or more different regular polygons. The criteria by which a polyform makes a polyhedra (based on the Platonic and or Archimedean polyhedra) would and should make for an interesting problem…. Here is a classic example of Recreational Mathematics (polyforms specifically) converging with science to become a genuine area of inquiry. At least in so far as polyform polyhedra are analogues for molecules. Polyform Geometry as a branch of Enumerative Geometry….defined as the study of polyforms in all dimensions and both within and between dimensions such as the application of planar polyforms to polyhedra. Finished the Pentagonal Antimetatori – Antimetanano tori. By extending polyforms to three dimensional analogues we have an example of game extension not of rule changes or novel game play (see below; topic Genius). Generalization is one way of extending and advancing play. Subsumption is likewise a form of generalization, but it can be simultaneously game changing. Geometry originated with the study of the surface of a planet; large scale matter. Now, it’s time to study matter at ever smaller scales i.e nanoscale phenomena….New Geometries are needed to do it (ala Mckay et al. in New Geometries for new materials) . There is so much “room at the bottom” (Feynman) for nanogeometry to grow….. I hope to facilitate that growth, not by providing the conceptual foundations for nanogeometry because that is only one possible metaphor (Foundationalism), rather by providing conceptual tools for its development and evolution. I come to Geometry by way of 2nd Generation Cognitive Science (i.e. the primacy of metaphor/anaology in the human conceptual system, embodied mathematics and creativity as a form of conceptual blending-especially in the work of philosopher Mark Johnson and linguist George Lakoff). Thus, I see myself as a cognitive geometer, experimental geometer perhaps even a scientific geometer, (not to mention a metageometer). On Euclidean Geometry: Profoundly influential, but flawed…conceptually conflated with so many different geometries. Definately a fertile data base from which I can mine ideas that in turn become their own geometries. It is time to evolve Geometry not vertically but laterally. In the sense that it has evolved into higher degrees of abstraction, instead lets focus on low dimensional Geometry and applied Geometry as it relates to other fields like Physics, Chemistry and Nanoscience. It is frustrating… I feel like John Henry…competing with computers i.e. 3D printers, CNC machines and Computational Geometers. I will lose in the end, but not before I make a dent in the conceptual machinary. Metageometry as I understand it and in the sense I will apply here is the application of geometric (and or scientific) methodologies to the study of Geometry itself. What would have been the Philosophy of Geometry is no more…Metageometry replaces it in the way that Metascience replaces the Philosophy of Science. It is Geometry as studied by geometers. The Philosophy of Science was the product of a conceptual blend and of methodologies i.e. the conceptual and linguistic analysis of science, but Philosophy from an evolutionary epistemological perspective is becoming extinct and so too is the Philosophy of Geometry. In its place will be Metageometry just as Metascience is in the process of replacing the Philosophy of Science. There are no sacred cows in Geometry….Even the conservation of quantity is subject to empirical question…and refutation (see Synergy and Dysergy in Geometry). Geometry is a mirror what you see in it is what you see in yourself. I’ve been polishing it for 25 years and now its transparent…I see myself from the inside out. Writing about Geometry (i.e. Reflections on Geometry) is an exercise of the Extended Mind (Andy Clark and David Chalmers). Thus, in the process of writing there is a dynamic feedback loop whereby the act of writing is itself part of the process of the creation of the ideas that I’m expressing. The act of writing is a form of active externalism in this sense. I am going to try to split my time between writing/thinking about Geometry and doing it i.e. creating polyhedra. I need both because I cannot do either without the other. They feed each other; another feedback strange loop (Douglas Hofstadter). I hate to do it, but I should change the name of Anti Geometry to Null Geometry its closer to the truth. 4/12/15 Geometry is the monster and it will swallow me whole…. One of the motifs on this web page is the idea that for every grouping of polyhedra there is a tessellation (thank you Prof. Dutch) and that for every polyhedra there is a tube. Tubes can be used either by themselves or with polyhedra to constuct other polyhedral structures and tessellations. I’ve worked out a dimensional analysis of geometric entities that allows for the classification of point polyhedra, skeletal polyhedra, hollow core (polygonal) polyhedra and solid polyhedra etc. A table can be generated to establish 16 classificatory categories, each leading to the identification of a unique geometric entity. These can then be applied to a combinatorial matrix for (Bidimensioal Geometry or Biontological Geometry) with a product of 196 possible combinations. I’ve been thinking of Syndimensional Geometry as if it were only one kind of Geometry, but it is really Bidimensional Geometry. Looked at in this way there can also be Nondimensional Geometry, Monodimensional Geometry, Bidimensional Geometry etc. What applies to Dimensional also applies to ontology so can have Non-ontological entities, Mono-ontological entities, Bi-ontological entities and Triontological entities perhaps this is where Enumerative Geometry enters the picture…..? Polyvalent Geometry subsumes Antigeometry and situates in a spectrum of Geometries like those on a number line; in the middle or at zero (thus and therefore neutral Geometry) between Positive and Negative Geometries. To think of it this way one gets both positive, neutral and negative Dimensional entities. Syndimensional Geometry can be subsumed by Antigeometry as an example of a heterodimensional ontology. Ontology preceeds theory and vice versa. For every Geometry there is an ontology and for every ontology there is a Geometry. 4/11/15 Changed the titles of all the pages (except the polyx pages) as well as their addresses to Archimedean and Platonic X…. This better reflects the nature of the polyhedra and the fact that they are all modelled on those 18 polyhedra. Now the web page is divided into roughly two sections; The Polyx pages that reflect my interest in the enumeration of polyhedra based on the Archimedean and Platonic Polyhedra and then all the Archimedean and Platonic pages. There is some overlap and no clear distinction at this time for the two. In fact, I could have just as easily titled them Archimedean and Platonic PolyX pages. This is a classifacatory problem because if I classified them all along Polyx lines (i.e. based on composition-(polygons)) it would not do justice to classes of polyhedra based on structural affinities like the Platonic and Archimedean polyhedra as classes of polyhedra. Perhaps there is a system of classification that combines both structure and composition similar to that found in Chemistry. Both are required to establish the identity of an object. Almost finished with the pentagonal antimetatubular tori (need one more anti icosidodecahedron as a vertex and more antitubes to complete. Finished the triangualr and square antimetatubular tori. Will start the hexagonal tori next. Reclassified the Extended Crown Polyhedra as Polyx Antipolyhedra. Classification is a monster. There are so many ways to do it (i.e. based on different kinds of Geometry) and each has consequences. If you do it one way i.e. based on dimension or morphology etc. then it suggests different possibilities for other kinds of polyhedra depending on which one you choose. Then there is the problem that a given geometric entity can be classified in multiple ways and also there is the related issue of multiple realizability or the ability to express the same polyhedra in a multiplicity of ways. All in all the monster can be tamed when when decides not to fight it, but to play with it. Classification is as much a creative process if not part of that process as creating polyhedra itself. As an artist (and to the extent that I identify myself as such) and having both painted and sculpted, I have to say that Geometry is by far the most creative activity I have ever engadged in! I have decided to reinvent myself as a nanodesigner. It better reflects my current interests and this is where I differ from Reticular Chemist, Omar Yaghi, he is an actual scientist. I on the otherhand am only a morphologist at worst and a Cognitive Geometer at best. Oh, I was notified today that two of my wooden sculptures have been accepted into the Bridges Conference Art Exhibition in Baltimore and sponsored by the AMA. Very pleased and excited to have been accepted and the opportunity to go. I look forward to talking with others who may have a common interest. I fear that the glue problem will detract from my sculptures, but there is nothing to be done. I cannot afford to make more. So, they will have to do.
4/10/15 Added lots of blank pages based on the idea that if you can have Polyhex polyhedra – Polyhex Fullerenes then you can also have: 1) Polytri Polyhedra – Polytri Nanopolyhedra 2) Polytetra Polyhedra – Polytetra Nanopolyhedra 3) Polypent Polyhedra – Polypent Nanopolyhedra 4) Polyoct Polyhedra – Polyoct Nanopolyhedra and Polydeca Polyhedra – Polydeca Nanopolyhedra Plus, the binary combination there of such as: 1) Polytritetra Polyhedra – Polytritetra Nanopolyhedra etc. I know this makes for a lot of pages, but the possibilities are staggering… So Now I am working on three things: 1) PolyN Polyhedra – PolyN Nanopolyhedra 2) The Theroretical Foundations of Antigeometry and its relationship to Syndimensional Geometry 3) The constructrion of two kinds of paper molecular models: A) Antipolyhedral and Antitubular Polyhedra Tori B) Antipolyhedral and Antitubular Polyhedra Tori etc…. What I’ve been talking about in terms of PolyN Polyhedra is actually the basis for Enumerative Geometry. The entire time I thought I was doing Toric (Discontinuous) Geometry (which is in a sense still true) I was also doing Enumerative Geometry with a special emphasis on Polyhedra. It need not be limited to polyhedra but applicabl to all geometric entities. What Reticular Chemist, Omar Yaghi is doing is appling enumerative methods to molecular networks (not systematically from a geometric point of view), while I have been working on enumerative methods on models of molecular polyhedra. I will use a taxonomy based on 2 dichotomies: 1) Ontological Homogeneity and Ontological Heterogeneity 2) Dimensional Homogeneity and Dimensional Heterogeneity with these in place, I can classify and generate a systematic approach to Enumerative Geometry. Enumerative Geometry affords an amazing outlet and opportunity for creativity. Let me also say that for every ontology there is a Geometry and for every Geometry an ontology. I can using the above classificatory system identify the Platonic and Archimedean polyhedra as examples of ontologically homogenous and ontologically heterogenous polyhedra respectively. Whether they are dimensionally homogenous or heterogenous is subject to the methody by which they were constructed/represented. i.e. solid/polygonal shell/wire frame/point net etc. I’ll detail how they are related when I can incorporate tables into this blog. Then the relationships will be clear. Photos of new polyhedral constructions coming sometime next week…. O.K. An internet search reveals that Enumerative Geometry already exists as a branch of Algebraic Geometry…. such is my luck… I just applied it to polyhedra with the Open Question (Divergent data set): What can be built with regular polygons…..and allowing for holes or gaps between polygons…? So what do you get with polyhex fullerenes when generalized to regular and semiregular Polygons and Poyhedra? Answer: The contents of this web page. So, I guess I’m back to Enumerative Antigeometry. That will take a while to articulate….
4/8/15 After 25 years of playing around with Geometry, I feel like I’m just getting warmed up. My new vision for Geometry and materials science is vast. I now know what I will do for the next 10-15 years-articulate that vision… I have catelogued over 300 New kinds of Geometry. More than I’ll ever be able to articulate. I must choose. I begin with Antigeometry and Syndimensional Geometry and apply it to materials science. Been thinking about Genius. I identify 3 kinds of game playing Genius’. 1) Those that advance the game. 2) Those that change the rules of the game, but it’s still the same game. 3) Those that invent new games, create the rules for them and advance the game. The last is its highest realization. Geometry is a game. Chemistry is a game. Nanoscience is a game. Let the games begin! The assumption thus far of computational and theoretical nanoscientists that model nanotubes and polyhedra (like fullerenes) is that the entire structure must be filled with regular polygons. This need not be the case. (I may be wrong about this? but…) What Reticular Chemist, Omar Yaghi, has done for porous networks can be applied to polyhedral molecules as well. Thus, Polyhex Fullerenes and Goldberg Fullerenes can be Micro-, Meso- and Macroporous structures as can their geometric analogs. These porous polyhedral molecules allow many novel possibilities including: 1) Embedding polyhedra in their surfaces 2) Creating porous nanopolyhedral and nanotublar structures of scalable design Scalability is a very important feature of molecular polyhedra and nanotubes that merits much consideration. I will touch on it as I go, but time constraints limit the number of models I can build to express this concept.
4/7/15 Updated this web page to reflect growing interest in representing paper models as both geometric and molecular models. Thus, the titles of each page (and the names of the models) contain both the geometric and the molecular category that the models are in. Worked extensively on this web page today. Made many discoveries on the internet i.e. 1) Polyhex Fullerenes (Nanopolyhedra) 2) Polyhex Nantubes and 3) Polyhex Nanotori by analogy I extended Polyhex analysis to: 1) Antinanopolyhedra 2) Antinanotubes and 3) Antinanotori 4) Antinanotesselations Will begin making models of them ASAP and invite others to do the same, the more the better! There is plenty of room for all at the table to sit and feast. Will update titles (on 4/8/15) of pages with a number representing the number of photos at each page, thus making it easier to navigate this web site.
4/5/15 Added photos to: Embedded Polyhedra, Mesoporous Polyhedra and Antipolyhedra. Finished adding Titles to all the photos. Will add Goldgerg variations and polyhedra with nanotubes combined next. It appears that many of my design ideas for combining nanopolyhedra and nanotubes have already been conceived of and constructed in beads by Mr. Horibe and Bin-Yaw Jin (see links to their work at the personal pages section). Their creations are amazing and much in the same spirit of Nanodesign and materials science as mine! I will draw on their work for further development of my own ideas, but I will also say that their work merits replicating in paper and wood so as to offer different kinds of representations (modeling) and with it different kinds of information.
4/3/15 I’ve been thinking a lot about the relationships between polyhedra and nanotubes. I’ve created a number of polyhedra based on Goldberg’s polyhedra that are in a sense generalizations of his and how they relate to nanotubes. I am now confident that I can build nanotubular polyhedra that can be scaled to the underlying polyhedra’s porosity. Will add a page on Embedded Polyhedra, Mesoporous Polyhedra and one on Metageometry (which I define as thinking about Geometry as apposed to doing Geometry of which the page on Creativity in Geometry is an example). I’ll be adding Titles for the photos as time permits.
4/1/15 I’ve been inventing a Variation of Goldberg polyhedra: http://en.wikipedia.org/wiki/Goldberg_polyhedron. Very excited about it. Lots to do! Will post photos in a new section entitled Variations on Goldberg Polyhedra soonish including the embedding of polyhedra in their surfaces! Stopped making the Skew Crown polyhedra (too boring and tedious/not enough creativity) will resume when I hit a brick wall elsewhere. I am now convinced of the the future of materials science and Geometry in the Rational Design of Molecular Polyhedra as the way to present my work. This conclusion is based on recent readings in materials science news and chemistry journals and especially after visiting Omar Yaghi’s web pages. I am hoping that the idea of making a construction toy out of the wooden sculptures does not undermine the significance of the rest of my work (or even presenting sculpture along with molecular models for that matter). I include the idea of such a possibility (of a construction toy) because I think its important to teach the youth of the world to think in terms of these and related structures (polydice and softballs not with standing).
3/28/15 I’ll be adding a Bibliography, Links and a Biographical Note (pages) soon. I’ve been working on Syndimensional Geometry and its inversion with its relationship to Reticular Chemistry/Geometry. Hope to hear from the Bridges Conference soon to see if my sculptures were accepted. Built a new Wooden Crown polyhedron: A Crown Truncated Dodecahedron and have plans for two more. Will take photos and post of more Crown Skew polyhedra when the weather permits.
2/28/15 Still working on the Crown Regular Skew Polyhedra. It is slow going as there have been some sociopolitical issues I have had to deal with that prevented me from working on them. Will post them when completed and therer is no snow and ice.
1/24/15 8 new photos- 5 in the Crown Polyhedra Section entitled Skew Crown Polyhedra these are 5 sets of 3 with Skew polyhedra being encapsulated along with the Archimedean polyhedra from which they are derived. Then also there are 3 photos of Mobius strips in that section as well. Enjoy Bert
1/10/15 Finally I’m over a bad cold and a sunny wind free day: 17 new photos. See Nanotubes, Antinanotubes, Antinanotubular Polyhedra and Nanotubular Polyhedra.
12/19/14 1) Still working on the Archimedeans have about 5 to go 2) Still raining/overcast weather so no photos today 3) Possible display at the local civic center on the horizon
12/18/14- Added two things to this web page today: 1) Synergy and Dysergy in Geometry page 2) Creativity in Geometry page (a short essay with examples) continue working on Archimedean polyhedra in colors for a general audience Maybe some stellated polyhedra too then…Photos???
12/17/14: completed the following: 1) Antinanotubular Polyhedra Cuboctahedron 2) Small (Coronene based) Antinanotubular Tetrahedron 3) (Nano) Tubular Tetrahedron, Octahedron, and Icosahedron 4)13 different models of nanotubes based on different Geometric patterns 5) Over 13 different antinanotubes based on similar geometries as those from #3 I’m waiting for a sunny day to take photos of them. Then, I will post them here before relocating them to the appropriate category on this site. Since, Ill be exhibiting some of my models at a local library, I will take some time to make colorful Platonic and Archimedean polyhedra, then I’ll start on more multicrown polyhedra. I’m not going to make any more tubular polyhedra because they can be modeled in Great Stella (software by Rob Webb) more easily than in paper or wood.