Upon following the work of Anton Betten

TO: Anton Betten, Colorado State University, Fort Collins
FM: Bruce E. Camber
RE: Homepage(s): CV, Google Scholar, Publications… On the problem of classification of combinatorial objects

URL (this page): https://81018.com/betten/

Second email: 21 May 2026

Dear Prof. Dr. Anton Betten:

My morning routine with my website  — https://81018.com — is first to check that is operational, and then I look at the overnight traffic. It’s usually not much, but I am looking for spikes or anything unusual. My page about your work had nine views (which is unusual). First, I couldn’t remember the focus of your work so went to the page to update it: https://81018.com/betten/  Second, I asked, “What is Anton doing that’s generating the interest?”

Any time you’d like an addition, a correction, or change to that page, just say the word. If you’d like me to delete it, that’s OK, too.

I went to GoogleSearch AI for update and posted it: https://81018.com/betten/#Anton

If our work is more right than wrong, the big bang theory will be re-written. I wondered if you have ever run that idea as a thought experiment to consider the upheaval in science it would cause.

At one time that idea would have ranked high on the Crackpot Index that John Baez (UC-Riverside) started in 1992! Today, with all its problems, the big bang theory might rank higher!

Might you comment?

Warmly,

Bruce

PS. I have attached an image from this morning’s top three pages! -BEC

First email: 1 December 2022 @ 10:11 AM (updated)

Dear Prof. Dr. Anton Betten:

You have the depth and breadth in geometry to advise us high school geometers who are stumbling around looking for someone to confirm or deny our “un-covering” of a fascinating geometry of five-tetrahedrons, over five-octahedrons, over five tetrahedrons: https://81018.com/15-2/

We are not finding it within any references. The five tetrahedral geometry has been studied a little, but not profoundly. The five octahedral geometry just appears to be overlooked. I had asked Hans Havlicek about it and he referred us to Michel Planat of The FEMTO-ST Institute in France. Michel has referred me to his work but not to any writings about this rather basic construction.

Are you familiar with it? If it is something that has been overlooked, might it be a good thing to introduce to your students at Colorado State University as well as the high school teachers within Fort Collins? 

I started my work on that five-octahedral unit in May. If it’s new, it needs to be in the hands of a professional and not a 75 year old, former high school teacher!  Thanks. 

Warmly,

Bruce

PS. My work on this modeling project actually began with an introduction to Aristotle’s mistake: https://81018.com/biased/#Aristotle back in 1998. It continued a visit with John Conway (2001) who couldn’t advise me about it. It got picked up with my correspondence with Zong and Lagarias (Aug 30, 2013), the authors of an AMS award winning publication, Mysteries in Packing Regular Tetrahedra. Six years later I finally asked (May 2022) asked myself, “What happens when five octahedron share a common centerpoint?” I quickly recognized it as a sister to the tetrahedral gap and it was all a result of playing with models!  https://81018.com/geometries/   -BEC

Google Search AI Assessment

The classification of combinatorial objects is a branch of discrete mathematics that deals with characterizing and listing finite structures (like codes, designs, and graphs) up to isomorphism. Pioneered by researchers like Anton Betten, this field relies on computational group theory to manage massive search spaces and eliminate isomorphic duplicates. [1, 2, 3, 4, 5]

Core Methodologies

The classification problem in combinatorics involves finding a complete, non-redundant set of isomorphism-invariant objects that satisfy specific constraints. Pioneered extensively by mathematician Anton Betten, this relies on computational group theory to manage massive combinatorial search spaces, identify automorphisms, and generate distinct equivalence classes. [1, 2, 3, 4]

The Core Problem

In combinatorics, the classification of objects (e.g., graphs, linear codes, designs, finite geometries) requires finding a transversal of the orbits of a group acting on a set of structures. Two primary computational challenges exist: [1, 2]

  • Isomorphism Problem: Determining whether two combinatorial objects are equivalent under a group action.
  • Combinatorial Explosion: The number of valid objects grows exponentially as parameters increase, making naive enumeration impossible. [1, 2]

Betten’s Methodological Approaches

To overcome these challenges, Betten’s research primarily uses and refines algorithmic generation techniques: [1]

  • Poset Classification: Rather than brute-force enumeration, algorithms examine a poset (partially ordered set) of combinatorial objects breadth-first, evaluating them level by level. This technique employs “isomorph rejection” to throw away isomorphic copies before they branch further into the search tree.
  • Backtrack Search and Orderly Generation: This approach involves defining a canonical or lexicographic order on the objects to rule out partial solutions that are not minimal.
  • Reduction to Exact Cover or Rainbow Clique: Complex combinatorial classification problems can be mapped into instances of Exact Cover or Rainbow Clique, making it easier to leverage high-performance solvers. [1, 2, 3, 4]

Orbiter Software

Betten developed and maintains Orbiter, a C++ based open-source software ecosystem designed specifically for the computational classification of combinatorial data. [1, 2]

  • Key Functionality: It calculates combinatorial objects up to isomorphism, maintains automorphism groups, and applies permutation group algorithms to fields like design theory, coding theory, and finite geometry.
  • Decoupled Architecture: Orbiter separates the permutation group’s logic from the specific combinatorial object’s logic, allowing highly adaptable generation across different algebraic actions. [1, 2, 3]

Broader Applications

The theoretical classification and computation of these discrete geometric objects form the mathematical foundation for modern post-quantum cryptography. Specifically, their properties are vital for designing code-based cryptosystems. [1]

To explore this methodology further, you can review Betten’s research publications on his Colorado State University Faculty Page or review the open-source software documentation directly via the ACM Digital Library. [1]

  • Anton Betten – Colorado State UniversityJul 16, 2020 — Summary. I am interested in the problem of classification of combinatorial objects. I consider applications of combinatorics to co…Colorado State University
  • Classifying Simplicial Dissections of Convex Polyhedra with SymmetryJun 6, 2020 — However, there are combinatorial objects for which this reduction is inefficient. Also, there is an interest in solving the isomor…National Institutes of Health (.gov)
  • Classifying Simplicial Dissections of Convex Polyhedra with SymmetryThe ranking of the poset introduces level sets, and the orbits partition these level sets. The efficiency of the orbit algorithm i…PubMed Central (PMC) (.gov)

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