# Michael Hartley's Publications

### A list of publications by Michael Hartley, including brief descriptions of the main results of each paper.

The main thread of my research has been abstract polytopes. I started by deriving faster methods for finding quotients of polytopes, then applied these techniques to a few special cases. Lately, I have focused more on finding new examples of regular polytopes. As well as publications along these lines, there have been a few publications in other mathematical areas, and some conference papers in various branches of computer science.

### Abstract Polytopes

(With E. Schulte and P. McMullen). "Symmetric Tesselations on Euclidean Spaceforms". Canadian Journal of Mathematics 51 pp1230-1239 (1999).
This article proves that a regular polytopes which is a euclidean spaceform (torus, or klein bottle, or other quotient of Euclidean space), must in fact be a torus. Request a Copy.

"All Polytopes are Quotients, and Isomorphic Polytopes are Quotients by Conjugate Subgroups", Discrete and Computational Geometry, 21 pp289-298 (1999)
The main result of this paper is stated in the title - Every polytope (regular or not) is a quotient of a regular polytope by a subgroup of the latter's automorphism group. Furthermore, isomorphic quotients are easy to identify. This article was based on work done for my PhD. Request a Copy.

"More on Quotient Polytopes", Aequationes Mathematicae, 57 pp108-120 (1999)
This article was also based on work done for my PhD. All Polytopes Are Quotients showed that every polytope is a quotient of a regular polytope by a subgroup N of the regular polytope's automorphism group W. More on Quotient Polytopes shows how important structures of the quotient are related to N and W, and derives a practical method for enumerating the quotients of a small regular polytope. Request a Copy.

"Polytopes of Finite Type", Discrete Mathematics, 218 pp97-108 (2000)
This article applies the results of APAQ and MOQP to the classical regular convex polytopes. The quotients of each are enumerated. Request a Copy.

"Quotients of Some Finite Universal Locally Projective Polytopes", Discrete and Computational Geometry, 29 pp435-443 (2003)
This article applies the results of APAQ and MOQP to various locally projective polytopes. The main result is to characterise the quotients of the {4,3,5} with hemiicosahedral vertex figures. The article also points out that Coxeter and Grunbaum's 11-cell (of type {3,5,3}) and Coxeter's 57-cell (type {5,3,5}) have no proper quotients. Request a Copy.

(With D. Leemans) "Quotients of a Universal Locally Projective Polytope of Type {5,3,5}", Mathematische Zeitschrift 247 pp663-674 (2004)
This article examines the universal polytope with dodecahedra facets, and hemi-icosahedral vertex figures. The polytope is discovered to be finite, with group isomorphic to J1 x L2(19). The quotients of the polytope are enumerated. Errors were made in the original enumeration, which were corrected in an errata published later in Math Z 253 pp433-434 (2006). Some auxiliary information appears on a web page for the article. Request a Copy.

"Locally Projective Polytopes of Type {4,3,...,3,p}", Journal of Algebra, 290 pp322-336 (2005)
This article, as the title implies, examines locally projective polytopes of type {4,3,...,3,p}. Some key results are that if p is odd, the facets cannot be hemicubes. Conversely, there do exist polytopes with hemicube facets whenever p is even. Examples are given of locally projective polytopes of type {4,3,3,5}. Request a Copy.

(With D. Leemans) "On the Thin Regular Geometries of Rank Four for the Janko Group J1", Innovations in Incidence Geometry, 1 pp181-190 (2005)
The Janko Group J1 has 6 thin regular rank 4 geometries. two of these are polytopes. This article shows how all the geometries may be constructed from one of the polytopes through application of some simple transformations. Request a Copy.

"Simpler Tests for Semisparse Subgroups", To Appear.
This article develops some more practical tests to apply to a subgroup of a polytope's automorphism group to see if the subgroup yields a quotient polytope. The results are applied to find an example of a locally toroidal polytope. The article was written much earlier than it was published. Request a Copy.

"An Exploration of Locally Projective Polytopes", Submitted.
This article completes and summarises the classification of finite universal locally projective polytopes. Interestingly, all rank 4 locally projective polytopes are finite. Request a Copy.

"An Atlas of Small Regular Polytopes", To Appear.
This article announces the publication online of the Atlas of Small Regular Polytopes. It also explains the methods used to construct the atlas. Request a Copy.

"A New Petrie-Like Construction for Abstract Polytopes", Submitted.
This article explores a generalisation of the Petrial to higher-ranked polytopes. Request a Copy.

"Polytopes Derived from Sporadic Simple Groups", Submitted.
This article explores the sporadic simple groups of order up to that of the Held group, and enumerates the polytopes arising from them. Detailed tables and permutation representations of the automorphism groups of the polytopes are given on this page. Request a Copy.

(With D. Leemans) "On Locally Spherical Polytopes of Type {5,3,5}", Submitted.
All locally projective polytopes of type {5,3,5} are covered by a minimal locally spherical polytope of this type, whose group is the direct product J1xJ1xL2(19) of two copies of the first Janko group and a projective special linear group. This article explores this polytope and some of its quotients. Request a Copy.

### Other Topics

(With B. M. Hartley) "A New Method for the Determination fo the Activity of Short Half-Life Descendants of Radon", Journal of Radiological Protection, 9 pp165-177 (1989).
This article uses least-squares methods to derive an algorithm for estimating the initial concentrations of certain radioactive isotopes in a sample. The method can be applied continuously, giving steadily more precise estimates as measurement progresses. Request a Copy.

"How to Build a Leaning Tower", Australian Mathematical Society Gazette, 26 pp9-14 (1999)
This article explores practical ways to balance planks in a stack that leans out an arbitrary horizontal distance from the base. Request a Copy.

(With Wong M. K.) "Loanwords from English to Malay in the Field of Mathematics", Language and Linguistics, 1 pp63-78 (2000)
This article explores patterns of borrowing into the Malay language mathematical lexicon. It hypothesises on reasons for the differences in these patterns between different fields of mathematics. Request a Copy.

"Partitions in the Prime Number Maze", Acta Arithmetica, 105, pp227-238 (2002).
The prime number maze is a maze whose rooms are prime numbers. There is a corridor from one room to another if the binary representations of the numbers differ by a single bit. Earlier authors noted that the maze appeared to be partitioned. This article proves that the partitions are real, and that in fact there are infinitely many partitions. Request a Copy.

(With S. Hoh, J. S. Tan) "Context-Aware Systems - A Primer for User-Centred Services", BT Technology Journal 24 pp186-194 (2006)
Proposed a framework for context-sensitive devices.

### Conference Proceedings (incomplete list)

(With A. Pourabdullah) "Gathering Unstructured Workflow Data into Relational Database Model Using Proces Definition Language", Conference Paper.
This work defined a "Process Definition Language" for describing workflows in a structured way suitable for non-technical people, and a tool for converting this PDL into a collection of database tables for use in a Workflow Management System. Request a Copy.