Can one design a geometry engine? On the (un)decidability of affine Euclidean geometries

We survey the status of decidabilty of the consequence relation in various axiomatizations of Euclidean geometry. We draw attention to a widely overlooked result by Martin Ziegler from 1980, which proves Tarski's conjecture on the undecidability of finitely axiomatizable theories of fields. We elaborate on how to use Ziegler's theorem to show that the consequence relations for the first order theory of the Hilbert plane and the Euclidean plane are undecidable. As new results we add: (A) The first order consequence relations for Wu's orthogonal and metric geometries (Wen-Ts\"un Wu, 1984), and for the axiomatization of Origami geometry (J. Justin 1986, H. Huzita 1991)are undecidable. It was already known that the universal theory of Hilbert planes and Wu's orthogonal geometry is decidable. We show here using elementary model theoretic tools that (B) the universal first order consequences of any geometric theory $T$ of Pappian planes which is consistent with the analytic geometry of the reals is decidable.Comment: 28 pages, revised version, May 25, 201

Axiomatizing Origami planes

We provide a variant of an axiomatization of elementary geometry based on logical axioms in the spirit of Huzita--Justin axioms for the Origami constructions. We isolate the fragments corresponding to natural classes of Origami constructions such as Pythagorean, Euclidean, and full Origami constructions. The sets of Origami constructible points for each of the classes of constructions provides the minimal model of the corresponding set of logical axioms. Our axiomatizations are based on Wu's axioms for orthogonal geometry and some modifications of Huzita--Justin axioms. We work out bi-interpretations between these logical theories and theories of fields as described in J.A. Makowsky (2018). Using a theorem of M. Ziegler (1982) which implies that the first order theory of Vieta fields is undecidable, we conclude that the first order theory of our axiomatization of Origami is also undecidable.Comment: 25 page

Unification in One Dimension

A physical theory of the world is presented under the unifying principle that all of nature is laid out before us and experienced through the passage of time. The one-dimensional progression in time is opened out into a multi-dimensional mathematically consistent flow, with the simplicity of the former giving rise to symmetries of the latter. The act of perception identifies an extended spacetime arena of intermediate dimension, incorporating the symmetry of geometric spatial rotations, against which physical objects are formed and observed. The spacetime symmetry is contained as a subgroup of, and provides a natural breaking mechanism for, the higher general symmetry of time. It will be described how the world of gravitation and cosmology, as well as quantum theory and particle physics, arises from these considerations.Comment: 498 pages, 63 figure

Proof and Proving in Mathematics Education

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Iowa State University, Courses and Programs Catalog 2009–2011

The Iowa State University Catalog is a two-year publication which lists all academic policies, and procedures. In addition, it includes information for fees, curriculum requirements and first-year courses of study for over 100 undergraduate majors; course descriptions for nearly 5000 undergraduate and graduate courses; and a listing of faculty members at Iowa State University.https://lib.dr.iastate.edu/catalog/1022/thumbnail.jp