6,352 research outputs found
Generalizing the SteinerâLehmus theorem using the Gröbner cover
In this note we present an application of a new tool (the Gröbner cover method, to discuss parametric polynomial systems of equations) in the realm of automatic discovery of theorems in elementary geometry. Namely, we describe, through a relevant example, how the Gröbner cover algorithm is particularly well suited to obtain the missing hypotheses for a given geometric statement to hold true. We deal with the following problem: to describe the triangles that have at least two bisectors of equal length. The case of two inner bisectors is the well known, XIX century old, SteinerâLehmus theorem, but the general case of inner and outer bisectors has been only recently addressed. We show how the Gröbner cover method automatically provides, while yielding more insight than through any other method, the conditions for a triangle to have two equal bisectors of whatever kind
Automated Generation of Geometric Theorems from Images of Diagrams
We propose an approach to generate geometric theorems from electronic images
of diagrams automatically. The approach makes use of techniques of Hough
transform to recognize geometric objects and their labels and of numeric
verification to mine basic geometric relations. Candidate propositions are
generated from the retrieved information by using six strategies and geometric
theorems are obtained from the candidates via algebraic computation.
Experiments with a preliminary implementation illustrate the effectiveness and
efficiency of the proposed approach for generating nontrivial theorems from
images of diagrams. This work demonstrates the feasibility of automated
discovery of profound geometric knowledge from simple image data and has
potential applications in geometric knowledge management and education.Comment: 31 pages. Submitted to Annals of Mathematics and Artificial
Intelligence (special issue on Geometric Reasoning
Using Automated Reasoning Tools in GeoGebra in the Teaching and Learning of Proving in Geometry
ABSTRACT: This document introduces, describes and exemplifies the technical features of some recently implemented automated reasoning tools in the dynamic mathematics software GeoGebra. The new tools are based on symbolic computation algorithms, allowing the automatic and rigorous proving and discovery of theorems on constructed geometric figures. Some examples of the use in the classroom of such commands are provided, including one describing how intuitive handling of GeoGebra automated reasoning tools may result in unexpected outputs. In all cases the emphasis is made in the potential utility of these tools as a guiding stick to foster student activities (exploration, reasoning) in the learning of elementary geometry. Moreover, a collection of appendices describing other, more sophisticated, low-level GeoGebra tools (Prove, ProveDetails), as well as instructions on how to obtain the translation of GeoGebra commands into other languages, and details about debugging, are included.Work partially supported by the grant MTM2017-88796-P from the Spanish MINECO and the ERDF (European
Regional Development Fund)
Automatic Deduction in Dynamic Geometry using Sage
We present a symbolic tool that provides robust algebraic methods to handle
automatic deduction tasks for a dynamic geometry construction. The main
prototype has been developed as two different worksheets for the open source
computer algebra system Sage, corresponding to two different ways of coding a
geometric construction. In one worksheet, diagrams constructed with the open
source dynamic geometry system GeoGebra are accepted. In this worksheet,
Groebner bases are used to either compute the equation of a geometric locus in
the case of a locus construction or to determine the truth of a general
geometric statement included in the GeoGebra construction as a boolean
variable. In the second worksheet, locus constructions coded using the common
file format for dynamic geometry developed by the Intergeo project are accepted
for computation. The prototype and several examples are provided for testing.
Moreover, a third Sage worksheet is presented in which a novel algorithm to
eliminate extraneous parts in symbolically computed loci has been implemented.
The algorithm, based on a recent work on the Groebner cover of parametric
systems, identifies degenerate components and extraneous adherence points in
loci, both natural byproducts of general polynomial algebraic methods. Detailed
examples are discussed.Comment: In Proceedings THedu'11, arXiv:1202.453
Generating perfect fluid spheres in general relativity
Ever since Karl Schwarzschild's 1916 discovery of the spacetime geometry
describing the interior of a particular idealized general relativistic star --
a static spherically symmetric blob of fluid with position-independent density
-- the general relativity community has continued to devote considerable time
and energy to understanding the general-relativistic static perfect fluid
sphere. Over the last 90 years a tangle of specific perfect fluid spheres has
been discovered, with most of these specific examples seemingly independent
from each other. To bring some order to this collection, in this article we
develop several new transformation theorems that map perfect fluid spheres into
perfect fluid spheres. These transformation theorems sometimes lead to
unexpected connections between previously known perfect fluid spheres,
sometimes lead to new previously unknown perfect fluid spheres, and in general
can be used to develop a systematic way of classifying the set of all perfect
fluid spheres.Comment: 18 pages, 4 tables, 4 figure
HR: A System for Machine Discovery in Finite Algebras
We describe the HR concept formation program which invents mathematical definitions and conjectures in finite algebras such as group theory and ring theory. We give the methods behind and the reasons for the concept formation in HR, an evaluation of its performance in its training domain, group theory, and a look at HR in domains other than group theory
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