Automated generation of geometric theorems from images of diagrams

Chen, Xiaoyu; Song, Dan; Wang, Dongming

doi:10.1007/s10472-014-9433-7

Cited by 23 publications

(12 citation statements)

References 26 publications

(39 reference statements)

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“…For the first method we must admit that in Wu's approach there is no way to express that the length of a segment is 3 2 . In fact, given a length q in general, we need to prove that the equation t = t 1 • t 2 = 0 is implied where t 1 and t 2 are the minimal polynomials of the expected q and −q, respectively.…”

Section: Some Properties Of a Regular Pentagonmentioning

confidence: 99%

“…One problem is that combining certain inference rules to the axioms usually produces an unmanageable big database of propositions, including trivial or uninteresting ones. There is already remarkable work done in this field, including [22] which is one of the first reports, and, in particular, producing proofs in elementary planar geometry, we refer to [1,3,5] where the combinatorial explosion is also addressed. The other problem is to identify which propositions are interesting enough to call them theorems, here we refer the reader to [6,9].…”

Section: Introductionmentioning

confidence: 99%

“…Instead, we need to use the minimal polynomial of the conjectured number. Actually, q 2 − 3q + 1 is the minimal polynomial of both3− q 2 + 3q + 1 is of − 3− √…”

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confidence: 99%

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Automated Detection of Interesting Properties in Regular Polygons

Kovács

2020

Math.Comput.Sci.

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We demonstrate a systematic, automated way of discovery of a large number of new geometry theorems on regular polygons. The applied theory includes a formula by Watkins and Zeitlin on minimal polynomials of $$\cos \frac{2\pi }{n}$$ cos 2 π n , and a method by Recio and Vélez to discover a property in a plane geometry construction. This method exploits Wu’s idea on algebraizing the geometric setup and utilizes the theory of Gröbner bases. Also a bijective function is given that maps the investigated cases to the first natural numbers. Finally, several examples are shown that are all previously unknown results in planar Euclidean geometry.

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Section: Some Properties Of a Regular Pentagonmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Automated Detection of Interesting Properties in Regular Polygons

Kovács

2020

Math.Comput.Sci.

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“…The above mentioned strategies have some similiarities to the ones introduced in [6], but here we focus on minimizing the number of objects that have to be compared in the process that practically compares all objects with all other objects.…”

Section: Mathematical Backgroundmentioning

confidence: 99%

Towards Automated Discovery of Geometrical Theorems in GeoGebra

Yu¹

2020

Preprint

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We describe a prototype of a new experimental GeoGebra command and tool Discover that analyzes geometric figures for salient patterns, properties, and theorems. This tool is a basic implementation of automated discovery in elementary planar geometry. The paper focuses on the mathematical background of the implementation, as well as methods to avoid combinatorial explosion when storing the interesting properties of a geometric figure.

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“…In , a method using a knowledge base is described in order to discover these relations in the case of Wernick's corpus and then to gather problems of this list in classes. Moreover, automatic methods for theorem discovery from a figure could also be used to find such relations, see for instance (Botana and Valcarce, 2006;Chen et al, 2015).…”

Section: Equivalent Systems In Connelly's Corpusmentioning

confidence: 99%

Using jointly geometry and algebra to determine RC-constructibility

Schreck

Mathis

2019

Journal of Symbolic Computation

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In most cases in geometry, applying analytic or algebraic tools on coordinates helps to solve some difficult problems. For instance, proving that a geometrical construction problem is solvable using ruler and compass is often impossible within a synthetic geometry framework. But in an analytic geometry framework, it is a direct application of Galois theory after performing triangularizations. However, these algebraic tools lead to a large amount of computation. Their implementation in modern Computer Algebra Systems (CAS) are still too time consuming to provide an answer in a reasonable time. In addition, they require a lot of memory space which can grow exponentially with the size of the problem. Fortunately, some geometrical properties can be used to setup the algebraic systems so that they can be more efficiently computed. These properties turn polynomials into new ones so as to reduce both the degrees and the number of monomials. The present paper promotes this approach by considering two corpora of geometric construction problems, namely Wernick's and Connely's lists. These lists contain about 280 problems. The purpose is to determine their status i.e. whether they are constructible or not with ruler and compass. Some of these problems had unknown status that will be settled in this paper. More generally, the status of all problems of these corpora are fully automatically given by an approach combining geometry and algebra.

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Automated generation of geometric theorems from images of diagrams

Cited by 23 publications

References 26 publications

Automated Detection of Interesting Properties in Regular Polygons

Automated Detection of Interesting Properties in Regular Polygons

Towards Automated Discovery of Geometrical Theorems in GeoGebra

Using jointly geometry and algebra to determine RC-constructibility

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