Tomas Recio

• Doctor of Philosophy
• Professor Emeritus at Universidad Antonio de Nebrija

We present the continuation of a study on an international MOOC for the training of mathematics teachers who were introduced to the methodology of outdoor mathematics through MathCityMap. Previously, we presented the immediate learning and professional development outcomes achieved by teachers who participated in the MOOC. Here, using the theoretic...

The 2025 ISSAC conference in Guanajuato, Mexico, marks the 50th event in this significant series, making it an ideal moment to reflect on the field's history. This paper reviews the formative years of symbolic computation up to 1975, fifty years ago. By revisiting a period unfamiliar to most current participants, this survey aims to shed light on o...

We present a new GeoGebra Discovery command, ShowProof , which provides a certified step-by-step algebraic proof and complexity grade of planar geometry theorems. The output is available in multiple languages and can be exported as plain text to Mathematica or Maple format for further analysis.

This paper explores the application of automated reasoning tools, specifically those implemented in GeoGebra Discovery, to the perimeter-area inequality in triangles. Focusing on the computational complex and real algebraic geometry methods behind these tools, this study analyzes a geometric construction involving a triangle with arbitrary side len...

At the ADG 2023 a protocol, implemented on an extension of Larus automated theorem prover, to automatically complete statements and proofs in Euclidean geometry, was presented by S. T. Gonzalez, P. Janičić, and J. Narboux. The approach is logic-based, in contrast to algebraic methods, and its performance was tested on five high-school level geometr...

In the context of Open Science, the importance of Borgman’s conundrum challenges that have been initially formulated concerning the difficulties to share Research Data is well known: which Research Data might be shared, by whom, with whom, under what conditions, why, and to what effects. We have recently reviewed the concepts of Research Software a...

We elaborate, as a benchmark, a list of geometry theorems that are regularly used in Hungarian high schools, in specialized tracks for mathematics. Then, these statements are implemented as GeoGebra files, and algebraically proven (with certification) in GeoGebra Discovery. Next, these files are imported in Java Geometry Expert (JGEX), and geometri...

Locus computation is an essential issue in mathematics education, and a traditional feature of Dynamic Geometry software (DGS). The rising of programs merging DGS and Computer Algebra software (CAS) has fostered a combined approach to locus computation, quite performing in standard examples, but demanding an extended theoretical, and the related al...

We present a new GeoGebra Discovery command, ShowProof, which provides • a certified step-by-step algebraic proof, and • complexity grade of planar geometry theorems. The output is available in multiple languages (English, Arabic, French, German, Hebrew, Hungarian, Italian, Spanish and Turkish) and can be exported as plain text to a • Mathematica,...

This paper explores the performance of ChatGPT and GeoGebra Discovery when dealing with automatic geometric reasoning and discovery. The emergence of Large Language Models has attracted considerable attention in mathematics, among other fields where intelligence should be present. We revisit a couple of elementary Euclidean geometry theorems discus...

We present a new GeoGebra Discovery command, ShowProof, which provides ▶ A certified step-by-step algebraic proof ▶ Complexity grade of planar geometry theorems ▶ Output in multiple languages, including English; Arabic, French, German, Hebrew, Hungarian, Italian, Spanish and Turkish ▶ Exportable to Mathematica, Maple, and Giac formats for further a...

• Addressing the same open-ended task as [GJN 2023] using the algebraic-based tools in GGb Discovery as a way to ✓ test the performance of our tool ✓ exhibit its new features ✓ show that using GeoGebra Discovery as a kind of “blind computer" offers many opportunities to develop and to improve human reasoning, but... allowing rather “to do (new) bet...

We compare the performance of two systems, ChatGPT 3.5 and GeoGebra 5, in a restricted, but quite relevant, benchmark from the realm of classical geometry: the determination of geometric loci, focusing, in particular, on the computation of envelopes of families of plane curves. In order to study the loci calculation abilities of ChatGPT, we begin b...

Background. Open Science seeks to render research outputs visible, accessible, reusable. The Open Science framework is currently evolving vigorously due, among others reasons, to the UNESCO Open Science Recommendation adopted in November 2021. In this context, it is relevant to better visualize and describe the relationships that hold among the dir...

In our contribution we describe some on-going improvements concerning the Automated Reasoning Tools developed in GeoGebra Discovery, providing different examples of the performance of these new features.

We address, through the automated reasoning tools in GeoGebra Discovery, a problem from a regional phase of the Austrian Mathematics Olympiad 2023.

Octic curves (i.e. algebraic plane curves of degree 8) appearing as geometric loci

GeoGebra Discovery (system entry)

We present some current improvements, implemented in the software package GeoGebra Discovery, that combine symbolic computation and graphics algorithms to faithfully visualize (semi-)algebraic expressions. Our implementation allows fluid animation of set of (semi-)algebraic sets of dimension 1 in a desktop application or a web browser. We use the T...

Computational algebraic geometry has already shown its great potential as the underlying tool for implementing automated reasoning algorithms in elementary geometry, in a complex numbers framework, currently available, and quite performant, at GeoGebra and GeoGebra Discovery. Although this complex-field approach is often fully operative for address...

We use a quite simple, yet challenging, elementary geometry statement, the so-called "never proved" (by a mathematician) theorem, introduced by Prof. Jiawei Hong in his communication to the IEEE 1986 Symposium on Foundations of Computer Science, to exemplify and analyze the current situation of achievements, ongoing improvements and limitations of...

We explore the construction of curves of degree 8 (octics) appearing as geometric loci of points defined by moving points on an ellipse and its director circle. To achieve this goal we develop different computer algebra methods, dealing with trigonometric or with rational parametric representations, as well as through implicit polynomial equations,...

We report on successfully porting the Tarski system for computing with Tarski formulas (Boolean combinations of polynomial sign conditions over the real numbers) to Javascript, thereby allowing it to run inside a browser. Tarski is an open-source software package, written in C/C++, that provides operations like formula simplification and quantifier...

In this paper, we describe an experience to test the predominant presence of Delaunay triangulations in the artwork of Okuda, a quite famous, young, contemporary Spanish artist. We addressed this task involving, as a citizen science activity in a STEAM (Science, Technology, Engineering, Art, Mathematics) education context, several hundreds of stude...

In this paper, we describe an educational experience in the context of the Master’s degree that is compulsory in Spain to become a secondary education mathematics teacher. Master’s students from two universities in Madrid (Spain) attended lectures that addressed—emphasizing the concourse of a dynamic geometry software package—some historical, didac...

We present the experimental command RealQuantifierElimination in GeoGebra Discovery. The command provides quantifier elimination over the reals and it is available on a fork, freely available, version of GeoGebra. We describe how this new command can be used in certain classroom situations. In our examples we focus on mathematical logic and element...

We introduce an experimental version of GeoGebra that successfully conjectures and proves a large scale of geometric inequalities by providing an easy-to-use graphical interface. GeoGebra Discovery includes an embedded version of the Tarski/QEPCAD B system which can solve a real quantifier elimination problem, so the input geometric construction ca...

Background: Research Software is a concept that has been only recently clarified. In this paper we address the need for a similar enlightenment concerning the Research Data concept. Methods: Our contribution begins by reviewing the Research Software definition, which includes the analysis of software as a legal concept, followed by the study of its...

Background: Open Science seeks to render research outputs visible, accessible and reusable. In this context, Research Data and Research Software sharing and dissemination issues provide real challenges to the scientific community, as consequence of recent progress in political, legal and funding requirements. Methods: We take advantage from the app...

We report on successfully porting and making available online TARSKI, a system for computing with Tarski formulas (Boolean combinations of polynomial sign conditions over the real numbers). TARSKI provides operations like formula simplification and quantifier elimination, and is able to read and write SMT-LIB syntax. The web version of TARSKI was c...

For a given angle α, the α-isoptic curve of a parabola is the geometric locus of points through which passes a pair of tangents to the parabola making an angle equal to α. We explore the inner isoptics of parabolas: they are the envelopes of the lines joining the points of contact of the parabola with the tangents through points on a given isoptic....

The present note intends to show some connections arising from an investigation around the origins of Computer Algebra, which starts from the first explicit appearances of this field in the decade of the 80’s of the past century, takes us back to previous work done by Charles Babbage and the Spanish engineer Torres–Quevedo in relation to their anal...

Our goal is to discuss the different issues that arise when attempting to visualize a joints-and-bars cube through GeoGebra, a widespread program that combines dynamic geometry (DGS) and computer algebra systems (CAS). As is standard in the DGS framework, the performance of the graphic model (i.e., the positions of the other vertices when dragging...

The COVID-19-induced distance education was perceived as highly challenging by teachers and students. A cross-national comparison of five European countries identified several challenges occurred during the distance learning period. On this basis, the article aims to develop a theoretical framework and design requirements for distance and online le...

Presentamos una experiencia de enseñanza STEAM (Science, Technology, Engineering, Art and Mathematics) llevada a cabo con los estudiantes de la especialidad de Matemáticas del Máster en Formación del Profesorado de Educación Secundaria y desarrollada con el objetivo de enfatizar la consideración del enfoque metodológico STEAM como instrumento para...

The popular dynamic mathematics program GeoGebraGeoGebra now includes tools for the mathematically rigorous proof and discoveryDiscover, discovery of general statementsStatement on geometric figures. This chapter will, first, present the current state of development of these automated reasoning toolsAutomated reasoning tools. Then the authors will...

Background: Open Science seeks to render research outputs visible, accessible and reusable. In this context, Research Data and Research Software sharing and dissemination issues provide real challenges to the scientific community, as consequence of recent progress in political, legal and funding requirements. Methods: We take advantage from the app...

Background: Research Software is a concept that has been only recently clarified. In this paper we address the need for a similar enlightenment concerning the Research Data concept. Methods: Our contribution begins by reviewing the Research Software definition, which includes the analysis of software as a legal concept, followed by the study of its...

In our contribution we will reflect, through a collection of selected examples, on the potential impact of the GeoGebra Discovery application on different social and educational contexts.

In our contribution we will reflect, through a collection of selected examples, on the potential impact of the GeoGebra Discovery application on different social and educational contexts.

We present a new method for the systematic and automated finding of theorems holding on a given elementary geometry figure. The process is illustrated by means of the software tool Automated Geometer, developed by the authors on top of GeoGebra, a dynamic geometry system with millions of users at high schools and universities, all over the world, t...

In this paper, we examine the impact of a massive open online course (MOOC) in the context of outdoor mathematics on the participating teachers’ professional development. We firstly introduce the theoretical background on outdoor mathematics, focusing on math trails with the digital tool MathCityMap and professional development to be accomplished u...

Bottema’s et al. classic book on Geometric Inequalities is a traditional benchmark for testing different automatic proving and discovery algorithms, such as the BOTTEMA or DISCOVERER programs, implemented on Maple since long ago. But, as we will show in our presentation, only very recently it has been available the mechanical finding of relations (...

We present some current achievements in the software package GeoGebra Discovery that provides several symbolic tools and commands to mechanically discover (and verify symbolically) relationships on planar geometry constructions. Our presentation includes the novel Discover tool and command, the Relation tool and command, and the Compare command. Ou...

The reflective narratives of three pre-service teachers are analysed from the perspective of discourse analysis and reflection on practice, applied to the acquisition of geometric competences and the role of dynamic geometry. In the context of the master's degree for future secondary school teachers in mathematics and a pedagogical innovation proje...

Recently developed GeoGebra tools for the automated deduction and discovery of geometric statements combine in a unique way computational (real and complex) algebraic geometry algorithms and graphic features for the introduction and visualization of geometric statements. In our paper we will explore the capabilities and limitations of these new too...

In our contribution we will reflect, through a collection of selected examples, on the potential impact of the GeoGebra Discovery application on different social and educational contexts.

We report, through different examples, the current development in GeoGebra, a widespread Dynamic Geometry software, of geometric automated reasoning tools by means of computational algebraic geometry algorithms. Then we introduce and analyze the case of the degeneracy conditions that so often arise in the automated deduction in geometry context, pr...

The goal of this document is to openly contribute with our comments to the EOSCArchitecture report: Scholarly Infrastructures for Research Software (SIRS), and thus, to participate in the European Open Science Cloud (EOSC) architecture design.

Background: Evaluation of the quality of research software is a challenging and relevant issue, still not sufficiently addressed by the scientific community. Methods: Our contribution begins by defining, precisely but widely enough, the notions of research software and of its authors followed by a study of the evaluation issues, as the basis for th...

In 2020, Germany and Spain experienced lockdowns of their school systems. This resulted in a new challenge for learners and teachers: lessons moved from the classroom to the children’s homes. Therefore, teachers had to set rules, implement procedures and make didactical–methodical decisions regarding how to handle this new situation. In this paper,...

We introduce GeoGebra Discovery's new feature: conjecturing and proving geometric inequalities and discuss if our implementation is ready to be used at school.

We present some current achievements in the software package GeoGebra Discovery that provides several symbolic tools and commands to mechanically discover (and verify symbolically) relationships on planar geometry constructions. Our presentation includes the novel Discover and Relation tools, and the Compare command. Our proposal successfully makes...

We present some current achievements in the software package GeoGebra Discovery that provides several symbolic tools and commands to mechanically discover (and verify symbolically) relationships on planar geometry constructions. Our presentation includes the novel Discover tool and command, the Relation tool and command, and the Compare command. Ou...

Supporting automated reasoning in the classroom has a long history in the era of computer algebra. Several systems have been developed and introduced as prototypes at various school levels during the last decades. A breakthrough in using computers to obtain automated proofs is still expected, even if some freely available systems offer easy access...

A branch of the Automated Deduction in Geometry (ADG) theory deals with the automatic proof and discovery of theses holding on a given collection of hypotheses. The mechanical proof and derivation of such statements, through computational complex algebraic geometry methods, will be exemplify in this paper through the performance of Ge-oGebra Automa...

We provide alternative solutions by using GeoGebra Discovery for both Problems 1 and 2 (Problem Corner, Oct. 2020). We also correct one of the statements of Problem 2. An interactive version of our solutions is available at https://matek.hu/zoltan/pc20oct.php.

We give an overview on possible computer aided proofs of Euler's inequality R ≥ 2r (where R stands for the circumradius and r for the inradius in a triangle). We summarize well-known algebraic approaches via factorization (by using Wu's characteristic set method or elimination with Gröbner bases), via searching for an implicit locus, and via a semi...

The goal of this document is to openly contribute with our comments to the EOSCArchitecture report: Scholarly Infrastructures for Research Software (SIRS) , and thus, to participate in the European Open Science Cloud (EOSC) architecture design.

We present two recent tools, integrated in the dynamic mathematics program GeoGebra, for automated proving and discovering in elementary geometry. First of all, the GeoGebra Discovery module, with the Relation, Prove, ProveDetails and LocusEquation commands. They are for humans because it is a human who must introduce the objects the human person w...

In this study, we explore automated reasoning tools (ART) in geometry education and we argue that these tools are part of a wider, nascent ecosystem for computer-supported geometric reasoning. To provide some context, we set out to summarize the capabilities of ART in GeoGebra (GGb), and we discuss the first research proposals of its use in the cla...

Summary • Automated Deduction in Geometry (ADG) deals with the automatic proof and discovery of theses holding on a given collection of hypotheses. • The dynamic mathematics program GeoGebra provides some extraordinary performing automated reasoning tools, through computational complex algebraic geometry methods. Some applications (linkages, augmen...

It is widely recognised nowadays that there is no single, accepted, unified definition of Open Science, which motivates our proposal of an Open Science definition as a political and legal framework where research outputs are shared and disseminated in order to be rendered visible, accessible, reusable is developed, standing over the concepts enhanc...

In the algebraic-geometry-based theory of automated proving and discovery, it is often required that the user includes, as complementary hypotheses, some intuitively obvious non-degeneracy conditions. Traditionally there are two main procedures to introduce such conditions into the hypotheses set. The aim of this paper is to present these two appro...

In this proposal we deal with a well-known Computational Geometry construction: the diagram of Delaunay. We will show how the GeoGebra implementation of the corresponding command brings the opportunity to explore, through Delaunay eyes, some recent artwork from two Spanish artists. This visual exploration leads, from our perspective, to a conjectur...

We describe a scenario in which mechanical intelligence can be used to visualize and discover new, hidden, geometric features of the creativity of artists: i.e. on drawings, paintings, monuments, etc. We summarily describe the recently implemented commands for automated reasoning on top of GeoGebra, a free, worldwide dynamic geometry environment, a...

El popular programa de matemática dinámica GeoGebra incluye herramientas para la verificación matemática rigurosa y el descubrimiento automático de proposiciones generales sobre figuras geométricas. En este trabajo se presenta, en primer lugar, una breve descripción de tales herramientas, para centrarse a continuación en una reflexión sobre su pote...

Este artículo tiene como objetivo ilustrar al lector, muy particularmente a los colegas de la comunidad matemática española, de la RSME, de la evolución de nuestro grupo de investigación y de su trabajo a lo largo de los últimos 30 años, en pos de la creación de un autómata geómetra. Para ello, en las secciones siguientes se trazan unas pinceladas...

In this short note we 1 describe a futuristic scenario in which mechanical intelligence could be used to visualize and discover new, hidden, geometric features in outdoor mathematics objects: i.e. on drawings, paintings, monuments, etc. Thus, we summarize some of the recently implemented commands for automated reasoning on top of GeoGebra, a free,...

We illustrate an on-going work on the crossroad of mechanical linkages, dynamic geometry and automated reasoning. In particular, we exemplify how some automated reasoning tools (ART) based on symbolic computation and recently implemented in GeoGebra, can be used for linkage exploration. For instance, when considering linkages for generating specifi...

MathCityMap (MCM) incorpora tecnología a los paseos matemáticos. En este artículo se describe cómo funciona MCM y su Aula Digital, así como sus posibilidades didácticas y su fundamentación teórica. Además, se recoge la relevancia de MCM en dos proyectos Erasmus+, a través de los cuales se abordan los aspectos curriculares que plantea el desarrollo...

Our proposal of an Open Science definition as a political and legal framework where research outputs are shared and disseminated in order to be rendered visible, accessible, reusable is developed, standing over the concepts enhanced by the Budapest Open Science Initiative (BOAI), and by the Free/Open Source Software (FOSS) and Open data movements....

Background: Evaluation of the quality of research software is a challenging and relevant issue, still not sufficiently addressed by the scientific community. Methods: Our contribution begins by defining, precisely but widely enough, the notions of research software and of its authors followed by a study of the evaluation issues, as the basis for th...

GeoGebra é um software de geometria dinâmica gratuito, com milhões de usuários em todo o mundo com um grande impacto na Educação Matemática. Esse software, agora, inclui recursos de raciocínio automatizado baseados em algoritmos de álgebra computacional que ampliam suas possibilidades de calcular e visualizar relações matemáticas entre alguns eleme...

In this chapter we describe a scenario in which automated reasoning and augmented reality are interrelated, yielding an Automatically Augmented Reality (in short: AAR). This is exemplified in the context of “math promenades”, in which GeoGebra’s recently implemented features for the automated manipulation of geometric facts are the basis for the au...

El popular programa de matemática dinámica GeoGebra1 ofrece, como cualquier otro sistema de geometría dinámica, algunas notables posibilidades y mejoras para apoyar la enseñanza de la geometría del plano euclídeo y para mejorar la construcción y la exploración de objetos geométricos, arrastrando elementos de la figura. Pero, además, recientemente,...

GeoGebra, a very popular software tool for dynamic mathematics, has recently been extended with an automated reasoning tool (ART). A description of the ART features and some examples and reflections regarding its prospective use in the classroom are the main goals of this chapter. ART is based on automatically algebraizing a given geometric constru...

Background: Evaluation of the quality of research software is a challenging and relevant issue, still not sufficiently addressed by the scientific community. Methods: Our contribution begins by defining, precisely but widely enough, the notions of research software and of its authors followed by a study of the evaluation issues, as the basis for th...

In this proposal we will briefly introduce a well-known Computational Geometry construction: the diagram of Delaunay, summarily describing some of its basic properties. The GeoGebra implementation of the corresponding command will bring the opportunity to explore, through Delaunay eyes, some recent artwork from two Spanish artists, yielding, once m...

We introduce, through a computational algebraic geometry approach, the automatic reasoning handling of propositions that are simultaneously true over some relevant collections of instances and false over some relevant collections of instances. A rigorous, algorithmic criterion is presented for detecting such cases, and its performance is exemplifie...

This paper reflects on the design of tasks that could take profit from the new features for Automated Reasoning, recently available in the dynamic geometry program GeoGebra. We report on some ongoing experiment, involving initial mathematics teacher students, aiming to develop and evaluate the characteristics of tasks that, with the concurrence of...

We present an algorithm to help converting expressions having non-negative quantities (like distances) in Euclidean geometry theorems to be usable in a complex algebraic geometry prover. The algorithm helps in refining the output of an existing prover, therefore it supports immediate deployment in high level prover systems. We prove that the algori...

The idea of envelope of a family of plane curves is an elementary notion in differential geometry. As such, its implementation in dynamic geometry environments is quite universal (Cabri, The Geometer’s Sketchpad, Cinderella, GeoGebra,...). Nevertheless, most of these programs return, when computing certain envelopes, both some spurious solutions an...

Along the last half century, automated deduction in elementary geometry has increasingly become one of the most successful achievements in the field of automated reasoning. Along these decades various methods and techniques have been studied and developed for automated proving and discovering of elementary geometry statements. On the other hand, dy...

We claim that the inquiry-based learning approach to geometry can be improved by considering a recently implemented algorithm for the automatic conjecturing and proving of elementary geometry statements. The new method we describe relies on some Automated Reasoning Tools commands, now available in the dynamic geometry software GeoGebra, that provid...

We report on preliminary work towards the automated finding of theorems in elementary geometry. The resulting system is being currently implemented on top of GeoGebra, a dynamic geometry system with millions of users at high schools and universities. Our system exploits GeoGebra's recently added new functionalities concerning automated reasoning t...

We report on preliminary work towards the automated finding of theorems in elementary geometry. The resulting system is being currently implemented on top of GeoGebra, a dynamic geometry system with millions of users at high schools and universities. Our system exploits GeoGebra's recently added new function alities concerning automated reasoning t...

We report on preliminary work towards the automated finding of theorems in elementary geometry. The resulting system is being currently implemented on top of GeoGebra, a dynamic geometry system with millions of users at high schools and universities. Our system exploits GeoGebra's recently added new functionalities concerning automated reasoning...

We present a deterministic algorithm for deciding if a polynomial ideal, with coefficients in an algebraically closed field K of characteristic zero, of which we know just some very limited data, namely: the number n of variables, and some upper bound for the geometric degree of its zero set in Kⁿ, is or not the zero ideal. The algorithm performs j...

We investigate and generalize to an extended framework the notion of true on components labeled by Zhou, Wang and Sun in their paper “Automated Reducible Geometric Theorem Proving and Discovery by Gröbner Basis Method”, J. Automat. Reasoning 59 (3), 331-344, 2017. A new, simple criterion is presented for a statement to be simultaneously not genera...

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 examp...

The software GeoGebra, as any other dynamic geometry system, offers some notorious ways to allow the student to perform his/her work as a mathematician, improving under certain conditions the exploration, the construction and the implementation of Euclidean plane geometry theorems. On the other hand, some specific features allowing the automatic an...