This thesis presents a visual introduction to hyperbolic geometry, aiming to involve the reader in an interactive way. Starting from the observation of Escher’s works to understand them intuitively, the thesis explores hyperbolic concepts using GeoGebra, a dynamic geometry software. Poincaré’s disk model is introduced as a way to translate hyperbolic concepts into a Euclidean representation. Indeed, to convince oneself of the validity of the theory, it is necessary to prove its relative non-contradiction by constructing a Euclidean model of hyperbolic geometry. In a Euclidean plane, a circle c of center O, circumference excluded, defines the Poincaré disk. The points of hyperbolic geometry are the points inside c, while the lines, which we will call P-lines to distinguish them from the Euclidean ones (the prefix P for each concept means hyperbolic), are the diameters of c and the arcs of circumferences orthogonal to c. Two circumferences are said to be orthogonal if they have at least one point E in common and their radii through E are orthogonal. In hyperbolic geometry Euclid’s V postulate, that of parallels, does not hold: given a line and a point outside it, there are an infinite number of parallel lines passing through the point. It follows that two distinct P-parallel lines can be: incident, if they meet at a point inside the disk; hyperparallel, if they meet at a point on the edge and have, that is, a common direction; ultraparallel if they do not even meet on the edge. P-angles are introduced as angles formed by P-lines and P-triangles as figures formed by three intersecting P-lines. The sum of the P-angles of a P-triangle is less than 180°, and two P-triangles that have three congruent P-angles are congruent: both consequences of the hyperbolic postulate. Hilbert’s axioms are satisfied in hyperbolic geometry, with the exception of the axiom of parallels: axioms of incidence, ordering and congruence. P-congruence is introduced as a notion of congruence specific to hyperbolic geometry: two P-segments, or more generally two P-figures, are said to be P-congruent if there is a P-isometry, i.e., a rigid motion that brings one into the other. Each P-isometry is the combination of a finite number of P-reflections, where the P-reflection with respect to a P-line l is a particular circular inversion that transforms into each other the two regions into which l divides the disk. In hyperbolic geometry, Euclidean distances are not maintained and the concept of similarity disappears: P-congruent figures become smaller as they move away from the center of the disk. Because of the abundance of P-reflections, the possible tessellations of the hyperbolic plane with regular P-polygons are infinite in great contrast to the Euclidean case. We say tessellations are all possible ways of covering the plane with geometric figures without overlapping or empty spaces. An example is the Coxeter figure, an important source of inspiration for Escher. From the angular defect of a P-triangle, obtained by subtracting the sum of the inner P-angles of the P-triangle from 180°, the concept of hyperbolic area is defined as directly proportional to it. The constant of direct proportionality is the curvature of the hyperbolic plane. Finally, the thesis emphasizes that hyperbolic geometry is not a theory for its own sake but has several practical applications, including its usefulness in representing complex physical concepts, such as in the theory of special relativity. Hyperbolic geometry is a vast field that is constantly being discovered and offers many opportunities for future research.
La tesi presenta un’introduzione visuale alla geometria iperbolica, con l’obiettivo di coinvolgere il lettore in modo interattivo. Partendo dalla loro osservazione delle opere di Escher, per comprenderli intuitivamente, la tesi esplora i concetti iperbolici utilizzando GeoGebra, un software di geometria dinamica. Il modello del disco di Poincaré viene introdotto come un modo per tradurre concetti iperbolici in una rappresentazione euclidea. Per convincersi della validità della teoria, è necessario infatti dimostrarne la non contraddittorietà relativa costruendo un modello euclideo della geometria iperbolica. In un piano euclideo, una circonferenza c di centro O, circonferenza esclusa, definisce il disco di Poincaré. I punti della geometria iperbolica sono i punti interni a c, mentre le rette, le cosiddette P-rette (dove il prefisso P per ogni concetto significa iperbolico), sono i diametri di c e gli archi di circonferenze ortogonali a c. Due circonferenze si dicono ortogonali se hanno almeno un punto E in comune e i loro raggi per E sono ortogonali. La geometria iperbolica si basa sulla negazione del V postulato di Euclide, quello delle parallele: date una retta e un punto esterno ad essa, esistono infinite rette parallele che passano per il punto. Segue che due P-rette distinte possono essere: incidenti, se si incontrano in un punto interno al disco; iperparallele, se si incontrano in un punto del bordo e hanno cioè un verso in comune; ultraparallele se non si incontrano nemmeno sul bordo. I P-angoli sono introdotti come angoli formati da P-rette e i P-triangoli come figure formate da tre P-rette intersecanti. La somma dei P-angoli di un P-triangolo è minore di 180° e due P-triangoli che hanno tre P-angoli congruenti sono congruenti: entrambe conseguenze del postulato iperbolico. Gli assiomi di Hilbert sono soddisfatti nella geometria iperbolica, ad eccezione dell’assioma delle parallele: assiomi di incidenza, ordinamento e congruenza. La P-congruenza viene introdotta come una nozione di congruenza specifica per la geometria iperbolica: due P- segmenti, o più in generale due P-figure, si dicono P-congruenti se esiste una P-isometria, ossia un movimento rigido che porti l’una nell’altra. Ogni P-isometria è la combinazione di un numero finito di P-riflessioni, dove la P-riflessione rispetto a una P-retta l è una particolare inversione circolare che trasforma l’una nell’altra le due regioni in cui l divide il disco. In geometria iperbolica le distanze euclidee non vengono mantenute e il concetto di similitudine scompare: figure P-congruenti rimpiccioliscono con l’allontanarsi dal centro del disco. Motivo per cui grazie a infinite P-riflessioni, le tassellazioni possibili del piano iperbolico con P-poligoni regolari sono infinite. Si dicono tassellazioni tutti i possibili modi di ricoprire il piano con figure geometriche senza sovrapposizioni o spazi vuoti. Un esempio è la figura di Coxeter, importante fonte d’ispirazione per Escher. A partire dal difetto angolare di un P-triangolo, ottenuto sottraendo a 180° la somma dei P-angoli interni del P-triangolo si definisce il concetto di area iperbolica come direttamente proporzionale ad esso. La costante di proporzionalità diretta è indicatrice della curvatura del piano iperbolico. La tesi sottolinea infine che la geometria iperbolica non è una teoria fine a se stessa ma ha diverse applicazioni pratiche, inclusa la sua utilità nella rappresentazione di concetti fisici complessi, come nella teoria della relatività ristretta. La geometria iperbolica è un campo vasto e in continua scoperta, che offre molte opportunità di ricerca futura.
Un'introduzione visiva alla geometria iperbolica
BARBATO, GIULIA
2022/2023
Abstract
This thesis presents a visual introduction to hyperbolic geometry, aiming to involve the reader in an interactive way. Starting from the observation of Escher’s works to understand them intuitively, the thesis explores hyperbolic concepts using GeoGebra, a dynamic geometry software. Poincaré’s disk model is introduced as a way to translate hyperbolic concepts into a Euclidean representation. Indeed, to convince oneself of the validity of the theory, it is necessary to prove its relative non-contradiction by constructing a Euclidean model of hyperbolic geometry. In a Euclidean plane, a circle c of center O, circumference excluded, defines the Poincaré disk. The points of hyperbolic geometry are the points inside c, while the lines, which we will call P-lines to distinguish them from the Euclidean ones (the prefix P for each concept means hyperbolic), are the diameters of c and the arcs of circumferences orthogonal to c. Two circumferences are said to be orthogonal if they have at least one point E in common and their radii through E are orthogonal. In hyperbolic geometry Euclid’s V postulate, that of parallels, does not hold: given a line and a point outside it, there are an infinite number of parallel lines passing through the point. It follows that two distinct P-parallel lines can be: incident, if they meet at a point inside the disk; hyperparallel, if they meet at a point on the edge and have, that is, a common direction; ultraparallel if they do not even meet on the edge. P-angles are introduced as angles formed by P-lines and P-triangles as figures formed by three intersecting P-lines. The sum of the P-angles of a P-triangle is less than 180°, and two P-triangles that have three congruent P-angles are congruent: both consequences of the hyperbolic postulate. Hilbert’s axioms are satisfied in hyperbolic geometry, with the exception of the axiom of parallels: axioms of incidence, ordering and congruence. P-congruence is introduced as a notion of congruence specific to hyperbolic geometry: two P-segments, or more generally two P-figures, are said to be P-congruent if there is a P-isometry, i.e., a rigid motion that brings one into the other. Each P-isometry is the combination of a finite number of P-reflections, where the P-reflection with respect to a P-line l is a particular circular inversion that transforms into each other the two regions into which l divides the disk. In hyperbolic geometry, Euclidean distances are not maintained and the concept of similarity disappears: P-congruent figures become smaller as they move away from the center of the disk. Because of the abundance of P-reflections, the possible tessellations of the hyperbolic plane with regular P-polygons are infinite in great contrast to the Euclidean case. We say tessellations are all possible ways of covering the plane with geometric figures without overlapping or empty spaces. An example is the Coxeter figure, an important source of inspiration for Escher. From the angular defect of a P-triangle, obtained by subtracting the sum of the inner P-angles of the P-triangle from 180°, the concept of hyperbolic area is defined as directly proportional to it. The constant of direct proportionality is the curvature of the hyperbolic plane. Finally, the thesis emphasizes that hyperbolic geometry is not a theory for its own sake but has several practical applications, including its usefulness in representing complex physical concepts, such as in the theory of special relativity. Hyperbolic geometry is a vast field that is constantly being discovered and offers many opportunities for future research.| File | Dimensione | Formato | |
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https://hdl.handle.net/10589/210312