Axioms for Absolute Geometry

1968 ◽  
Vol 20 ◽  
pp. 158-181 ◽  
Author(s):  
J. F. Rigby
Keyword(s):  
Hyperbolic Geometry ◽  
Euclidean Geometry ◽  
Absolute Geometry

The axioms of Euclidean geometry may be divided into four groups: the axioms of order, the axioms of congruence, the axiom of continuity, and the Euclidean axiom of parallelism (6). If we omit this last axiom, the remaining axioms give either Euclidean or hyperbolic geometry. Many important theorems can be proved if we assume only the axioms of order and congruence, and the name absolute geometry is given to geometry in which we assume only these axioms. In this paper we investigate what can be proved using congruence axioms that are weaker than those used previously.

scholarly journals KONSISTENSI AKSIOMA-AKSIOMA TERHADAP ISTILAH-ISTILAH TAKTERDEFINISI GEOMETRI HIPERBOLIK PADA MODEL PIRINGAN POINCARE

EDUPEDIA ◽  
2018 ◽  
Vol 2 (2) ◽  
pp. 161
Author(s):  
Febriyana Putra Pratama ◽  
Julan Hernadi
Keyword(s):  
Half Plane ◽  
Hyperbolic Geometry ◽  
Euclidean Geometry ◽  
Cross Ratio ◽  
Literature Study ◽  
Disk Model ◽  
Non Euclidean Geometry ◽  
The Cross ◽  
Definition Of ◽  
Poincaré Disk

This research aims to know the interpretation the undefined terms on Hyperbolic geometry and it’s consistence with respect to own axioms of Poincare disk model. This research is a literature study that discusses about Hyperbolic geometry. This study refers to books of Foundation of Geometry second edition by Gerard A. Venema (2012), Euclidean and Non Euclidean Geometry (Development and History)  by Greenberg (1994), Geometry : Euclid and Beyond by Hartshorne (2000) and Euclidean Geometry: A First Course by M. Solomonovich (2010). The steps taken in the study are: (1) reviewing the various references on the topic of Hyperbolic geometry. (2) representing the definitions and theorems on which the Hyperbolic geometry is based. (3) prepare all materials that have been collected in coherence to facilitate the reader in understanding it. This research succeeded in interpret the undefined terms of Hyperbolic geometry on Poincare disk model. The point is coincide point in the Euclid on circle . Then the point onl γ is not an Euclid point. That point interprets the point on infinity. Lines are categoried in two types. The first type is any open diameters of   . The second type is any open arcs of circle. Half-plane in Poincare disk model is formed by Poincare line which divides Poincare field into two parts. The angle in this model is interpreted the same as the angle in Euclid geometry. The distance is interpreted in Poincare disk model defined by the cross-ratio as follows. The definition of distance from  to  is , where  is cross-ratio defined by  . Finally the study also is able to show that axioms of Hyperbolic geometry on the Poincare disk model consistent with respect to associated undefined terms.

The Nature of Arguments Provided by College Geometry Students With Access to Technology While Solving Problems

2010 ◽  
Vol 41 (4) ◽  
pp. 324-350 ◽  
Author(s):  
Karen F. Hollebrands ◽  
AnnaMarie Conner ◽  
Ryan C. Smith
Keyword(s):  
Hyperbolic Geometry ◽  
Euclidean Geometry ◽  
Dynamic Geometry ◽  
Disk Model ◽  
Geometric Objects ◽  
Access To Technology ◽  
Poincaré Disk ◽  
Support Students

Prior research on students' uses of technology in the context of Euclidean geometry has suggested it can be used to support students' development of formal justifications and proofs. This study examined the ways in which students used a dynamic geometry tool, NonEuclid, as they constructed arguments about geometric objects and relationships in hyperbolic geometry. Eight students enrolled in a college geometry course participated in a task-based interview that was focused on examining properties of quadrilaterals in the Poincaré disk model. Toulmin's argumentation model was used to analyze the nature of the arguments students provided when they had access to technology while solving the problems. Three themes related to the structure of students' arguments were identified. These involved the explicitness of warrants provided, uses of technology, and types of tasks.

On the Medians of a Triangle in Hyperbolic Geometry

1958 ◽  
Vol 10 ◽  
pp. 502-506 ◽  
Author(s):  
O. Bottema
Keyword(s):  
Hyperbolic Geometry ◽  
Euclidean Geometry ◽  
Common Point ◽  
Non Euclidean Geometry ◽  
Image Position

In non-Euclidean geometry the three medians of a triangle A 1A2A3 (each joining a vertex A i with the internal midpoint G i of the opposite side) are concurrent; their common point is the centroid G. But the Euclidean theorem ' which depends on similarity, does not hold.

scholarly journals XXIX.—Semi-regular Networks of the Plane in Absolute Geometry

1906 ◽  
Vol 41 (3) ◽  
pp. 725-747 ◽  
Author(s):  
Duncan M. Y. Sommerville
Keyword(s):  
Hyperbolic Plane ◽  
Euclidean Geometry ◽  
Regular Polygons ◽  
Ordinary Space ◽  
Absolute Geometry ◽  
Regular Polyhedra ◽  
Elliptic Geometry ◽  
Regular Networks

§ 1. The networks considered in the following paper are those networks of the plane whose meshes are regular polygons with the same length of side.When the polygons are all of the same kind the network is called regular, otherwise it is semi-regular.The regular networks have been investigated for the three geometries from various standpoints, the chief of which may be noted.1. The three geometries can be treated separately. For Euclidean geometry we have then to find what regular polygons will exactly fill up the space round a point. For elliptic geometry we have to find the regular divisions of the sphere, or, what is the same thing, the regular polyhedra in ordinary space. The regular networks which do not belong to either of these classes are then those of the hyperbolic plane.

scholarly journals An Unsupervised Learning Method for Attributed Network Based on Non-Euclidean Geometry

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 905
Author(s):  
Wei Wu ◽  
Guangmin Hu ◽  
Fucai Yu
Keyword(s):  
Ricci Curvature ◽  
Hyperbolic Geometry ◽  
Euclidean Geometry ◽  
Structural Information ◽  
Information Aggregation ◽  
Network Embedding ◽  
Heterogeneous Information ◽  
Non Euclidean Geometry ◽  
Attributed Network ◽  
Attribute Information

Many real-world networks can be modeled as attributed networks, where nodes are affiliated with attributes. When we implement attributed network embedding, we need to face two types of heterogeneous information, namely, structural information and attribute information. The structural information of undirected networks is usually expressed as a symmetric adjacency matrix. Network embedding learning is to utilize the above information to learn the vector representations of nodes in the network. How to integrate these two types of heterogeneous information to improve the performance of network embedding is a challenge. Most of the current approaches embed the networks in Euclidean spaces, but the networks themselves are non-Euclidean. As a consequence, the geometric differences between the embedded space and the underlying space of the network will affect the performance of the network embedding. According to the non-Euclidean geometry of networks, this paper proposes an attributed network embedding framework based on hyperbolic geometry and the Ricci curvature, namely, RHAE. Our method consists of two modules: (1) the first module is an autoencoder module in which each layer is provided with a network information aggregation layer based on the Ricci curvature and an embedding layer based on hyperbolic geometry; (2) the second module is a skip-gram module in which the random walk is based on the Ricci curvature. These two modules are based on non-Euclidean geometry, but they fuse the topology information and attribute information in the network from different angles. Experimental results on some benchmark datasets show that our approach outperforms the baselines.

A Note on Absolute Geometry

1968 ◽  
Vol 11 (5) ◽  
pp. 719-722
Author(s):  
Roland Brossard
Keyword(s):  
Congruence Relation ◽  
Euclidean Geometry ◽  
Absolute Geometry ◽  
Suitable Definition ◽  
The Absolute

Metric axioms have been given in [3] for space euclidean geometry. If we replace the "similarity axiom" by the "congruence axiom", where congruence is defined to be a similarity of ratio one, the resulting structure is absolute geometry. In order to show this we choose a suitable definition for absolute geometry. The P a s c h system of axioms, given in an improved formulation by H. S. M. Coxeter in [4], is particularly suitable; the primitive notions are points, betweenness relation, and congruence relation. We can verify that every axiom for the absolute geometry in [4] in a theorem in [3] where the similarity axiom has been replaced by the congruence axiom. The only case for which it is not obvious is axiom 15.15 in [4] which says that if ABC and A' B' C' are two triangles with BC ≡ B'C' CA ≡ C'A1, AB ≡ A ' B ', while D and D' are two further points such that [B, C, D] and [B', C' D'] and BD ≡ B' D', then AD ≡ A' D'. In that case we first prove that if two triangles ABC and A'B C are such that AB/A'B' ≡ BC/B'C' ≡ CA/C'A' ≡ 1 then they are congruent; a proof of this, independent of the similarity axiom, can be found in [2]. The proof of 15. 15 in [4] is then obvious. As every axiom in the weakened structure of [3] is a theorem of absolute geometry we have a definition for this geometry.

Note on the Geometries in which Straight Lines are represented by Circles

1909 ◽  
Vol 28 ◽  
pp. 81-94
Author(s):  
Duncan M. Y. Sommerville
Keyword(s):  
Hyperbolic Geometry ◽  
Euclidean Geometry ◽  
Present Note ◽  
Spherical Geometry ◽  
Point Of View ◽  
Elementary Geometry ◽  
Straight Lines

In a recent paper read before the Society, Professor Carslaw gave an account, from the point of view of elementary geometry, of the well-known and beautiful concrete representation of hyperbolic geometry in which the non-Euclidean straight lines are represented by Euclidean circles which cut a fixed circle orthogonally. He also considered the case in which the fixed circle vanishes to a point, and showed that this corresponds to Euclidean geometry. The remaining case, in which the fixed circle is imaginary and which corresponds to elliptic or spherical geometry, is not open to the same elementary geometrical treatment, and Professor Carslaw therefore omitted any reference to it. As this might be misleading, the present note has been written primarily to supply this gap. It has been thought best, however, to give a short connected account of the whole matter from the foundation, from the point of view of analysis, omitting the detailed consequences which properly find a place in Professor Carslaw's paper.

The Perils of Parallels!

2020 ◽  
pp. 156-166
Author(s):  
Nicholas Mee
Keyword(s):  
Nineteenth Century ◽  
Hyperbolic Geometry ◽  
Euclidean Geometry ◽  
Spherical Geometry ◽  
Space And Time ◽  
The Earth

In the nineteenth century, three mathematicians—Bolyai, Gauss, and Lobachevsky—almost simultaneously discovered the possibility of non-Euclidean or hyperbolic geometries. These geometries rest on axioms that do not include the parallel postulate. This means that many results of Euclidean geometry do not hold. Spherical geometry is considered as a model to illustrate why this is the case. The mathematician Donald Coxeter inspired artist M. C. Escher to produce remarkable artworks based on the hyperbolic geometry of the Poincaré disc. Gauss attempted to measure the curvature of the space around the Earth. Since Einstein, we know that gravity curves space and time.

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