The equidistant involution of the hyperbolic plane and two models of the Euclidean plane geometry

2013 ◽  
Vol 104 (2) ◽  
pp. 201-212
Author(s):  
Miroslava Antić
Keyword(s):  
Hyperbolic Plane ◽  
Euclidean Plane ◽  
Plane Geometry

scholarly journals Formalization of the arithmetization of Euclidean plane geometry and applications

2019 ◽  
Vol 90 ◽  
pp. 149-168 ◽  
Author(s):  
Pierre Boutry ◽  
Gabriel Braun ◽  
Julien Narboux
Keyword(s):  
Euclidean Plane ◽  
Plane Geometry

Frieze Patterns in the Hyperbolic Plane

1974 ◽  
Vol 17 (1) ◽  
pp. 45-50 ◽  
Author(s):  
C. W. L. Garner
Keyword(s):  
Hyperbolic Plane ◽  
Euclidean Plane ◽  
Symmetry Groups ◽  
Euclidean Case ◽  
Parallel Lines

AbstractIt is well known that in the Euclidean plane there are seven distinct frieze patterns, i.e. seven ways to generate an infinite design bounded by two parallel lines. In the hyperbolic plane, this can be generalized to two types of frieze patterns, those bounded by concentric horocycles and those bounded by concentric equidistant curves. There are nine such frieze patterns; as in the Euclidean case, their symmetry groups are and

Regular compound tessellations of the hyperbolic plane

1964 ◽  
Vol 278 (1373) ◽  
pp. 147-167 ◽  
Keyword(s):  
Hyperbolic Plane ◽  
Euclidean Plane ◽  
Regular Polyhedron ◽  
Alternating Group ◽  
Platonic Solids ◽  
Icosahedral Group ◽  
Analogous Compound ◽  
Equilateral Triangles ◽  
Two Parameter

Kepler, in his Harmonice mundi of 1619, extended the idea of a regular polyhedron in at least two directions. Observing that two equal regular tetrahedra can interpenetrate in such a way that their twelve edges are the diagonals of the six faces of a cube, he called this combination stella octangida . It is occasionally found in nature as twinned crystals of tetrahedrite, Cu 10 (Zn, Fe, Cu) 2 Sb 4 S 13 . In addition to this ‘compound’ of two tetrahedra inscribed in a cube, there are several other compound polyhedra, the prettiest being the compound of five tetrahedra inscribed in a dodecahedron. The icosahedral group of rotations may be described as the alternating group on these five tetrahedra. Kepler observed also that the tessellation of squares (or regular hexagons, or equilateral triangles), filling and covering the Euclidean plane, may be regarded as an infinite analogue of the spherical tessellations which are ‘blown-up’ versions of the Platonic solids. Putting these two ideas together, one naturally regards the compound polyhedra as compound tessellations of the sphere. The analogous compound tessellations of the Euclidean plane (18 two-parameter families of them) were enumerated in 1948. The present paper describes many compound tessellations of the hyperbolic plane: five one-parameter families and seventeen isolated cases. It is conjectured that this list is complete, but there remains the possibility that a few more isolated cases may still be discovered.

Euclidean constructions

1954 ◽  
Vol 47 (4) ◽  
pp. 231-233
Author(s):  
Robert C. Yates
Keyword(s):  
Euclidean Plane ◽  
Plane Geometry

In the spirit of the old-time revival and the spring tonic, I feel it periodically necessary to “reaffirm the faith” and refresh myself in the fundamental constructions of Euclidean plane geometry. It seems always such a satisfying experience that I wish to share it. My refreshment takes the following form.

scholarly journals The Double Six of Lines and a Theorem in Euclidean Plane Geometry

1952 ◽  
Vol 1 (1) ◽  
pp. 1-7 ◽  
Author(s):  
John Dougall
Keyword(s):  
Projective Space ◽  
Euclidean Plane ◽  
Natural Extension ◽  
Plane Geometry ◽  
Three Dimensions ◽  
Classical Theorem ◽  
Invariant Relation ◽  
Geometrical Form

The object of the present paper is to establish the equivalence of the well-known theorem of the double-six of lines in projective space of three dimensions and a certain theorem in Euclidean plane geometry. The latter theorem is of considerable interest in itself for two reasons. In the first place, it is a natural extension of Euler's classical theorem connecting the radii of the circumscribed and the inscribed (or the escribed) circles of a triangle with the distance between their centres. Secondly, it gives in a geometrical form the invariant relation between the circle circumscribed to a triangle and a conic inscribed in the triangle. For a statement of the theorem, see § 13 (4).

The Elements of Non-Euclidean Plane Geometry and Trigonometry

1919 ◽  
Vol 9 (141) ◽  
pp. 374
Author(s):  
Philip E. B. Jourdain ◽  
H. S. Carslaw
Keyword(s):  
Euclidean Plane ◽  
Plane Geometry

scholarly journals A Mechanical Verification of  the Independence of Tarski's  Euclidean Axiom

2021 ◽  
Author(s):  
◽  
Timothy James McKenzie Makarios
Keyword(s):  
Projective Plane ◽  
Hyperbolic Plane ◽  
Axiom System ◽  
Plane Geometry ◽  
Mechanical Verification ◽  
The Real ◽  
Cartesian Plane ◽  
Proof Verification

<p>This thesis describes the mechanization of Tarski's axioms of plane geometry in the proof verification program Isabelle. The real Cartesian plane is mechanically verified to be a model of Tarski's axioms, thus verifying the consistency of the axiom system. The Klein–Beltrami model of the hyperbolic plane is also defined in Isabelle; in order to achieve this, the projective plane is defined and several theorems about it are proven. The Klein–Beltrami model is then shown in Isabelle to be a model of all of Tarski's axioms except his Euclidean axiom, thus mechanically verifying the independence of the Euclidean axiom — the primary goal of this project. For some of Tarski's axioms, only an insufficient or an inconvenient published proof was found for the theorem that states that the Klein–Beltrami model satisfies the axiom; in these cases, alternative proofs were devised and mechanically verified. These proofs are described in this thesis — most notably, the proof that the model satisfies the axiom of segment construction, and the proof that it satisfies the five-segments axiom. The proof that the model satisfies the upper 2-dimensional axiom also uses some of the lemmas that were used to prove that the model satisfies the five-segments axiom.</p>

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