Hyperbolic Escher GeoGebra
A look at the Escher artwork related to the Hyperbolic Geometry by means of GeoGebra: Some of the yet know and some of the unknow as hyperbolic, also.
Table des matières
- Escher`s mathematical work
- Escher`s mathematical art
- Preliminaries works on infinity representation
- Hyperbolic Geometry and Poincaré Disk model
- Hyperboic Geometry Models
- GeoGebra Poincaré Disk model
- Tesselation on the Hyperbolic plane
- Circle Limit I
- Circle Limit III
- The Halfplane model and the Escher work
- Halfplane model with GeoGebra
- Escher first works over the Halfplane model
- The spiral and the circular inversion
- The circular inversion
- The spiral and the double spiral
- References
- References
Escher`s mathematical art
The mathematical ideas behind their work
[size=150]The dutch artist M.C. Escher, is well know round the world, by their artistic works with a strong inspiration on mathematical concepts. Some of them are based in their own studies, and some others in the studies and suggestions of mathematicians like [b][i]Roger Penrose[/i][/b], [b][i]George Pólya[/i][/b] and [b][i]H.S.M. Coxeter[/i][/b]. With the last one he maintained fruitful contacts.[br][br]In their extensive graphic work they appear multiple mathematical ideas: Symmetries, regular or stellated polyhedra, tessellations, topology, spherical geometry and impossible objects, between others. But something that he always pursued was the representation of the idea of infinity. This guided his steps towards hyperbolic geometry.[/size]
M.C. Escher, looking at one of his works
Chasing non-euclidean universes
[size=150]One of their first encounters with hyperbolic geometry was at 1956. From that date, he produces several artworks based on the representation of the hyperbolic plane. And also some more with a less clear mathematical idea behind that, but also pursuing the representation of infinity. [br][br]In this geometry, as in the spherical geometry also, they are not valid the same principles that are valid in euclidean geometry. He achieved more and more perfection, with graphic works in a mixture of styles, [i]Op-art[/i] (Optical Art) and [i]Surrealism. [/i][br][br]He maintained other constant of their work, that of tessellation, which he learned from Pólya, but now in the new spaces. The idea of infinity is represented in the limit as a smaller and smaller drawings, which could not have been done without great virtuosity on their part.[/size]
Hyperboic Geometry Models
Hyperbolic Geometry
[size=150]In this geometry we impose the condition that for each point outside a straight line more than one parallel can pass to it. In fact, there will be infinite parallel lines to a given one. If the lines intersect at infinity, we will say that they are parallel, but if they do not even intersect there, then we will say that they are ultraparallel.[br][br]A triangle in this geometry has angles that always add less than 180º.[/size]
[size=150]There are 3 models that can be represented in the Euclidean plane for this geometry. They are first devised by Beltrami. The 3 models are:[br][br][list][*]Poincaré Disk[br][/*][*]Klein Disk[br][/*][*]Upper Halfplane[/*][/list][br]We will first use the Poincaré Disk model. This model is conformal (it respects the value of angles in original space), although it isn't isometric (it distorts the distances between points in original and the model).[/size]
Comparing among the 3 models
Halfplane model with GeoGebra
The Upper Halfplane model
[size=150]In the Upper Halfplane model, the points are the points of the upper real half plane (y> 0), not including the line y = 0. The lines in this are either semicircunferences perpendicular to the x-axis, or semirects perpendicular to it.[/size]
The circular inversion
The inversion application
[size=150]The well know inversion map plays an important role when restricted to the Hyperbolic disk. So the hyperbolic inversion maps circles and lines in circles and/or lines, playing in the hyperbolic world a similar figure that plays the reflection in the euclidean one.[br]In the Poincaré Disk model, hyperbolic lines (arc perpendicular to de disk, are mapped to the complementary arc and circles tangent to the disk are mapped into also tangent circles by that point, but inside. Straight lines outsides the disc are mapped into circles inside that pases by the center, corresponding by inversion to infinity point[/size]
References
[list=1][*]ABAR, Celina A.P. [i]O uso[/i][i]do GeoGebra na investigaçao da geometría elíptica.[/i] Actas del VI Congreso Uruguayo de Educación Matemática.[/*][*][color=#000000]COXETER, H.S.M. [/color][i]The Non-Euclidean Symmetry of Escher's Picture 'Circle Limit III'[/i]. Leonardo, Vol. 12, No. 1, pp. 19-25. Published by The MIT Press.[/*][*] COXETER, H.M.S. The trigonometry of hyperbolic tessellations.[url=https://cms.math.ca/cmb/]Canad. Math. Bull.[/url] [url=https://cms.math.ca/cmb/v40/]40[/url][color=#666666](1997), 158-168. 1997.[/color][/*][*][color=#333333]DUNHAM, Douglas. [/color][i]Some Maths behind the Circle Limit's Patterns.[/i] University of Minessotta.[/*][*][color=#333333]DUNHAM, Douglas. [/color][color=#666666][i]A Circle Limit III Calculation. [/i][/color]University of Minessotta.[/*][*]GREENBERG, M. [i]Euclidean and Non-Euclidean Geometry. Third edition.[/i] W.H. Freeman Inc. New York, 1993.[/*][*]RODRÍGUEZ TRÍAS, Ujué. Herramientas Hiperbólicas y Proyectivas con GeoGebra. TFM, Universidad de Cantabria.[/*][/list][br][br][br][br][br][br]