Hyperbolic Planes
"For God's sake, please give it up. Fear it no less than the sensual passions, because it, too, may take up all your time and deprive you of your health, peace of mind and happiness in life." Wolfgang Bolyai (1775-1856) to his son Janos Bolyai regarding the study of hyperbolic geometry
Anyone interested in geometry? How about crocheting? Pondering the shape of the universe ?
Until the ninteenth century, mathematicians knew about only two kinds of geometry, the Euclidean plane and the sphere. The discovery of hyperbolic space in the 1820s and 1830s innitiated the formal study of non -Euclidean geometry.
In Euclid's geometry, all surfaces are flat and parallel lines always stay the same distance apart, never meeting and never diverging. However in curved non-Euclidean geometries, lines that start off parallel eventually cross each other in a positive curvature and these same lines can diverge from from each other in a negative curvature.
A sphere is an example of a surface where the lines curve in on itself and is closed. A hyperbolic plane is a surface in which the the space curves away from itself at every point but like a Euclidean plane is open and infinite.
It is easy to understand a sphere because we have experience in our everyday lives with them. But how does one fully grasp the geometric opposite, a hyperbolic plane? For over 100 years, mathematicians searched for a physical surface with hyperbolic geometry without sucess. Until in 1997 Daina Taimina, a mathematician from Cornell University made the first useable hyperbolic model out of crochet.
You can see more examples of hyperbolic planes at Cabinet Magazine and I really enjoy The Institute For Figuring which has yet more models as well as interesting lectures on "The Physics of Snowflakes" , "The Mathematics of Paper Folding" and "The Logic Alphabet".
Anyone interested in geometry? How about crocheting? Pondering the shape of the universe ?Until the ninteenth century, mathematicians knew about only two kinds of geometry, the Euclidean plane and the sphere. The discovery of hyperbolic space in the 1820s and 1830s innitiated the formal study of non -Euclidean geometry.
In Euclid's geometry, all surfaces are flat and parallel lines always stay the same distance apart, never meeting and never diverging. However in curved non-Euclidean geometries, lines that start off parallel eventually cross each other in a positive curvature and these same lines can diverge from from each other in a negative curvature.
A sphere is an example of a surface where the lines curve in on itself and is closed. A hyperbolic plane is a surface in which the the space curves away from itself at every point but like a Euclidean plane is open and infinite.
It is easy to understand a sphere because we have experience in our everyday lives with them. But how does one fully grasp the geometric opposite, a hyperbolic plane? For over 100 years, mathematicians searched for a physical surface with hyperbolic geometry without sucess. Until in 1997 Daina Taimina, a mathematician from Cornell University made the first useable hyperbolic model out of crochet.
You can see more examples of hyperbolic planes at Cabinet Magazine and I really enjoy The Institute For Figuring which has yet more models as well as interesting lectures on "The Physics of Snowflakes" , "The Mathematics of Paper Folding" and "The Logic Alphabet".
posted by ljmcinnis at 4:32 PM
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