Regular Pentagonal Tilings in Non-Euclidean Geometry
A dodecahedron can be considered a regular tiling of 12 pentagons on the surface of a sphere, with Schlafli symbol {5,3}, having 3 pentagons around reach vertex.
In the hyperbolic plane, there are tilings of regular pentagons, for instance order-4 pentagonal tiling, with Schlafli symbol {5,4}, having 4 pentagons around reach vertex. Higher order regular tilings {5,n} can be constructed on the hyperbolic plane, ending in {5,∞}.
| Sphere | Hyperbolic plane | |||||
|---|---|---|---|---|---|---|
{5,3} |
{5,4} |
{5,5} |
{5,6} |
{5,7} |
{5,8} |
...{5,∞} |
Read more about this topic: Pentagon Tiling
Famous quotes containing the words regular and/or geometry:
““I couldn’t afford to learn it,” said the Mock Turtle with a sigh. “I only took the regular course.”
“What was that?” inquired Alice.
“Reeling and Writhing, of course, to begin with,” the Mock Turtle replied; “and then the different branches of Arithmetic—Ambition, Distraction, Uglification, and Derision.”
“I never heard of ‘Uglification,’” Alice ventured to say.”
—Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)
“... geometry became a symbol for human relations, except that it was better, because in geometry things never go bad. If certain things occur, if certain lines meet, an angle is born. You cannot fail. It’s not going to fail; it is eternal. I found in rules of mathematics a peace and a trust that I could not place in human beings. This sublimation was total and remained total. Thus, I’m able to avoid or manipulate or process pain.”
—Louise Bourgeois (b. 1911)