hyperbolic-geometry

Types of Spirals.

Logarithmic Spiral - self-similar spiral curve which often appears in Nature. Spira Mirabilis, Latin for “miraculous spiral”, is another name for the Logarithmic Spiral. The size of the spiral increases but its shape is unaltered with each successive curve, a property known as Self-Similarity. Possibly as a result of this unique property, the Spira Mirabilis has evolved in Nature, appearing in certain growing forms such as nautilus shells and sunflower heads. 

Fermat’s Spiral
- in the sunflower and daisy, the mesh of spirals occurs in Fibonacci Numbers because Divergence (angle of succession in a single spiral arrangement) approaches the Golden Ratio. The shape of the spirals depends on the growth of the elements generated sequentially. In mature-disc phyllotaxis, when all the elements are the same size, the shape of the spirals is that of Fermat Spirals - ideally. That is because Fermat’s Spiral traverses equal annuli in equal turns. 

Archimedean Spiral - it is the locus of points corresponding to the locations over time of a point moving away from a fixed point with a constant speed along a line which rotates with constant angular velocity. The Archimedean Spiral has the property that any ray from the origin intersects successive turnings of the spiral in points with a constant separation distance, hence the name “Arithmetic Spiral”.

Hyperbolic Spiral - transcendental plane curve also known as a Reciprocal Spiral. A Hyperbolic Spiral is the opposite of an Archimedean Spiral. It begins at an infinite distance from the pole in the center (for θ starting from zero r = a/θ starts from infinity), and it winds faster and faster around as it approaches the pole; the distance from any point to the pole, following the curve, is Infinite.

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Circle Limit - Wood Engravings by M.C. Escher

Around 1956, M.C. Escher explored the concept of representing infinity on a two-dimensional plane. Discussions with Canadian mathematician H.S.M. Coxeter inspired Escher’s interest in hyperbolic tessellations, which are regular tilings of a hyperbolic plane. Escher’s wood engravings Circle Limit I–IV demonstrate this concept. In 1959, Coxeter published his finding that these works were extraordinarily accurate: “Escher got it absolutely right to the millimeter.”

Hyperbolic planes are difficult to explain. In fact, hyperbolic geometry is an extremely huge topic. Many visualizations of hyperbolic planes have been discovered (including the circle limits afore mentioned). Taking Circle Limit III for example (the one with the fishes), here is the gist of what these artworks have to do with hyperbolic geometry:

  • The number of fishes within a distance of n from the center rises exponentially.
  • The fishes have equal hyperbolic area. Yes, the tiny fishes on the very edge of the circle are the same size as the fishes in the center (on an actual hyperbolic plane, anyway).
  • So, the area of a ball of radius n must rise exponentially in n.

Learn more. (x) (x) (x)

flickr

Brunnian Link Tiling by Ross Hilbert
Via Flickr:
A hyperbolic tiling created using the Fractal Science Kit fractal generator. See www.fractalsciencekit.com/ for details. A Hyperbolic Tiling replicates a polygon over the hyperbolic plane represented by the Poincare disk in such a way as to form a hyperbolic tiling pattern. The Poincare disk is a model for hyperbolic geometry that maps the hyperbolic plane onto the unit disk.

The Cayley graph of the free group on two generators (in red and blue), embedded in the Poincaré hyperbolic disk model.

Since the free group has no relations, its Cayley graph has no cycles. The resulting structure, an infinite tree where every node is connected to a fixed number of neighbours, is also called a Bethe lattice.

I absolutely adore this picture: it combines so many interesting mathematical ideas (group theory, graph theory, hyperbolic geometry) into a single image!

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Experiments in Processing: curious eels in hyperbolic sea.

I love hyperbolic geometry, heptagons, and Escher.

The eels move on the sides of a triheptagonal tiling of the hyperbolic plane in the Poincaré disk model. You better appreciate this because getting the coordinates is a real pain in the *** (done with a python script, <a href=“https://github.com/rantonels/poincare“>source</a>).

BONUS: nauseating Moebius translation.