Hyperbolic Geometry

The man made world we see around us is constructed using straight lines, from the houses we live in to the skyscrapers we admire. However, in nature we observe wonderful shapes such as the beautiful undulations of coral and the crinkled surface of lettuce.

These natural phenomena follow hyperbolic geometry, where the plane is not necessarily flat, as opposed to the conventional Euclidean geometry.

Hyperbolic Geometry

Hyperbolic geometry is not considered Euclidean as it violates one of the axioms called the parallel postulate:

“If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.”

Put more simply, this axiom says that there is only one possible straight line through a point that does not meet the original line, as depicted by the image below.

Corals, crochet and the cosmos: how hyperbolic geometry pervades the universe

Source: phys.org

 

However, this is untrue in hyperbolic geometry – there can be more than one line passing through this point that do not meet.

Corals, crochet and the cosmos: how hyperbolic geometry pervades the universe

Source: phys.org

When this non-Euclidean geometry was discovered in the early 19th century, it was not easily accepted by mathematicians. Wolfgang Bolyai, a Hungarian mathematician, said to his son János Bolyai: “for God’s sake, please give it up!”

Hyperbolic Geometry in our Universe

In nature, we can see a variety of different manifestations of hyperbolic geometry, including corals and reef organisms like kelp. Scientists believe that this is due to the fact in organisms that have to maximise their surface area (such as in filter feeding animals), hyperbolic shapes provide an excellent solution.

In 1997, physicist Andrei Rode also built hyperbolic surfaces at a molecular scale from carbon atoms to create carbon nano-foams.

Furthermore, although the shape of our universe is still unknown, it is a topic of great research with instruments such as the Hubble Space telescope. Some evidence suggests that our universe is an Euclidean structure, however new evidence indicates that we might just live in a hyperbolic world.

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I’m sorry that today’s post is quite a short one. I will resume my ‘Forgotten Mathematicians’ series on Friday! Also, I’m trying to add more of my sources in my posts, so let me know what you think! M x

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