Tag Archives: euclidean geometry

Style: University of Virginia Lawn

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There are three manmade UNESCO world heritage sites in the United States: The Liberty Bell, The Statue of Liberty, and The University of Virginia Lawn with Monticello. The UNESCO designation basically means there is something noteworthy of distinctive about the site. I happen to live near to the University of Virginia, so I get to take a lot of photos. (As of this post, I just discovered that all the modern photos on the lawn Wikipedia page are mine. I love to see where the creative commons take my works. Side note: check out my very large Flickr collection of mostly creative commons images.)

Many years ago, Benoit Mandelbrot, the creator of fractal geometry, visited the university to give a talk. He said it was like walking into the lion’s den of Euclidean geometry. I always liked this description; everything about the university is columns and arches and perspective points. Monticello and the University were laid out by Thomas Jefferson, who one gets the feeling never actually died living around here. He was the ambassador to France for a while, and greatly admired the architecture. He came back to the states with those architectural inspirations.

The UVA lawn, shown below, has the rotunda at one end (the second one… the first one burned down and blew up when a professor tried to save it with TNT) and is lined by ten pavilions. Between the pavilions are dorm rooms that distinguished fourth year students still live in. Each of the ten pavilions is architecturally different, and behind each is a garden in a different style which no doubt will be the topic of a future post. Pavilion 2 is pictured below. Professors still live in the pavilions. The pavilions were built in a strange order, to ensure that diminished funds would not diminish the scope of the project.

It’s very easy to find plenty of reading material on Jefferson and the University if you are interested, so I won’t try to write a tome here. However I’ll include a few of my pictures that may hopefully spark your interest.

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Monticello, i.e. the back of a nickel

 

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Fractals in Life: photos

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Fractals are a branch of math that better describes nature. Before fractals, there was Euclidean geometry, the geometry of lines, planes, and spaces. Euclidean geometry cannot give the length of a rough coastline; neither can it give the surface area of shaggy tree bark. The answer you arrive at in Euclidean geometry depends upon the scale at which you examine an object– intuitively the distance an ant travels over rough terrain is different from the distance we cover walking. The ant interacts with the terrain at a different length scale than we do. Fractal geometry is designed to handle objects with multiple meaningful length scales.

Fractal objects are sometimes called “scale-free“. This means that the object looks roughly the same even if examined at very different zooms. Many natural objects look similar at multiple zooms. Below, I include a few. The craters on the moon are scale-free. If you keep zooming in, you could not tell the scale of the image.

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Terrain is often scale-free in appearance– a few years ago there was a joke photo with a penny for scale. The owners of the photo subsequently revealed that the “penny” was actually 3 feet across. I couldn’t find the photo, but you cannot tell the difference. The reveal was startling. Below is a picture of Mount St Helens. The top of this mountain is five miles (3.2ish km) away. Streams carry silt away from the mountain, as you can see more towards the foreground. They look like tiny streams. These are full-sized rivers. Mount St. Helens lacks much of the vegetation and features we would usually use to determine scale. The result is amazingly disorienting, and demonstrates scale invariance.

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Plants are often scale free too. Small branches are very similar in appearance to large branches. Ferns look very fractal. Below is a picture of Huperzia phlegmaria. Each time this plant branches, there are exactly two branches. Along its length, it branches many times. It is a physical realization of a binary tree.

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For more about fractals, read my posts about fractal measurement and the mandelbrot set.