Fractal Geometry

Last winter in my travels to Tucson and the Grand Canyon, I had a conversation with a guy named David about fractal geometry.  The subject came up when we were discussing agate formation.  A fractal is a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole -- a property called self-similarity. The use of fractals enables the prediction of patterns. Roots of the idea of fractals go back to the 17th century.

There are several examples of fractals, which are defined as portraying exact self-similarity, quasi self-similarity, or statistical self-similarity. While fractals are a mathematical construct, they are found in nature, too. Natural objects that are approximated by fractals to a degree include clouds, mountain ranges, lightning bolts, river water sheds, snow flakes, various vegetables (cauliflower and broccoli), animal coloration patterns, and in some cases -- agates.

Many cacti have fractal patterns.

So do tree limbs...

And ice crystals.....

And leaves....

And flowers.....

And river drainage patterns.....

Of course, a kaleidoscope is sort of a way to cheat the fractal creation process -- using mirrors.

One of the most intricate and beautiful images in all of mathematics is the Mandelbrot set, discovered by Benoit Mandelbrot in 1980. Most people within the mathematics community, and many people outside of the discipline, have seen this image and have marveled at its geometric intricacy. 

The Mandelbrot set is a particular mathematical set of points, whose boundary generates a distinctive and easily recognisable two-dimensional fractal shape. The Mandelbrot set is generated by iteration. Iteration means to repeat a process over and over again. In mathematics this process is most often the application of a mathematical function. For the Mandelbrot set, the function involved is the simplest nonlinear function imaginable, namely x2 + c, where c is a constant. To iterate x2 + c, we begin with a seed for the iteration. This is a (real or complex) number.

Many agates also demonstrate fractal patterns.

Charlevoix Castle Farms Show -- 3rd Post

Today I'll include the last pictures from Castle Farms. As I continued walking around the castle, I captured pictures of the Garden Railroad. It opened in July 208 and was expanded in 2010. Up to 35 trains can be run simultaneously on seven multi-tiered tracks. The railroad is set among the gardens and in two sections includes trains, buildings, vehicles and people from the 1890s and 1950s. There is more than 1,600 feet of track, 50 buildings, 200 people, a 26-foot bridge,a double helix tornado tower, a river with waterfalls, and five level twin coaster. The track is stainless steel and there are both steam and diesel locomotives. The entire railroad is built at a 1:29 ratio.

Then I continued walking around the castle and took pictures of some of the incredible flowers.

Do you think this brass sculpture was inspired by flowers?

Charlevoix Castle Farms Art Show -- Post 2

Today I will post the second of three sets of photos from the art show I was at over the weekend in Charlevoix. I would like to thank all of you who have given me positive feedback about this photo journal blog. I enjoy taking the photos and sharing them with you, but the feedback helps to keep me going!

Before I opened my booth on Sunday morning, I walked around the grounds.  There were several different areas, including this hedge-contained butterfly garden.

In the back of the castle there was an area set up for a wedding, that must have taken place the previous evening.  What a great place for a wedding!

In a couple of places there were large chess sets.

I love the copper statutes guarding one of the main entrances.

Some of the art booths were quite colorful!

The aisle ways at this show accommodate a lot of people!

Charlevoix Castle Farms Art Show -- Post 1

Last Friday my son and I headed south out of Grand Marais first thing in the morning. I dropped Kevin off in Traverse City, where he rented a car to head down state to visit with relatives before returning to Vermont. Of course, we had to drive over Big Mac to get to the lower peninsula.

The art fair was held at Castle Farms in Charlevoix.

I only took 10 minutes to walk around the grounds immediately next to the area of the art show.  A fountain greets visitors....

Beyond the fountain is a pond.

This white duck made for interesting pictures.

Here are a few shots of my booth, which was located in the east wing.

Camping East of Grand Marais and Agates

I have been so busy working the museum and getting ready for my art show on Labor Day weekend, that I decided to celebrate the last day of August and go camping with my son, Kevin. We headed east of town and camped on state land located on the bluff above the Lake Superior beach. Although it was cloudy and sprinkled a few times, the lake was calm and the temperatures were pleasant. We cooked over the camp fire and just relaxed.

I woke up and just missed the sun rise.  I was still able to get some nice shots.

I received this photo of an agate found by Rich near the Two Hearted River.