In graph theory a graph is a collection of nodes where connections between the two are shown by lines. This can easily be applied to polygons in order to draw non convex figures. You start off with a regular convex polygon and then to draw the regular star polygons you connect each vertex to another by skipping a certain number of nodes.

If you connect up each node to every other node you get what is called a simple graph which is actually the collection of every regular star polygon drawn on top of each other. Here is that graph for a 16-gon. I then filed in the resulting tiling so that no tile is ever touching another filled in tile along an edge. In the first one I then shaded all of those tiles in ROYGBIV color order. In the next two I took the original image and shaded in what ever was left white in ROYGBIV and VIBGYOR color order.

I finally got smart and decided to make some polygonal tilings that were completely made using a computer instead of doing hand work. You can download a cool free java program called geogebra that allows you to do a bunch of cool dynamic geometry. I made each shape once using geogebra and then copied pasted and rotated to form the tesselation in photoshop. I like the second one which really shows the strong connections between triangles, hexagons and circles.

Squares:



Equilateral triangles:

I decided to try playing with different angles for making hyperbolas (as in the previous post) and decided to make the angles form regular polygons. The red objects are hexagons, the yellow triangles, and the blue are squares. There are 3 regular and 8 semi-regular tilings of the plane made out of these regular polygons. This is one of them.

The art teacher at my school has students make geometric drawings in which curved shapes are formed by having lots of little lines cross each other. In differential calculus you learn that all smooth curves can be formed using these so called tangent lines. As you get more and more lines the curves become smoother and in the limit as the number of lines goes to infinity you can form any smooth curve.

I decided to combine this idea with that of a fractal where if you zoom into the shape you see a similar one. Below you will see the result. It was all done using a ruler and pencils and colored markers. I then scanned it and cloned it into 4 to make the final piece. Everything is using a 1/16 inch grid and the larges piece is 8 inches meaning it has 128 lines on each side. I figure I colored in somewhere around 5,000 to 10,000 squares all together. There are 6 smaller iterations inside it and each is half the size of the one before it. The smallest is 1/4 inch by 1/4 inch. I colored them in the rainbow sequence (ROYGBIV) since there were 7 of them. You really need to click on these to see the details.

The main fractal image.



Copied Scanned and tiled

Today is the vernal equinox. I'm so happy that it is finally springtime. Here was the view from my backyard this morning. Unfortunately, I really should have cleaned the camera lens after using it to take pictures for our cooking blog.

I finally finished a papercraft project I've been working on for about a month. It's made up of parts of 29 sheets of cardstock. It is a model of a cathedral that is being built in Barcelona called La Sagrada Familia (The sacred family.) The designer is Antoni Gaudi who built a lot of cool and crazy looking buildings. He started working on this project in 1883 and worked on it until his death in 1926. It is currently slated to be finished in 2020. The completed cathedral will have 18 towers,12 for each of the apostles, 4 for the messengers, 1 for Mary, and the tallest at about 560 feet for Jesus Christ. My model is only about a foot tall and doesn't do the real building justice...but it still is amazingly detailed and is by far the hardest papercrafting project I've embarked on that doesn't involve moving pieces (still haven't finished up the watt governor for the steam engine that has all moving parts.)

Everything needed to create the model (except for materials of course!) can be downloaded here:
http://cp.c-ij.com/en/contents/3154/sagrada_familia/index.html

Here's some pictures of the model, and a video of the actual cathedral as it is now.





















Video of actual cathedral.