I've always been interested in maps and different projections and how they make such a difference in the way the world looks when a sphere is projected onto a flat surface.  Yesterday my wife sent me this link to an issue of Life Magazine from March 1, 1943.  On page 41 of this issue they discuss a new (at the time) projection from Buckminster Fuller, a famous inventor who is now most well-known for his geodesic spheres and houses made out of them.  This projection is called The Dymaxion World, and can be folded up to form a geometric object known as a Cubeoctahedron.  While the point of the projection is to have a flat-map with desirable properties such as low distortion and the ability to have all of the landmasses contiguous, I just think it looks cool folded up to form a bizzare globe.   Note: Fuller eventually decided on another polyhedron, the icosahedron, to use for The Dymaxion World.  It is a much "rounder" shape that actually looks like a globe. I've made those before and I don't think they look nearly as cool.

The images of the templates at the very bottom of this post, obtained from the original article, are high enough resolution to make about a 10 inch globe on standard paper if you click on the pictures and download the higher resolution version.  I used thick cardstock, and a tacky glue to construct mine.  All you have to do is glue Tab A to Tab A, Tab B to Tab B, Tab C to Tab C...

Here's a picture of the one I built:

   

 

 

I've been a big fan of M.C. Escher's Art since I was a little kid. His impossible figures drawings are mindblowing and his tesselations are fantastic. In the last couple of months, I've actually done a couple of projects that involve his work.

The first I did on Thanksgiving break. I had become very interested in making jigsaw puzzles on my scroll saw and after doing a couple of interesting puzzles that I made by printing pictures I found online and gluing them to the piece to be cut, I decided that Escher's tesselations would make fantastic puzzles. The one I settled on making is called Mosaic II and is a very interesting piece that is made from 40 different animals that fit perfectly together. It is colored in such a way that dark pieces and light pieces only touch the other at "corners." For my version this isn't quite true because I left out the tongue of the snake because I didn't want it to be easy to break. It's a fun puzzle that takes people between a half hour and an hour if they have no prior knowledge of what the picture looks like.

The original Escher:




Initial outline cut on my scroll saw (The whole piece is about the size of a 8.5 x 11 sheet of paper):



Well I've never tried wood burning before...but hey it looks like fun. I used wood burning tips on my soldering iron.



A few hours later!




Even later....Wood burning is finished!




The Finished Product after about sanding, staining, and coating with polyeurethane. The total project was about 10 hours worth of work:






The second project I did in the morning of one of my days off during Christmas break. It's a paper version of Escher's Relativity that is cut and folded out of one piece of paper and stays together using only tabs and no glue. The technique is called Origamic Architecture and at some point I will probably put up a post on all the different pieces I've copied using this technique. I found the diagram for Relativity on a flikr and you can download it here. Note: my version is mirrored from both what the diagram suggests and the original Escher piece. I "decided" to fold it the other direction.

The original Escher drawing:





The piece I made by downloading the diagram, printing it on a piece of paper, and then cutting and folding as indicated.



What Escher's would look like if it was mirrored like mine:

A year and a half ago at Christmas time I started making a series of modular origami structures that are known as polypolyhedra since they are made up of an "orderly tangle" of other polygons or polyhedra. They are only held together by frictional locks (ok, ok, I did use tape a couple of times...but that's just because so many people have handled them!) and are all made up of a single type of modular origami unit. Since it was Christmas time, I decided to build them out of large (12 inch) metallic cardstock. Normally, these are hanging up in my classroom, but I took them down to use my classroom for a location to film part of a webvideo series that I'm working on with my brother, a cousin and a bunch of my students.

To learn more about these geometric figures check out: http://www.langorigami.com/science/polypolyhedra/polypolyhedra.php4 and while you're at it check out more of Dr. Robert Lang's Origami. By the way he's a Ph.D. Physicist!

The five intersecting tetrahedra is a fun little puzzle to solve. There are probably millions of ways to put the object together but only a few that work without causing the paper to bend a lot! I think that I could probably build and assemble it in a couple of hours now...but the first time I did it, it probably took about 10!


4 intersecting triangles





6 intersecting squares





6 intersecting pentagons





five intersecting tetrahedra







All together now

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.

This post combines two of my hobbies, making things out of paper (papercrafting) and exploring and building fractals. For those of you that are not familiar with fractals, they are objects that have repeating patterns at different sizes. In other words when you zoom in on a part of the object it looks like a smaller copy of the entire object. Here are 3 different fractals that I have built out of paper.

Sierpinski Tetrahedron (Tetrix). This is the 3D analog of the Sierpinski Triangle. It is formed by taking 4 tetrahedrons and gluing them together vertex to vertex forming a single hollow tetrahedon. You then repeat with 4 or these units, and then repeat with 4 of those units and so on forever. This tetrahedron is iteration 3 with 64 individual tetrahedrons and is about 9 inches tall. One interesting thing about this fractal is that if you look at it from the correct angle it looks like the Sierpinski Triangle. Another is that if you look at it from 2 certain angles it actually looks like a completely filled in 2D surface. (I actually want to put an image on it so that you only see it from the proper angle)


























3D fractal tree. This is a 3D fractal tree of my own design. Every branch is a triangular prism and each branches into 3 smaller branches at a defined angle. It has 4 stages of branches meaning there are 81 of the smallest branches, 27 of the next smallest, 9 of the next, 3 of the next, and finally 1 trunk. If it went another iteration then it would intersect itself...meaning I didn't design it properly.













Koch Surface. This is a 2D analog of a variation of a 1D Koch curve or Koch Snowflake. Each main box has 5 boxes that are 1/3 the length, width, and height placed in the center of each face. It is also surrounded by 8 other boxes 1/3 the length, width, and height. This process is then repeated. The one I made has an initial box that is about 4 inches on a side, 5 more boxes 1/3 this size (I didn't surround it by the 8 other boxes), 5x13=65 1/9 this size and finally 65x13= 845 1/27 this size. The smalles boxes are actually made from 1/8 bass wood and painted.