Math Craft: Hot

We've all made them. I remember making hundreds of paper snowflakes when I was in elementary school. You take a piece of paper and fold it in half, then fold it in half again. You now have a piece that is one fourth the size of the original. Now you fold it in half diagonally. You then cut slices out of the edges of the paper, and unfold to find that you have created a snowflake. The resulting snowflake has four lines of symmetry and looks something like this:

I have a lot more images at

After becoming addicted to basic sonobe modular origami, I decided to make ornaments for relatives as Christmas gifts.

Computer Science Professor Francesco De Comité has a fantastic gallery of mathematical images on Flickr. As part of this collection, he has a few hundred images of real or rendered polyhedra made out of paper or playing cards which he calls "slide togethers." These are constructed by making cuts and then sliding one component into the other, creating a shape without using any glue. He constructed the entire set of the platonic solids—the cards form their edges—which can be seen in the image...

These drawings were made with Google SketchUp. There is a dodecahedral model, icosahedral model, and a third I don't know the name of, made of rhombic faces obtained by connecting vertices of the other two. The final image is all three models together. I'll use a ShopBot CNC router to cut out the pieces this week.

This three dimensional Sierpinski tetrahedral structure was created with a lot of help from my Year 10, 12 and 13 classes.

Here's a great excuse to play with your food—and learn some math while you're at it. We've all seen a hexaflexagon folded out of paper, but how about a burrito?

You may remember string art from your elementary school days. If so, it probably makes you think of the 2D geometrical designs that took every ounce of patience you had as a kid. Or those laboriouscurve stitchdrawings, which string art was actually birthed from. But thanks to some innovative modern artists, string art has gotten a lot more interesting. Here are some of the most creative applications so far.

This is a new line of work I've started - inspired by string art of Archimedean Lines, these are 3-dimensional sculptures made using Electro-Luminescent Wire weaved around a clear acrylic frame. They hang on the wall, but each has a sense of depth so their look alters from different angles.

I'm a noob to this so I apologize for the ignorance.

Mario Marín has made an incredible collection of models and sculptures based on polyhedra, often using everyday and readily available items. The site is in Spanish, but click on the links on the left and there are plenty of photographs, and more can be seen in Mario's blog.

Hey!

Below, my construction of a Platonic Solid made from playing cards. To make your own, templates can be found at George Hart's site; there are also full step-by-step instructions here.

Oobject put together a neat compilation of the famous telephone inventor's love for tetrahedrons. Scroll down to see his collection of pyramids, building towers, buildings, boats, kites and planes—all made completely out of tiny tetrahedrons. Amazing.

Erik Demaine is a Professor of Electronic Engineering and Comp Sci at MI, but he is also an origami folder who has had work displayed at the Museum of Modern Art in NYC. He makes some beautiful models and intricate puzzles, but in my opinion the really inspirational work is the curved creased models.

Last Thursday's post demonstrated how to Make Yin-Yang Pillow boxes, which were based on equilateral triangles and squares. The units for making these boxes were created by Phillip Chapman-Bell, who runs an amazing origami blog and has a spectacular flickr photostream. Using these units, you can make also make 4 of the 5 platonic solids. I made an additional template based on the regular pentagon so that the dodecahedron can be built completing the set.

This is probably the least "Mathy" thing I will ever post. In my opinion, it's impossible to have architecture that isn't mathematical in some sense, so I am posting it anyway. Two years ago, I made a papercraft version of a cathedral in Christchurch New Zealand (It was severely damaged in an earthquake earlier this year) and cut holes for all of the windows and lit it with LED lights. I gave it to my Mom as a Christmas gift. I thought it made for a pretty amazing "Christmas Village" piece.

Looking into mathematical quilting, I came across a community of mathematical knitters. Check out Dr. Sarah-Marie Belcastro's (research associate at Smith college and lecturer at U Mass Amherst) mathematical knitting resource page.

It's once again Monday, which means it's time to highlight some of the most recent community submissions posted to the Math Craft corkboard.  Since two of these posts were on polyhedral versions of M.C. Escher's tessellations, I thought we'd take a look at building a simple tessellated cube based off of imitations of his imagery.

I was browsing Reddit.com yesterday and noticed this post.  User guyanonymous (yes I am really crediting him regardless of his name!) had posted up this string-art picture which has parabolic curves created from straight lines and gave me permission to post it up here on the corkboard.  I love the repeating "flower" pattern.

Below, polyhedron animation test #1. The model was folded using Cory Poole's modular origami tutorial.

Vladimir Bulatov makes sculptures of fantastic variations on polyhedra and other geometric objects. His site is full of incredible metal, glass, and wooden geometric sculptures, including a full section on pendants and bracelets. Here are just a dozen or so of the hundreds of beautiful objects that he has produced.

Here's a Math Craft project that takes less than 20 minutes, has an attractive, practical result, and is at least a little mind-blowing due to folding along curves.

Imatfaal's awesome post on Escher's tessellations on Polyhedra reminded me of some ornaments I made this summer.  I made some of Escher's square tessellations onto cubes and then reprojected them onto spheres.  I actually used a 60 sided Deltoidal hexecontahedron since that net is fairly easy to fold and looks pretty round.

Snow Angels:

I spent a little bit more time making 6 sided Kirigami Snowflakes using the method of this post.  I'm really happy with how all of these turned out.  I'd love to see other people post up some snowflakes.  They're easy and a lot of fun.  And I could use some more inspiration!

I've already posted a brief roundup of interesting models folded by Michal Kosmulski, expert orgami-ist and IT director at NetSprint. However, I didn't include my favorite model, because I felt it deserved its own post. Kosmulski folded an elaborate and large Sierpinski tetrahedron, which he deems "level 3" in difficulty. (Translation: hard). It is constructed with 128 modules and 126 links, based on Nick Robinson's trimodule.

If you take two flat mirrors and place them front to back and look at them, you can see an infinite number of reflections. While this is a self-replicating pattern and can be somewhat mesmerizing, it isn't anywhere as interesting as looking at the chaotic scattering of light that can occur between 3 or 4 spheres.

I wish there was more information about this impressively massive sonobe model, but all I can glean is that it appears to have been made by Imogen Warren, and was posted by Room 3. So awesome.