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 b...
Last post, the Sonobe unit was introduced as a way to use multiple copies of a simply folded piece of paper to make geometric objects. In this post, we are going to explore that concept further by making two more geometric models. The first is the truncated icosahedron, which is a common stitching pattern for a soccer ball. The second was supposed to be the pentakis dodecahedron, but through systematic errors last night, I actually built a different model based off of the rhombic triacontahed...
It's Monday, which means once again, it's time to highlight some of the recent community submissions posted to the Math Craft corkboard. I also thought that we'd try and create something known as an "Orderly Tangle" or "Polylink".
In honor of the new Astronomy World, I thought we should look at a few planetary icosahedrons. The icosahedron is the most round of the Platonic solids with twenty faces, thus has the smallest dihedral angles. This allows it to unfold into a flat map with a reasonably acceptable amount of distortion. In fact, Buckminster Fuller tried to popularize the polyhedral globe/map concept with his Dymaxion Map.
Math Craft Monday: Community Submissions (Plus How to Make a Modular Origami Intersecting Triangles Sculpture)
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. I also thought we'd take a look at building a model that has appeared in numerous posts. It's the simplest of the intersecting plane modular origami sculptures: The WXYZ Intersecting Planes model.
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.
Reuben Margolin builds large scale kinetic sculptures based off of mechanical waves. Some of his sculptures contain hundreds of pulleys all working in harmony with each other to create sinusoidal waves and their resulting interference patterns. He designs them all on paper and does all of the complicated trigonometric calculations by hand. Everything is mechanical; there are no electronic controllers.
One of my favorite simple projects is building two circle wobblers. I love how such a simple object amazes with its motion. The two circle wobbler is an object made out of two circles connected to each other in such a way that the center of mass of the object doesn't move up or down as it rolls. This means that it will roll very easily down a slight incline. It will also roll for a significant distance on a level surface if you start it by giving it a small push or even by blowing on it!
Modular origami is a technique that can be used to build some pretty interesting and impressive models of mathematical objects. In modular origami, you combine multiple units folded from single pieces of paper into more complicated forms. The Sonobe unit is a simple example unit from modular origami that is both easy to fold and compatible for constructing a large variety of models. Below are a few models that are easy to make using this unit.
Welcome to Math Craft World! This community is dedicated to the exploration of mathematically inspired art and architecture through projects, community submissions, and inspirational posts related to the topic at hand. Every week, there will be approximately four posts according to the following schedule:
Curve stitching is a form of string art where smooth curves are created through the use of straight lines. It is taught in many Junior High and High School art classes. I discovered it when my math students started showing me the geometric art they had created.
Using only a circle and straight lines, it's possible to create various aesthetic curves that combine both art and mathematics. The geometry behind the concentric circle, ellipse, and cardioid dates back centuries and is easily found in the world around us. From an archery target to an apple, can you name these geometric shapes?
In Monday's post, we created a sliceform model of a hyperbolic paraboloid. In today's post, we will create a similar model using skewers. The hyperbolic paraboloid is a ruled surface, which means that you can create it using only straight lines even though it is curved. In fact, the hyperbolic paraboloid is doubly ruled and is one of only three curved surfaces than can be created using two distinct lines passing through each point. The others are the hyperboloid and the flat plane.
The "slide-together" paper construction method is a fun and satisfying way to build 3D geometric objects. It only requires paper, scissors or an exacto knife, and some patience.
Math Craft admin Cory Poole provided quite a few recipes for sonobe models in his blog, and I followed one to make the pentakis dodecahedron here.
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 laborious curve stitch drawings, 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.
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.
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: If you fold it in half diagonall...
Math Craft admin Cory Poole posted instructions on How to Make a Cube, Octahedron & Icosahedron from Sonobe Units, plus some great complex models in his article, How to Make a Truncated Icosahedron, Pentakis Dodecahedron & More. These models use the standard sonobe unit and a coloured variant.
This three dimensional Sierpinski tetrahedral structure was created with a lot of help from my Year 10, 12 and 13 classes. It is inspired by the Sierpinski triangle fractal.
Compound of two cubes with a Minecraft theme.
Back in August, Scientific American posted a slideshow fitting for Math Craft. Click through to check out a slideshow depicting beauty found in mathematical structures—including a beautiful knot theory chart befitting of this week's project.
In mathematics, a knot is a closed circle in a three-dimensional space that crosses itself multiple times. Since it is closed, it has no ends to tie, meaning you can't actually create such a knot. However, if you tie the ends together after you create a knot in the standard way, you will have something that is close to the mathematical description. In this post, we will explore the creation of mathematical knot sculputures using copper tubing and solid solder wire.
This is a zigzag tensegrity based on a small stellated dodecahedron. There are string pentagons on the outside of the model where the vertices have opened. It is made of thirty units, consisting of a barbecue stick pair with a loop of elastic. The stick pairs are all "floating", and weave through the model without contacting any other stick pairs. It is quite tricky to assemble, but can be done entirely by hand.
I got hooked on origami sometime after Math Craft admin Cory Poole posted instructions for creating modular origami, but I had to take a break to finish a quilt I've been working on for a while now. It's my first quilt, and very simple in its construction (straight up squares, that's about it), but it got me thinking about the simple geometry and how far you could take the design to reflect complex geometries. Below are a few cool examples I found online.
If you thought the last post on Two Circle Wobblers was wild, then wait until you see what happens when you build wobblers out of two half circles or two ellipses. In both of these cases, the center of gravity still remains constant in the vertical direction, allowing them to roll down the slightest of inclines or even travel a significant distance on a level surface if given a push or even when blown on.
You can do some pretty cool stuff with the golden ratio. The image above is made from taking each quarter-circle in the golden spiral and expanding it into a full circle. In the second image, the spiral and the golden rectangles are overlaid on the the first image, showing how it works.
Download the Software Go to the Antiprism downloads page. Download and install Antiprism 0.20.
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. In Erik's own words describing the above models: "Each piece in this series connects together multiple circular pieces of paper (between two and three full circles) to make a large circular ramp ...
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.
I wondered how silly you could get with sonobe, and had a bash at a buckyball, which is a fullerene (technically a truncated isocahedron; you can see a simple model here). It's twelve pentagons—each surrounded by 5 hexagons (20 in total)—making a football shape in England or a soccer ball shape in the USA.
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.
After Cory Poole posted some great Escher snowflakes, and Cerek Tunca had the great idea of using it as a base for a tetrahedron, well, I just had to give it a go. I will post a few more pictures and variants later (I think this was what Cerek was envisaging—if not let me know!)
This week's post on creating 6-sided Kirigami Snowflakes got me interested in seeing whether I could use the process to create tessellation snowflakes using the method. I still haven't succeeded, but I did decide to make some ornaments based off a few of the tessellations by M.C. Escher that have a 6 sided symmetry.
Here's my Sonobe Jasmine Dodecahedron built from Imatfaal's instructions.
My first attempt with the sonobe unit! I'm now addicted. I'll be spending the holiday folding. Instructions here.
Scrabble is definitely my pastime addiction of choice, but it's not the only game I frequent. I'm a big chess fan, crossword lover, and hooked on puzzles—any kind of puzzles. Logic puzzles, sudoko, and... the Rubik's Cube.
Since today (11/11/11) is the last 6 digit binary date this century, I thought we should look at some kinetic binary calculators.
It's another Monday, which means once again, it's time to highlight some of the recent community submissions posted to the Math Craft corkboard. Additionally, I thought we'd take a look at the Mobius Strip.