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.
Fractals and stars are two of the most beautiful and complicated-looking classes of geometric objects out there. We're going to explore these objects and how to carve them on a pumpkin. Unlike the last one on carving polyhedral pumpkins, where we used the entire pumpkin to carve a 3 dimensional shape, the pumkin carving in this post will involve two-dimensional images on a small part of the pumpkin's surface.
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.
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.
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.
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...
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...
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.
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:
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?
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".
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.
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.
This is how my version of an origami Christmas tree turned out based on the instructions I posted awhile back. Cory also made a version from white glossy paper, which looks great. I opted for the green and brown look, but it wasn't easy.
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.
I spent the holiday weekend becoming fluent in the basics of modular origami. With practice, you can churn out the below models surprisingly quickly.
Richard Sweeney is an incredible artist whose body of work consists mainly of sculptures made from paper. His art is often related to origami, and much of his work is related to geometrical forms. I personally really love his modular forms in paper. Many of them are based off of the platonic solids, which have been discussed in previous posts this week. Below are a small number of his sculptures, which are very geometric in nature.
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.
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.
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.
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 ...
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.
Here's my version of his icosahedron: I colored it in this one so that you can see the pentagonal faces of a dodecahedron:
Tom Friedman is one of my favorite artists. He's got a great sense of humor, and his work is meticulous and beautiful. He forays into Math Art, and from a partisan perspective, he seems to be inspired by mathematics, but the end results are more of a whimsical twist than a mathematically "correct" execution. But I could be totally wrong. Comment below and fill me in.
It's Monday, which means once again, 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 sliceform model of a hyperbolic paraboloid.
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.
These boxes are inspired by a comment from Imaatfal Avidya on a corkboard post on Platonic polyhedra from sonobe units. Imaatfal was commenting about how the cube and octahedron are related to each other.
This is just too cool. As soon as I saw this, I thought, "Math Craft!"
Torus knots are beautiful knots formed by wrapping a line around a torus and tying the ends together to form a loop. The resulting knot has a star-like appearance when viewed from above. The 36 examples with the least number of crossings can be seen at the Knot Atlas's page on torus knots.
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.
Thanksgiving. It's sadly over. But happily replaced by the Christmas season!
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.
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.
Much more complex than I had to make it- that's why I posted it. I think it looks cool...
After becoming addicted to basic sonobe modular origami, I decided to make ornaments for relatives as Christmas gifts. I tried using fancy paper from stores like Paper Source, and cutting it to proper origami size, but I could never get the tight folds I wanted with non-traditional, non-origami paper. I ended up using this metallic origami paper that folds beautifully, and I'm pretty happy with the tiny models I ended up with. Forgive these pictures (iPhone/Instagram), I don't have my regular...
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.
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...
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.
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.