The hyperboloid of one sheet is a fascinating shape that turns up in many places. It was therefore a great example to take for a test of thearender which I recently purchased. This shows off its double ruled nature:
Hyperboloid lighting
November 6, 2011
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Posted by gelada
CAMel
September 29, 2011
CAMel is a project to develop Rhino Grasshopper components for CAM (Computer Aided Manufacturing). Hence the silly name. It is very much work in progress, but if you are brave enough, here is a first release. All images and the video on this page are of a machine running GCode generated by CAMel.
Download CAMel 0.12
Download Rhino file (only needed if you want to see the example setup).
At present the components are just clusters with scripted components written within Grasshopper. The next major step will be to convert this into a proper grasshopper plug-in. This release has a grasshopper component with some documentation (there is a little more inside the clusters). All the code is CC-BY-SA licensed, and of course it should be noted that this is very much “use at your own risk”! My belief is that Grasshopper provides a natural environment to experiment with creating your own toolpaths. The purpose of CAMel is to make this process as easy as possible by giving the tools to convert simple toolpath ideas into usable paths and then exporting the GCode that will drive a machine.
The main components are as follows:
- GCode Writer: Converts lists of points, vectors and feed rates into GCode for the machine.
- GCode Checker: Reads GCode and checks and optimises it. For example a 5-axis machine can usually obtain any tool angle in two different ways. This selects the better angle. It will also give warnings of undesirable behavior in the GCode.
- Surfacing: Creates a toolpath to cut an arbitrary surface (very rough version, designed to test others)
- Swarf cutting: Creates toolpath from information about the movement of the tip of the tool and the point in which the tool enters the surface. For a 5-axis machine these paths can be quite different.
The code is currently set up for a single machine, I am happy to try to help adapt it to other machines (other commitments allowing) so get in touch if you are interested.
These components and code were developed with Santiago R Perez 21st Century Chair of Integrated Practice at the Fay Jones School of Architecture, University of Arkansas. I work in the Mathematics Department at the same university.
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Posted by gelada
Hexayurt dome details and models
August 7, 2011People are now starting to build my tri-dome and quad-dome versions of the hexayurt, so it is time to give some of the technical details. To start, however, here is an application of the intermediate value theorem!
When I started working on the details for the tri-dome I realised I had made a bad assumption (thinking that the form was geometrically pure). This means that some of the details in my original write up were wrong. All a little embarrassing. Ironically, I might have missed a form that does actually work, had I not made the bad assumption. The shape, like the hexayurt, starts with a hexagonal based pyramid. In a traditional hexayurt this lies on top of a hexagon of vertical walls. Instead of this we attach a square to three of the edges and the classic hexayurt triangle (isocoles triangle with base and height the same length) to the other three. We can look at what happens as the pyramid is moved away from the ground, while the edges of the shapes rest on it:
This does not give a great building; there are holes. The holes are triangles and two of the sides have a fixed length. The final edge changes length, starting long, and ending short. We know we can fill the holes with classic hexayurt triangles. Two of the edges are the right length we just need the third. The length changes smoothly as we raise the roof, and starts shorter and ends longer than we want. Here we can apply the intermediate value theorem, so the correct length must be passed. As a mathematician I would stop there, the system works; however people are building the things…
So to calculate the correct angle for the square sides of the model we can look vertically down, calling the angle of the square face θ, (and assuming that the boards we are using are 8′ by 4′) needing as the classic maths problem asks to “find x”.
In this case
we want 
Solving the quadratic:
Which gives an angle of about 49°, and the height of the roof (assuming 4′x8′ panels) is 
Technical details for TriDome: angles to nearest half degree, lengths to nearest inch (assuming 4'x8' panels). On the left the angles between faces and point heights, on the right lengths and angles of the base.
Technical details for QuadDome: angles to nearest half degree, lengths to nearest inch (assuming 4'x8' panels). On the left the angles between faces and point heights, on the right lengths and angles of the base.
Finally here are the hexayurt models (rhino 3dm and vrml formats) of the hexayurt, H13, TriDome, QuadDome, plus a couple of others, including a very large one.
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Posted by gelada
Why not knot wire?
May 23, 2011
I have been thinking quite a bit recently about ideas of knotting and weaving. There will probably be another post on the theme soon. As a mathematician it brought me straight back to Knot theory, I love looking at the strange images that appear on the blackboards in the lectures and offices of topolgists, many of which contain knots. This video lecture from Elvis Zap is a classic example (even if you cannot follow, just sit back and enjoy the drawing!). Not to forget the beautiful uses knotted designs have been put to outside of mathematics.
At some point during this I needed something made out of metal, and decided to bend some wire into a trefoil. It was satisfying, so I though I would look online to see if I could find collections of physically made knots. These were surprsingly hard to find. There were plenty of examples to be found (even the Museum of Mathematics‘s famous knotted bagel), but I could not find any systematic collections. So I decided to make my own, using this knot zoo for reference. Here are the knots that can be drawn with seven crossings or less, using Conway’s tangle notation:
It was great fun making the knots and I encourage anyone who studies them, even idly, to have a go. I felt the knots themselves come alive in my head as I made them. I started to think how the knots could be put together out of sections of twists, further study of this lead me to tangles and Conway’s notation. You might notice that this came late as the written labels on the knots are the more commonly used Alexander–Briggs notation. That is a lot less satisfying as after the number of crossings the numbers to not refer to the properties of the knots.
In addition making sure that the wire holds naturally in shape without touching itself is great for 3d intuition. One thing that struck me as I started to bend the wire was how 2 dimensional most knot images are. The crossing number is a classic example of this as it is a 2d not a 3d property. There are, of course, good reasons for this both in design and exposition, but it was interesting feeling how the knots changed as you allow to move more freely. Of course this had some issues when I came to present the knots here, of course in 2d (I hope I managed to get all the pictures so spare crossings are easy to remove!). A video might work slightly better:
2 Comments | Art, Mathematics | Tagged: Art, geometry, John Conway, Knot Theory, Mathematics, models, Topology, visual maths, Weaving | Permalink
Posted by gelada
