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.

Earlier experiments with swarf cutting.

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Posted by gelada

Hexayurt dome details and models

August 7, 2011

People 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

x = 4 \sqrt{4 \cos(\theta)^2+1+2\sqrt{3}\cos(\theta)},

we want x = 4\sqrt{5} so:

4\sqrt{5} = 4 \sqrt{4 \cos(\theta)^2+1+2\sqrt{3}\cos(\theta)}

5 = 4 \cos(\theta)^2+1+2\sqrt{3}\cos(\theta)

0 = 2 \cos(\theta)^2 + \sqrt{3}\cos(\theta) - 2

Solving the quadratic:

\cos(\theta) = \frac{-\sqrt{3} \pm \sqrt{19}}{4}

Which gives an angle of about 49°, and the height of the roof (assuming 4′x8′ panels) is 8 \sin(\theta), just over 6′ at the edge and 10′ in the centre. We can use these, and useful facts about general tetrahedra to calculate all the angles between faces by using the lengths of their edges. If you want to follow the details yourself, you need to add vectors to get some of the edge lengths, then use the Cayley-Menger determinant to find the volume of the tetrahedron, and then the generalised Sine rule to (halfway down this page) to give the angle.

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

Islamic Geometry

December 20, 2010

Marc Pelletier is a geometric artist, one of the visionaries behind the amazing Zometool system and the designer and builder of 120-cell models including one given to John Conway at Princeton  and one at the Fields institute (given on the occasion of Coxeter’s 95th birthday). More recently he has been working on Islamic tiling patterns, drawing on the work of Jay Bonner, an expert on the geometric art of the Middle East. Marc has created an elegant and general system to generate such tilings with fine control over the symmetry and structures that come out. Here are a couple of sample designs. Marc, Chaim Goodman-Strauss and I were discussing the methods and how they can be put into a mathematical framework.

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Posted by gelada

Unscheduled Post: The Silver Ratio

May 20, 2009

John Cook on The Endeavour has just mentioned the wonderful silver ratio. As this is probably my favourite number I can’t resist the chance to put up some pictures. The silver ratio: \Psi = 1+\sqrt{2}, is as John mentions the value of the continued fraction with just 2′s, it is also the larger solution of the equation \Psi^2 - 2 \Psi - 1 = 0. This goes directly into its geometric interpretations, as the diameter of an octagon and the size of a rectangle that gives a smaller version of itself when you remove two squares:

Oct_and_rect

Geometric interpretations of the Silver Ratio.

In terms of tiling the golden ratio of course has the Penrose tiling, with its five fold rotational symmetry, the silver ratio plays the same role for the Ammann-Beenker tiling, with 8-fold rotational symmetry:

Ammann_Scaling_bb

Version of my Ammann-scaling artwork

In fact if you find things with 8-fold rotation (Islamic art for example) the silver ratio will be lurking around. I have a personal theory that the silver ratio was as much in Christopher Wren‘s work as the golden. I have not studied it in depth, the floor under the great dome of St. Paul’s has an giant octagon. Anyone know any good studies that might mention this, it would be good to have evidence!

Silver_circles

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Posted by gelada

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