Polynomials in Wood

December 4, 2011

What has 1-x/2-6x^2+11x^3-7x^4+3/2x^5 got to do with wood? Like you until a few days ago I would have said “Probably nothing” then I came across this chart:

Where it relates to how the bending strength of wood changes depending on the number of knots. From this lovely book, that I found at the local second hand book shop during Samuel Hansen’s recent visit to Fayetteville:

Which, is full of other equations and models, such as this one:

N = \frac{PQ}{P sin^n \theta + Q cos^n \theta}

which is then explored for several values of n.

Some of the tables caught my eye just for beautiful way that they present information:

Finally, its not just equations, there is also a collection of patterns, along with the intriguing chapter on Structural Design of Sandwich Construction (probably not what I am thinking about):

All this points out to me, once again how mathematics can be a powerful tool to help study anything. I know that when it comes down to it this is really just the well established link between mathematics and engineering, but, as a material, wood is so much more accessible and visceral than, say, concrete. For some a book on wood might even answer the eternal question of “How am I going to use this?” but it does at least show that quintic polynomials really do come up in real situations!

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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

Arrange whatever pieces come your way

May 14, 2011

(with apologies to Virginia Wolff)

A simple, classic puzzle is to give two shapes and ask if there is a way to cut one up so the pieces can be rearranged into the other. This game might seem to become silly if both shapes are the same;  if we insist that the new arrangement must be different the game becomes interesting again. Think about it, can you come up with ways to cut up a square so that the pieces can be formed into two different squares? Here is an example, not with a square, but with a rhombus:Having the same shape has an advantage. Think about the letter p below, it is part of the blue trapezium, when we rearrange the tiles the p moves with the shape. As the two shapes are the same we can think of this new p within the original rhomb. We can now repeat the process as many times as we want. In this case, it might be a little unsatisfying, however, as the next step for our p would cut it into two different pieces, as it lies on the edge. So where is it safe to put a p so that it will never get cut up? To answer this we have to follow the cutting lines, and a beautiful pattern emerges:The p would be safe within any of the pentagons, but if it crosses any of the edes it will, eventually be cut apart.

Puzzle: Can you work out the difference between the green and the blue pentagons? (Hint: it relates to the dotted and solid lines in the earlier pictures).

Studying what happens when we can move points or objects around in a space (in this case moving p around a rhomb) is studied in a part of mathematics called Dynamical systems the particular example here is called a Piecewise Isometry  (see this paper for a more formal account of their history and study). I have studied these systems myself, and recently submitted a paper looking at the behaviour and number theory that occurs within the pentagon generating system shown above (take a look! It has lots of pictures as well as more formal mathematics).

As you might have guessed from my preoccupations part of my interest in these systems is the pretty images that they produce; this system is particularly rich. This leads to the image at the top. You can take any rhombus and cut it up in a similar way. Take any rhomb (as shown below) and rotate until the side of the rhomb lines up with the top. This will leave a triangle and a trapezium that can be moved back on top of the original rhomb:Additionally this gives a system where the rotation on the two parts is the same, just around different points. You have to be a little careful, but you can use this to give a system for any angles. For any of these systems we can ask the question: Where is it safe to write p? Every angle gives a different pattern, and tiny changes in the angle leads to large changes in the pattern, however the patterns do relate to one another in some ways, as you can see in this video:

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Posted by gelada

I find myself looking for a job…

January 22, 2011

I have a weird collection of skills. Mathematics, talking about mathematics, art, making…

I am certainly missing opportunities, maybe because few know the skill set even exists! So its time to advertise myself. Perhaps you are looking for someone who can…

  • Do mathematics at a research level, especially:
    • Geometry, understanding the spaces we live in and more exotic ones.
    • Tilings and patterns.
    • History and culture of Mathematics
  • Talk maths in public.
  • Teach (and be creative at it)
  • Program
  • Use computer manufacturing tools, Laser cutters, 3d printers, 3/5/n-axis routers.
  • Make Art and do Design

You need more evidence? I guess that makes sense. More details are below. If you still need to know more get in touch. I can provide references! (edmund.harriss at mathematicians.org.uk)

More details and evidence…

Mathematics: The heart of what I do, I have been an academic mathematician since getting my PhD from Imperial College in 2004. I have written papers, and been invited far and wide to talk about my work. See my CV for the gory details.

Geometry, Tilings and Patterns: I have a very strong understanding of the space we live in (and more exotic spaces). As this is a mathematical understanding I also have the tools to make this concrete, putting it into the equations and other things that computers can play with. My mathematical research has looked at tilings and patterns. Especially substitution tilings a sort of scaling symmetry, I probably know as much about the Penrose tiling than anyone else alive or dead!

History of Mathematics: It is mostly an amateur interest, though I nearly started a PhD with David Fowler before beginning one on tilings. I also think about the role of mathematics as a subject in the world and its relationship to art.

Talking maths in public: You can understand what I have to say without specialist training! I have explained the beauty and wonder of mathematics from the sacred halls of the Royal Society to primary schools. You can even hear me on the radio (and of course read this blog!). Or dive into the geekiness of prime birthdays.

Teaching: I want to teach people to actually think mathematically, not just get the rules that can be followed to a right answer, and have had success with it. Of course I can teach a traditional maths course and these are often necessary to get the bulk of material across, however I have also worked with more innovative courses. That is why I came to Arkansas. I wanted to teach MATH 2033 the conspiracy or mathematics course designed to corrupt people into the subject by giving a glimpse of  undecidability, game theory, 4 dimensional geometry, hyperbolic geometry, topology, codes, sphere packings… The students then have to come up with their own projects and, as could be expected often get incredibly creative.

3d cog spirographs

Art and Design: I can make pretty pictures, normally using maths. I am on the board of the new Art and Science masters at Central St Martins school of Art in London, and designed the screens for the new Mathematics Learning Centre at Imperial College London. I can do graphic design in 2d and make models and render them in 3d. I can use all the standard software, Adobe Illustrator and Photoshop, Rhino 3d (especially with Grasshopper) etc.

Making: I make things, normally focussing on explaining mathematics. I even have my own Laser cutter! I designed some larger versions of the hexayurt, a simple building made, without waste from 12 sheets of plywood or other materials. I am currently working with the FabLab at the architecture school here at the University of Arkansas, and am writing software to drive their 5-axis router.


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Posted by gelada

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