The 2×1 rectangle and Domes

March 26, 2012

Next week I am going to be at the Gathering for Gardner, an exciting meeting of mathematicians, magicians, puzzlers and others inspired by the life and work of Martin Gardner. This post is a version of the talk I will be giving.

The 2×1 rectangle is not one of mathematics most celebrated shapes.

Yet it is so much more flexible than the more common square.

Even better you can cut it in half on the diagonal to make a 2×1 right triangle,

which has the beautiful property that it is a 5-reptile. Five copies of it come together to make a larger version. Repeating this gives the Conway Pinwheel tiling, which has triangles occurring in an infinite number of directions.

Yet the 2×1 rectangle is a lot more common in life, just go into your local hardware store:

Using the diagonal cut triangle and uncut rectangles, Vinay Gupta designed the hexayurt,

a small house that can be built from 12 sheets, without waste. In contrast to geodesic domes, that cannot be made from sheet materials without making many cuts or wasting material. Here is one:

and a plywood one:

Hexayurts have become one of the standard accommodations at Burning Man:

or look at this map, the red dots show the location of the hundreds of hexayurts at last years event.

Vinay set me the challenge of making larger domes using these shapes. The hexayurt itself suggests that hexagons will be important, and we can put two 2×1 rectangles together to make a square. Squares and hexagons come together to form the truncated octahedron.

This obviously would not work as a dome, so we must cut it. There are two natural cuts that can be made. One perpendicular to the 4-fold axis, and one perpendicular to the 3-fold:

So we have two new larger domes, the tri-dome and the quad-dome:

What is really cool is that both of these domes were made for Burning Man last year:

Tri-dome:

Quad-dome:

One neat thing about the truncated Octahedron is that it is a space-filler. You can use them to tile 3d space. We can therefore bring quad-domes together to make even larger structures, like this one:

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Posted by gelada

Prime Phyllotaxis Spirals

March 18, 2012

The phyllotaxis spiral is one of the classical forms of mathematics, and there is a wonderland of resources available online both images and explanations. The basic idea is to put points round in a spiral with the same angle between each point. This gives a family of forms:

Note that, as the angle changes the dots sometimes pack in better than others, this can actually be studied and the best packing is related to the golden ratio. The points in this spiral are placed down in order, so we can associate each to a whole number:

Now when I see a lit of numbers like this, I want to pull out the primes, see what pattern they make:

There are some hints at patterns, lets expand out, and look just at dots:

There seem to be spiral arms which are richer in prime numbers than others. We can analyse things further by colouring each number depending on its prime factors. The more prime factors the lighter the number, giving the image for the start of this post:

Now there is a clear pattern, light and dark arms spiraling out. Can we understand this pattern?

Think about the construction of a phyllotaxis pattern we turn the same angle every time, that means within a particular pattern we can find other phyllotaxis patterns. The one at twice the speed, three times the speed and so on.  For example we could dive our pattern into two patterns each with twice the rotation angle. This gives:

All prime number (other than 2) are odd, so they must lie on the subspiral corresponding to the odd numbers. In addition it turns out that the spiral arms that we see are related to the Fibonacci numbers (themselves closely linked to the Golden Ratio). The particular curves we see relate to 144. Here is the spiral given by multiples of 144, pulling out just one such curve:

Note that in the prime factor picture this curve gives a very light line as every number in it is a multiple of 144, and 144 itself has 6 prime factors (three twice and two four times). Taking the multiples of 6 instead of 144 (which gives us several of these curves as 6 divides 144) we see another pattern of lines that are light in the image:

More importantly the curves next to these ones give numbers one more or less than a multiple of six. Every prime number has this form (all other numbers are multiples of 2 or 3 or both). This gives the curves of prime numbers we saw.

So by considering the construction of the initial image it begins to reveal its secrets. Yet, just as with the primes on their own there seems to be plenty of mystery left for investigation…

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Posted by gelada

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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