This post is merely to provide a link to a video of a mathematical machine that I made over the Christmas holidays. Even if maths isn’t your thing, you might enjoy the movements and the rhythms.
Sarah Kenchington and I made this machine for the Full of Noises festival in Barrow-in-Furness in August 2018.
Sarah designed and made the bicycly bits that raise the table-tennis balls from the pit into the hoppers at the top, and I made the two devices that the balls descend through on their way to the cow bells and glockenspiel.
The complete machine also included other noise-making devices and an exercise-bike powered drive system, both made by Sam Underwood. It was housed in a greenhouse. Here’s a video of the whole thing in action at Full of Noises.
We shot the video in this post in a hurry on a dark damp Tuesday morning before packing the machine up to take it to Barrow, so it comes with apologies for the poor lighting in places.
The peg board (Galton board) that appears from 1:13 to 1:31 is an established classic (see below if you want to make one). The swinging-ramp ball-feeding device (2:09 to 2:18) is a revival of something I designed for the Chain Reactor.
What’s new from me is the arrangement for feeding the balls from the wire chute into the swinging-ramp assembly (1:56 to 2:18). Its operation should be clear from the video, except perhaps for one detail. Because this device may jam if it tries to collect a ball that has not quite arrived at the bottom of the wire chute, and because the timing of the arrival of the balls is erratic, it’s necessary to maintain a queue of balls in the chute to guarantee that there’s always a ball in place at the bottom to be collected. To achieve this, we arranged that the average rate of ball delivery into the chute (determined by the number of spoons on the bicycle chain) was greater than the rate of collection of balls out of the chute, and had an overflow route for the excess balls. Once three balls have accumulated in the chute, any further balls are diverted back into the ball pit (2:30-2:40).
Making the Galton board
Chris Wallace and I discovered while making the Chain Reactor that the horizontal spacing of the pegs on a Galton Board is important. If the spacing is too great, a ball that sets off rightwards will tend to keep going rightwards, and vice versa. To get good randomisation, the ball should rattle between each pair of pegs, and to get this to happen, the gap between the pegs should be only slightly greater than the diameter of the balls. This in turn means that the pegs need to be precisely placed to avoid there being pairs of pegs that don’t let the balls through at all.
In that project we achieved the necessary precision by making the position of each peg (a bolt) adjustable, but with something like 100 bolts, this difficult job was very tedious and sorely tried Chris’s patience.
This time round, I developed a system that let me get every hole in the right place first time. Firstly, I cut the board into four strips so that all parts of it were accessible to a pillar drill. This guaranteed that every hole was accurately perpendicular. Secondly, I made a drilling jig (top right) to get the hole spacing correct. After drilling each hole, I put the peg (the bolt on the right-hand part of the jig) into the just-drilled hole, and the drill for the next hole into the drill hole on the left-hand part of the jig. The spacing between the peg and drill hole is adjustable using the long bolt. Thirdly, I made a large custom table for the pillar drill (bottom right), with a fence arrangement so that each row of holes was straight.
When I was doing the drilling, the only measurements I had to make were to get the first hole in each row in the right place with respect to the previous row. It took me a few hours to perfect the drilling arrangements, but then only an hour or so to drill 90 holes, all exactly where I wanted them.
You’re walking along a woodland path. Suddenly you hear a recipe for yogurt being recited somewhere in the bushes. A short while later, you hear an apology for kicking you in a tender place emerging from the trees. What is going on?
Cut Adrift is an installation by Edinburgh artist Mark Haddon. Over the past 20 years or so, Mark has collected handwritten notes that he has found lying on the ground. The notes are a varied mix: letters, recipes, instructions, apologies… Mark recorded people reading these notes out loud. As part of Sanctuary, a 24-hour art event in southern Scotland, he and I arranged that the passing of people along a path would trigger the playing of these recordings from a sound system hidden in the undergrowth.
I helped Mark with the technical side of the project. We used a Teensy microcontroller board with an audio adaptor board audio shield to store, sequence, and play the audio clips. The signal went to a 12 V audio amplifier and thence to a pair of loudspeakers. We used a PIR (passive infra-red) motion detector to detect people walking by. The output from the detector was connected to one of the input pins of the Teensy. The whole thing was powered using a 12 V lead-acid battery. The battery, Teensy, and amplifier all went in a plastic storage crate to protect them from the weather.
The passage of a person triggered one audio clip. The choice of which clip to play was random, but subject to a rule that made a clip more and more likely to be chosen the longer it was since it was last played. My program held the clips in a queue. When a clip was needed, there was a 50% probability that the first clip in the queue would be chosen, a 25% probability that the second clip would be chosen, and so on as far as the 5th clip. Clips lower in the queue than this would never be chosen. Once the chosen clip had been played, it was put at the bottom of the queue and all the other clips moved up one place.
I’m grateful to Jen Sykes of Glasgow School of Art for pointing me in the direction of the Teensy and its audio board.
When you stir a cup of tea, the surface of the rotating liquid develops a dip in the middle. The faster you stir, the deeper the dip. But the liquid surface is quite uneven; to get a smooth surface, throw the spoon away and spin the whole cup continuously. Once the liquid inside has caught up with the cup, and everything is turning at the same speed, the liquid surface forms a beautiful smooth curve known as a paraboloid of revolution.
Sarah McLeary and I are applying this idea to make thin paraboloidal plaster shells. We spin a bucket on a potter’s wheel (Sarah is a potter), and pour plaster into it. The plaster rapidly flows until its surface forms a deep paraboloidal curve, and then sets. We now use this cast, still spinning, as a mould, and cast a thin layer of plaster inside it, to make a paraboloidal shell.
That’s our first attempt above. It’s about 20 cm across and 3 mm thick. It might look like a part of a sphere, but in a profile view it’s easy to see that the curvature is tightest at the base and gradually decreases up the sides, as you’d expect for a paraboloid.
I love the fact that we didn’t decide what shape this shell was going to be: physics did.
There’s lots of experimentation ahead. When we’ve got the hang of it, I’ll explain our methods in more detail. But as this picture of our third attempt shows, we haven’t quite cracked it yet.
I made the machine in the video below for no other reason than that I felt like making a vehicle that was propelled in a non-standard way. The idea arose when I wondered about setting my Product Design Engineering students the challenge of making a vehicle that wasn’t driven through its wheels. In the end we did a different project, but I still couldn’t resist having a go myself.
Originally there was going to be a separate weight that was shifted back and forth to make the vehicle tip. I was rather pleased when I realised that I could make the batteries, the Arduino, and one of the motors do double duty as the weight.
(I didn’t choose the carpet.)
I’ve just spent the last day and a half making another irregular polyhedron to use as a lampshade in the room where I work. As I worked, I was reminded how useful it is to have a table with motorised height adjustment. The thing that I was making started off flat, and finished 40 cm in diameter, so a table that was the right height at the beginning would be far too high towards the end of the construction.
But it’s not just about workpieces that grow. Different assembly operations are best done at different heights: cutting and folding need a lower table than glueing, for example, and if I need to take a close look at something, it’s useful to be able to raise it as close as possible to eye level. (I do nearly all making tasks standing up, by the way.)
There are many different sit-stand desks out there. Mine is an Ikea BEKANT electric sit-stand desk. The height adjusts electrically from 65cm to 120 cm, and it takes about 20 seconds to cover the full range. It seems to be reasonably solidly built, though I wouldn’t try to do woodwork on it.
It seemed a bit of an extravagance when I bought it, but I love it. By enabling me to keep a better posture, it’s much more comfortable to work at than a fixed-height desk, and the ability to move the workpiece to the best height for any given operation materially improves the quality of the things I make.
How could it be improved? Foot switches (or even better, speech control!) would be handy for those stressful times when you need to change the height of the desk quickly, but have both hands occupied holding something together. And being able to tilt the tabletop would be wonderful – a project for the future, maybe…
The lilac chaser is a remarkable visual phenomenon that is normally seen as a computer animation. Dr Rob Jenkins of York University wanted to show people that the effect works with real, honest-to-goodness, physical lights, so he asked me to make him the equipment to do this. The video below shows you how the apparatus works. Note that the limitations of my camera mean that the effect is not as strong in the video as it is in real life.
I used an Arduino microcontroller board to control the LEDs.
One useful technique that I developed here was a way of producing an an even spot of light from an LED. Diffused LEDs give an even spread of light but send light in all directions, which is wasteful if you want only a small bright spot. Clear LEDs are available which direct the light in quite a narrow beam, but the distribution of light is very uneven. I found that shining the light from a clear LED down a short white tube, about 10 mm internal diameter and 60 mm long, did a very good job of producing a sharp-edged even spot on a piece of tracing paper placed at the end of the tube. I assume that the many reflections inside the tube thoroughly mix up the light. I found the tubing in the plumbing section of a hardware shop, and lightly roughened the inside of it using fine sandpaper.
To get a spot with a blurred edge, I placed a second tracing-paper screen a short distance away from the end of the tube. By varying the distance of this screen I was able to vary how blurred the patch of light on it was.
In a previous post I showed some examples of irregular polyhedra that I’ve been making out of paper. These polyhedra were all based on points distributed quasi-randomly over a sphere. At each point, my program placed a plane tangent to the sphere. The result of the intersection of all of those planes was an irregular polyhedron.
The 47-hedron in the picture above was made in a rather different way. It’s certainly irregular, but the process that created it was not at all random. As in the previous polyhedra, the starting point was a sphere with a number of points distributed over it. But this time, the points were placed according to a simple and perfectly regular rule.
Imagine you you draw a line from one pole of a sphere to the opposite pole, winding sort-of-helically around the sphere. Then place a number of points at exactly equal spacings along this line. In the examples below, there are 43 points.
If your quasi-helix had 2.5 turns, the result would look like the diagram on the right. The numbers are where the points are placed on the imaginary sphere; the resulting polyhedron is also drawn. The helical structure is very clear, but notice the irregularity of the faces: for example, the lower edges of faces 26, 27, and 28 are all different. This is because the radius of the quasi-helix is not constant, so turns of the helix near the poles will have fewer faces on them than turns near the equator, which means that the way the numbers on one turn line up with the numbers on the previous and subsequent turns keeps changing.
If we increase the number of turns in the quasi-helix to 5, the helical structure is still clear:With 7.5 turns in the quasi-helix (below), adjacent-numbered faces are now so far apart that the faces on the turns above and below them are starting to intrude into the spaces between them. See, for example, how faces 27 and 28 are almost pushed apart by faces 19 and 35.By the time we have 10 turns (below), consecutively-numbered faces are so far apart that faces from the turns above and below often meet between them, making the helical structure hard to discern (see, for example, how faces 34 and 22 squeeze in between faces 28 and 29). The polyhedron is beginning to look quite irregular, but note that consecutively-numbered faces are still more similar to each other than they are to the other faces.
The 47-hedron on the right (same as at the top of the post) was based on a quasi-helix with about 12.5 turns. At first sight, it looks quite random, and it’s certainly irregular, but there are still visible similarities between faces if you look hard enough. Compare, for example, the mid-blue and light-blue faces at roughly 10 o’clock and 2 o’clock respectively. And what about the dark-blue and very dark-blue faces just below the centre? One looks suspiciously like a rotated version of the other – and it is! In fact the whole polyhedron has an axis of 180° rotational symmetry. This suprised me when I spotted it, but it shouldn’t have. A helix has 180° rotational symmetry (turn a corkscrew upside down and it looks the same) so it’s inevitable that shapes based on a helix will have that symmetry also.
This summer Chris Wallace and I spent a week renovating the chemistry-themed Chain Reactor machine that we built for the Edinburgh International Science Festival in 2011. Since then the machine has appeared every year, not just in Edinburgh but in Abu Dhabi, where it was installed next to a beach.
Unfortunately the salt air in Abu Dhabi meant that we got more chemistry going on than we’d intended, as the rusty condition of the nuts and bolts above shows (pictured not in Abu Dhabi but instead on a chilly November morning on the beach at Portobello in Edinburgh, where I live). Thus a big part of the job was replacing hundreds of zinc-plated nuts and bolts with stainless steel equivalents.
As we did this, I noticed a clear pattern to the corrosion. Surfaces facing upwards suffered much more badly than surfaces facing downwards, and if a bolt was sheltered from above by an overhanging part of the machine, it didn’t rust very much, even though it was completely open to the air.
This would make sense if the corrosive agent (tiny droplets of brine or particles of salt, maybe) was predominantly falling downwards through the air, which such droplets/particles must do if the air is reasonably calm.
You can see evidence of something similar in kitchens, where frying fills the air with an invisible mist of fat particles, each slowly falling through the air. You may not realise this until you see the tacky layer that gradually develops on the top sides of your shelves and cupboards, but not on the undersides. And if you wear glasses, it’s the inside of the lenses that you need to clean after frying; this is the side that faces upwards as you look down to cook.
This makes me wonder whether we’d need to clean or redecorate our kitchen walls less often if they were slightly overhanging so that grease particles were less likely to land on them. A slope of a degree or so from the vertical wouldn’t be too conspicuous and might make all the difference.
It’s a remarkable fact that there are only five regular convex polyhedra, that is, solid shapes whose flat faces are all identical regular polygons. The most familiar example is the cube, with 6 identical square faces. There are other, less regular but still orderly polyhedra, such as the truncated icosahedron, seen in a bloated form in some footballs.
But what, I wondered, about truly irregular polyhedra, where every face is a different irregular polygon? What would they look like? Last January I set out to make some and find out.
My irregular polyhedra are all based on spheres. Imagine placing a number of dots on a sphere. At each dot, let there be a plane just touching the sphere. These planes will all intersect each other, and if we remove the parts of each plane cut off by the neighbouring planes, we’re left with a polyhedron, with one face for each dot that we placed on the sphere. The nature of this polyhedron depends upon how the points are distributed over the sphere. In the polyhedra shown here, the placement of dots was random but subject to certain constraints.
I wrote some software that allowed me to choose how many faces I wanted, and to regulate how evenly spread over the sphere the dots were. I could preview the resulting polyhedron, and when I saw one that I liked, the program produced a set of images of the faces of the polyhedron, with numbered tabs on them. Then all I had to do was print them, cut them out, fold the tabs, and glue them all together. This is not a task for the impatient: the 83-hedron at the top took about 3 days.
The polyhedra shown here differ principally in how evenly the dots were spread over the imaginary internal sphere. After randomly placing the dots, the program simulated repulsion between them (as if they were electric charges). The longer this repulsion process went on, the more evenly distributed the dots became. For the 29-hedron, no repulsion process was done, and for the 43-hedron, the process was allowed to continue until the dots stopped moving; presumably this is now (in some undefined sense) as regular as a 43-hedron can get.
For the 29-hedron (right), I took advantage of the four-colour map theorem, which tells me that if I want to colour the faces such that no two neighbouring faces have the same shade, 4 shades of card are all I need. I leave it to you to convince yourself that if this theorem is true for flat maps, it must be true for maps on balls too. (If you don’t see where maps come into it, think of each face of the polyhedron as a country on a political map.)
None of this would have happened had it not been for a visit to Jenny Dockett to talk about her Illuminating Geometry project. It was while talking to Jenny that the idea for making irregular polyhedra came to me.
I wrote the software in Python. The internally-illuminated shapes are made out of layout paper (very thin paper), and the 29-hedron is made of thin card. I used UHU Office Pen glue (not to be confused with UHU Pen glue). This glue doesn’t wrinkle the thin paper, and dries at about the right speed, but has proved to be somewhat unreliable in humid weather.