Although we live in a 3D world, we aren’t always very good at judging the volumes of things. A few years ago I had the idea of exploring our (mis)judgements of volume by making a collection of differently-shaped objects, all of which had a volume of a pint. I didn’t do anything about it at the time, but when I discovered recently that Pint of Science in Glasgow was holding Creative Reactions, an art exhibition, I decided to take the hint and get to work.
Cuboids
These cuboids all have a volume of a pint.
Optimal shapes
These shapes not only have a volume of a pint, but they are all optimal in terms of surface area:
Of all the cuboids with a volume of a pint, a cube has the smallest surface area. Of all the cylinders with a volume of a pint, a cylinder whose height and diameter are equal has the smallest surface area. Of all the cones with a volume of a pint, a cone whose height is the square root of 2 times its base diameter has the smallest surface area.
And of all solid shapes with a volume of a pint, a sphere has the smallest surface area.
I made the cylinder, sphere and cone on a lathe, and the sphere on a bandsaw.
A quiet pint
Here we have the lowest (most distal) pint of my arm and hand about to pour a pint of beer into the front pint of my face. All the beer-glass shapes are casts of the interiors of pint glasses. I was slightly disappointed by the beer-glass casts; I was hoping that they might seem strikingly small compared to the actual filled beer glasses, but they don’t.
Casting was a new venture for me. My thanks to Amy Grogan and Alys Owen of the casting workshop at Glasgow School of Art for their help and advice.
My thanks also to Laura McCaughey, Marta Campillo Poveda, and Danielle Leibnitz, who organised the exhibition.
Before Christmas, my enterprising friend Clare decided to brighten the dark nights of a Scottish winter by turning her greenhouse into an illuminated art gallery. She asked friends to produce translucent artworks that could be hung in the greenhouse and lit from within.
My contribution is a representation of the movement across the sky of Jupiter and Saturn (and some smaller planets) in the two years bracketing the recent Great Conjunction. It’s made from a sheet of wallpaper, painted black, with holes cut out and with coloured filters placed behind the holes.
A photograph of the piece. It’s just over 50 cm wide. The variations in brightness of the discs aren’t part of the plan; the illumination wasn’t perfectly even.
The piece is divided into 30 rows. All but one of these rows contain a large red disc (representing Jupiter) and a large yellow disc (representing Saturn). A row may also contain smaller discs, representing Mars in pink, Venus in white and Mercury in blue. The purple discs represent the ex-planet Pluto. All of the discs hugely exaggerate the size of their planets.
Each row represents the same strip of the sky, in the sense that if I had included stars on the piece, the same stars would appear in the same positions on every row. From top to bottom, the rows show that strip of sky at 25-day intervals, covering a period roughly from roughly one year before the Great Conjunction to one year after. The discs in each row indicate the positions of any planets that are in that strip of sky at the time.
Concentrating on Jupiter (red) and Saturn (yellow) first, we see that they have a general leftward motion, but with periods of rightward motion. Jupiter’s overall leftward motion is faster than Saturn’s: it starts to the right of Saturn and finishes to the left. Because Jupiter overtakes Saturn, there comes a point where they are at the same place in the sky. This is the Great Conjunction: in this row, both Jupiter and Saturn are represented by a single large white disc.
Mars, Venus and Mercury move much faster. Mars crosses our field of view in only 4 rows (roughly 100 days) and Venus and Mercury make repeat visits. Pluto wavers back and forth without appearing to make much leftward progress at all.
The FAQ
Why do the planets move along the same line? They don’t exactly, but it’s pretty close. All of the planets, including the Earth, move around the Sun in roughly circular orbits. Except for Pluto’s, these orbits are more or less in the same plane (like circular stripes on a dinner plate). Because our viewpoint (the Earth) is in this plane, we look at all the orbits edge on, and the planets appear to follow very similar straightish paths across the sky. I have chosen to neglect the slight variations in path and depict the planets as following one another along exactly the same straight line
Why do Jupiter and Saturn move mainly right to left? Looking down from the North, all of the planets orbit anticlockwise. Mars, Jupiter, and Saturn have bigger orbits than the Earth, we’re observing them from inside their orbits (and from the Earth’s northern hemisphere). Thus their general movement is leftwards. (If you don’t get it, whirl a conker around your head on a string, so that it moves anticlockwise for someone looking down. The conker will move leftwards from your point of view.) The orbits of Venus and Mercury are inside the Earth’s orbit; their movements as seen from the Earth are rather complicated.
Why do Jupiter, Saturn, and Pluto sometimes move from left to right? Earth is in orbit too, so we’re observing the planets from a moving viewpoint. If you move your head from side to side, nearby objects appear to move back and forth against the background of distant objects. Exactly the same effect happens with our view of the outer planets as the Earth moves around its orbit from one side the Sun to the other – they appear to move back and forth once a year against the background of distant stars. But at the same time, they are also really moving leftwards (as we look at them). The sum of the planet’s real motion with their apparent back-and-forth motion gives the lurching movement that we see: mainly leftwards but with episodes of rightward motion. Note that the planets never actually move backwards: they just appear to. The same thing happens to Mars, but none of its periods of retrograde motion coincided with its visit to our strip of the sky.
Why do some planets move faster across the sky than others? The larger a planet’s orbit, the more slowly it moves. For the outer planets, a larger orbit also means that we’re watching it from a greater distance, so it appears to move more slowly still. Saturn’s orbit is about twice as big as Jupiter’s, so it moves more slowly across the sky than Jupiter. Jupiter “laps” Saturn about once every 20 years: these are the Great Conjunctions. Mars’ orbit is smaller than Jupiter’s, so it moves more quickly across the sky. Meanwhile lonely Pluto plods around its enormous orbit so slowly that the leftward trend of its motion is barely discernible; all we see is the side-to-side wobble caused by our own moving viewpoint. As for Mercury and Venus: it’s complicated.
Please could you stop being evasive about the movements of Venus and Mercury? It really is complicated. The orbits of Venus and Mercury are smaller than the Earth’s: we observe them from the outside. If the Earth was stationary, we’d see Venus and Mercury moving back and forth from one side of the Sun to the other. Returning to our conker-whirling experiment, it’s like watching a conker being whirled by somebody else rather than whirling it yourself. But the Earth is moving around its orbit too. And then Venus and Mercury are also moving rather fast: Mercury orbits the Sun 4 times for each single orbit made by the Earth. Combine all of these things and it becomes very confusing. Whereas the outer planets’ episodes of retrograde (backwards) movement across the sky occur less than once a year, Mercury is retrograde about three times a year.
Do the planets really follow a horizontal path across the sky? This question doesn’t have an answer. We’re using the pattern of stars, all inconceivably distant compared to the planets, as the fixed background against which we view the movement of the planets. You may have noticed that the stars move in arcs across the sky during the night; this is due to the Earth’s rotation on its axis. So our strip of sky moves in an arc too, and turns as it moves. So if it ever is horizontal, it is only briefly so, and when and if it is ever horizontal will depend upon your latitude.
Jupiter and Saturn never exactly lined up, did they? No, they didn’t (see the answer to the first question). On this scale, at the Great Conjunction the discs representing Jupiter and Saturn should be misaligned vertically by about a millimetre. With our hugely over-sized planets, this means almost total overlap, which still misrepresents the actual event, where the planets were separated by many times their own diameter. And for all other rows, where the two discs don’t overlap, a millimetre’s misalignment would be imperceptible. A final and maybe more compelling reason for my neglect of the misalignment of the planets’ paths is that I don’t know how to calculate it.
Anything else to confess? Yes. There’s a major element of fiction about the piece in that it’s not physically possible to see all of these arrangements of the planets. The reason is that for some of these snapshots, the Earth is on the opposite side of the Sun from most or all of the planets, and Sun’s light would drown out the light from the planets. In other words, it would be daytime when the planets are above the horizon, and therefore in practice they would be invisible. This was almost the case for the Great Conjunction, where there was only a short period of time between it becoming dark enough for Jupiter and Saturn to be visible, and them disappearing over the horizon.
A further element of fiction is that, even in the depths of a Scottish winter’s night, Pluto is far too faint to be seen with the naked eye, not to mention not being regarded by the authorities as a planet any more. But it was passing at the time of the Great Conjunction and it seemed a pity to miss it out.
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.
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).
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.
