Turn Binomials into Gaussians with Polymer Clay

I’ve been playing a bit further with the visual demonstrations of the binomial/ central limit theorems. I love polymer clay as a medium, it’s extremely easy to work with, colourful, and you can even make fractal patterns with it.

Here’s my first approach, it turned out a bit less intuitive than I was hoping, but it was still fun to make.

The first step was to make a series of square stacks, encoding all possible 3-bit values (000, 001, 010…… 111):

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Then slice them in quarters and make long strands from each:

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Turn each one into a long rope:

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And assemble into a Galton board style device, with each path represented by the colour ordering of the strand (e.g. 010 = Left, Right, Left)

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You can immediately see that at the final clusters at the bottom, the number of each colour bands is the same (e.g. three blues, two blues and an orange, etc).

Representing that in the Galton board left-left-right leads to the same result as right-left-left.

I was fairly happy with that, but I wanted a more detailed model, and I decided to drop the encoded-strings idea as being to difficult to implement without distorting. Here’s my results for simple strings:

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This I’m pretty happy with. You can see at a glance why the middle result is 6x more probable than the right hand result (since six paths lead there versus only one), and you also don’t have to wait for the balls to collect as with the conventional display.

After baking the models in the oven for half an hour they’re now robust and permanant, and I’ll keep them on the shelf for the next time the central limit theorem comes up.

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Braided Galton Boards – Visualizing Probabilities with String

I’ve been playing around with Galton Boards recently. They’re an awesome demonstration of the central limit theorem, and how several independent 50/50 events sum together to make a Gaussian/Bell curve.

Seeing the standard Galton board gives an intuitive feel for how random variables behave, but it takes time to run the experiment and see the results accumulate.

I wanted a more immediate visual demonstration, and after a few prototypes I settled on using a thread to represent each possible path through the system:

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I used 12.5mm dowels cut to size, and hammered into the lasercut wood.

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The 64 and 32 string versions

The strings are cut to length, then held in place with a screw clamp and threaded through one at a time:

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Stats and craft therapy all in one

It might be because I finished this late at night, but I found it very difficult to thread without making a mistake. The strategy I ended up using was to hold the previous thread in my left hand and the next thread in my right. By jiggling my left hand I saw the old thread bounce, which gave a good reminder of where I was in the sequence, and hence to put the new thread in the right spot.

 

Files here for anyone that wants to make their own:

https://www.thingiverse.com/thing:2362584

It took a bit of prototyping before I got a form factor I was happy with. Here’s two of the failed attempts, that I avoided because they either weren’t practical, or they were slightly too opaque a display of the concept:

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Lasercut Yarn Bobbins

I’m working on a project that involves lots of different coloured cords, and I wanted a way to avoid tangles while transporting & working with them.

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Files here:

https://www.thingiverse.com/thing:2360450

 

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Bayes’ Ruler – A Bayesian Slide Rule

I’m a big fan of Bayesian statistics, and also old analog computers. It occured to me one day that I hadn’t ever seen a slide rule for calculating Bayes’ theorem…

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Behold a fine posterior

Here’s how to use it. Let’s say you’re a doctor, and treating a patient with a 80% chance of having gallstones. You order a new test done, and it comes back positive. What’s the probability they have gallstones?

After looking in the medical texts, you find the test is 5:1 more likely to be positive if the person has gallstones than if they don’t.

So, we align the arrow with our ‘prior’ on the left at 80% (4:1 odds), and look for the 5:1 diagnostic odds, which gives us a result of 20:1, or 95.2% probable the patient has gallstones.

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There’s far better explanations out there than this, please check out Wikipedia for what’s almost certainly better worded than mine.

I didn’t have the heart to design something like this by hand in inkscape. Instead I used Python and the DXFwrite library  to generate the files, and then lasercut it from acrylic.

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I also made a wooden version with a clear acrylic top, which should be less fragile in a bag:

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Files are here for anyone that wants to make their own:

https://www.thingiverse.com/thing:2347920

 

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Pachinko Crystal Modelling

I saw this video recently, and fell in love with the Atomix toy. Sadly it seems you can’t get them anymore. However, I remembered that in the garage  I had over a thousand Japanese pachinko balls left over from a previous project…

This was a quick one evening build, I just wanted to get it done before I got distracted. Some sawing and cutting later, and here’s what I have:

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Can for scale, as with all my projects

I used thick 12mm thick plywood I had laying around the space, and a bandsaw-box like technique to cut out the inside of the frame. The front is lasercut 3mm acrylic, which I was originally worried might not be up to the job of keeping 3kg of agitated steel contained, but after shaking it a few times it seems to be sturdy enough.

You can hold the frame and shake it by hand, but I wanted something to be a bit more play-friendly. I had a lazy-susan bearing kicking around after I did various rebuilds of  a robot several years ago, and I decided that would allow some gentle motion.

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4×2 lovingly cut, and expertly screwed together in whatever fashion got it done before bedtime. 

With two wood pieces acting as a tilted base, the whole assembly sits tilted on the table and can be gently rotated with a light touch.

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Finished piece

It’s certainly very  satisfying to play around with, you can see the fault lines, grain boundaries, different orientations, gaps and other nifty crystallographic features. And for a one evening build I’m very happy with it.

Also, words can’t express how noisy the 11mm pachinko balls are compared to the ~1mm ones Steve used in his model. However provided you’re the one using it, it’s relaxing and soothing like a rain machine. (However if someone else is doing the playing, for some reason it doesn’t seem quite so serene…)

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Demonstration ROS Coordinate Frames

I spent a bit of time recently help a friend set up a robotics lab. One of the things I wanted to nail down was a consistent set of coordinate frames for all the equipment.

The Robot Operating System is awesome, and provided you set up your TF models correctly, it’ll do all the work for you. However it’s sometimes counterintuitive how it should be set up. ROS has a standard way of arranging coordinates (REP0103), but it’s different for ordinary coordinate frames and camera frames. (For very good historical reasons; the camera frame has Z pointing forwards, which means X is lined up with image columns, and Y is image rows).

I decided to make some tangible models for people to use when they’re discussing transformations. When troubleshooting robots you constantly find yourself in front of a whiteboard making “finger axes”, I wanted something we could use instead, and that had the same colouring and labels as the software.

The idea is first, to have something tangible to play with while brainstorming, and second, as a reminder of the convention, to allow errors to be detected. If there’s an error, I really want someone to ask “Wait, why is the green arrow pointing in a different direction on screen?”.

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Making the ‘Ultimate’ Pochade Boxes

The previous  writeup was getting a bit long, so I thought I’d put the steps used for making the boxes in their own post.

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I started with the frame, using pine and biscuit joints for securing it:

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