Slide Rule By Helix

June 4, 2025 by Al Williams No comments

It is no secret that we like slide rules around the Hackaday bunker, and among our favorites are the cylindrical slide rules. [Chris Staecker] likes them, too, and recently even 3D printed a version. But spurred by comments on his video, he decided to try something that might be unique: a helical slide rule. You can see how it works in the video below.

With a conventional slide rule, the scale is rotated around a cylinder so that it is the same length as a much longer linear scale. However, this new slide rule bends the entire rule around a cylinder and allows the slide to move, just like a conventional slide rule. If you have a 3D printer, you can make your own.

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Multi-Divi book with hand thumbing through it

Math, Optimized: Sweden’s Maximal Multi-Divi

March 28, 2025 by Heidi Ulrich 30 Comments

Back in the early 1900s, before calculators lived in our pockets, crunching numbers was painstaking work. Adding machines existed, but they weren’t exactly convenient nor cheap. Enter Wilken Wilkenson and his Maximal Multi-Divi, a massive multiplication and division table that turned math into an industrialized process. Originally published in Sweden in the 1910’s, and refined over decades, his book was more than a reference. It was a modular calculating instrument, optimized for speed and efficiency. In this video, [Chris Staecker] tells all about this fascinating relic.

What makes the Multi-Divi special isn’t just its sheer size – handling up to 9995 × 995 multiplications – but its clever design. Wilkenson formatted the book like a machine, with modular sections that could be swapped out for different models. If you needed an expanded range, you could just swap in an extra 200 pages. To sell it internationally, just replace the insert – no translation needed. The book itself contains zero words, only numbers. Even the marketing pushed this as a serious calculating device, rather than just another dusty math bible.

While pinwheel machines and comptometers were available at the time, they required training and upkeep. The Multi-Divi, in contrast, required zero learning curve – just look up the numbers for instant result. And it wasn’t just multiplication: the book also handled division in reverse, plus compound interest, square roots, and even amortizations. Wilkenson effectively created a pre-digital computing tool, a kind of pocket calculator on steroids (if pockets were the size of briefcases).

Of course, no self-respecting hacker would take claims of ‘the greatest invention ever’ at face value. Wilkenson’s marketing, while grandiose, wasn’t entirely wrong – the Multi-Divi outpaced mechanical calculators in speed tests. And if you’re feeling adventurous, [Chris] has scanned the entire book, so you can try it yourself.

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Circuit Impedance Calculations Without Cumbersome Simulations

March 2, 2021 by Bryan Cockfield 11 Comments

Using circuit simulating software like SPICE can be a powerful tool for modeling the behavior of a circuit in the real world. On the other hand, it’s not always necessary to have all of the features of SPICE available all the time, and these programs tend to be quite expensive as well. To that end, [Wes Hileman] noticed an opportunity for a specific, quick method for performing impedance calculations using python without bulky, expensive software and came up with a program which he calls fastZ.

The software works on any network of passive components (resistors, capacitors, and inductors) and the user can specify parallel and series connections using special operators. Not only can the program calculate the combined impedance but it can perform frequency analysis at a specified frequency or graph the frequency response over a wide range of frequencies. It’s also running in python which makes it as simple as importing any other python package, and is also easy to implement in any other python program compared to building a simulation and hoping for the best.

If you find yourself regularly drawing Bode plots or trying to cobble together a circuit simulation to work with your python code, this sort of solution is a great way to save a lot of headache. It is possible to get the a piece of software like SPICE to to work together with other python programs though, often with some pretty interesting results.

Run The Math, Or Try It Out?

January 30, 2021 by Elliot Williams 39 Comments

I was reading Sonya Vasquez’s marvelous piece on the capstan equation this week. It’s a short, practical introduction to a single equation that, unless you’re doing something very strange, covers everything you need to know about friction when designing something with a rope or a cable that has to turn a corner or navigate a wiggle. Think of a bike cable or, in Sonya’s case, a moveable dragon-head Chomper. Turns out, there’s math for that! Continue reading “Run The Math, Or Try It Out?” →

Gravity Simulations With An FPGA

January 2, 2017 by Lewin Day 25 Comments

Gravity can be a difficult thing to simulate effectively on a traditional CPU. The amount of calculation required increases exponentially with the number of particles in the simulation. This is an application perfect for parallel processing.

For their final project in ECE5760 at Cornell, [Mark Eiding] and [Brian Curless] decided to use an FPGA to rapidly process gravitational calculations. This allows them to simulate a thousand particles at up to 10 frames per second. With every particle having an attraction to every other, this works out to an astonishing 1 million inverse-square calculations per frame!

The team used an Altera DE2-115 development board to build the project. General operation is run by a Nios II processor, which handles the VGA display, loads initial conditions and controls memory. The FPGA is used as an accelerator for the gravity calculations, and lends the additional benefit of requiring less memory access operations as it runs all operations in parallel.

This project is a great example of how FPGAs can be used to create serious processing muscle for massively parallel tasks. Check out this great article on sorting with FPGAs that delves deeper into the subject. Video after the break.

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Grinding Gears: Figuring Out The Ratio

September 19, 2016 by Gerrit Coetzee 53 Comments

Practically any combination of motor and gearbox can be mathematically arranged to fit all sorts of problems. You could lift a crane with a pager motor, it just might take a few hundred years. However, figuring out exactly what ratio you need can feel bit backwards the first time you have to do it.

A gear is nothing more than a clever way to make two circles rotate in concert with each other as if they were perfectly joined at their circumferences. Rather than relying on the friction between two rotating disks in contact, the designer instead relies on the strength of a gear tooth as the factor limiting the amount of torque that can be applied to the gear.

Everything is in gearing is neatly proportional. As long as your point of reference is correct, and some other stuff. Uh, it gets easier with practice.

$\frac{radius_1}{radius_2} = \frac{velocity_1}{velocity_2} = \frac{tangentialforce_1}{tangentialforce_2} = \frac{torque_1}{torque_2}$

Now as my physics professors taught me to do, let’s skip the semantics and spare ourselves some pedantics. Let us assume that all gears have a constant velocity when you’ve averaged it all out. Sure there is a perceptible difference between a perfect involute and a primitive lantern gear, but for the sake of discussion it doesn’t matter at all. Especially if you’re just going to 3D print the thing. Let’s say that they’re sitting on perfect bearings and friction isn’t a thing unless we make it so. Also we’ll go ahead and make them perfectly aligned, depthed, and toleranced.

Typically, a gearbox is used for two things. You have a smaller torque that you’d like to make into a bigger one or you have one rotational velocity that you’d like to exchange for another. Typically torque is represented with a capital or lowercase Tau (Ττ) and rotational velocity likes to have a lowercase omega (ω). It also doesn’t matter at all; it just makes your equations look cooler.

Now a lot of tutorials like to start with the idea of rolling a smaller circle against a bigger one. If the smaller circle is a third as large as the big one, it will take three rotations of the small circle to make the big one rotate twice. However, it is my opinion that thinking it in terms of the force applied allows a designer to think about the gearing more effectively.