# Lego Technic Binary Expansion of Pi

This Pi Day video demonstrates how to approximate a gearing ratio of Pi by representing Pi as a binary number and using differentials arranged in a chain formation to create the binary representation. Each differential implements 1 binary digit. If the binary digit is zero the differential can be replaced by a gearing ratio of a half to reduce the number of differentials required. This model has 23 binary digits and approximates Pi to 7 significant figures beating last year's model which approximated Pi to 6 significant figures using the continued fraction 355/113 and implemented using a differential pair. See the video https://youtu.be/fGg11gqZy8M.

This model has 202 gears and is almost 90cm long! The entire model is driven by a medium sized PF motor.

If you are interested in learning how to create a gearing ratio of N/P using differentials then check out this video https://youtu.be/ORvnzx36j2g

# More to explore

## Lego Technic New 12/20 Gear High Speed Friction Test

The Lego Technic set 42140 App-Controlled Transformation Vehicle introduces two new gears - a flat version of the 12 tooth bevel gear and a flat version of the 20 tooth bevel gear. These gears are designed to have less friction.

In this follow up video I measure the high speed difference ...read more

## Lego Technic Pi Day Machine

Here I present a Pi Day special Lego Technic machine that produces a gearing ratio of Pi/8 using a 56mm wheel driving a 13x13 square wheel.

The correctness of the ratio is confirmed by matching it with output of a 2 differential gearing system that produces a ratio of 355/9...read more

## Lego Technic torque in practice - how to work with torque in automatic gearbox design

n this video I present a ten level Lego Technic torque creation gadget for testing lego power function motors and automatic gear switching mechanisms. The effect of torque on a large lego PF motor is tested against the theoretical power curve to compare the optimal gear switching point in practic...read more

## New Flat 12/20 Gear Friction

The Lego Technic set 42140 App-Controlled Transformation Vehicle introduces two new gears - a flat version of the 12 tooth bevel gear and a flat version of the 20 tooth bevel gear. These gears are designed to have less friction.

In this video I measure the difference in gearing efficiency...read more

### Gearing Ratio Calculator Tool Help

## Measuring Gearbox Efficiency

This video describes a method for measuring the efficiency of a gearbox using the Lego Technic Powered Up hub. Gearbox efficiency is defined as the ratio of the output power to the input power and here I show how to measure each of these values to allow you to calculate the overall efficiency.read more

## Lego Technic Gearing Ratio Calculator Tool

Here I present my new Lego Technic Gearing Ratio Calculator tool. This tool allows you to upload your Lego Technic .ldr (Lego Draw) file and it will simulate the gearing system and visualize it as well as showing the gearing ratios relative to the input component.

The tool can be accessed...read more

## Lego Technic Applied Gearbox Theory Improved

This Lego Technic video demonstrates an improved design of a Lego technic 2 speed automatic gearbox based on a theoretical approach. Typically a Lego automatic gearbox consists of two stages namely a gear selector and a torque detector. In this video, both of these components are improved upon.read more

## Lego Technic Gearbox Theory Applied - how to apply theory to help design 2 speed automatic gearboxes

This video applies lego technic gearbox, torque, and gearing ratio theory to design a prototype lego 2 speed automatic gearbox. First I describe the main components of an automatic gearbox, namely a gearing ratio selector and a torque detector. The ideal gear switch over torque point is described...read more

## Lego Technic 3 Speed Automatic Gearbox

Here a Lego Technic 3 speed automatic gearbox is presented. The gearbox automatically switches between one of 3 gear ratios, namely a ratio of 1, 1/3rd and 1/9th depending on the load on the output. It uses two 2 speed automatic transmissions in series to create the three speed automatic ge...read more