### School projects

We just started a series of workshops in schools! The first subjects are *Trigonometry and Navigation* and *Linear Equations and Differential Gears*.

We just started a series of workshops in schools! The first subjects are *Trigonometry and Navigation* and *Linear Equations and Differential Gears*.

The world's first replica of Tchebyshev's ingenious mechanical calculator from 1876 with any construction set. The original calculator is on display at the State Museum of the History of St. Petersburg. In all previous calculating machines the carry happened suddenly. The carry mechanisms were the central design and manufacturing problem. Tchebychev showed how epicyclic gear trains could be used to obtain a continuous carry mechanism that works reliably with arbitrary many digits. My fischertechnik replica shares this quality: Arbitrary many of the shown addition units can be composed to obtain arbitrary high precision. The transmission of 10:1 for the continuous carry cannot be realized directly with fischertechnik spur gears, so I had to redesign the gear train from scratch. I also added a simple multiplication unit that should make the machine an ideal and fun tool for learning or rediscovering the basic laws of arithmetic.

Differential gears are an ideal tool to illustrate the use of coordinate transformations. The mathematical setting is as simple as possibe and students experience a real world application. In a differential gear three rotational motions around one common axis are coupled mechanically. Usually, there are two sun gears and a carrier with a third gear meshing the two sun gears each under an angle of 90°. An observer who turns synchronously with the carrier measures the angles *x'* and *y'* between the carrier mark and the two marks on the sun gears. These angles clearly obey the equation *x'* + *y'* = 0. A general observer measures three angles *x*, *y*, *z* between a fourth mark and the three previous marks. Turning the whole mechanism by the angle *-z* of the carrier one sees the relations *x'* = *x* - *z* and *y'* = *y* - *z*. Thus, for the general observer, all possible positions of the mechanism are described by the linear equation *x* + *y* - 2*z* = 0. This can convincingly be illustrated by keeping one of the sun gears fixed and turning the other one. The carrier will turn halfway.

Binary arithmetic is one of the core concepts of modern computer technology. A 4-bit binary adder is constructed from only a few fischertechnik parts. It consists of four ball flip-flops, a concept invented by John Thomas Godfrey in 1965. The calculator can be used to explore binary addition, subtraction, multiplication and division in an interactive and memorizable way. It was designed to be used in classroom: It can be built by students in less than one hour, works reliably without any adjustment, and the required parts are cheap.

Kelvin is a string calculator for the solution of simultaneous linear equations. It is built with the fischertechnik construction set. This calculator roughly models devices invented by Thomson in 1878 and developed by Wilbur in 1934. The solution process has been altered to make a smooth operation possible within the limits of a construction set and to involve the user more actively. The calculator can be used to explore a mathematical core concept in an interactive and truly hands-on way.