By Eben van Tonder, 22 April 25
1. The Involute Blade and Euler
Involute geometry is the shape traced by the end of a taut string unwinding from a circle. This seemingly simple curve has enormous implications in precision engineering, especially in gear systems and high-speed slicers such as those made by Treif.
- Who invented the math?
The foundation of calculus was independently developed in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz. However, it was Leonhard Euler (1707–1783) who gave calculus much of its usable form and introduced functions such as f(x), differential equations, and curvature calculations that underpin the modern use of involute shapes. - Euler and the Involute:
Euler formalised the mathematics of curvature and parametric geometry, which made it possible to model the involute curve and predict its contact properties — essential for tools like blades and gears. - Why is the involute blade excellent?
It maintains a constant angle of engagement with the product being cut. This minimizes resistance and maximizes slicing efficiency. - Self-sharpening property:
The angle at which the involute blade meets the product surface changes along the curve, causing the contact point to move across the blade during each cut. This natural movement distributes wear and avoids constant contact on one edge, keeping the blade sharp longer. It’s the same principle used in gear teeth, where the involute ensures smooth, wear-resistant engagement.
2. Servo Systems and Feedback Mechanisms
Modern Treif slicers, portioners, and vacuum fillers operate with servo motors, which allow high-precision motion control based on continuous feedback.
How servo systems work:
- Input: A command (e.g., position, speed, volume) is issued by the controller.
- Movement: The motor acts.
- Sensor feedback: Encoders measure actual movement.
- Comparison and correction: The system adjusts if output differs from input — this is a closed-loop system.
Historical roots:
- The principle began with James Watt’s centrifugal governor (1788), a steam engine mechanism that adjusted steam flow to maintain a stable speed.
- This concept matured with James Clerk Maxwell’s 1868 paper “On Governors”, which mathematically described dynamic feedback systems.
- Later, Norbert Wiener coined the field of cybernetics, defining control and feedback in biological and mechanical systems (Wiener, 1948).
3. Feedback in Biology and Ecology
Nature has long employed feedback systems. Biological and ecological systems operate through negative feedback loops — constantly adjusting and correcting rather than stabilising in static equilibrium.
- Examples:
- Thermoregulation in mammals (body temperature control)
- Predator-prey cycles (e.g., lynx and hare populations)
- Homeostasis in cell environments
These systems are not based on equilibrium but on adaptive oscillation. Real stability arises from dynamic response to changing conditions, not from remaining unchanged.
4. How These Principles Shape Modern Society
All modern high-speed portioning and filling technology rests on these cornerstones:
- Euler’s mathematics allows us to model, design, and optimise mechanical and natural systems.
- The involute blade, grounded in calculus and geometry, enables precise, low-resistance cutting.
- Feedback systems make servo fillers and slicers intelligent — they learn, adjust, and correct in real time.
Societal Impacts:
- Factory automation: Precision fillers dose meat accurately in grams, even with soft materials.
- Medical tech: Infusion pumps use servo-controlled feedback to deliver drugs.
- Climate response systems: Feedback modelling is used to simulate tipping points.
- Cybernetics and AI: Adaptive response lies at the heart of self-learning algorithms.
We still live in a world profoundly shaped by these early discoveries — and every clean slice of sausage or portioned steak pays quiet homage to Euler, Newton, Watt, and Wiener.
References
- Euler, L. (1748). Introductio in analysin infinitorum.
- Newton, I. (1666). Method of Fluxions (published posthumously in 1736).
- Leibniz, G. W. (1684). Nova methodus pro maximis et minimis.
- Maxwell, J. C. (1868). “On Governors.” Proceedings of the Royal Society of London, 16, 270–283.
- Wiener, N. (1948). Cybernetics: Or Control and Communication in the Animal and the Machine. MIT Press.
- Slocum, A. H. (1992). Precision Machine Design. Society of Manufacturing Engineers.
- Resnick, M. (1994). Turtles, Termites, and Traffic Jams: Explorations in Massively Parallel Microworlds. MIT Press.
- Dörner, D. (1996). The Logic of Failure: Recognizing and Avoiding Error in Complex Situations. Basic Books.
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