Mathematics reveals the hidden elegance of geometric forms, where even the simplest shapes can display surprising and complex properties. Among the most fascinating of these are the sphericon, the Möbius strip and the rattleback, each with distinct mathematical significance that continues to intrigue mathematicians, scientists and engineers.
Take the sphericon, for instance, a solid that combines characteristics of both cones and spheres, resulting in a unique rolling behaviour. Constructed by slicing a bicone (two cones joined at their bases) in half, rotating one half by 90 degrees and reconnecting the pieces, the sphericon has a smooth, continuous curvature.
Unlike spheres or cylinders that roll smoothly along a predictable circular path, the sphericon’s motion causes it to wobble as it rolls. Every part of its surface contacts the ground at some point during this movement, giving rise to complex rolling dynamics. Its volume happens to be exactly half of that of a sphere with the same radius but looks nothing like one.
The Möbius strip is the most popular of the bunch. You might have seen Iron Man interacting with one, when he figures out how to time-travel in Avengers: Endgame. By taking a strip of paper, twisting it by 180 degrees and joining the ends, one creates a surface with just one side and one boundary.
This non-orientable surface challenges conventional geometry: tracing a path along the strip allows a point to cover the entire surface, returning to the starting position without crossing an edge. If you lived on a Möbius strip, you wouldn’t be able to consistently distinguish clockwise from anticlockwise turns! Nowadays, however, they’re actually used to make conveyor belts that wear evenly on both sides during use.
Then there is the rattleback, a solid that defies our intuitive understanding of rotational dynamics. You might have seen this as a novel children’s toy. This semi-ellipsoidal object has the peculiar ability to spin smoothly in one direction but, when spun in the opposite direction, quickly decelerates, rattles and reverses its spin. This behaviour arises from the object’s asymmetric mass distribution and interaction with angular momentum. The rattleback’s motion can be described through differential equations that capture the instability when spun against its preferred direction.
Together, the sphericon, Möbius strip and rattleback highlight the intricate relationship between geometry, physics and mathematics. Their unique properties challenge our understanding of stability, motion and structure, demonstrating how simple shapes can reveal profound mathematical truths.
Sound Bites
• In 2023, mathematicians discovered the first shape capable of covering a plane without any gaps or overlaps, while producing a pattern that never repeats ‒ a truly remarkable achievement. This shape, commonly known as ‘The Hat’ or ‘The Einstein Hat’, is not named after the famous scientist, but rather because the word ‘einstein’, derived from the German words ‘ein’ and ‘stein’, translates to ‘one stone’.
• Peter Borg (University of Malta), alongside Carl Feghali and Remi Pellerin (both from the University of Lyon), solved a problem originally posed by world-renowned mathematician Gyula O. H. Katona regarding cliques and independent sets in graphs. Their breakthrough was published last year in the journal Discrete Applied Mathematics, ranked 24th globally in this field.
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DID YOU KNOW?
• In 1879, Georg Cantor demonstrated that not all infinities are the same size. While the set of counting numbers (0, 1, 2, ...) and whole numbers (…, -2, -1, 0, 1, 2, …) share the same type of infinity, other sets, like the real numbers, represent a much larger infinity.
• There are over 300 unique proofs for Pythagoras’ Theorem, the most of any theorem in mathematics.
• Euler’s Identity, where e raised to the power of i times pi equals -1, beautifully connects the important numbers e, i (the square root of -1), and pi – none of which are negative or whole numbers – to give the result of -1. Many people consider it the most beautiful equation in mathematics.
• The number ‘zero’ was formally introduced in India in the fifth century. Arab mathematicians later spread the concept, along with the Arabic numeral system, to the Western world.
For more trivia, see: www.um.edu.mt/think.