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Aristotle’s Wheel Paradox

Imagine a large wheel with a smaller wheel fixed inside it, both sharing the same center. When this composite wheel rolls along a surface without slipping, it appears that both the outer and inner wheels cover the same linear distance in one full rotation. This seems counterintuitive—the inner wheel has a smaller circumference, so how can both trace the same path length?

Understanding the Mechanics

The explanation lies in how each wheel interacts with the surface:

  • Outer Wheel: In contact with the ground, it rolls without slipping. The distance it covers in one full rotation equals its circumference.
  • Inner Wheel: It rotates with the outer wheel but doesn’t touch the ground. Instead, it’s dragged along, combining rotation with slipping. Its motion is not true rolling.

This difference becomes evident in the paths traced by points on each wheel. A point on the outer edge follows a cycloidal path. In contrast, a point on the inner wheel traces a curtate cycloid—a shorter, looping trajectory due to its proximity to the center.

The Illusion of Equal Distance in the Video

In the old video often used to illustrate this idea, a single wheel features two concentric circles drawn on it—one large, one small. As the wheel rolls without slipping, only the outer edge touches the ground. Both drawn circles rotate together, but neither rolls; they’re simply carried along.

Visually, both circles appear to move the same distance across the screen. This creates the illusion of equal travel. In reality, the smaller circle slips more than the larger one due to its tighter arc around the center. As you may have noticed yourself, this so-called paradox arises from a misunderstanding of how rotation and slipping function when only the outer rim is in contact with the surface.

A Mathematical Perspective

Mathematically, the paradox shows that a one-to-one correspondence between points on two rotating paths doesn’t imply equal arc lengths. While each point on the smaller circle aligns with a point on the larger one, their trajectories differ due to the nature of their motion.

Conclusion

Aristotle’s Wheel Paradox isn’t a true paradox, but a reminder that intuition can mislead when dealing with motion and geometry. The apparent equal travel of both wheels results not from identical rolling behavior, but from the interplay between rotation, slipping, and perception.

⇨ Further reading.

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The Trapezoid Trap

Here’s a rather tricky puzzle—perfect for the classroom or a fun activity with your kids (and possibly an excuse to sharpen your own spatial skills).

Print and cut out the five puzzle pieces (see Fig. A), then try to fit them all into the larger shape (Fig. B) without overlapping. Yes, it’s possible… As you may have noticed, all the pieces—including the final shape—are similar right trapezoids. They do, however, vary in scale, just to keep things interesting.

Cut out the 5 puzzle pieces (right trapezoids; fig. A) in order to fit them all into the larger shape (fig. B) without overlapping.

You can download the full template in PDF format here.

The first person to post a correct solution will receive a set of our original postcard designs.

And if you find yourself strangely fascinated by these slanted quadrilaterals, you’re not alone. Even ancient Greek mathematicians toyed with shapes like these to explore proportions and symmetry. Curious about trapezoids (or wondering if they’re secretly out to get you)? Here’s a helpful read: https://en.wikipedia.org/wiki/Trapezoid

Happy puzzling—and remember, if it feels impossible, you’re probably getting close.

(Hint: Some pieces may need to be flipped over, as if seen through a mirror.)

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Benford’s Law: Why 1 Comes First

Benford’s Law is a curious mathematical rule that describes how often different digits (1–9) appear as the first digit in many real-life datasets. Surprisingly, lower digits (like 1) show up much more frequently than higher ones (like 9).

Simple Formula

The probability of a digit d being the first digit is:

📌 P(d) = log₁₀(1 + 1/d)

For example, the number 1 appears as the first digit about 30% of the time, while 9 appears only about 5% of the time! This pattern shows up in finance, science, populations, and even street addresses.

A fascinating rule of nature—numbers aren’t as random as they seem!

benford's law

⇨ Further reading.

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The Symbiotic Relationship Between Physics and Mathematics

Physics created mathematics, or was it mathematics that created physics? Back in ancient times, it is evident that gravity gave rise to what we now call a 90-degree angle, a symbol of perpendicularity in two or three dimensions. This marked our hominid ancestors’ first concrete encounter with mathematics. Other rediscoveries followed, as mathematics is, at its core, a constant RE-discovery of the hidden facets of the world around us. In the natural world, physics shaped the mathematical instincts of countless builders, from bees constructing their hives to spiders weaving their webs, both sensing forces and symmetry with innate precision.
Today, however, the realms of the infinitely small and the infinitely large can only be understood through mathematics. Our senses and intuition fall short when faced with these new dimensions. Concepts such as the EPR effect, gauge invariance, and spin would remain invisible to us without the crutches provided by mathematics.

· EPR effect: A quantum phenomenon where two particles become instantaneously linked, regardless of distance.
· Gauge invariance: A principle that certain physical properties remain unchanged under specific transformations.
· Spin: A fundamental quantum property of particles that represents intrinsic angular momentum.

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The Metallic Ratios: Beyond the Golden Ratio

Many people are familiar with the Golden Ratio (φ), but it is just one member of a broader family known as the Metallic Ratios. These ratios describe a recursive relationship between the sides of a rectangle. Given a rectangle with side lengths A and B (B > A), the Metallic Ratios satisfy the equation:

A / B = (B – n × A) / A

where n is a fixed integer.

For the Golden Ratio (n = 1):
A / B = (B – A) / A
which leads to:
φ² – φ – 1 = 0
solving this gives:
φ = (1 + √5) / 2 ≈ 1.618…

For the Silver Ratio (n = 2):
A / B = (B – 2A) / A
which leads to:
ψ² – 2ψ – 1 = 0
solving this gives:
ψ = 1 + √2 ≈ 2.414…

Different values of n define other Metallic Ratios. The table below presents some of them.

Curious minds, don’t stop here—explore the fascinating properties of the other Metallic Ratios and see what patterns and surprises you can uncover!

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The Art of Puzzling: A Mind-Bending Philosophy

Puzzles have always been an essential part of human culture, not just as entertainment but as a means of sharpening the mind. From the ancient Rhind Papyrus to modern brain teasers, they challenge our perceptions, forcing us to see beyond the obvious. They teach us that even the simplest elements—dots, lines, shapes—can hold extraordinary meaning, just as they have throughout history, from astronomy to the birth of the decimal system.

At their core, puzzles embody a philosophy: they cultivate curiosity, nurture self-irony, and encourage a critical yet playful detachment from reality. Each challenge is an invitation to think differently, to question, to explore. They remind us that solutions often emerge not from brute logic but from creative leaps, from seeing patterns where others see chaos.
For those who wish to dive deeper into this world, I invite you to experiment and create your own puzzles with our online tutorials. There, you can build, solve, and enjoy the endless possibilities that come from playing with perception. After all, life itself is nothing but a grand puzzle—one we’re all trying to piece together.

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UMBRELLA ILLUSION

One of my illusions from the late ’90s. Take a look at the colorful umbrellas in Figures A and B—are they the same or different? About 80% of people will say that Umbrella A has jagged, zigzag edges, while Umbrella B has smooth, wavy lines. But here’s the trick—you’ve been fooled by the brightness contrast of the rays inside the umbrellas. In reality, both umbrellas are identical in shape, perfectly congruent.

This illusion shows a phenomenon called curvature blindness, which was rediscovered in 2017 by Japanese psychologist Kohske Takahashi. He created a powerful variant and studied its impact on how we perceive shapes.

© Kohske Takahashi – The wavy lines appear different depending on the background and how the repetitive dark color is applied to them.

⇨ Read further

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The Zaniest Word in Math: Zenzizenzizenzic

One of the most peculiar numeral words in English, zenzizenzizenzic (/’zɛnziːzɛnziːzɛnzik/), denotes the square of the square of a number’s square. It appeared only once in English, in Robert Recorde’s The Whetstone of Wit (1557). The term derives from the obsolete zenzic, meaning the square of a number. Zenzic was borrowed from German, where mathematicians of the 14th and 15th centuries adopted it from the medieval Italian censo, itself a descendant of Latin census. Italian algebraists used censo to translate the Arabic māl (literally “possessions” or “property”), the standard term for a squared number. This association arose because early mathematicians, including the Arabs, conceptualized squared numbers as representing areas, particularly land—hence, property.

Notably, zenzizenzizenzic is the only English word with six Zs.