The Kitāb al-Fuṣūl fī al-Ḥisāb al-Hindī (كتاب الفصول في الحساب الهندي), or The Book of Chapters on Hindu Arithmetic, authored by Abū al-Ḥasan Aḥmad ibn Ibrāhīm al-Uqlīdisī in 952 CE, is the earliest known Arabic treatise detailing Indian arithmetic and the use of Hindu-Arabic numerals. A unique manuscript of this work is preserved in the Yeni Cami Library in Istanbul. The treatise also offers the earliest documentation of numerals in use in Damascus and Baghdad.
Another significant reference is found in Talqīḥ al-Afkār bi-Rusūm Ḥurūf al-Ghubār (تلقيح الأفكار برُسوم حروف الغبار), or Fertilization of Thoughts with the Help of Dust Letters, by the Berber mathematician Ibn al-Yāsamīn (ابن الياسمين), who died in 1204. In the excerpt shown below, he presents the Indian numerals, stating:
“Know that specific forms have been chosen to represent all numbers; they are called ‘ghubār’ (dust), and they are these (first row). They may also appear like this (second row). However, among us, people use the first type of forms.”
An intriguing anecdote about Ibn al-Yāsamīn is that he composed mathematical poems, such as the Urjūza fī al-Jabr wa al-Muqābala, to make algebra more accessible. These poetic works were not only educational tools but also reflected the rich interplay between mathematics and literature in the Islamic Golden Age.
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!
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.
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!
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.
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.
Sometimes, shifting our view is the key to seeing things more clearly.
This quincuncial Antarctic-centered projection offers a fresh take on the South Pole, allowing us to appreciate Antarctica from a different perspective, as well as the vastness of the world’s oceans.
In the 18th century, Swiss scientist Horace-Bénédict de Saussure invented the “cyanometer,” a simple yet poetic device— a circular chart with 52 shades of blue, ranging from white to dark blue, to measure the sky’s blue hues. Inspired by his love for the Alps, especially Mont Blanc, Saussure climbed to great heights to study the deeper blue skies at higher altitudes.
Saussure believed the sky’s color was influenced by atmospheric particles. He tested the cyanometer at different elevations, noting that the summit of Mont Blanc matched the 39th shade. Later, explorer Alexander von Humboldt set a new record of 46 during his Andean expeditions.
Although Saussure’s theory linking sky color to atmospheric moisture didn’t pan out, his invention captured imaginations. Though it faded from science, the cyanometer lives on as an artistic and symbolic nod to our curiosity about nature. Modern versions even track air quality while celebrating the beauty of the ever-changing sky.
Next time you look up, think of Saussure and his ingenious little tool!
Ever wondered what happens when you shoot a bullet in water? The deeper the water, the faster the bullet slows down. Water’s higher density causes much more resistance than air, rapidly draining the bullet’s kinetic energy. In just a few meters, the bullet can come to a complete stop! Why? Water creates a drag force that decelerates the bullet. The formula behind this? Drag Force (Fₑ) = ½ * Cₔ * ρ * A * v² Where: Cₔ = Drag coefficient (depends on the bullet shape) ρ = Water’s density (about 1000 kg/m³) A = Bullet’s cross-sectional area v = Bullet’s velocity
As the bullet travels, drag slows it down and uses up its energy quickly. In just a few meters, the bullet is stopped dead in its tracks!
