Category: Science

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Conway’s Pinwheel Tiling

John Conway uncovered a right triangle whose sides follow the ratio 1 : 2 : √5. This triangle can be subdivided into five smaller triangles similar to the original. By making the whole triangle the central piece of a larger one and repeating the process, the pattern grows step by step, producing an aperiodic tiling of the plane with triangles appearing in infinitely many orientations.

Conway’s Pinwheel Tiling

The tiling is attributed to Conway, but Charles Radin was the first to formally publish it in Annals of Mathematics (1994), giving Conway full credit for the discovery.

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Tangram Gone Wild

A hands-on geometry challenge
This mega-tangram combines five Sam Loyd–style sets: four identical ones (with one missing its triangular piece) and a fifth that’s slightly larger.

Each set can be arranged to form six geometric shapes: a square, a truncated triangle, a parallelogram, a rectangle, a cross, and a right triangle.

Use this activity to explore geometric concepts such as symmetry, area, and spatial reasoning. You can also create your own rules or design new shapes, making it an interactive way to develop problem-solving and critical-thinking skills in a classroom or group setting.

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Eyespot Mimicry

The cat in the picture was my most loyal assistant, Sylvester, a beautiful Abyssinian who for years made our studio his home. He had adopted a comfortable chair as his pied-à-terre, and while he slept there, something curious often caught the eye: if you stared at his closed eyelids, didn’t it seem as though they suddenly opened?

Eye cat camouflage
Image taken from my book World of Visual Illusions, available from Amazon.

The clear and dark stripes around his eyes (Fig. A) roughly trace the outlines of real cat eyes (Fig. B). In the animal world, eyes are powerful signals—used to warn, deceive, or intimidate. These “subjective eyes,” known to scientists as ocelli, are a kind of passive defense, deterring potential threats even in sleep. When awake, the same markings act like natural eyeliners, making his eyes appear larger and more striking. I was the first to study this phenomenon in cats, observing how these markings function as a subtle form of visual automimicry.*

This visual strategy, known as automimicry, is widespread in nature. Many butterflies, such as Smerinthus ocellatus (Fig. C), display prominent eyespots on their wings—patterns that echo the gaze of larger animals, enough to startle or mislead predators.

*Automimicry is most often studied in wild species

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Why Our Year Is the Way It Is

The Origin of ‘Bissextile’— and Why September Isn’t Month Seven Anymore

The Earth takes 365 days, 5 hours, 48 minutes, and 46 seconds to orbit the Sun. That little extra time is why we have leap years.

In 46 BCE—known as the Year of Confusion—the Roman calendar had drifted badly out of sync with the seasons. To fix this, Julius Caesar made that year an epic 445 days long to catch up, then set a simple rule: add one extra day every four years. The Romans slipped it in before sexto calendas Martias (February 24), calling it bis sexto—the origin of “bissextile,” or leap year.

The Julian calendar slightly overshot the true solar year by about 11 minutes. By 1582, the drift had pushed dates 10 days ahead of the Sun. Pope Gregory XIII trimmed those days in October (October 4 was followed by October 15) and fine-tuned the leap-year rule: century years aren’t leap years unless divisible by 400. That’s why 2000 had a February 29, but 1900 didn’t.

This change keeps our calendar so close to the Sun’s timing that it will take more than 3,000 years to be off by a single day.

The Gregorian calendar rolled out in Catholic countries in 1582, spread to Protestant nations over the next century, and reached Russia in 1918. Many Orthodox churches still use the old Julian dates for religious feasts—Orthodox Christmas falls on January 7.

Ancient Rome started the year on March 1. Charlemagne’s empire began it on Christmas Day. France’s King Charles IX moved it to January 1 in 1564—leaving September (septem, 7), October (octo, 8), November (novem, 9), and December (decem, 10) stuck with names that no longer match their place in the year.

old calendar
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The Two Faces of a Spoon

At Archimedes’ Lab, we love discovering how even the most ordinary objects can reveal extraordinary truths. Take a spoon—yes, just a regular kitchen spoon. Hold it up and take a look… Why is your reflection upside-down on one side, and upright on the other?

That’s not magic—it’s optics! The concave side (the scooping part) acts like a converging lens with a focal point. When your face is close enough, the reflected light rays cross at that point, flipping your image. Voilà—an upside-down version of you. Now flip to the convex side, and your image stays upright, just a little smaller and bent around the edges.

During our workshops, we like to turn this into a playful moment. We joke with kids that the spoon is magical—it reveals who’s telling the truth. “If your face is upside-down,” we say with a grin, “the spoon knows you’re fibbing!” The reaction? Giggles, wide eyes, and just the right moment to sneak in a quick optics lesson.

One humble spoon. Two faces. A world of curious learning.

Reflections on a spoon
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Hands-On Wonders: A Mathemagical Collection

Ever wondered what happens when math puts on a magician’s hat? These books are the distilled magic of my hands-on math workshops across Europe — from Paris to Palermo, Geneva to Ghent — where paper folded, minds twisted, and logic sparkled in English, French, and Italian!

Impossible Folding Puzzles

1) “Impossible Folding Puzzles and Other Mathematical Paradoxes” — a playful dive into mind-bending problems where nothing is quite what it seems. Can a puzzle have no solution… or too many? Dare to fold your brain.

Still available on Amazon.

2) “Pliages, découpages et magie : Manuel de prestidi-géométrie” — where math meets illusion to spark curiosity and creativity.
Perfect for teachers, students, and curious minds: touch, fold, cut… and let the magic unfold!
Available on Amazon.

2) “Pliages, découpages et magie : Manuel de prestidi-géométrie” — un livre où maths et illusion se rencontrent pour éveiller curiosité et créativité.
Pour enseignants, élèves et esprits joueurs : touchez, pliez, découpez… la magie opère!
Dispo sur Amazon.

Pliage decoupages

3) “MateMagica” —  They say there’s enough carbon in the human body to make 900 pencils… but just one is all you need for these clever puzzles!
Fun, surprising, and thought-provoking — because, as Martin Gardner put it, “Mathematics is just the solution of a puzzle.”
Now on Amazon.

3) “MateMagica” —  Si dice che nel corpo umano ci sia abbastanza carbonio per 900 matite… ma per questi rompicapi ne basta una!
Sorprendenti, divertenti e stimolanti — perché, come diceva Martin Gardner, “la matematica è nient’altro che la soluzione di un rompicapo.”
Disponibile su Amazon.

I write and illustrate my own books in five languages: English, French, Italian, German, and Spanish.
If you’re a publisher or literary agent seeking original, high-quality educational content that blends creativity with clarity, I’d be pleased to explore potential collaborations.

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Hidden in Plain Sight


Many prey animals, like deer and wild boars (ungulates), are dichromats — they have only two types of cone cells, sensitive to:
• short wavelengths (blue)
• medium wavelengths (green)
They lack red-sensitive cones, so they can’t tell red or orange from green or brown.To their eyes, a tiger’s vivid orange coat blends seamlessly into the forest — it looks greenish or brownish, like the surrounding foliage.

tiger color camouflage

So, a tiger’s color isn’t designed to fool us — it’s meant to fool them.
To us, it’s a blazing orange predator.
To a deer, it’s a silent shadow in the grass.
Camouflage, it turns out, is all about who’s watching.

Further reading: https://www.archimedes-lab.org/what_is_seeing.html

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Roman Numerals vs Hindu-Arabic Numbers in Psychology

What Our Brains See vs What They Read

The way our brain interprets Roman numerals and Hindu-Arabic numbers reveals an interesting distinction in psychology—especially when viewed through the lens of communication theory.

According to the psychologist and communication theorist Paul Watzlawick, signs and symbols can be divided into two categories: analogical and digital.

· Analogical signs resemble what they represent. They are intuitive and visually descriptive.

· Digital signs are symbolic. They rely on learned codes and have no visual connection to what they signify.

In this sense, Roman numerals (like I, II, III) are analogical. When we see “II”, we can immediately see two units. The visual repetition reflects the quantity directly—our brain interprets the number almost as a drawing of its value.

On the other hand, Hindu-Arabic numbers (like 2, 3, 4) are digital. The symbol “2” doesn’t visually resemble two objects—it’s abstract. Understanding it depends on prior learning and decoding. The brain treats it more like language than image.

This distinction matters. Roman numerals engage perception in a way that mimics reality. Arabic numerals, by contrast, engage abstract reasoning. The first shows, the second tells.

In daily life, we may not notice the difference—but psychologically, the visual nature of Roman numerals connects us to meaning more directly, while the efficiency of Arabic numerals supports speed, calculation, and abstraction.

In short:
Roman numerals speak to the eye.
Arabic numbers speak to the mind.

⇨ More about numbers.

roman numbers vs arabic numbers
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GHOST COLORS (2)

Take a closer look at the image below—you’re in for a mind-bending surprise! There’s absolutely NO yellow, and not even red or green in sight. (Zoom in if you don’t believe it!) The only actual colors used are blue, cyan, and magenta.
What you’re seeing is a fascinating phenomenon known as “simultaneous color contrast” and “color assimilation”.—effects that ‘trick’ the brain into perceiving colors that aren’t really there.

© Gianni A. Sarcone

When you magnify a portion of the image in Photoshop, as seen below, to the right, you see a series of black bars. Some gaps that appeared yellow at first are actually pure WHITE, and the eyedropper tool confirms that only CYAN and MAGENTA are present.

The green tint perceived in some areas is a result of the interaction between black and cyan, just as the appearance of red is due to the interplay of black and magenta. As for the yellow-looking circle, it’s actually an optical effect caused by the white space between the black bars reacting to the surrounding dark blue lines — a classic case of simultaneous contrast.

© GIanni A. Sarcone

⇨ Learn more.

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The Origins of Our Numerals

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.