A curve is called “isotrepent” (or “syntrepent”) if a copy of itself can roll along it without slipping, maintaining tangency at all points.
In mathematics, isotrepent curves describe a specific dynamic relationship between two curves that roll over each other while rotating around a fixed point. This concept is particularly relevant in the study of planar motion and point trajectories in the plane.
Isotrepent curves are often examined in differential geometry, the branch of mathematics that explores the relationships between curves and surfaces. In essence, isotrepency captures the interaction between two curves via rolling and rotation, offering a rich field of study in both geometry and mathematical dynamics.
Mathematician Kasper Muller explored Goldbach’s Conjecture, which states: “Every even positive integer greater than 2 can be written as the sum of two primes.”
This has been verified for all even numbers up to 4×10¹⁸.
Muller proposed a striking geometric model. He arranged all even numbers consecutively along a horizontal line as blue dots and drew circles centered on each even number. Remarkably, the circumference of each circle always intersects two primes on the line—for example:
Center 4 → intersects 3 and 5 (sum = 4 × 2)
Center 6 → intersects 5 and 7 (sum = 6 × 2)
Center 8 → intersects 5 and 11 (sum = 8 × 2)
Center 10 → intersects 7 and 13 (sum = 10 × 2)
Center 12 → intersects 7 and 17 (sum = 12 × 2)
…
This visual approach gives an intuitive geometric perspective on a classic number theory problem.
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.
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.
Illustration from the Brockhaus and Efron Encyclopedic Dictionary (Энциклопедический словарь Брокгауза и Ефрона, 1890–1907)
Long before computers and digital graphics, scholars relied on meticulously crafted diagrams to visualize ideas. This monumental Russian encyclopaedia—published in 35 small or 86 large volumes—contains over 121,000 articles, 7,800 illustrations, and 235 maps. Contributions came from some of Russia’s greatest minds, including Dmitri Mendeleev and Vladimir Solovyov. Originally a joint project of Leipzig and St. Petersburg publishers.
There’s something striking about the Kaktovik Iñupiaq numerals; they make numbers immediate and tangible. Created in 1994 by middle school students in Kaktovik, Alaska, this base-20 system mirrors the Iñupiaq language’s counting structure. Each symbol visually represents quantity, making arithmetic intuitive and culturally grounded—a rare instance where numbers truly speak the language of the people.
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?
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
This vector field is defined by the function: F(x, y) = ( -y – x(x² + y²) , x – y(x² + y²) ) Each arrow indicates both the direction and the magnitude of the field at different points in space. Similar to gravity, the structure of the field draws everything inward, spiraling toward the center.
Mathematics not only models the mechanics of the universe—it unveils dimensions of reality that lie beyond direct perception.
The Moon never ceases to amaze. It appears almost the same angular size as the Sun in our sky (both ≈ 0.5°), a remarkable coincidence caused by the Sun being roughly 400 times farther from Earth than the Moon—and about 400 times larger—allowing the phenomenon of perfect solar eclipses.
Thanks to tidal locking, a precise 1:1 spin-orbit resonance, the Moon always presents the same face to Earth. Meanwhile, our planet rotates once every 24 hours* with an axis tilted by nearly 24°, giving rise to the seasons.
About 13 lunar cycles fit into one solar year, though more precisely it’s closer to 12.4. To reconcile this, ancient calendars alternated between 12 and 13 lunar months to stay in step with the solar year. Even more remarkable is the Moon’s Metonic cycle, or enneadecaeteris (from the Ancient Greek ἐννεακαιδεκαετηρίς, meaning ‘nineteen’), a period of 19 years after which the lunar phases repeat on the same dates.
Another intriguing visual coincidence involves simple geometry: take the diameter of Earth, multiply it by 4, divide by π, then subtract the Earth’s diameter—you get almost exactly the Moon’s diameter:
(4 × D_Earth / π) – D_Earth ≈ D_Moon
In other words, imagine Earth inside a perfect square. Transform the square’s perimeter into a circle of the same length (red circle). The difference between this circle’s diameter and Earth’s closely matches the Moon’s diameter—a subtle, elegant harmony of nature and mathematics.
At about 3,500 km—roughly a quarter of Earth’s diameter—the Moon is unusually large, leading some astronomers to view Earth–Moon as a double planet.
Of course, a bit of poetic license has been taken with the numbers here. A day is not exactly 24 hours down to the second, and few people outside specialized circles will mention that a sidereal day is slightly shorter than a solar one—about 23 hours, 56 minutes, and 4 seconds.
Communication is less a matter of nitpicking than of reaching the right audience. My aim is not to satisfy grumpy math teachers or know-it-alls eager to flaunt their knowledge, but to speak to the curious—those with sharp critical thinking who still preserve their sense of wonder at the world we inhabit.
This leads to a philosophical question: how far can a number that interprets reality—length, space, time, degrees, and so on—be rounded while still remaining faithful to that reality? The artist, the craftsman, the poet senses it instinctively. The mathematician, on the other hand, needs elaborate formulas to determine it.
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.
The three green strips of ten numbers share a curious property: each number tells how often the red digit on its row appears in the other two strips. Take the digit 2 in strip A, for example—it sits next to red 7, meaning that there’s one 7 in strip B and one in strip C. The 1 in strip B (row 7) confirms there’s one 7 in strips A and C, and the same goes for strip C. This reciprocal logic applies all the way through, for every digit.
