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Building Perfect Circles from Ellipses

A curve is called “isotrepent” (or “syntrepent”) if a copy of itself can roll along it without slipping, maintaining tangency at all points.

Building Perfect Circles from Ellipses

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.

Building Perfect Circles from Ellipses - 2

☞ Further reading.

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Circles, Primes, and Goldbach’s Mystery

THE ADDITIVE SYMMETRY

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)

…

Goldbach Conjecture with circles

This visual approach gives an intuitive geometric perspective on a classic number theory problem.

👉 Further reading: https://www.cantorsparadise.com/different-ways-of-viewing-the-goldbach-conjecture-d4c224f5008d

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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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Shades of Gray Experiments

Our perception of brightness is far from absolute. Subtle shifts in context, contrast, and boundaries can make identical tones appear strikingly different. These experiments invite you to explore how the eye and brain work together—sometimes in conflict—to construct the world of light and shade we think we see.

1. The Great Gray Deception
Although the two sets A and B appear different, they are in fact identical—set B is simply inverted. This is the Chevreul illusion, where the visual system amplifies contrasts at edges, generating apparent boundaries and brightness variations that do not physically exist.

2. Gray Discs
All four gray circles share the same luminance and hue, yet they seem distinct. This effect, known as simultaneous brightness contrast, occurs when the perceived lightness of a surface changes according to the brightness of its surroundings.

3. Argyle Pattern Illusion
At first glance, the pattern appears to alternate between light and dark diamonds. In reality, every diamond has the same luminance as the one on the far left. This illusion illustrates how contextual contrast and spatial arrangement strongly influence visual interpretation.

4. Sheen and Halo Effects
By presenting identical gray gradient squares—with and without contour lines—we can observe how outlines alone alter the perceived texture and luminosity. This demonstrates how edge information enhances the impression of sheen, while its absence produces diffuse halos within the same tonal range.

Note
The images in this collection were developed during my workshops and have appeared in several of my books. You can find some of them here.

Publishers interested in projects related to visual perception and optical art are welcome to contact me—I’m always open to collaboration and new creative ventures.

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Mathematical Diagrams – Before Computers

Illustration from the Brockhaus and Efron Encyclopedic Dictionary (Энциклопедический словарь Брокгауза и Ефрона, 1890–1907)

Efron Encyclopedic Dictionary

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.

More: Digitized copies .

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A Vigesimal Visual System by the Kaktovik Iñupiaq

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.

☞  More number facts.

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Numerals for the Abacus

The Ancient Romans preferred concrete, tangible concepts over abstraction. They always attached numbers to things, which explains why they ignored the concept of zero.

Familiar with the abacus for calculation, the Romans conceived numbers as sets or groups of objects. For example, the abstract equation 3 × 4 = 12 was understood as “three times a set of four objects gives twelve,” in Latin: ter quaterna duodecim sunt. Similarly, a division like 100 ÷ 5 = 20 was seen as “one hundred items divided into groups of five gives twenty parts,” or si centum in quinos partimur, fiunt eorum viginti partes.

This idea of grouping remained central in Latin. You do not say “in couple” but bini, not “in threes” but terni, not “in single file” but singuli. Romans did not think in terms of abstract numbers as we do today. To express multiplication or division, they used three categories: number adjectives (the result), number adverbs (iterations of a set), and number distributives (the set itself). [See: http://www.informalmusic.com/latinsoc/latnum.html]

Some Roman Fractions
Although lacking a zero, the Romans had a sophisticated fraction system, derived from weights and measures of land. Romans used symbols such as S for ½ and dot patterns like the quincunx for fractions. Most fraction names came from the as—a bronze coin or pound—divided into twelve parts (unciae):

  • deunx (11/12)
  • decunx (10/12)
  • nonuncium / dodrans (9/12)
  • bes / bessis (2/3)
  • septunx (7/12)
  • semis (1/2)
  • quincunx (5/12)
  • triens (1/3)
  • quadrans / teruncius (1/4)
  • sextans (1/6)
  • sesuncia (1/8)
  • uncia (1/12)
  • semiuncia (1/24)
  • binae sextulae / duella (1/36)
  • siculus (1/48)
  • sextula (1/72)
  • dimidia sextula (1/144)
  • scripulum (1/288)

Curiosity: The abacus lives on. In Japan, the soroban is still taught in primary schools, not only as a historical tool but as a means to develop rapid mental calculation. With the visual imagery of a soroban, one can sometimes calculate as quickly—or faster—than with a calculator.

☞ Discover the fascinating history of numerals,

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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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Black Hole of Mathematics

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.

Black Hole of Mathematics

Mathematics not only models the mechanics of the universe—it unveils dimensions of reality that lie beyond direct perception.

☞ Vector Fields.