## Illusory Structures

That’s what happens when you fall down a Penrose staircase…

## Nearly Right

Did you know? When you calculate 45)/e6, you get around 1! This means a triangle with sides π2, e3, and √π5 is nearly a right triangle…

## Balance & Unity: Hexagonal-Heptagonal Harmony

This heptagonal radial tessellation with hexagonal tiles not only serves as an aesthetically pleasing visual creation but also stands as a testament to the harmonious coexistence of mathematical precision and artistic expression.

## Ship in a Klein Bottle

Embarking on a journey in a Klein bottle? Absolutely. A Klein bottle is a mind-bending non-orientable surface, defying the usual inside-outside norms. Technically, the ship’s navigating the interior…

## Perception in Motion: Illusion, Confusion, and Zen Insight

Many perceive the two 3D cross-like shapes as moving significantly, though they remain stationary!

The interplay of color shades (light/dark) on the edges and body of the cross-like wire frames creates the illusion of motion. The alternating shadings simulate “motion blur,” leading some researchers to attribute these illusory movements to delays in luminance processing, producing a signal that deceives the motion system and induces “kinetopsia” (motion perception)..

This brings to mind an anecdote: Two Zen monks debated a flag moved by the wind. One claimed, ‘The flag is moving…’ while the other countered, ‘The wind is moving!’ The monastery’s prior intervened, stating, ‘Not the wind, not the flag; the mind is moving…’

This short anecdote serves to explain that the concept and perception of motion is sometimes ambiguous.

## Perceptual Puzzle

Size Distortion: The length and curvature of the blue curves A and B in the diagram are highly deceptive. However, the curves are congruent! This presents an intriguing variation of the Delboeuf illusion, wherein size judgment is distorted by peripheral context.

## Impossible Paper Turn-up

Let’s explore a captivating paper-folding puzzle, part of the impossible origami family. It is simple to carry out and can be done at any moment.

Cut a 10-centimeter-wide strip from a two-colored sheet (in our case, yellow and red). Make sure the paper strip is at least 24-25 cm long! By gluing the ends together, you may turn this strip into a ring, as shown below. The goal is to create a turn-up in the ring without tearing it… Impossible? Many of your friends will attempt, but the outcome will always be negative.

This is how the trick works:

Flatten the paper ring and fold the top edge by 2-3 cm (see fig. 1). Unfold only the single top layer of the fold, creating two triangular folds at each corner of the flattened ring (fig. 2). Next, fold the two outer edges along the base of the flattened triangles (fig. 3). Then, fold the outer edge upstream. This way, a folded edge is now on each face of the flattened ring.

Now, insert your fingers inside the ring, firmly holding one corner between your thumbs and index fingers. Carefully separate your hands, pulling the sheet at the corner, releasing the excess paper hidden between the folds (fig. 5a and 5b). Repeat on the other corner. Finally, rearrange the paper ring, smoothing out the visible folds. The top edge of the ring is now completely turned on itself, creating a perfect turn-up without tears. Who would have thought? Congratulations, magician!

## Egguation

Solve for the perfect omelette with this eggcellent formula!

In fact, if you graph x2 + y2 = 2y, you will obtain a flawless egg shape:

However, there are also other methods to create a perfect ovoid shape using a compass and ruler, as illustrated below.

## Cat Tessellation

Explore the captivating world of tessellations! Immerse your surroundings in the charm of feline grace and geometric perfection. Available as prints and merchandise in our online gallery.

There’s a story behind this geometric drawing; it depicts our late cat, Sylvestre. He was an Abyssinian cat with a fawn-colored coat, mirroring the illustration. Sylvestre was our daily companion in the studio for nearly 20 years.

## Timeless Trigonometry: Plimpton 322’s Revolutionary Legacy in Mathematics

Plimpton 322,” a clay tablet originating from ancient Mesopotamia during the Old Babylonian period (1900-1600 BCE), precedes Hipparchus by over 1,000 years. This artifact not only provides novel avenues for contemporary mathematical research but also holds implications for mathematics education. The trigonometry revealed in “Plimpton 322” presents a more straightforward and precise approach, showcasing distinct advantages compared to our current methods.