A neat animated tribar! It’s worth noting that the tribar, or Penrose triangle (aka Reutersvärd triangle), attributed to British mathematician Roger Penrose, was not technically ‘invented’ or ‘discovered’ by him. The geometric principles underlying its existence were already evident in Greek and Arabic ornamentation, including tiling and friezes…

## Circles & Roots

Delve into the realm of Sacred Geometry, where circles unveil the elegance of successive square roots from 1 to 6. Extend your exploration with the enigmatic charm of the square root of *Phi*.

Picture available as prints and merchandise from our **online gallery**.

## Apparent size ≠ Real size

Simple demonstration of apparent size and distance… See how the color rings (*annuli*, in mathematical language) fit snugly.

## Nested Penrose Triangles

This is an illusory geometric structure that cannot exist in our 3D world. Let’s Explore its captivating depths and intrigue.

Here’s how to create this impossible structure. Start by drawing two parallel lines spaced apart from each other and divide them into 7 equally spaced lines.

Then follow the visual steps A, B, C, and D illustrated below. At the beginning (fig. A), you will need to replicate the alignment of the 9 parallel lines three times while applying a 60-degree rotation to each one, finally arranging them to form a triangle. Subsequently, follow the visual directions in B and C to obtain the figure shown in fig. D.

At last, you can add color and gradients to the structure as illustrated below.

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## Topological Oddity: A Picture-Hanging Puzzle

Imagine the linear pattern as a hanging rope. Now, removing any one of these four nails will cause the entire rope to fall.

## Illusory Structures

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

## Nearly Right

Did you know? When you calculate (π^{4}+π^{5})/e^{6}, you get around 1! This means a triangle with sides π^{2}, e^{3}, and √π^{5} 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.