## 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.

Discover prints and merchandise featuring this op art masterpiece at my online gallery

## 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.

## 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.

## Amazing Disentanglement Puzzle

How to transform ordinary rulers into captivating feats of magic? Equip yourself with two standard 30 cm rulers, made of wood or other materials. Attach a 20 cm string to each ruler by threading it through the hole at one end (see Fig. 1). Form a cord loop around one ruler, knotting the loose end of the cord, as depicted in figures 2 and 3. Ensure the string ring is not too tight, allowing it to glide smoothly along the ruler.

Repeat with the second ruler, threading the cord through the loop of the first ruler, as shown in figure 4.

The challenge is to separate the rulers without cutting or unraveling the cords. Despite the apparent difficulties, the solution unfolds seamlessly.

This string puzzle can also be build using two plastic pipes and two curtain rings (see figure 5).

show solution

## Throwing a Curve

When the plate turns, the umbrella moves through a vertical plane. What shape does the umbrella make in the plane? Surprisingly, it’s a hyperbola (with the umbrella handle included).

## Psychology of Shapes

At times, we encounter challenges in discerning geometric forms. For instance, a perfect hexagon eludes perception as its alternating sides are “absorbed” by parallel segments.

## Knot, knot, who’s there? Topology…

Lebrecht Goeritz was a German mathematician who designed some trivial knots almost a century ago. His most famous unknot has eleven crossings.

## Visual Calculus

Mamikon A. Mnatsakanian (Armenian: Մամիկոն Մնացականյան) devised in 1959 a visual method to show that the areas of two annuli with the same chord length are the same regardless of inner and outer radii.
As an undergraduate, Mamikon specialized in the development of geometric methods for solving calculus problems by a visual approach that makes no use of formulas, which he later developed into his system of visual calculus.