Picture this: a mug and a doughnut walk into a café. The barista squints and says, “Wow, I’ve never seen twins so alike!” The doughnut grins, “Topology—it’s a hole new world!”

## The Intrigue of Simplicity

“A world without problems is an illusion, so is a world without solutions.” – Gianni A. Sarcone

According to the second rule of Sarcone & Waeber’s puzzle-solving principles, “*nothing is ever as simple as it seems.*” This is why we enjoy deceptively simple puzzles that seem almost impossible to solve. Here’s a classic topological puzzle you can create and enjoy with your kids.

You can explore the full set of puzzle-solving rules by Sarcone & Waeber here: https://www.archimedes-lab.org/sarcone_rules.html

## Exploring Impossible Figures: A Philosophical Twist on Reality

Impossible figures are intriguing forms that defy the reality of our 3D world. In the 90s, I began experimenting with representations of cylinders featuring fictitious right angles—an artistic play on geometry. These twisting impossible figures, which defy mathematics and physics, are ideal for interior decoration or merchandise like t-shirts or mugs because they captivate the viewer’s imagination and provoke thought. Their striking, paradoxical nature draws attention, making them perfect conversation pieces, while also offering a philosophical twist on the idea that what seems real may not always be.

By stripping those imaginary geometric form down to their essence, I instill them with greater power.

These drawings are taken from my book* Drawing Optical Illusions*, still available on **Amazon**.

And are available as prints and t-shirts and merch from my online gallery!

*Buy prints and apparels here:*https://redbubble.com/shop/ap/164477598

*y prints and apparels here:*https://redbubble.com/shop/ap/164403619

## Impossible Escape: A Topological Illusion

This is a fun and surprising escape trick for kids and family, that uses simple props:

· 1 karabiner clip,

· 1 metal ring,

· 1 loop of cord.

To begin:

1. The ring and the green karabiner clip are securely attached to the cord and cannot be removed.

2. Now, watch closely as I attach the green karabiner to the metal ring, making the setup even more complex.

3. Surprisingly, I can now simply pull the cord free.

Can you believe your eyes? Let’s try it together!

More topological magic tricks:

– https://archimedes-lab.org/2021/06/29/magic-topology-2/

– https://archimedes-lab.org/2020/06/09/magic-topology/

– https://archimedes-lab.org/2017/12/14/how-to-magically-untie-a-shoelace-double-knot/

## Magic Square for Gunners

A magic square is a grid where the sum of the numbers in each row, column, and diagonal is the same, creating a harmonious balance. A “geomagic” square, on the other hand, is a grid of geometrical shapes where each row, column, or diagonal can be assembled into an identical shape known as the “target shape”. Like numerical magic squares, all shapes in a geomagic square must be distinct.

The postage stamp below, issued by Macau Post on October 9, 2014, pays tribute to Lee Sallows, the creator of geomagic squares.

## Topology: A Hole Through a Hole in a Hole…

Have you ever pondered the nature of holes? These peculiar void-like spaces that seem to exist in the fabric of reality, yet defy simple categorization… Are they real objects that we can interact with, like a donut or a Swiss cheese? Are they abstract mathematical entities, part of the strange world of topology? Or do they exist only in our minds, as metaphysical concepts that arise from the limitations of our perception?

Explore these intriguing questions through our specially crafted posters and merchandise, which delve into the fascinating nature of holes. Discover them in **our online gallery**.

Hang the poster to spark endless curiosity, or wear the t-shirt to carry a conversation starter wherever you go.

## 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.

*© Giannisarcone.com,*source

*.*

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

*© Giannisarcone.com,*source

*.*

## 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).