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…

## Joyful Cubes!

Unraveling the mathematical euphoria of N = 2³+3³+4³+5³+6³+7³+8³+9³

Continue reading “Joyful Cubes!”## Autokinetic Illusion

Immerse yourself in the mesmerizing experience as blue droplets seemingly sway gracefully, creating an illusion of gentle motion. The yellow horizontal lines contribute to a wave-like dance, enhancing the visual allure.

This op art piece embodies a peripheral drift illusion (PDI), wherein a sawtooth luminance grating in the visual periphery induces the illusion of movement.

Fascinatingly, studies by vision researchers reveal that the illusory motion activates brain regions akin to those triggered by actual movement.

Noteworthy accolades include a feature on Google Science Fair (@googlescifair):

Explore and acquire “Hold On Tight” as prints and posters through our online gallery.

## 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 x^{2} + y^{2} = 2^{y}, 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.

## 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).

## Is it possible to create objects out of nothing indefinitely?

Yes, but only with a geometric trick that combines perpetual motion and “magic”. All you need is a simple sheet of graph paper, which you’ll cut into three distinct pieces after going through a step-by-step procedure that allows you to create confetti indefinitely from nothing. The game can be played indefinitely in a cyclical fashion.

Continue reading “Is it possible to create objects out of nothing indefinitely?”## Magic Multiplication Tables

A neat math trick to perform: ask someone to sketch an hexagon on a multiplication table, then instruct them to sum the numbers at its vertices. By sharing the result, you can deduce the central number of the hexagon. How? Simply divide the sum by 6.

Additionally, here’s a secret: it works with pentagons too!

## Geometric Art Mobile

The Magic of Projective Geometry: Installation by Johannes Langkamp (German/Dutch, born 1985).

## Aperiodic Tiling

An aperiodic monotile, humorously dubbed an ‘einstein’ (from the German term “einstein,” meaning “one stone” or “one tile”), is a single tile that covers a surface without repeating patterns. This posed a challenging question for some time: could such a tile exist, or was it impossible?

In 2023, David Smith and his team provided an answer. They discovered a simple tile called a “hat” that can achieve this aperiodic tiling. The geometric shape of the “hat” tile is based on the symmetry and edges of a hexagon, as shown in the picture. According to Smith, this tile, along with its reflection (shown in blue), enables an array of unique, non-repeating tile arrangements. The “hat” falls within the broader category of Smith–Myers–Kaplan–Goodman-Strauss tiles.