Discover the Golden Ellipse

A golden ellipse is one where the axes are in golden proportion, meaning the ratio of the major axis (a) to the minor axis (b) is the golden ratio:
φ = (1 + √5)/2.
To visualize this, draw a golden ellipse along with its inscribed and circumscribed circles: the smallest circle fitting inside the ellipse and the largest circle surrounding it.
Interestingly, the area of the ellipse matches the area of the “annulus” formed between these two circles!
Here’s how it works:
Let a be the semi-major axis and b the semi-minor axis, with a = φb.
The area of the annulus is:
π(a² − b²) = πb²(φ² − 1)
The area of the ellipse is:
πab = πφb²
And as φ² − 1 = φ, then πb²(φ² − 1) = πφb².


Isn’t it fascinating how geometry intertwines with the golden ratio?

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

Ambigram Magic Squares

When using standard characters, the digits 0, 1, and 8 are symmetrical around the horizontal axis, while 6 and 9 are interchangeable when rotated 180 degrees.

With these digits, we can create magic squares that maintain their constant sum even when flipped, as illustrated below.

Interestingly, when these numbers are represented in LCD style, we can also include the digit 2, which resembles a 5 when inverted. This allows for the creation of magic squares with additional properties related to both 2D and 3D symmetry—whether flipped or mirrored—such as the remarkable example created by Chris Wardle.

This isn’t the first magic square to exhibit such fascinating properties; there are many variations out there. I encourage you to explore and discover your own creations online. If you have original ideas for magic squares with these unique characteristics, we would love for you to share them with us!

For those interested in learning more about the history and mathematics behind magic squares, check out this fascinating article from the Royal Institution: The Fascination of Magic Squares.

More links of interest:
[1] https://www.rigb.org/explore-science/explore/blog/fascination-magic-squares
[2] https://math.hmc.edu/funfacts/magic-squares-indeed/
[3] https://patcherymenagerie.blogspot.com/2019/07/magic-squares.html
[4] https://www.geeksforgeeks.org/magic-squares-fun-fact-and-more/
[5] https://mathcommunities.org/magic-squares/
[6] https://www.magischvierkant.com/links-eng/
[7] https://chelekmaths.com/2020/06/30/cracking-the-cryptic-joy-and-magic-squares/
[8] https://www.byrdseed.tv/magic-squares/

Cylindrical Anamorphosis of Hand With Reflecting Sphere

M.C. Escher’s 1935 lithograph Hand With Reflecting Sphere inspired artist Kelly M. Houle to create her own interpretation in charcoal on illustration board. When a cylindrical mirror is placed at the center, it produces a striking reflection. Houle explains, “When the original image is bent and stretched into a circular swath, the shadows seem to fall in all directions. When the curved mirror is used to reflect the anamorphic distortion, the forms take on the familiar rules of light and shading, making them appear three-dimensional” (Kelly M. Houle, “Portrait of Escher: Behind the Mirror,” in D. Schattschneider and M. Emmer, eds., M.C. Escher’s Legacy, 2003).

The original work.
Circular anamorphosis of the original work as seen by an observer.
Final result: 3D cylindrical anamorphosis.

Cylindrical anamorphosis is an art technique that creates distorted images that appear normal when viewed through a cylindrical mirror, manipulating perspective and light to produce a three-dimensional effect from a two-dimensional surface.

About Kelly M. Houle

Kelly M. Houle is known for her work in anamorphic art and illuminated manuscripts. Her projects often blend artistic expression with scientific themes, such as her illuminated manuscript based on Darwin’s On the Origin of Species. She has exhibited her work widely and continues to explore innovative techniques in contemporary art.

For more information about her work, visit Kelly M. Houle’s website.

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.

Concept by Lee Sallows.

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

Further reading.

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.

© Giannisarcone.com, source.

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.