Four Dimensional Hats: A Visual Wonder!

[Image courtesy of]

The mobius strip is one of the simplest objects in the world, and yet, the most mind-bending. If you take a strip of paper, add a single twist, and tape the ends together, you transform a two-sided object into a single-sided object. It becomes one continuous surface.

(We’ve discussed the concept briefly in the blog before, but in bagel form.)

But did you know that you can take that idea a step further and end up with this?

[Image courtesy of]

This is a Klein bottle, an object with one continuous surface. If you trace a path along the surface, you will traverse from the “inside” to the “outside” and back again without breaking stride.

Yes, the word “bottle” is a bit of a misnomer, since this won’t actually hold any liquids; they would just flow along the surface, going “inside” and back “out” without pooling anywhere. This is a result of a mistranslation, as the German word “flache” (surface) was translated as “flasche” (bottle).

This limerick sums up the Klein bottle nicely:

A German topologist named Klein
Thought the Mobius Loop was divine.
Said he, “If you glue
The edges of two,
You get a weird bottle like mine.”

[Image courtesy of Pinterest.]

Although the Klein bottle can’t quite exist as a three-dimensional object — since the object has to pass through itself, which can only happen in four dimensions — we can come close enough to create some impressive approximations, like the glass “Klein bottles” pictured above.

YouTuber and physics student Toby Hendy has even managed to create a technique to knit yourself a Klein bottle hat! Check it out:

Although it’s not an optical illusion, it’s still a visual puzzle for the eyes and the mind, one that has captured the imaginations of mathematicians, artists, and many others throughout the years.

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A logic puzzle with an artistic twist!

Puzzles are truly a worldwide phenomenon. So many different cultures and groups have created fantastic, long-lasting puzzle styles that continue to resonate across decades and even centuries.

In the past I’ve endeavored to make PuzzleNation Blog a bit more PuzzleInternational by sharing overseas puzzle flavors from German and Spanish puzzle books that’ve been passed to me by fellow puzzlers.

And I’m so excited that another friend of the blog has shared an absolute treasure trove of international puzzle books with me, ensuring that our puzzly world tour will continue!

So today, instead of examining a single puzzle book and getting a glimpse into a particular culture’s brand of puzzles, I’ve picked a particular type of puzzle and we’ll be exploring magazines from several different countries dedicated to that puzzle!

Let’s take a global look at Logic Art!

Logic Art puzzles (also known as Pixel Puzzles, Pic-a-Pix, Illust-Logic, Griddlers, Hanjie, and Picture Puzzles) are a wonderfully artistic take on deduction-style logic puzzles.

Essentially, you’re given an empty grid with numbers along the top and left-hand side. These numbers indicate black squares to color in and white squares to leave alone. By deducing where to place the black squares and white squares, a pixelated picture will emerge!

(For more complete rules and solving tips, check out this helpful guide from our friends at Penny/Dell Puzzles.)

So, the difficulty of the solve and creativity of the solution image are only limited by the puzzle constructor’s imagination and your own puzzle savvy.

Some magazines, like these German puzzle books, stick to the simple black square/white square mechanic…

… while others, like this Cyrillic magazine with several colors and this Hungarian magazine with splashes of red, encourage greater use of color in your Logic Artwork.

These smaller, digest-sized Cyrillic magazines offer multiple grids per page with simpler solution images.

But look at the level of detail some of the larger grids offer!

I must admit, though, I’m partial to these Japanese puzzle books, if only for this particular solution image:

Logic Art is obviously a puzzle with global appeal. Although not as universal as Sudoku (or as intuitively easy to solve), it clearly strikes a chord with solvers across the world.

It’s always a treat to explore puzzles from another culture’s perspective. Thanks for taking this journey with me today.

Thanks for visiting PuzzleNation Blog today! You can share your pictures with us on Instagram, friend us on Facebook, check us out on TwitterPinterest, and Tumblr, and be sure to check out the growing library of PuzzleNation apps and games!