The fractal is a marvel of science, architecture, mathematics, and nature. If you're reading this, then we guarantee you have already seen one of these patterns in your everyday life. Fractals can be snowflakes, patterns on the wallpaper, designs from your Geometry class in grade school, or even a random doodle you made today while drifting off into space.
In simplest terms, a fractal is a geometric pattern that is repeated over and over on smaller scales. After continually zooming into the design, you will see the same pattern, over and over, growing continually smaller and more intricate until your viewing lens is infinitesimally small. Now, your eyes can't really zoom in that far, so a regular human would not be able to create a perfect fractal just by drawing. But, it's perfectly acceptable to imitate the style of a fractal for as far as the human eyes can see, and anything beyond that you won't have to worry about! Follow this link here to see a POV example of zooming in on a fractal (the Koch snowflake) to see how it is made up of infinite, smaller “self-similar” versions of the original. From Kubiya’s point of view, Mechanical Puzzles and fractals have always gone hand in hand because of their intricate, lovely patterns. The self-similarity of a fractal lends itself well to puzzle design because of how the human eye is trained to look for patterns within patterns, especially patterns containing cycles or repetition. The patterns we see in nature are the patterns that our brain is trained to pick up on and will want to replicate with the objects we interact with. That being said….
Can Fractals Help You Solve Puzzles??
The short answer is - yes! Ever since we were babies, our brain has been learning how to decode patterns, discover similarities between objects, and look for cause and effect relationships in the world. This is exactly what happens when a person starts to solve a puzzle: which pieces can move? Does the pattern or order of pieces mean anything? Does the same thing happen each time I push this lever, or do I get different results depending on how I have this other level positioned? For more specifics on puzzle solving, check out our blog post from earlier this year with some helpful tips. For now, we’re focused on fractals.
To give you an idea of some design patterns, and to put pictures to the description, check out these two puzzles from Kubiya, the Plus 022 Puzzle: Wooden Geometric Jigsaw Puzzle and the Terdragon 019 Puzzle: Wooden Jigsaw Puzzle. The designers at Kubiya were inspired by fractal imagery when creating these puzzles, proving once again how often the seemingly polar opposites of mathematics and design/artistry can intersect. The Plus 022 is an abstract geometric puzzle made of 25 jigsaw puzzle pieces. The object of the puzzle is to fit all 25 uniquely shaped pieces into the tray. Each piece fits with every other piece, but there's only one way for all the pieces to fit into the tray. The puzzle pieces can be assembled in thousands of different ways which allows for endless creativity to create new shapes…. or new “fractals”. The second puzzle, the Terdragon, is also an abstract geometric puzzle. This set is made up of 19 jigsaw puzzle pieces, and the object is to fit all 19 pieces into the tray. Again, it may look easy at the beginning but it's actually very tricky to solve once you get started!
Fractals in Nature and History
Fractal designs can even be seen in snowflakes that fall to earth, as soon as scientists had evolved the necessary technology to zoom in far enough and check out the never-ending intricacies of each edge. That's why most people grow up hearing about how no two snowflakes are exactly alike - the patterns are so complicated that it could never be perfectly copied. The hardcore science websites will tell you that there is a chance of two being copies, but the chance is so low it might as well be nothing. Also, because of the fragility of snowflakes, good luck proving it! Designs like these that appear first in nature can be some of the most beautiful, because they all appear inseparable from that feeling that they are supposed to be here. And yet, with the sheer vastness of possibilities in ways that the universe might have formed, who's to say that intricate fractals are not just a part of many chaotic patterns in nature that, given enough time, would appear regardless? Whatever you think the answer might be, there's no denying the beauty and mystery of a well-constructed fractal.
Finally, a fascinating story to end on, there is a very real and really baffling historical example of a puzzle that fractals were able to shed some light on. As with a lot of things in science, one of the earliest initial studies (of what would be later described as fractals) came from a surprisingly difficult question: How long is the coastline of England? This is the exact sort of problem that interests mathematicians and puzzle solvers - particularly when the answer turns out to be more convoluted than anticipated. It turned out, when experts went to measure, that quantifying the length of a coastline could depend on the length of the ruler they were using(huh?), or the pixel quality of the map or photo being measured. Now, most distances people measure don't change like that, so this seems like a suspicious result. However, keep in mind that coastlines can have large amounts of bends and odd edges, and so a smaller ruler or scale being used can measure a longer overall length of a coastline (since the smaller ruler can capture more of those twists). Coastlines, of course, are not true fractals, but they do appear to be at times. Some of the most interesting developments in math, puzzles, and mechanics come out of the ebb and flow between theoretical concepts and actual physical things that exist in the real world. Fractals like the Koch snowflake are relics of pure geometry, but it took a mundane coastline query to kick off the study of this gorgeous branch of science.
In spirit, humans should always remember the fractal when solving puzzles; there is no correct angle or position from which to view it. Whether you zoom in on the details and describe what you see, or squint and describe the big picture - you are correct! However, you might never fully grasp the beauty and mystery of a great puzzle or fractal if you were to only view it one way, at one time, in one context. Cheers as always, and happy puzzling.
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