MAKE readers enjoy making things that are useful in innovative or just plain weird ways. Some of these great ideas work well and reach the pages of this magazine. Some just don’t work out in the real world. And some ideas are downright misguided.

For a change of pace, having recently neglected “teasers” in these pages, I put some thought into things you simply can’t make, no matter how clever and ingenious you are. There are still folks who think, “Anything is possible if you tinker with it enough.” Allow me to demolish that naive notion.

Nature behaves according to fundamental laws. We can discover some of these and express them in the universal language of mathematics, elevating them to the status of “laws” of nature. Newton’s laws, conservation laws, and most importantly the laws of geometry describe how nature works. There’s good reason to suppose that all laws of nature arise from the limitations of the geometry of the universe. When we have formulated and thoroughly tested a law about how nature works and are certain it doesn’t contradict other known laws, that same law is also telling us how nature doesn’t work.

Another diabolical thing about nature is how interrelated its basic laws of physics are — you can’t violate one of them without violating many others. Nature blocks us from violating any one of its fundamental laws, those that are so thoroughly tested that they will likely survive any future advances in science.

### Futile Projects

Can you cut a triangle from a flat piece of paper that has exactly equal sides but unequal angles? Silly idea, you say. That would violate laws of Euclidean geometry, and we know that our neighborhood of the universe obeys Euclidean geometry at least to the precision that we are capable of measuring.

Can you design a walking path that is downhill all the way around — in either direction, clockwise or counterclockwise? It’s an old joke: “When I was a kid I had to walk 2 miles to and from school everyday, uphill both ways.” We laugh because we know it’s impossible. This says something about the laws that operate around closed paths. Perpetual-motion machine inventors ought to take this seriously.

Could you cut from wood a cube with 7 faces? That violates the logical definition of a cube. That’s like making a 4-sided triangle. But a sphere with surface area of only 4 times the square of its radius is another class of impossibility, one that isn’t merely a logical paradox, but a violation of a geometric property of the real world.

Try putting 3 genuine hen’s eggs inside a wooden box and closing the lid. When you reopen it, you find 4 hen’s eggs, which, when broken, are found to be perfectly genuine. No gimmicks in the box, no sleight of hand. Good trick if you could do it, but nature doesn’t work that way.

### Apparent Impossibilities

Ever since craftsmen made wooden models of sailing ships inside glass bottles, people have delighted in making “seemingly impossible” artifacts. The peach inside a glass bottle of peach brandy is one example. The neck of the bottle is far too small for a peach, but just right for the branch and pollinated blossom of a peach tree. Time in the summer sunshine does the rest.

You can buy bottles with small necks with a complete pack of playing cards inside, still sealed with the tax label. The mystery of how it’s done would be answered by watching someone painstakingly make one, but then the finished object would seem less fascinating.

And of course there are the tiny bottles with a genuine coin inside, the bottle having been blown around the coin. Search “impossible bottle” on Wikipedia.

Visual illusions are usually 2-dimensional, artistic depictions of things not possible in 3-dimensional space. Figure A is one of mine. At casual glance it seems to depict 2 flat washers interlocked, but then you notice that the interlocking would be impossible if the washers were really flat. And even worse, each washer is self-contradictory in its depiction, taking advantage of the depth ambiguity of the near and far edges of ellipses. The colored faces and edges give the game away — the front faces seem to connect with back edges of each washer.

Many illusion pictures, rendered on a flat surface, succeed by such artistic ambiguities of perspective. The illusion would be destroyed if depicted from a different angle. Some clever artists make sculptures that appear to be entirely different things when seen from different angles, and the next step is sculptures that appear to be different illusions when seen from different angles.

Gears are useful for those who make machinery, but pesky friction is ever-present. This inspired me to design the 3-gear system in Figure B. As the gears mesh, gear teeth are transformed to an indeterminate quantum state where they are neither here nor there, and it’s impossible to determine which gear they belong to. It’s well known that there’s no friction in quantum mechanics, so these gears must be perfectly efficient. At least that’s my theory. I’ve colored the faces of the teeth to make it easier for you to understand the principle. I hoped someone more skilled than I might produce an animated version showing the gears rotating and the gear teeth morphing where they mesh. It’s a challenge!

Often the geometric peculiarities of isometric perspective are the trick that makes these illusion pictures perplexing. Isometric perspective removes the vanishing points of photographic perspective, so that parallel lines remain parallel and don’t converge in the picture. Engineering drawings use this technique so there’s no size reduction with distance and all dimensions may be measured with a ruler on a flat page.

So, could such an illusion be photographed? My experimental shot of an impossible triangle made from Legos (shown in Figure C) was achieved with a digital camera and some simple lenses arranged in a telecentric system, which renders reality in isometric perspective.

Notice how the far edge of the green base plate seems larger than the near edge. Yet careful measurement of the picture, using a ruler, shows that it isn’t. If this were classical photographic perspective, the far edge would be shorter in the picture.

You can take such pictures yourself. To learn more, see my article on telecentric lens systems at makezine.com/go/telecentric.

Can such pictures work in stereo 3D photo-graphy? I’ve not done it yet, but it may be possible using some other deceptions. Can ordinary artistic illusions be rendered in 3D? Some can. For examples see makezine.com/go/3dphoto.

We mustn’t forget that these are only lines and color on a flat page. It’s our brains that, from long experience, condition us to interpret them as if they were familiar objects like washers, gears, and rectangular blocks.

We can imagine many things that nature doesn’t allow us to make; we just don’t always know in advance what they are. But sometimes nature has already told us all we need to conclude that they’re impossible, and pursuing them would be futile. Some things can be faked with deceptive illusions. And some things, like a 3-headed flat coin, will take considerably longer to achieve. Like forever.

I challenge you to try your hand at these impossibilities and take them a step further. If you do, email me your ideas and responses ([email protected]), and I’ll add them to my website.