Penrose Triangle and Perpetual Motion

What does an octahedron, perpetual motion, Penrose triangle and a waterfall have in common? We would have watched the Spielberg movie The Close Encounters of the Third Kind; Why Third Kind? Why not first or second or fourth?

Answer for the trivia first. Alien encounters are classified into three kinds by Dr. Allen J. Hynek in his 1972 book The UFO experience: A Scientific Study. The First kind is where one simply "sights" the UFO and goes gaga about it; the Second kind is where one sights physical evidence of alien presence like crop circles or flaming footprints; and the Third kind is where one, sights an actual alien (and not New Yorkers) and in a sub division G of the third kind, er, makes physical contact with the alien (abducted).

Similar to that trivia, Perpetual Motion Machines (PMM) are of three kinds, if we also include Prof. Zemansky's classification. The perpetual motion machine of the first kind (PMM1) violates the principle of energy conservation or the First Law of Thermodynamics; the device generates more energy than it uses or somehow "creates" energy out of nothing. It actually offers free lunch.

The PMM2 is subtler. It is a device that, by operating in a thermodynamic cycle, to be precise, continuously cools a place without any side effects. For instance, our air conditioner or refrigerator cools a place or region, by receiving energy from electricity but also throws out heat to the "surrounding" of the cooled region. In this sense, an air conditioner or refrigerator are not PMMs but obey the Second Law.

PMM2 is a device that converts completely heat energy to useful work energy. PMM2 need not violate the First Law of Thermodynamics; that is, it can conserve energy, but nevertheless remain fictitious in its ambition to continuously cool a spot colder than its surroundings or extract mechanical work from a single heat reservoir. It violates the Second Law of Thermodynamics.

Now to answer the other question of "What does an octahedron, perpetual motion, Penrose triangle and a waterfall have in common?"

Maurits C. Escher is one of my favorite creators of art. He was a pioneer of Mathematical Art, involving interesting mathematical ideas including, among other things, unique vanishing points, hyperbolic tessellations (tiling of hyperbolic planes), ideas from topology, symmetry groups, Platonic solids and the Mobius strip.

Waterfall is a lithograph M. C. Escher made in 1961 and is shown above. This involves two impossible triangles, an idea proposed independently by the mathematician Roger Penrose (It was first created by the Swedish artist Oscar Reutersverd in 1934). The impossible triangle or the tri-bar (shown on the right), although can be depicted as a two dimensional pattern, is impossible to be constructed practically (try it with slate wood frames, if you want to).

The waterfall shown in Escher's picture utilizes this impossible triangle twice in the water flow loop, which is "endless" and relies on our brain's compulsion to see three dimensional objects out of a two dimensional pattern. The impossibility of the waterfall is obvious if you follow the route of the water, which is in a "source-less" continuous loop. Notice, the waterfall even drives the Pelton wheel like device at the bottom of the fall.This is a perfect example of a Perpetual Motion Machine of the First kind, where Energy is generated or "created" out of nowhere.

Any such claim for a PMM (and there is a web page full of them documented with chronology) immediately reminds me of this masterpiece of Escher and all its implications. Recently I had a chance to get reminded of this picture through a PMM1 claim, which spun the thoughts of this essay-let.

Only the octahedron remains to be explained in the question that started this note. Observe at the top of the two big columned structures in the Waterfall picture. There are some Platonic solids intersecting each other there.

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