Necker cube

In 2-dimensional geometry, a necker cube is a simple self-intersecting 12-sided polygon that looks remarkably like a standard 3-dimensional cube when viewed from 53.7 centimeters away in a precisely orthogonal direction; but, in spite of that, is flatter than a pancake. The only known way of distinguishing between the two is to feel around them with your fingersCAUTION: Watch out for sharp corners! Remember, you only have nine fingers left.. 
==Euclid's Pythagoras's nightmare==
The first necker cube was accidentally constructed by Euclid of Samos Pythagoras of Samos in 495 BCE using nothing but a straightedge, a compass, and a dozen emptied-out Pixie Sticks™ as part of an ecologically-sound recycling program. Pythagoras's original intention was to build a portable model of his humble abode (a 6x6x6-cubit cargo container sitting on the beach on the outskirts of Athens), but it unexpectedly and catastrophically collapsed upon itself overnight. The cargo container also collapsed upon itself several days later, but no serious injuries were reported.Other than Pythagoras being killed outright, of course. However, history records that he was going to die anyway. All the king's horses and all the king's menDuring the lifetime of Pythagoras, Ancient Greece reverted from a full-fledged democracy to its original monarchical and horse-dominated state. pulled Pythagoras's mangled corpse out of the wreckage and argued amongst themselves for years, trying like hell to figure out which sides of the flattened model (and the flattened cargo container) were supposed to be the forward-facing side, the backward-facing side, the top side, the bottom side, the hot side, the cold side, the left side, the right side, the north side, the south side, the inside, or even the outside. None, including Pythagoras, could adequately solve the problem at hand, showing just how stupid and useless ancient Greek logic is. And thus everybody forgot about the ancient Greeks. 
==The Renaissance==
Long after everybody forgot about the ancient Greeks, the necker cube was independently rediscovered during the Renaissance by the Renaissance painter Leonardo Da Vinci. Leonardo was trying to figure out how in hell he could add the appearance of depth to his portrait of the famed Lisa del Giocondo, in particular to enhance her non-so-ample femininistic proportions. Leonardo first attempted to paint two (2) Monas Lisa side by side facing in slightly different directions, but the resulting painful eye strain was not really worth the effort. He then tried plotting the contours of each breast with an intricate network of intersecting lines and triangles using nothing but a straightedge, a compass, and his eager bare hands; and that did the trickAnd thus Leonardo Da Vinci is recognized the world over to this day as the Founding Father of Cubism.. The resulting optical illusion was so stunning and life-like that bootleg copies of the new-and-improved Mona Lisa swamped the market within weeks, and Italian women everywhere started wearing bras shaped like cubes just to show off be like her. Unfortunately, this resulted in mass confusion when people in general couldn't adequately determine whether or not the new bras were either being worn inside-out or outside-in. It wasn't until the discovery of the klein bottle and the wonderbra many centuries later that this sort of thing no longer mattered to anybody. 
==The underlying geometric problem at hand==
Meanwhile, the underlying geometric problem at hand refused to go away so easily, until the chance discovery of the Necker Group (named after the original necker cube and some otherwise-obscure mathematician whose last name, coincidentally, was Necker). What with the powerful formalism of group theory now in their (the mathematician's) toolboxesUnlike carpenters and surgeons in the real world, mathematicians do not ordinarily carry physical toolboxes. This is simply a metaphorical expression, not meant to be taken literally., the elusive solution seemed immediately at hand. Or so everybody thought, before their hopes were mercilessly dashed because group theory is too hard to understand, even for mathematicians. Remember what they said about Newtonian gravity before it fell victim to the 3-body problem? Or Einsteinian gravity before it fell victim to the 4-body problem? Or physics in general before it fell victim to Murphy's Law? How many times does this sort of thing need to happen, people???The world may never know. Anyway, it was only after an additional billions of person-hours of grueling labor on abaci, adding machines, and quantum computers that all twelve (12) of the confusing sides of the industry-standard necker cube were properly indexed, cataloged, and labelled for all time to come. 
==Modern applications==
The necker cube and its closest geometric relatives, such as the necker square, the necker tesseract, the necker staircase, and the necker murphy bed, are today heavily incorporated into both Pythagorean Euclidean and non-Euclidean architectures. Necker cubes also form the skeletal frameworks for isometric blueprints, non-isometric blueprints, modern video games, and artsy-fartsy wallpaper. 
===Footnotes in small print===
category:Geometrycategory:Cubes