YESTERDAY IN “I Was Reading Wittgenstein the Other Day….,” LRB reviewer A.W. Moore cited “the two ways of seeing a Necker cube” appearing in Wittgenstein’s Proposition 5.5423. Though this cube was new to me, it is not unrelated to other objects of discussion here at SimanaitisSays. Here are tidbits about these, some real objects, others Wittgenstein “non-sense,” still others shape-shifting in the imagination.
The Necker Cube (Not to be Confused with the Brodie Knob aka Necker’s Knob.) Wikipedia describes, “The Necker cube is an optical illusion that was first published as a rhomboid in 1832 by Swiss crystallographer Louis Albert Necker.”
Above, the Necker cube. Below, its two possible interpretations. Images by Gauravjuvekar in Wikipedia.
Wikipedia continues, “It is a simple wire-frame, two dimensional drawing of a cube with no visual cues as to its orientation, so it can be interpreted to have either the lower-left or the upper-right square as its front side.” Thus, “The Necker cube is an ambiguous drawing.”
“Each part of the picture, Wikipedia notes, “is ambiguous by itself, yet the human visual system picks an interpretation of each part that makes the whole consistent. The Necker cube is sometimes used to test computer models of the human visual system to see whether they can arrive at consistent interpretations of the image the same way humans do.”
Another Ambiguous Drawing. Wikipedia says, “Sir John Tenniel‘s illustration of the Caterpillar for Lewis Carroll‘s Alice’s Adventures in Wonderland is noted for its ambiguous central figure, whose head can be viewed as being a human male’s face with a pointed nose and chin, or as being the head end of an actual caterpillar, with the first two right ‘true’ legs visible.”
Alice meets the Caterpillar in Wonderland. Illustration by Sir John Tenniel from Wikipedia.
An Escher Necker. “Maurits Cornelis Escher, (1898–1972)” Wikipedia recounts, “was a Dutch graphic artist who made woodcuts, lithographs, and mezzotints, many of which were which were inspired by mathematics.”
“The impossible cube or irrational cube,” Wikipedia describes, “is an impossible object invented by M.C. Escher for his print. It is a two-dimensional figure that superficially resembles a perspective drawing of a three-dimensional cube, with its features drawn inconsistently from the way they would appear in an actual cube.”
Above, Escher’s impossible cube. Image by 4C from Wikipedia. Below, his 1958 lithograph Belvedere. Image from etsy.com via Wikipedia.
Wikipedia observes, “There is a man seated at the foot of the building holding an impossible cube. He appears to be constructing it from a diagram of a Necker cube at his feet, with the intersecting lines circled…. The woman about to climb the steps of the building is modeled after a figure from the right-hand panel of Hieronymus Bosch‘s 1500 triptych The Garden of Earthly Delights.”
The Cubed Cube. The Necker cube might be called a binary ambiguity, with two contrasting views imaginable. By contrast, the image below is tertiary.
A cube in/notched out of/or off of a cube.
As I noted back in “The Eyes Have It—0r D0 Th3y?,” is it a small box in a corner? A large box with a cubical notch? Or a large box soaring upward to the northwest with a cubical nose pointing off in a southeasterly direction?
All in good optical fun. ds
© Dennis Simanaitis, SimanaitisSays.com, 2024