Wednesday, July 24, 2013

Visualizing n dimensions

Quantum, physics, Charlene Brown
(click on image to enlarge)

4D object + its 3D shadow
Watercolour and crayon
©2013 Charlene Brown

In devising a geometry of  more than three dimensions, mathematicians have determined that a four-dimensional cube would consist of eight cubes, just as a three-dimensional cube is made up of six squares. The hypercube (also called a tesseract) in Salvador Dali’s ‘Crucifixion’  has been suggested as an ‘aesthetically pleasing’ visualization of this concept… I don’t think so.
When I referred to the difficulty most people, including most geniuses, have in visualizing more than the usual four dimensions, in my blog post on Cubism on June 19, I wasn’t specifically referring to Dali, but now I am. I prefer my interpretation in 4D object + its 3D shadow.  The required eight cubes are aligned along the four extended diagonals of the purple cube, the four blue ones on the diagonals projecting out the back four corners of the cube and the orange ones on those emerging out the front.  Just as a three-dimensional object casts a two-dimensional shadow, a four-dimensional object casts a three-dimensional shadow, in this case the purple cube.