Works by Scott CarterCube Descending a Stair Case | Etudesworks in situ
 
Etude 4a and Etude 5Scott Carter, 2007 UltraChrome K3 on Piezo Pro Matte Canvas media 41.75 x 39.25 in. and 37.75 x 33.25 in. From Scott Carter: The collection of Etudes are an homage to artist Tony Robbin who initiated a serious artistic discussion of 4-dimensional space following the computer revolution. They are studies from several points of view. First, they are studies in the functionality of the Adobe Illustrator software. What can the software do? How are transparencies affected? How do colors blend? What are the effects of distortion palettes? Second, they are reproducible. A student trying to understand these aspects of 4-dimensional space can use the figures to reproduce and modify these views. Third, they are a depiction of a fundamental 4-dimensional figure and its projection onto the plane. The planar projection is achieved by the matrix 
The advantage in using this matrix is that it keeps vertices on the drawing lattice and the distances make a hypercube appear to fit onto a regular octagon. The figure that is the fundamental form is the cartesean product of two right isosceles triangles. In 4-space, this figure is the intersection of all convex sets that contain the set of vertices: 
The figure is translated in the 4 coordinate directions to checkerboard color 4-space. Within each copy of the figure several squares and triangles are colored. Our lower dimensional eyes perceive surface as the boundary of solid. To see solids, or indeed, hyper-solids, we draw those surface that overlap. The solid objects overlap as well. Various features are emphasized by opacity or line thickness. These choices are made via an aesthetic sieve, and do not necessarily carry mathematical meaning. Hold an orange before your eyes. You see a single orange disk -- the front of the orange. The back of the orange projects to the same disk and is invisible to the eye in a static view. The entire flesh of the orange all projects to the same disk on the retina. rays of sight intersect the orange, usually in segments. This is how three dimensional space is perceived. When 4-space is projected to the canvas, planes of sight are projected to the same point in the plane. The matrix above has such a plane in its null space. An unbounded 3-dimensional figure in 4-space, such as the boundary of a hypercube, will project to 3-space with a surface of folds (analogous to the apparent edge of the orange disk), and with two 3-dimensional figures occupying the same space in the projection (just as the front and back of the orange peel occupy the same disk on the retina. The challenge for the illustrator of 4-dimensional objects is to depict the simultaneity of space. top 
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