Sol LeWitt and Voxel Ambient Occlusion

I've previously written about the relationship between the work of artist Sol LeWitt and computer science. Some representative geometric works by Lewitt include:

Note the exploration of the regular grid, mathematical patterns, tone and contrast, and light and shadow. The image that I created below follows in this vein of patterns in (exhaustive) combinatorial art and, I fancy, echoes LeWitt's interests.

Each tile in this image is a top view of a square of a white plane. The square is akin to the center of a Tic-Tac-Toe (UK: Naughts and Crosses) surrounded by up to eight white cubes in the adjacent squares. Those boxes are cropped out, so that only the effects of shadowing on the central square are present. That is, this is the complete set of ambient occlusion masks needed in a voxel system (such as Minecraft).

Each of the surrounding squares on the 3x3 grid of the Tic-Tac-Toe board may have a shadowing cube present or not. If you think of "there is a cube here" as the digit 1 and "there is no cube here" as the digit 0, then these patterns correspond to the possible values of an 8-digit binary number...there are 2⁸=256 of them. Of course, many of these are duplicates.  For example, once the horizontal and vertical squares are filled, no additional shadowing can come from corners.  So, there are 2⁴=32 tiles that look like the one in the lower-right.

I created these tiles using the G3D Innovation Engine to generate all possible scenes of this description and then applied the Scalable Ambient Obscurance algorithm (which is built into G3D) to efficiently generate the shadowing. I then used Image Magick's montage tool to combine them to produce this image.  The idea for generating tiles in this way is due to Mike Mara at NVIDIA. Below is an image showing their application for efficient illumination effects for the codeheart.js expert "orthovoxel" example. The left image has no ambient occlusion.  The right image uses these tiles to produce dynamic ambient occlusion in real-time.

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Morgan McGuire is a professor of Computer Science at Williams College and a professional game developer. He is the author of The Graphics Codex, an essential reference for computer graphics that runs on iPhone, iPad, and iPod Touch.