William Bricken

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Our understanding of a concept is tightly connected to the way we represent that concept. Traditionally, mathematics is presented textually. As a consequence novice errors, in elementary algebra for example, are due as much to misunderstandings of the nature of tokens as they are to miscomprehensions of the mathematical ideas represented by the tokens. This(More)
Instead of traditional mathematical notation, we can describe formal mathematical systems in visual form. While traditional notation uses a linear sequence of symbols, visual mathematics uses boundary notation, which is comprised of objects and boundaries to enclose objects. Boundary notation is abstract, decoupling the underlying mathematics of a system(More)
Psychological immersivity is the most important performance measure of effectiveness for media experiences, from watching a computer generated animation to having an interactive experience in a virtual reality (VR) environment. To offer and foster the best value hardware systems and the most effecive software media, we need to know what determines(More)
Boundary logic is a formal diagrammatic system that combines Peirce's Entitative Graphs with Spencer Brown's Laws of Form. Its conceptual basis includes boundary forms composed of non-intersecting closed curves, voidsubstitution (deletion of irrelevant structure) as the primary mechanism of reduction, and spatial pattern-equations that define valid(More)
The intuitive properties of configurations of planar non-overlapping closed curves (boundaries) are presented as a pure boundary mathematics. The mathematics, which is not incorporated in any existing formalism, is constructed from first principles, that is, from empty space. When formulated as patternequations, boundary algebras map to elementary logic and(More)
By now, almost everyone in computer graphics has read, seen, or experienced virtual worlds--"artificial realities" generated by computers which surround the user. Virtual.world systems are the focus of the media, grist for the TV mill, and everyone's pick as "The Big Idea of the 90s." But what does it all mean? Is virtual-worlds technology really that(More)
Peirce's Alpha Existential Graphs (AEG) is combined with Spencer Brown's Laws of Form to create an algebraic diagrammatic formalism called boundary logic. First, the intuitive properties of configurations of planar nonoverlapping closed curves are viewed as a pure boundary mathematics, without conventional interpretation. Void representational space(More)
The 18th International Conference on Computer Graphics and Interactive Techniques, SIGGRAPH'91, featured a juried selection of virtual reality and hypermedia demonstrations entitled Tomorrow's Realities. Conference participants were treated to twenty-six exhibits which represented the state of the art in virtual world techniques. The SIGGRAPH'91 Conference(More)
I’ve spent over 20 years implementing algorithms for Boolean minimization, with particular application to area optimization of large semiconductor circuits. This exploration has included almost all published algorithms, dozens of conventional and exotic data structures, over two dozen programming languages and styles, and almost every level of(More)